pyqtgraph/pyqtgraph/functions.py

# -*- coding: utf-8 -*-
"""
functions.py -  Miscellaneous functions with no other home
Copyright 2010  Luke Campagnola
Distributed under MIT/X11 license. See license.txt for more information.
"""

from __future__ import division

import decimal
import re
import struct
import sys
import warnings
import math

import numpy as np
from .util.cupy_helper import getCupy
from .util.numba_helper import getNumbaFunctions

from . import debug, reload
from .Qt import QtGui, QtCore, QT_LIB, QtVersion
from . import Qt
from .metaarray import MetaArray
from collections import OrderedDict

# in order of appearance in this file.
# add new functions to this list only if they are to reside in pg namespace.
__all__ = [
    'siScale', 'siFormat', 'siParse', 'siEval', 'siApply',
    'Color', 'mkColor', 'mkBrush', 'mkPen', 'hsvColor',
    'CIELabColor', 'colorCIELab', 'colorDistance',
    'colorTuple', 'colorStr', 'intColor', 'glColor',
    'makeArrowPath', 'eq',
    'affineSliceCoords', 'affineSlice',
    'interweaveArrays', 'interpolateArray', 'subArray',
    'transformToArray', 'transformCoordinates',
    'solve3DTransform', 'solveBilinearTransform',
    'clip_scalar', 'clip_array', 'rescaleData', 'applyLookupTable',
    'makeRGBA', 'makeARGB',
    # 'try_fastpath_argb', 'ndarray_to_qimage',
    'makeQImage',
    # 'ndarray_from_qimage',
    'imageToArray', 'colorToAlpha',
    'gaussianFilter', 'downsample', 'arrayToQPath',
    # 'ndarray_from_qpolygonf', 'create_qpolygonf', 'arrayToQPolygonF',
    'isocurve', 'traceImage', 'isosurface',
    'invertQTransform',
    'pseudoScatter', 'toposort', 'disconnect', 'SignalBlock']


Colors = {
    'b': QtGui.QColor(0,0,255,255),
    'g': QtGui.QColor(0,255,0,255),
    'r': QtGui.QColor(255,0,0,255),
    'c': QtGui.QColor(0,255,255,255),
    'm': QtGui.QColor(255,0,255,255),
    'y': QtGui.QColor(255,255,0,255),
    'k': QtGui.QColor(0,0,0,255),
    'w': QtGui.QColor(255,255,255,255),
    'd': QtGui.QColor(150,150,150,255),
    'l': QtGui.QColor(200,200,200,255),
    's': QtGui.QColor(100,100,150,255),
}  

SI_PREFIXES = 'yzafpnµm kMGTPEZY'
SI_PREFIXES_ASCII = 'yzafpnum kMGTPEZY'
SI_PREFIX_EXPONENTS = dict([(SI_PREFIXES[i], (i-8)*3) for i in range(len(SI_PREFIXES))])
SI_PREFIX_EXPONENTS['u'] = -6

FLOAT_REGEX = re.compile(r'(?P<number>[+-]?((((\d+(\.\d*)?)|(\d*\.\d+))([eE][+-]?\d+)?)|((?i:nan)|(inf))))\s*((?P<siPrefix>[u' + SI_PREFIXES + r']?)(?P<suffix>\w.*))?$')
INT_REGEX = re.compile(r'(?P<number>[+-]?\d+)\s*(?P<siPrefix>[u' + SI_PREFIXES + r']?)(?P<suffix>.*)$')

    
def siScale(x, minVal=1e-25, allowUnicode=True):
    """
    Return the recommended scale factor and SI prefix string for x.
    
    Example::
    
        siScale(0.0001)   # returns (1e6, 'μ')
        # This indicates that the number 0.0001 is best represented as 0.0001 * 1e6 = 100 μUnits
    """
    
    if isinstance(x, decimal.Decimal):
        x = float(x)
    try:
        if not math.isfinite(x):
            return(1, '')
    except:
        raise
    if abs(x) < minVal:
        m = 0
    else:
        m = int(clip_scalar(math.floor(math.log(abs(x))/math.log(1000)), -9.0, 9.0))
    if m == 0:
        pref = ''
    elif m < -8 or m > 8:
        pref = 'e%d' % (m*3)
    else:
        if allowUnicode:
            pref = SI_PREFIXES[m+8]
        else:
            pref = SI_PREFIXES_ASCII[m+8]
    m1 = -3*m
    p = 10.**m1
    return (p, pref)


def siFormat(x, precision=3, suffix='', space=True, error=None, minVal=1e-25, allowUnicode=True):
    """
    Return the number x formatted in engineering notation with SI prefix.
    
    Example::
        siFormat(0.0001, suffix='V')  # returns "100 μV"
    """
    
    if space is True:
        space = ' '
    if space is False:
        space = ''
        
    
    (p, pref) = siScale(x, minVal, allowUnicode)
    if not (len(pref) > 0 and pref[0] == 'e'):
        pref = space + pref
    
    if error is None:
        fmt = "%." + str(precision) + "g%s%s"
        return fmt % (x*p, pref, suffix)
    else:
        if allowUnicode:
            plusminus = space + "±" + space
        else:
            plusminus = " +/- "
        fmt = "%." + str(precision) + "g%s%s%s%s"
        return fmt % (x*p, pref, suffix, plusminus, siFormat(error, precision=precision, suffix=suffix, space=space, minVal=minVal))


def siParse(s, regex=FLOAT_REGEX, suffix=None):
    """Convert a value written in SI notation to a tuple (number, si_prefix, suffix).

    Example::

        siParse('100 µV")  # returns ('100', 'µ', 'V')

    Note that in the above example, the µ symbol is the "micro sign" (UTF-8
    0xC2B5), as opposed to the Greek letter mu (UTF-8 0xCEBC).

    Parameters
    ----------
    s : str
        The string to parse.
    regex : re.Pattern, optional
        Compiled regular expression object for parsing. The default is a
        general-purpose regex for parsing floating point expressions,
        potentially containing an SI prefix and a suffix.
    suffix : str, optional
        Suffix to check for in ``s``. The default (None) indicates there may or
        may not be a suffix contained in the string and it is returned if
        found. An empty string ``""`` is handled differently: if the string
        contains a suffix, it is discarded. This enables interpreting
        characters following the numerical value as an SI prefix.
    """
    s = s.strip()
    if suffix is not None and len(suffix) > 0:
        if s[-len(suffix):] != suffix:
            raise ValueError("String '%s' does not have the expected suffix '%s'" % (s, suffix))
        s = s[:-len(suffix)] + 'X'  # add a fake suffix so the regex still picks up the si prefix

    # special case: discard any extra characters if suffix is explicitly empty
    if suffix == "":
        s += 'X'

    m = regex.match(s)
    if m is None:
        raise ValueError('Cannot parse number "%s"' % s)

    try:
        sip = m.group('siPrefix')
    except IndexError:
        sip = ''

    if suffix is None:
        try:
            suf = m.group('suffix')
        except IndexError:
            suf = ''
    else:
        suf = suffix

    return m.group('number'), '' if sip is None else sip, '' if suf is None else suf


def siEval(s, typ=float, regex=FLOAT_REGEX, suffix=None):
    """
    Convert a value written in SI notation to its equivalent prefixless value.

    Example::
    
        siEval("100 μV")  # returns 0.0001
    """
    val, siprefix, suffix = siParse(s, regex, suffix=suffix)
    v = typ(val)
    return siApply(v, siprefix)

    
def siApply(val, siprefix):
    """
    """
    n = SI_PREFIX_EXPONENTS[siprefix] if siprefix != '' else 0
    if n > 0:
        return val * 10**n
    elif n < 0:
        # this case makes it possible to use Decimal objects here
        return val / 10**-n
    else:
        return val
    

class Color(QtGui.QColor):
    def __init__(self, *args):
        QtGui.QColor.__init__(self, mkColor(*args))
        
    def glColor(self):
        """Return (r,g,b,a) normalized for use in opengl"""
        return self.getRgbF()
        
    def __getitem__(self, ind):
        return (self.red, self.green, self.blue, self.alpha)[ind]()
        
    
def mkColor(*args):
    """
    Convenience function for constructing QColor from a variety of argument 
    types. Accepted arguments are:
    
    ================ ================================================
     'c'             one of: r, g, b, c, m, y, k, w
     R, G, B, [A]    integers 0-255
     (R, G, B, [A])  tuple of integers 0-255
     float           greyscale, 0.0-1.0
     int             see :func:`intColor() <pyqtgraph.intColor>`
     (int, hues)     see :func:`intColor() <pyqtgraph.intColor>`
     "#RGB"          hexadecimal strings prefixed with '#'
     "#RGBA"         previously allowed use without prefix is deprecated and 
     "#RRGGBB"       will be removed in 0.13
     "#RRGGBBAA"     
     QColor          QColor instance; makes a copy.
    ================ ================================================
    """
    err = 'Not sure how to make a color from "%s"' % str(args)
    if len(args) == 1:
        if isinstance(args[0], str):
            c = args[0]
            if len(c) == 1:
                try:
                    return Colors[c]
                except KeyError:
                    raise ValueError('No color named "%s"' % c)
            have_alpha = len(c) in [5, 9] and c[0] == '#'  # "#RGBA" and "#RRGGBBAA"
            if not have_alpha:
                # try parsing SVG named colors, including "#RGB" and "#RRGGBB".
                # note that QColor.setNamedColor() treats a 9-char hex string as "#AARRGGBB".
                qcol = QtGui.QColor()
                qcol.setNamedColor(c)
                if qcol.isValid():
                    return qcol
                # on failure, fallback to pyqtgraph parsing
                # this includes the deprecated case of non-#-prefixed hex strings
            if c[0] == '#':
                c = c[1:]
            else:
                warnings.warn(
                    "Parsing of hex strings that do not start with '#' is"
                    "deprecated and support will be removed in 0.13",
                    DeprecationWarning, stacklevel=2
                )
            if len(c) == 3:
                r = int(c[0]*2, 16)
                g = int(c[1]*2, 16)
                b = int(c[2]*2, 16)
                a = 255
            elif len(c) == 4:
                r = int(c[0]*2, 16)
                g = int(c[1]*2, 16)
                b = int(c[2]*2, 16)
                a = int(c[3]*2, 16)
            elif len(c) == 6:
                r = int(c[0:2], 16)
                g = int(c[2:4], 16)
                b = int(c[4:6], 16)
                a = 255
            elif len(c) == 8:
                r = int(c[0:2], 16)
                g = int(c[2:4], 16)
                b = int(c[4:6], 16)
                a = int(c[6:8], 16)
            else:
                raise ValueError(f"Unknown how to convert string {c} to color")
        elif isinstance(args[0], QtGui.QColor):
            return QtGui.QColor(args[0])
        elif np.issubdtype(type(args[0]), np.floating):
            r = g = b = int(args[0] * 255)
            a = 255
        elif hasattr(args[0], '__len__'):
            if len(args[0]) == 3:
                r, g, b = args[0]
                a = 255
            elif len(args[0]) == 4:
                r, g, b, a = args[0]
            elif len(args[0]) == 2:
                return intColor(*args[0])
            else:
                raise TypeError(err)
        elif np.issubdtype(type(args[0]), np.integer):
            return intColor(args[0])
        else:
            raise TypeError(err)
    elif len(args) == 3:
        r, g, b = args
        a = 255
    elif len(args) == 4:
        r, g, b, a = args
    else:
        raise TypeError(err)
    args = [int(a) if np.isfinite(a) else 0 for a in (r, g, b, a)]
    return QtGui.QColor(*args)


def mkBrush(*args, **kwds):
    """
    | Convenience function for constructing Brush.
    | This function always constructs a solid brush and accepts the same arguments as :func:`mkColor() <pyqtgraph.mkColor>`
    | Calling mkBrush(None) returns an invisible brush.
    """
    if 'color' in kwds:
        color = kwds['color']
    elif len(args) == 1:
        arg = args[0]
        if arg is None:
            return QtGui.QBrush(QtCore.Qt.BrushStyle.NoBrush)
        elif isinstance(arg, QtGui.QBrush):
            return QtGui.QBrush(arg)
        else:
            color = arg
    elif len(args) > 1:
        color = args
    return QtGui.QBrush(mkColor(color))


def mkPen(*args, **kargs):
    """
    Convenience function for constructing QPen. 
    
    Examples::
    
        mkPen(color)
        mkPen(color, width=2)
        mkPen(cosmetic=False, width=4.5, color='r')
        mkPen({'color': "#FF0", width: 2})
        mkPen(None)   # (no pen)
    
    In these examples, *color* may be replaced with any arguments accepted by :func:`mkColor() <pyqtgraph.mkColor>`    """
    color = kargs.get('color', None)
    width = kargs.get('width', 1)
    style = kargs.get('style', None)
    dash = kargs.get('dash', None)
    cosmetic = kargs.get('cosmetic', True)
    hsv = kargs.get('hsv', None)
    
    if len(args) == 1:
        arg = args[0]
        if isinstance(arg, dict):
            return mkPen(**arg)
        if isinstance(arg, QtGui.QPen):
            return QtGui.QPen(arg)  ## return a copy of this pen
        elif arg is None:
            style = QtCore.Qt.PenStyle.NoPen
        else:
            color = arg
    if len(args) > 1:
        color = args
        
    if color is None:
        color = mkColor('l')
    if hsv is not None:
        color = hsvColor(*hsv)
    else:
        color = mkColor(color)
        
    pen = QtGui.QPen(QtGui.QBrush(color), width)
    pen.setCosmetic(cosmetic)
    if style is not None:
        pen.setStyle(style)
    if dash is not None:
        pen.setDashPattern(dash)
    return pen


def hsvColor(hue, sat=1.0, val=1.0, alpha=1.0):
    """Generate a QColor from HSVa values. (all arguments are float 0.0-1.0)"""
    return QtGui.QColor.fromHsvF(hue, sat, val, alpha)
    
# Matrices and math taken from "CIELab Color Space" by Gernot Hoffmann
# http://docs-hoffmann.de/cielab03022003.pdf
MATRIX_XYZ_FROM_RGB = np.array( (
    ( 0.4124, 0.3576, 0.1805),
    ( 0.2126, 0.7152, 0.0722),
    ( 0.0193, 0.1192, 0.9505) ) )
    
MATRIX_RGB_FROM_XYZ = np.array( (
    ( 3.2410,-1.5374,-0.4985),
    (-0.9692, 1.8760, 0.0416),
    ( 0.0556,-0.2040, 1.0570) ) )

VECTOR_XYZn = np.array( ( 0.9505, 1.0000, 1.0891) ) # white reference at illuminant D65

def CIELabColor(L, a, b, alpha=1.0):
    """
    Generates as QColor from CIE L*a*b* values.
    
    Parameters
    ----------
        L: float
            Lightness value ranging from 0 to 100
        a, b: float
            (green/red) and (blue/yellow) coordinates, typically -127 to +127.
        alpha: float, optional
            Opacity, ranging from 0 to 1

    Notes
    -----
    The CIE L*a*b* color space parametrizes color in terms of a luminance `L` 
    and the `a` and `b` coordinates that locate the hue in terms of
    a "green to red" and a "blue to yellow" axis.
    
    These coordinates seek to parametrize human color preception in such a way
    that the Euclidean distance between the coordinates of two colors represents
    the visual difference between these colors. In particular, the difference
    
    ΔE = sqrt( (L1-L2)² + (a1-a2)² + (b1-b2)² ) = 2.3
    
    is considered the smallest "just noticeable difference" between colors.
    
    This simple equation represents the CIE76 standard. Later standards CIE94
    and CIE2000 refine the difference calculation ΔE, while maintaining the 
    L*a*b* coordinates.
    
    Alternative (and arguably more accurate) methods exist to quantify color
    difference, but the CIELab color space remains a convenient approximation.
    
    Under a known illumination, assumed to be white standard illuminant D65 
    here, a CIELab color induces a response in the human eye
    that is described by the tristimulus value XYZ. Once this is known, an
    sRGB color can be calculated to induce the same response.
    
    More information and underlying mathematics can be found in e.g.
    "CIELab Color Space" by Gernot Hoffmann, available at
    http://docs-hoffmann.de/cielab03022003.pdf .
    
    Also see :func:`colorDistance() <pyqtgraph.colorDistance>`.
    """ 
    # convert to tristimulus XYZ values
    vec_XYZ = np.full(3, ( L +16)/116 )  # Y1 = (L+16)/116
    vec_XYZ[0] += a / 500                # X1 = (L+16)/116 + a/500
    vec_XYZ[2] -= b / 200                # Z1 = (L+16)/116 - b/200 
    for idx, val in enumerate(vec_XYZ):
        if val > 0.20689:
            vec_XYZ[idx] = vec_XYZ[idx]**3
        else:
            vec_XYZ[idx] = (vec_XYZ[idx] - 16/116) / 7.787
    vec_XYZ = VECTOR_XYZn * vec_XYZ # apply white reference
    # print(f'XYZ: {vec_XYZ}')

    # convert XYZ to linear RGB
    vec_RGB =  MATRIX_RGB_FROM_XYZ @ vec_XYZ
    # gamma-encode linear RGB
    arr_sRGB = np.zeros(3)
    for idx, val in enumerate( vec_RGB[:3] ):
        if val > 0.0031308: # (t) RGB value for linear/exponential transition
            arr_sRGB[idx] = 1.055 * val**(1/2.4) - 0.055
        else:
            arr_sRGB[idx] = 12.92 * val # (s)
    arr_sRGB = clip_array( arr_sRGB, 0.0, 1.0 ) # avoid QColor errors
    return QtGui.QColor.fromRgbF( *arr_sRGB, alpha )

def colorCIELab(qcol):
    """
    Describes a QColor by an array of CIE L*a*b* values.
    Also see :func:`CIELabColor() <pyqtgraph.CIELabColor>` .

    Parameters
    ----------
    qcol: QColor
        QColor to be converted

    Returns
    -------
    NumPy array 
        Color coordinates `[L, a, b]`.
    """
    srgb = qcol.getRgbF()[:3] # get sRGB values from QColor
    # convert gamma-encoded sRGB to linear:
    vec_RGB = np.zeros(3)
    for idx, val in enumerate( srgb ):
        if val > (12.92 * 0.0031308): # coefficients (s) * (t)
            vec_RGB[idx] = ((val+0.055)/1.055)**2.4
        else:
            vec_RGB[idx] = val / 12.92 # (s) coefficient
    # converted linear RGB to tristimulus XYZ:
    vec_XYZ = MATRIX_XYZ_FROM_RGB @ vec_RGB
    # normalize with white reference and convert to L*a*b* values
    vec_XYZ1 = vec_XYZ / VECTOR_XYZn 
    for idx, val in enumerate(vec_XYZ1):
        if val > 0.008856:
            vec_XYZ1[idx] = vec_XYZ1[idx]**(1/3)
        else:
            vec_XYZ1[idx] = 7.787*vec_XYZ1[idx] + 16/116
    vec_Lab = np.array([
        116 * vec_XYZ1[1] - 16,              # Y1
        500 * (vec_XYZ1[0] - vec_XYZ1[1]),   # X1 - Y1
        200 * (vec_XYZ1[1] - vec_XYZ1[2])] ) # Y1 - Z1
    return vec_Lab

def colorDistance(colors, metric='CIE76'):
    """
    Returns the perceptual distances between a sequence of QColors.
    See :func:`CIELabColor() <pyqtgraph.CIELabColor>` for more information.

    Parameters
    ----------
        colors: list of QColor
            Two or more colors to calculate the distances between.
        metric: string, optional
            Metric used to determined the difference. Only 'CIE76' is supported at this time,
            where a distance of 2.3 is considered a "just noticeable difference".
            The default may change as more metrics become available.
    
    Returns 
    -------
    List
        The `N-1` sequential distances between `N` colors.
    """
    metric = metric.upper()
    if len(colors) < 1: return np.array([], dtype=np.float)
    if metric == 'CIE76':
        dist = []
        lab1 = None
        for col in colors:
            lab2 = colorCIELab(col)
            if lab1 is None: #initialize on first element
                lab1 = lab2 
                continue
            dE = math.sqrt( np.sum( (lab1-lab2)**2 ) )
            dist.append(dE)
            lab1 = lab2
        return np.array(dist)
    raise ValueError(f'Metric {metric} is not available.')

def colorTuple(c):
    """Return a tuple (R,G,B,A) from a QColor"""
    return c.getRgb()

def colorStr(c):
    """Generate a hex string code from a QColor"""
    return ('%02x'*4) % colorTuple(c)


def intColor(index, hues=9, values=1, maxValue=255, minValue=150, maxHue=360, minHue=0, sat=255, alpha=255):
    """
    Creates a QColor from a single index. Useful for stepping through a predefined list of colors.
    
    The argument *index* determines which color from the set will be returned. All other arguments determine what the set of predefined colors will be
     
    Colors are chosen by cycling across hues while varying the value (brightness). 
    By default, this selects from a list of 9 hues."""
    hues = int(hues)
    values = int(values)
    ind = int(index) % (hues * values)
    indh = ind % hues
    indv = ind // hues
    if values > 1:
        v = minValue + indv * ((maxValue-minValue) // (values-1))
    else:
        v = maxValue
    h = minHue + (indh * (maxHue-minHue)) // hues
    
    return QtGui.QColor.fromHsv(h, sat, v, alpha)


def glColor(*args, **kargs):
    """
    Convert a color to OpenGL color format (r,g,b,a) floats 0.0-1.0
    Accepts same arguments as :func:`mkColor <pyqtgraph.mkColor>`.
    """
    c = mkColor(*args, **kargs)
    return c.getRgbF()

    

def makeArrowPath(headLen=20, headWidth=None, tipAngle=20, tailLen=20, tailWidth=3, baseAngle=0):
    """
    Construct a path outlining an arrow with the given dimensions.
    The arrow points in the -x direction with tip positioned at 0,0.
    If *headWidth* is supplied, it overrides *tipAngle* (in degrees).
    If *tailLen* is None, no tail will be drawn.
    """
    if headWidth is None:
        headWidth = headLen * math.tan(math.radians(tipAngle * 0.5))
    path = QtGui.QPainterPath()
    path.moveTo(0,0)
    path.lineTo(headLen, -headWidth)
    if tailLen is None:
        innerY = headLen - headWidth * math.tan(math.radians(baseAngle))
        path.lineTo(innerY, 0)
    else:
        tailWidth *= 0.5
        innerY = headLen - (headWidth-tailWidth) * math.tan(math.radians(baseAngle))
        path.lineTo(innerY, -tailWidth)
        path.lineTo(headLen + tailLen, -tailWidth)
        path.lineTo(headLen + tailLen, tailWidth)
        path.lineTo(innerY, tailWidth)
    path.lineTo(headLen, headWidth)
    path.lineTo(0,0)
    return path
    

def eq(a, b):
    """The great missing equivalence function: Guaranteed evaluation to a single bool value.
    
    This function has some important differences from the == operator:
    
    1. Returns True if a IS b, even if a==b still evaluates to False.
    2. While a is b will catch the case with np.nan values, special handling is done for distinct
       float('nan') instances using math.isnan.
    3. Tests for equivalence using ==, but silently ignores some common exceptions that can occur
       (AtrtibuteError, ValueError).
    4. When comparing arrays, returns False if the array shapes are not the same.
    5. When comparing arrays of the same shape, returns True only if all elements are equal (whereas
       the == operator would return a boolean array).
    6. Collections (dict, list, etc.) must have the same type to be considered equal. One
       consequence is that comparing a dict to an OrderedDict will always return False.
    """
    if a is b:
        return True

    # The above catches np.nan, but not float('nan')
    if isinstance(a, float) and isinstance(b, float):
        if math.isnan(a) and math.isnan(b):
            return True

    # Avoid comparing large arrays against scalars; this is expensive and we know it should return False.
    aIsArr = isinstance(a, (np.ndarray, MetaArray))
    bIsArr = isinstance(b, (np.ndarray, MetaArray))
    if (aIsArr or bIsArr) and type(a) != type(b):
        return False

    # If both inputs are arrays, we can speeed up comparison if shapes / dtypes don't match
    # NOTE: arrays of dissimilar type should be considered unequal even if they are numerically
    # equal because they may behave differently when computed on.
    if aIsArr and bIsArr and (a.shape != b.shape or a.dtype != b.dtype):
        return False

    # Recursively handle common containers
    if isinstance(a, dict) and isinstance(b, dict):
        if type(a) != type(b) or len(a) != len(b):
            return False
        if set(a.keys()) != set(b.keys()):
            return False
        for k, v in a.items():
            if not eq(v, b[k]):
                return False
        if isinstance(a, OrderedDict) or sys.version_info >= (3, 7):
            for a_item, b_item in zip(a.items(), b.items()):
                if not eq(a_item, b_item):
                    return False
        return True
    if isinstance(a, (list, tuple)) and isinstance(b, (list, tuple)):
        if type(a) != type(b) or len(a) != len(b):
            return False
        for v1,v2 in zip(a, b):
            if not eq(v1, v2):
                return False
        return True

    # Test for equivalence. 
    # If the test raises a recognized exception, then return Falase
    try:
        try:
            # Sometimes running catch_warnings(module=np) generates AttributeError ???
            catcher =  warnings.catch_warnings(module=np)  # ignore numpy futurewarning (numpy v. 1.10)
            catcher.__enter__()
        except Exception:
            catcher = None
        e = a==b
    except (ValueError, AttributeError): 
        return False
    except:
        print('failed to evaluate equivalence for:')
        print("  a:", str(type(a)), str(a))
        print("  b:", str(type(b)), str(b))
        raise
    finally:
        if catcher is not None:
            catcher.__exit__(None, None, None)
    
    t = type(e)
    if t is bool:
        return e
    elif t is np.bool_:
        return bool(e)
    elif isinstance(e, np.ndarray) or (hasattr(e, 'implements') and e.implements('MetaArray')):
        try:   ## disaster: if a is an empty array and b is not, then e.all() is True
            if a.shape != b.shape:
                return False
        except:
            return False
        if (hasattr(e, 'implements') and e.implements('MetaArray')):
            return e.asarray().all()
        else:
            return e.all()
    else:
        raise TypeError("== operator returned type %s" % str(type(e)))


def affineSliceCoords(shape, origin, vectors, axes):
    """Return the array of coordinates used to sample data arrays in affineSlice().
    """
    # sanity check
    if len(shape) != len(vectors):
        raise Exception("shape and vectors must have same length.")
    if len(origin) != len(axes):
        raise Exception("origin and axes must have same length.")
    for v in vectors:
        if len(v) != len(axes):
            raise Exception("each vector must be same length as axes.")
        
    shape = list(map(np.ceil, shape))

    ## make sure vectors are arrays
    if not isinstance(vectors, np.ndarray):
        vectors = np.array(vectors)
    if not isinstance(origin, np.ndarray):
        origin = np.array(origin)
    origin.shape = (len(axes),) + (1,)*len(shape)

    ## Build array of sample locations. 
    grid = np.mgrid[tuple([slice(0,x) for x in shape])]  ## mesh grid of indexes
    x = (grid[np.newaxis,...] * vectors.transpose()[(Ellipsis,) + (np.newaxis,)*len(shape)]).sum(axis=1)  ## magic
    x += origin

    return x

    
def affineSlice(data, shape, origin, vectors, axes, order=1, returnCoords=False, **kargs):
    """
    Take a slice of any orientation through an array. This is useful for extracting sections of multi-dimensional arrays
    such as MRI images for viewing as 1D or 2D data.
    
    The slicing axes are aribtrary; they do not need to be orthogonal to the original data or even to each other. It is
    possible to use this function to extract arbitrary linear, rectangular, or parallelepiped shapes from within larger
    datasets. The original data is interpolated onto a new array of coordinates using either interpolateArray if order<2
    or scipy.ndimage.map_coordinates otherwise.
    
    For a graphical interface to this function, see :func:`ROI.getArrayRegion <pyqtgraph.ROI.getArrayRegion>`
    
    ==============  ====================================================================================================
    **Arguments:**
    *data*          (ndarray) the original dataset
    *shape*         the shape of the slice to take (Note the return value may have more dimensions than len(shape))
    *origin*        the location in the original dataset that will become the origin of the sliced data.
    *vectors*       list of unit vectors which point in the direction of the slice axes. Each vector must have the same 
                    length as *axes*. If the vectors are not unit length, the result will be scaled relative to the 
                    original data. If the vectors are not orthogonal, the result will be sheared relative to the 
                    original data.
    *axes*          The axes in the original dataset which correspond to the slice *vectors*
    *order*         The order of spline interpolation. Default is 1 (linear). See scipy.ndimage.map_coordinates
                    for more information.
    *returnCoords*  If True, return a tuple (result, coords) where coords is the array of coordinates used to select
                    values from the original dataset.
    *All extra keyword arguments are passed to scipy.ndimage.map_coordinates.*
    --------------------------------------------------------------------------------------------------------------------
    ==============  ====================================================================================================
    
    Note the following must be true: 
        
        | len(shape) == len(vectors) 
        | len(origin) == len(axes) == len(vectors[i])
        
    Example: start with a 4D fMRI data set, take a diagonal-planar slice out of the last 3 axes
        
        * data = array with dims (time, x, y, z) = (100, 40, 40, 40)
        * The plane to pull out is perpendicular to the vector (x,y,z) = (1,1,1) 
        * The origin of the slice will be at (x,y,z) = (40, 0, 0)
        * We will slice a 20x20 plane from each timepoint, giving a final shape (100, 20, 20)
        
    The call for this example would look like::
        
        affineSlice(data, shape=(20,20), origin=(40,0,0), vectors=((-1, 1, 0), (-1, 0, 1)), axes=(1,2,3))
    
    """
    x = affineSliceCoords(shape, origin, vectors, axes)

    ## transpose data so slice axes come first
    trAx = list(range(data.ndim))
    for ax in axes:
        trAx.remove(ax)
    tr1 = tuple(axes) + tuple(trAx)
    data = data.transpose(tr1)
    #print "tr1:", tr1
    ## dims are now [(slice axes), (other axes)]

    if order > 1:
        try:
            import scipy.ndimage
        except ImportError:
            raise ImportError("Interpolating with order > 1 requires the scipy.ndimage module, but it could not be imported.")

        # iterate manually over unused axes since map_coordinates won't do it for us
        extraShape = data.shape[len(axes):]
        output = np.empty(tuple(shape) + extraShape, dtype=data.dtype)
        for inds in np.ndindex(*extraShape):
            ind = (Ellipsis,) + inds
            output[ind] = scipy.ndimage.map_coordinates(data[ind], x, order=order, **kargs)
    else:
        # map_coordinates expects the indexes as the first axis, whereas
        # interpolateArray expects indexes at the last axis. 
        tr = tuple(range(1, x.ndim)) + (0,)
        output = interpolateArray(data, x.transpose(tr), order=order)
    
    tr = list(range(output.ndim))
    trb = []
    for i in range(min(axes)):
        ind = tr1.index(i) + (len(shape)-len(axes))
        tr.remove(ind)
        trb.append(ind)
    tr2 = tuple(trb+tr)

    ## Untranspose array before returning
    output = output.transpose(tr2)
    if returnCoords:
        return (output, x)
    else:
        return output


def interweaveArrays(*args):
    """
    Parameters
    ----------

    args : numpy.ndarray
           series of 1D numpy arrays of the same length and dtype
    
    Returns
    -------
    numpy.ndarray
        A numpy array with all the input numpy arrays interwoven

    Examples
    --------

    >>> result = interweaveArrays(numpy.ndarray([0, 2, 4]), numpy.ndarray([1, 3, 5]))
    >>> result
    array([0, 1, 2, 3, 4, 5])
    """

    size = sum(x.size for x in args)
    result = np.empty((size,), dtype=args[0].dtype)
    n = len(args)
    for index, array in enumerate(args):
        result[index::n] = array
    return result


def interpolateArray(data, x, default=0.0, order=1):
    """
    N-dimensional interpolation similar to scipy.ndimage.map_coordinates.
    
    This function returns linearly-interpolated values sampled from a regular
    grid of data. It differs from `ndimage.map_coordinates` by allowing broadcasting
    within the input array.
    
    ==============  ===========================================================================================
    **Arguments:**
    *data*          Array of any shape containing the values to be interpolated.
    *x*             Array with (shape[-1] <= data.ndim) containing the locations within *data* to interpolate.
                    (note: the axes for this argument are transposed relative to the same argument for
                    `ndimage.map_coordinates`).
    *default*       Value to return for locations in *x* that are outside the bounds of *data*.
    *order*         Order of interpolation: 0=nearest, 1=linear.
    ==============  ===========================================================================================
    
    Returns array of shape (x.shape[:-1] + data.shape[x.shape[-1]:])
    
    For example, assume we have the following 2D image data::
    
        >>> data = np.array([[1,   2,   4  ],
                             [10,  20,  40 ],
                             [100, 200, 400]])
        
    To compute a single interpolated point from this data::
        
        >>> x = np.array([(0.5, 0.5)])
        >>> interpolateArray(data, x)
        array([ 8.25])
        
    To compute a 1D list of interpolated locations:: 
        
        >>> x = np.array([(0.5, 0.5),
                          (1.0, 1.0),
                          (1.0, 2.0),
                          (1.5, 0.0)])
        >>> interpolateArray(data, x)
        array([  8.25,  20.  ,  40.  ,  55.  ])
        
    To compute a 2D array of interpolated locations::
    
        >>> x = np.array([[(0.5, 0.5), (1.0, 2.0)],
                          [(1.0, 1.0), (1.5, 0.0)]])
        >>> interpolateArray(data, x)
        array([[  8.25,  40.  ],
               [ 20.  ,  55.  ]])
               
    ..and so on. The *x* argument may have any shape as long as 
    ```x.shape[-1] <= data.ndim```. In the case that 
    ```x.shape[-1] < data.ndim```, then the remaining axes are simply 
    broadcasted as usual. For example, we can interpolate one location
    from an entire row of the data::
    
        >>> x = np.array([[0.5]])
        >>> interpolateArray(data, x)
        array([[  5.5,  11. ,  22. ]])

    This is useful for interpolating from arrays of colors, vertexes, etc.
    """
    if order not in (0, 1):
        raise ValueError("interpolateArray requires order=0 or 1 (got %s)" % order)

    prof = debug.Profiler()

    nd = data.ndim
    md = x.shape[-1]
    if md > nd:
        raise TypeError("x.shape[-1] must be less than or equal to data.ndim")

    totalMask = np.ones(x.shape[:-1], dtype=bool) # keep track of out-of-bound indexes
    if order == 0:
        xinds = np.round(x).astype(int)  # NOTE: for 0.5 this rounds to the nearest *even* number
        for ax in range(md):
            mask = (xinds[...,ax] >= 0) & (xinds[...,ax] <= data.shape[ax]-1) 
            xinds[...,ax][~mask] = 0
            # keep track of points that need to be set to default
            totalMask &= mask
        result = data[tuple([xinds[...,i] for i in range(xinds.shape[-1])])]
        
    elif order == 1:
        # First we generate arrays of indexes that are needed to 
        # extract the data surrounding each point
        fields = np.mgrid[(slice(0,order+1),) * md]
        xmin = np.floor(x).astype(int)
        xmax = xmin + 1
        indexes = np.concatenate([xmin[np.newaxis, ...], xmax[np.newaxis, ...]])
        fieldInds = []
        for ax in range(md):
            mask = (xmin[...,ax] >= 0) & (x[...,ax] <= data.shape[ax]-1) 
            # keep track of points that need to be set to default
            totalMask &= mask
            
            # ..and keep track of indexes that are out of bounds 
            # (note that when x[...,ax] == data.shape[ax], then xmax[...,ax] will be out
            #  of bounds, but the interpolation will work anyway)
            mask &= (xmax[...,ax] < data.shape[ax])
            axisIndex = indexes[...,ax][fields[ax]]
            axisIndex[axisIndex < 0] = 0
            axisIndex[axisIndex >= data.shape[ax]] = 0
            fieldInds.append(axisIndex)
        prof()

        # Get data values surrounding each requested point
        fieldData = data[tuple(fieldInds)]
        prof()
    
        ## Interpolate
        s = np.empty((md,) + fieldData.shape, dtype=float)
        dx = x - xmin
        # reshape fields for arithmetic against dx
        for ax in range(md):
            f1 = fields[ax].reshape(fields[ax].shape + (1,)*(dx.ndim-1))
            sax = f1 * dx[...,ax] + (1-f1) * (1-dx[...,ax])
            sax = sax.reshape(sax.shape + (1,) * (s.ndim-1-sax.ndim))
            s[ax] = sax
        s = np.product(s, axis=0)
        result = fieldData * s
        for i in range(md):
            result = result.sum(axis=0)

    prof()

    if totalMask.ndim > 0:
        result[~totalMask] = default
    else:
        if totalMask is False:
            result[:] = default

    prof()
    return result


def subArray(data, offset, shape, stride):
    """
    Unpack a sub-array from *data* using the specified offset, shape, and stride.
    
    Note that *stride* is specified in array elements, not bytes.
    For example, we have a 2x3 array packed in a 1D array as follows::
    
        data = [_, _, 00, 01, 02, _, 10, 11, 12, _]
        
    Then we can unpack the sub-array with this call::
    
        subArray(data, offset=2, shape=(2, 3), stride=(4, 1))
        
    ..which returns::
    
        [[00, 01, 02],
         [10, 11, 12]]
         
    This function operates only on the first axis of *data*. So changing 
    the input in the example above to have shape (10, 7) would cause the
    output to have shape (2, 3, 7).
    """
    data = np.ascontiguousarray(data)[offset:]
    shape = tuple(shape)
    extraShape = data.shape[1:]

    strides = list(data.strides[::-1])
    itemsize = strides[-1]
    for s in stride[1::-1]:
        strides.append(itemsize * s)
    strides = tuple(strides[::-1])
    
    return np.ndarray(buffer=data, shape=shape+extraShape, strides=strides, dtype=data.dtype)


def transformToArray(tr):
    """
    Given a QTransform, return a 3x3 numpy array.
    Given a QMatrix4x4, return a 4x4 numpy array.
    
    Example: map an array of x,y coordinates through a transform::
    
        ## coordinates to map are (1,5), (2,6), (3,7), and (4,8)
        coords = np.array([[1,2,3,4], [5,6,7,8], [1,1,1,1]])  # the extra '1' coordinate is needed for translation to work
        
        ## Make an example transform
        tr = QtGui.QTransform()
        tr.translate(3,4)
        tr.scale(2, 0.1)
        
        ## convert to array
        m = pg.transformToArray()[:2]  # ignore the perspective portion of the transformation
        
        ## map coordinates through transform
        mapped = np.dot(m, coords)
    """
    #return np.array([[tr.m11(), tr.m12(), tr.m13()],[tr.m21(), tr.m22(), tr.m23()],[tr.m31(), tr.m32(), tr.m33()]])
    ## The order of elements given by the method names m11..m33 is misleading--
    ## It is most common for x,y translation to occupy the positions 1,3 and 2,3 in
    ## a transformation matrix. However, with QTransform these values appear at m31 and m32.
    ## So the correct interpretation is transposed:
    if isinstance(tr, QtGui.QTransform):
        return np.array([[tr.m11(), tr.m21(), tr.m31()], [tr.m12(), tr.m22(), tr.m32()], [tr.m13(), tr.m23(), tr.m33()]])
    elif isinstance(tr, QtGui.QMatrix4x4):
        return np.array(tr.copyDataTo()).reshape(4,4)
    else:
        raise Exception("Transform argument must be either QTransform or QMatrix4x4.")

def transformCoordinates(tr, coords, transpose=False):
    """
    Map a set of 2D or 3D coordinates through a QTransform or QMatrix4x4.
    The shape of coords must be (2,...) or (3,...)
    The mapping will _ignore_ any perspective transformations.
    
    For coordinate arrays with ndim=2, this is basically equivalent to matrix multiplication.
    Most arrays, however, prefer to put the coordinate axis at the end (eg. shape=(...,3)). To 
    allow this, use transpose=True.
    
    """
    
    if transpose:
        ## move last axis to beginning. This transposition will be reversed before returning the mapped coordinates.
        coords = coords.transpose((coords.ndim-1,) + tuple(range(0,coords.ndim-1)))
    
    nd = coords.shape[0]
    if isinstance(tr, np.ndarray):
        m = tr
    else:
        m = transformToArray(tr)
        m = m[:m.shape[0]-1]  # remove perspective
    
    ## If coords are 3D and tr is 2D, assume no change for Z axis
    if m.shape == (2,3) and nd == 3:
        m2 = np.zeros((3,4))
        m2[:2, :2] = m[:2,:2]
        m2[:2, 3] = m[:2,2]
        m2[2,2] = 1
        m = m2
    
    ## if coords are 2D and tr is 3D, ignore Z axis
    if m.shape == (3,4) and nd == 2:
        m2 = np.empty((2,3))
        m2[:,:2] = m[:2,:2]
        m2[:,2] = m[:2,3]
        m = m2
    
    ## reshape tr and coords to prepare for multiplication
    m = m.reshape(m.shape + (1,)*(coords.ndim-1))
    coords = coords[np.newaxis, ...]
    
    # separate scale/rotate and translation    
    translate = m[:,-1]  
    m = m[:, :-1]
    
    ## map coordinates and return
    # nan or inf points will not plot, but should not generate warnings
    with warnings.catch_warnings():
        warnings.simplefilter("ignore", RuntimeWarning)
        mapped = (m*coords).sum(axis=1)  ## apply scale/rotate
    mapped += translate
    
    if transpose:
        ## move first axis to end.
        mapped = mapped.transpose(tuple(range(1,mapped.ndim)) + (0,))
    return mapped
    
    

    
def solve3DTransform(points1, points2):
    """
    Find a 3D transformation matrix that maps points1 onto points2.
    Points must be specified as either lists of 4 Vectors or 
    (4, 3) arrays.
    """
    import numpy.linalg
    pts = []
    for inp in (points1, points2):
        if isinstance(inp, np.ndarray):
            A = np.empty((4,4), dtype=float)
            A[:,:3] = inp[:,:3]
            A[:,3] = 1.0
        else:
            A = np.array([[inp[i].x(), inp[i].y(), inp[i].z(), 1] for i in range(4)])
        pts.append(A)
    
    ## solve 3 sets of linear equations to determine transformation matrix elements
    matrix = np.zeros((4,4))
    for i in range(3):
        ## solve Ax = B; x is one row of the desired transformation matrix
        matrix[i] = numpy.linalg.solve(pts[0], pts[1][:,i])  
    
    return matrix
    
def solveBilinearTransform(points1, points2):
    """
    Find a bilinear transformation matrix (2x4) that maps points1 onto points2.
    Points must be specified as a list of 4 Vector, Point, QPointF, etc.
    
    To use this matrix to map a point [x,y]::
    
        mapped = np.dot(matrix, [x*y, x, y, 1])
    """
    import numpy.linalg
    ## A is 4 rows (points) x 4 columns (xy, x, y, 1)
    ## B is 4 rows (points) x 2 columns (x, y)
    A = np.array([[points1[i].x()*points1[i].y(), points1[i].x(), points1[i].y(), 1] for i in range(4)])
    B = np.array([[points2[i].x(), points2[i].y()] for i in range(4)])
    
    ## solve 2 sets of linear equations to determine transformation matrix elements
    matrix = np.zeros((2,4))
    for i in range(2):
        matrix[i] = numpy.linalg.solve(A, B[:,i])  ## solve Ax = B; x is one row of the desired transformation matrix
    
    return matrix

def clip_scalar(val, vmin, vmax):
    """ convenience function to avoid using np.clip for scalar values """
    return vmin if val < vmin else vmax if val > vmax else val

def clip_array(arr, vmin, vmax, out=None):
    # replacement for np.clip due to regression in
    # performance since numpy 1.17
    # https://github.com/numpy/numpy/issues/14281

    if vmin is None and vmax is None:
        # let np.clip handle the error
        return np.clip(arr, vmin, vmax, out=out)

    if vmin is None:
        return np.core.umath.minimum(arr, vmax, out=out)
    elif vmax is None:
        return np.core.umath.maximum(arr, vmin, out=out)
    elif sys.platform == 'win32':
        # Windows umath.clip is slower than umath.maximum(umath.minimum)
        if out is None:
            out = np.empty_like(arr)
        out = np.core.umath.minimum(arr, vmax, out=out)
        return np.core.umath.maximum(out, vmin, out=out)
    else:
        return np.core.umath.clip(arr, vmin, vmax, out=out)


def _rescaleData_nditer(data_in, scale, offset, work_dtype, out_dtype, clip):
    """Refer to documentation for rescaleData()"""
    data_out = np.empty_like(data_in, dtype=out_dtype)

    # integer clip operations are faster than float clip operations
    # so test to see if we can perform integer clipping
    fits_int32 = False
    if data_in.dtype.kind in 'ui' and out_dtype.kind in 'ui':
        # estimate whether data range after rescale will fit within an int32.
        # this means that the input dtype should be an 8-bit or 16-bit integer type.
        # casting to an int32 will lose the fractional part, therefore the
        # output dtype must be an integer kind.
        lim_in = np.iinfo(data_in.dtype)
        # convert numpy scalar to python scalar to avoid overflow warnings
        lo = offset.item(0) if isinstance(offset, np.number) else offset
        dst_bounds = scale * (lim_in.min - lo), scale * (lim_in.max - lo)
        if dst_bounds[1] < dst_bounds[0]:
            dst_bounds = dst_bounds[1], dst_bounds[0]
        lim32 = np.iinfo(np.int32)
        fits_int32 = lim32.min < dst_bounds[0] and dst_bounds[1] < lim32.max

    it = np.nditer([data_in, data_out],
            flags=['external_loop', 'buffered'],
            op_flags=[['readonly'], ['writeonly', 'no_broadcast']],
            op_dtypes=[None, work_dtype],
            casting='unsafe',
            buffersize=32768)

    with it:
        for x, y in it:
            y[...] = x
            y -= offset
            y *= scale

            # Clip before converting dtype to avoid overflow
            if clip is not None:
                if fits_int32:
                    # converts to int32, clips back to float32
                    np.core.umath.clip(y.astype(np.int32), clip[0], clip[1], out=y)
                else:
                    clip_array(y, clip[0], clip[1], out=y)

    return data_out


def rescaleData(data, scale, offset, dtype=None, clip=None):
    """Return data rescaled and optionally cast to a new dtype.

    The scaling operation is::

        data => (data-offset) * scale
    """
    if dtype is None:
        out_dtype = data.dtype
    else:
        out_dtype = np.dtype(dtype)

    if out_dtype.kind in 'ui':
        lim = np.iinfo(out_dtype)
        if clip is None:
            # don't let rescale cause integer overflow
            clip = lim.min, lim.max
        clip = max(clip[0], lim.min), min(clip[1], lim.max)

        # make clip limits integer-valued (no need to cast to int)
        # this improves performance, especially on Windows
        clip = [math.trunc(x) for x in clip]

    if np.can_cast(data, np.float32):
        work_dtype = np.float32
    else:
        work_dtype = np.float64

    cp = getCupy()
    if cp and cp.get_array_module(data) == cp:
        # Cupy does not support nditer
        # https://github.com/cupy/cupy/issues/5021

        data_out = data.astype(work_dtype, copy=True)
        data_out -= offset
        data_out *= scale

        # Clip before converting dtype to avoid overflow
        if clip is not None:
            clip_array(data_out, clip[0], clip[1], out=data_out)

        # don't copy if no change in dtype
        return data_out.astype(out_dtype, copy=False)

    numba_fn = getNumbaFunctions()
    if numba_fn and clip is not None:
        # if we got here by makeARGB(), clip will not be None at this point
        return numba_fn.rescaleData(data, scale, offset, out_dtype, clip)

    return _rescaleData_nditer(data, scale, offset, work_dtype, out_dtype, clip)


def applyLookupTable(data, lut):
    """
    Uses values in *data* as indexes to select values from *lut*.
    The returned data has shape data.shape + lut.shape[1:]
    
    Note: color gradient lookup tables can be generated using GradientWidget.

    Parameters
    ----------
    data : ndarray
    lut : ndarray
        Either cupy or numpy arrays are accepted, though this function has only
        consistently behaved correctly on windows with cuda toolkit version >= 11.1.
    """
    if data.dtype.kind not in ('i', 'u'):
        data = data.astype(int)

    cp = getCupy()
    if cp and cp.get_array_module(data) == cp:
        # cupy.take only supports "wrap" mode
        return cp.take(lut, cp.clip(data, 0, lut.shape[0] - 1), axis=0)
    else:
        return np.take(lut, data, axis=0, mode='clip')
    

def makeRGBA(*args, **kwds):
    """Equivalent to makeARGB(..., useRGBA=True)"""
    kwds['useRGBA'] = True
    return makeARGB(*args, **kwds)


def makeARGB(data, lut=None, levels=None, scale=None, useRGBA=False, output=None):
    """
    Convert an array of values into an ARGB array suitable for building QImages,
    OpenGL textures, etc.
    
    Returns the ARGB array (unsigned byte) and a boolean indicating whether
    there is alpha channel data. This is a two stage process:
    
        1) Rescale the data based on the values in the *levels* argument (min, max).
        2) Determine the final output by passing the rescaled values through a
           lookup table.
   
    Both stages are optional.
    
    ============== ==================================================================================
    **Arguments:**
    data           numpy array of int/float types. If 
    levels         List [min, max]; optionally rescale data before converting through the
                   lookup table. The data is rescaled such that min->0 and max->*scale*::
                   
                      rescaled = (clip(data, min, max) - min) * (*scale* / (max - min))
                   
                   It is also possible to use a 2D (N,2) array of values for levels. In this case,
                   it is assumed that each pair of min,max values in the levels array should be 
                   applied to a different subset of the input data (for example, the input data may 
                   already have RGB values and the levels are used to independently scale each 
                   channel). The use of this feature requires that levels.shape[0] == data.shape[-1].
    scale          The maximum value to which data will be rescaled before being passed through the 
                   lookup table (or returned if there is no lookup table). By default this will
                   be set to the length of the lookup table, or 255 if no lookup table is provided.
    lut            Optional lookup table (array with dtype=ubyte).
                   Values in data will be converted to color by indexing directly from lut.
                   The output data shape will be input.shape + lut.shape[1:].
                   Lookup tables can be built using ColorMap or GradientWidget.
    useRGBA        If True, the data is returned in RGBA order (useful for building OpenGL textures). 
                   The default is False, which returns in ARGB order for use with QImage 
                   (Note that 'ARGB' is a term used by the Qt documentation; the *actual* order 
                   is BGRA).
    ============== ==================================================================================
    """
    cp = getCupy()
    xp = cp.get_array_module(data) if cp else np
    profile = debug.Profiler()
    if data.ndim not in (2, 3):
        raise TypeError("data must be 2D or 3D")
    if data.ndim == 3 and data.shape[2] > 4:
        raise TypeError("data.shape[2] must be <= 4")
    
    if lut is not None and not isinstance(lut, xp.ndarray):
        lut = xp.array(lut)
    
    if levels is None:
        # automatically decide levels based on data dtype
        if data.dtype.kind == 'u':
            levels = xp.array([0, 2**(data.itemsize*8)-1])
        elif data.dtype.kind == 'i':
            s = 2**(data.itemsize*8 - 1)
            levels = xp.array([-s, s-1])
        elif data.dtype.kind == 'b':
            levels = xp.array([0,1])
        else:
            raise Exception('levels argument is required for float input types')
    if not isinstance(levels, xp.ndarray):
        levels = xp.array(levels)
    levels = levels.astype(xp.float64)
    if levels.ndim == 1:
        if levels.shape[0] != 2:
            raise Exception('levels argument must have length 2')
    elif levels.ndim == 2:
        if lut is not None and lut.ndim > 1:
            raise Exception('Cannot make ARGB data when both levels and lut have ndim > 2')
        if levels.shape != (data.shape[-1], 2):
            raise Exception('levels must have shape (data.shape[-1], 2)')
    else:
        raise Exception("levels argument must be 1D or 2D (got shape=%s)." % repr(levels.shape))

    profile('check inputs')

    # Decide on maximum scaled value
    if scale is None:
        if lut is not None:
            scale = lut.shape[0]
        else:
            scale = 255.

    # Decide on the dtype we want after scaling
    if lut is None:
        dtype = xp.ubyte
    else:
        dtype = xp.min_scalar_type(lut.shape[0]-1)

    # awkward, but fastest numpy native nan evaluation
    nanMask = None
    if data.dtype.kind == 'f' and xp.isnan(data.min()):
        nanMask = xp.isnan(data)
        if data.ndim > 2:
            nanMask = xp.any(nanMask, axis=-1)
    # Apply levels if given
    if levels is not None:
        if isinstance(levels, xp.ndarray) and levels.ndim == 2:
            # we are going to rescale each channel independently
            if levels.shape[0] != data.shape[-1]:
                raise Exception("When rescaling multi-channel data, there must be the same number of levels as channels (data.shape[-1] == levels.shape[0])")
            newData = xp.empty(data.shape, dtype=int)
            for i in range(data.shape[-1]):
                minVal, maxVal = levels[i]
                if minVal == maxVal:
                    maxVal = xp.nextafter(maxVal, 2*maxVal)
                rng = maxVal-minVal
                rng = 1 if rng == 0 else rng
                newData[...,i] = rescaleData(data[...,i], scale / rng, minVal, dtype=dtype)
            data = newData
        else:
            # Apply level scaling unless it would have no effect on the data
            minVal, maxVal = levels
            if minVal != 0 or maxVal != scale:
                if minVal == maxVal:
                    maxVal = xp.nextafter(maxVal, 2*maxVal)
                rng = maxVal-minVal
                rng = 1 if rng == 0 else rng
                data = rescaleData(data, scale/rng, minVal, dtype=dtype)

    profile('apply levels')

    # apply LUT if given
    if lut is not None:
        data = applyLookupTable(data, lut)
    else:
        if data.dtype != xp.ubyte:
            data = xp.clip(data, 0, 255).astype(xp.ubyte)

    profile('apply lut')

    # this will be the final image array
    if output is None:
        imgData = xp.empty(data.shape[:2]+(4,), dtype=xp.ubyte)
    else:
        imgData = output

    profile('allocate')

    # decide channel order
    if useRGBA:
        dst_order = [0, 1, 2, 3]    # R,G,B,A
    elif sys.byteorder == 'little':
        dst_order = [2, 1, 0, 3]    # B,G,R,A (ARGB32 little endian)
    else:
        dst_order = [1, 2, 3, 0]    # A,R,G,B (ARGB32 big endian)
        
    # copy data into image array
    fastpath = try_fastpath_argb(xp, data, imgData, useRGBA)

    if fastpath:
        pass
    elif data.ndim == 2:
        # This is tempting:
        #   imgData[..., :3] = data[..., xp.newaxis]
        # ..but it turns out this is faster:
        for i in range(3):
            imgData[..., dst_order[i]] = data
    elif data.shape[2] == 1:
        for i in range(3):
            imgData[..., dst_order[i]] = data[..., 0]
    else:
        for i in range(0, data.shape[2]):
            imgData[..., dst_order[i]] = data[..., i]
        
    profile('reorder channels')
    
    # add opaque alpha channel if needed
    if data.ndim == 3 and data.shape[2] == 4:
        alpha = True
    else:
        alpha = False
        if not fastpath:    # fastpath has already filled it in
            imgData[..., dst_order[3]] = 255

    # apply nan mask through alpha channel
    if nanMask is not None:
        alpha = True
        # Workaround for https://github.com/cupy/cupy/issues/4693
        if xp == cp:
            imgData[nanMask, :, dst_order[3]] = 0
        else:
            imgData[nanMask, dst_order[3]] = 0

    profile('alpha channel')
    return imgData, alpha


def try_fastpath_argb(xp, ain, aout, useRGBA):
    # we only optimize for certain cases
    # return False if we did not handle it
    can_handle = xp is np and ain.dtype == xp.ubyte and ain.flags['C_CONTIGUOUS']
    if not can_handle:
        return False

    nrows, ncols = ain.shape[:2]
    nchans = 1 if ain.ndim == 2 else ain.shape[2]

    Format = QtGui.QImage.Format

    if nchans == 1:
        in_fmt = Format.Format_Grayscale8
    elif nchans == 3:
        in_fmt = Format.Format_RGB888
    else:
        in_fmt = Format.Format_RGBA8888

    if useRGBA:
        out_fmt = Format.Format_RGBA8888
    else:
        out_fmt = Format.Format_ARGB32

    if in_fmt == out_fmt:
        aout[:] = ain
        return True

    npixels_chunk = 512*1024
    batch = int(npixels_chunk / ncols / nchans)
    batch = max(1, batch)
    row_beg = 0
    while row_beg < nrows:
        row_end = min(row_beg + batch, nrows)
        ain_view = ain[row_beg:row_end, ...]
        aout_view = aout[row_beg:row_end, ...]
        qimg = QtGui.QImage(ain_view, ncols, ain_view.shape[0], ain.strides[0], in_fmt)
        qimg = qimg.convertToFormat(out_fmt)
        aout_view[:] = imageToArray(qimg, copy=False, transpose=False)
        row_beg = row_end

    return True


def ndarray_to_qimage(arr, fmt):
    """
    Low level function to encapsulate QImage creation differences between bindings.
    "arr" is assumed to be C-contiguous. 
    """

    # C++ QImage has two kind of constructors
    # - QImage(const uchar*, ...)
    # - QImage(uchar*, ...)
    # If the const constructor is used, subsequently calling any non-const method
    # will trigger the COW mechanism, i.e. a copy is made under the hood.

    if QT_LIB.startswith('PyQt'):
        # PyQt5          -> non-const
        # PyQt6 >= 6.0.1 -> non-const
        img_ptr = int(Qt.sip.voidptr(arr))  # or arr.ctypes.data
    else:
        # bindings that support ndarray
        # PyQt5          -> const
        # PyQt6 >= 6.0.1 -> const
        # PySide2        -> non-const
        # PySide6        -> non-const
        img_ptr = arr

    h, w = arr.shape[:2]
    bytesPerLine = arr.strides[0]
    qimg = QtGui.QImage(img_ptr, w, h, bytesPerLine, fmt)
    qimg.data = arr
    return qimg


def makeQImage(imgData, alpha=None, copy=True, transpose=True):
    """
    Turn an ARGB array into QImage.
    By default, the data is copied; changes to the array will not
    be reflected in the image. The image will be given a 'data' attribute
    pointing to the array which shares its data to prevent python
    freeing that memory while the image is in use.
    
    ============== ===================================================================
    **Arguments:**
    imgData        Array of data to convert. Must have shape (height, width),
                   (height, width, 3), or (height, width, 4). If transpose is
                   True, then the first two axes are swapped. The array dtype
                   must be ubyte. For 2D arrays, the value is interpreted as 
                   greyscale. For 3D arrays, the order of values in the 3rd
                   axis must be (b, g, r, a). 
    alpha          If the input array is 3D and *alpha* is True, the QImage 
                   returned will have format ARGB32. If False,
                   the format will be RGB32. By default, _alpha_ is True if
                   array.shape[2] == 4.
    copy           If True, the data is copied before converting to QImage.
                   If False, the new QImage points directly to the data in the array.
                   Note that the array must be contiguous for this to work
                   (see numpy.ascontiguousarray).
    transpose      If True (the default), the array x/y axes are transposed before 
                   creating the image. Note that Qt expects the axes to be in 
                   (height, width) order whereas pyqtgraph usually prefers the 
                   opposite.
    ============== ===================================================================    
    """
    ## create QImage from buffer
    profile = debug.Profiler()
    
    copied = False
    if imgData.ndim == 2:
        imgFormat = QtGui.QImage.Format.Format_Grayscale8
    elif imgData.ndim == 3:
        # If we didn't explicitly specify alpha, check the array shape.
        if alpha is None:
            alpha = (imgData.shape[2] == 4)
            
        if imgData.shape[2] == 3:  # need to make alpha channel (even if alpha==False; QImage requires 32 bpp)
            if copy is True:
                d2 = np.empty(imgData.shape[:2] + (4,), dtype=imgData.dtype)
                d2[:,:,:3] = imgData
                d2[:,:,3] = 255
                imgData = d2
                copied = True
            else:
                raise Exception('Array has only 3 channels; cannot make QImage without copying.')
        
        profile("add alpha channel")
        
        if alpha:
            imgFormat = QtGui.QImage.Format.Format_ARGB32
        else:
            imgFormat = QtGui.QImage.Format.Format_RGB32
    else:
        raise TypeError("Image array must have ndim = 2 or 3.")
        
    if transpose:
        imgData = imgData.transpose((1, 0, 2))  # QImage expects row-major order

    if not imgData.flags['C_CONTIGUOUS']:
        if copy is False:
            extra = ' (try setting transpose=False)' if transpose else ''
            raise Exception('Array is not contiguous; cannot make QImage without copying.'+extra)
        imgData = np.ascontiguousarray(imgData)
        copied = True
        
    profile("ascontiguousarray")
    
    if copy is True and copied is False:
        imgData = imgData.copy()
        
    profile("copy")

    return ndarray_to_qimage(imgData, imgFormat)


def ndarray_from_qimage(qimg):
    img_ptr = qimg.bits()

    if img_ptr is None:
        raise ValueError("Null QImage not supported")

    h, w = qimg.height(), qimg.width()
    bpl = qimg.bytesPerLine()
    depth = qimg.depth()
    logical_bpl = w * depth // 8

    if QT_LIB.startswith('PyQt'):
        # sizeInBytes() was introduced in Qt 5.10
        # however PyQt5 5.12 will fail with:
        #   "TypeError: QImage.sizeInBytes() is a private method"
        # note that sizeInBytes() works fine with:
        #   PyQt5 5.15, PySide2 5.12, PySide2 5.15
        img_ptr.setsize(h * bpl)

    memory = np.frombuffer(img_ptr, dtype=np.ubyte).reshape((h, bpl))
    memory = memory[:, :logical_bpl]

    if depth in (8, 24, 32):
        dtype = np.uint8
        nchan = depth // 8
    elif depth in (16, 64):
        dtype = np.uint16
        nchan = depth // 16
    else:
        raise ValueError("Unsupported Image Type")

    shape = h, w
    if nchan != 1:
        shape = shape + (nchan,)
    arr = memory.view(dtype).reshape(shape)
    return arr


def imageToArray(img, copy=False, transpose=True):
    """
    Convert a QImage into numpy array. The image must have format RGB32, ARGB32, or ARGB32_Premultiplied.
    By default, the image is not copied; changes made to the array will appear in the QImage as well (beware: if
    the QImage is collected before the array, there may be trouble).
    The array will have shape (width, height, (b,g,r,a)).
    """
    arr = ndarray_from_qimage(img)

    fmt = img.format()
    if fmt == img.Format.Format_RGB32:
        arr[...,3] = 255
    
    if copy:
        arr = arr.copy()
        
    if transpose:
        return arr.transpose((1,0,2))
    else:
        return arr
    
def colorToAlpha(data, color):
    """
    Given an RGBA image in *data*, convert *color* to be transparent. 
    *data* must be an array (w, h, 3 or 4) of ubyte values and *color* must be 
    an array (3) of ubyte values.
    This is particularly useful for use with images that have a black or white background.
    
    Algorithm is taken from Gimp's color-to-alpha function in plug-ins/common/colortoalpha.c
    Credit:
        /*
        * Color To Alpha plug-in v1.0 by Seth Burgess, sjburges@gimp.org 1999/05/14
        *  with algorithm by clahey
        */
    
    """
    data = data.astype(float)
    if data.shape[-1] == 3:  ## add alpha channel if needed
        d2 = np.empty(data.shape[:2]+(4,), dtype=data.dtype)
        d2[...,:3] = data
        d2[...,3] = 255
        data = d2
    
    color = color.astype(float)
    alpha = np.zeros(data.shape[:2]+(3,), dtype=float)
    output = data.copy()
    
    for i in [0,1,2]:
        d = data[...,i]
        c = color[i]
        mask = d > c
        alpha[...,i][mask] = (d[mask] - c) / (255. - c)
        imask = d < c
        alpha[...,i][imask] = (c - d[imask]) / c
    
    output[...,3] = alpha.max(axis=2) * 255.
    
    mask = output[...,3] >= 1.0  ## avoid zero division while processing alpha channel
    correction = 255. / output[...,3][mask]  ## increase value to compensate for decreased alpha
    for i in [0,1,2]:
        output[...,i][mask] = ((output[...,i][mask]-color[i]) * correction) + color[i]
        output[...,3][mask] *= data[...,3][mask] / 255.  ## combine computed and previous alpha values
    
    #raise Exception()
    return np.clip(output, 0, 255).astype(np.ubyte)

def gaussianFilter(data, sigma):
    """
    Drop-in replacement for scipy.ndimage.gaussian_filter.
    
    (note: results are only approximately equal to the output of
     gaussian_filter)
    """
    cp = getCupy()
    xp = cp.get_array_module(data) if cp else np
    if xp.isscalar(sigma):
        sigma = (sigma,) * data.ndim
        
    baseline = data.mean()
    filtered = data - baseline
    for ax in range(data.ndim):
        s = sigma[ax]
        if s == 0:
            continue
        
        # generate 1D gaussian kernel
        ksize = int(s * 6)
        x = xp.arange(-ksize, ksize)
        kernel = xp.exp(-x**2 / (2*s**2))
        kshape = [1,] * data.ndim
        kshape[ax] = len(kernel)
        kernel = kernel.reshape(kshape)
        
        # convolve as product of FFTs
        shape = data.shape[ax] + ksize
        scale = 1.0 / (abs(s) * (2*xp.pi)**0.5)
        filtered = scale * xp.fft.irfft(xp.fft.rfft(filtered, shape, axis=ax) *
                                        xp.fft.rfft(kernel, shape, axis=ax),
                                        axis=ax)
        
        # clip off extra data
        sl = [slice(None)] * data.ndim
        sl[ax] = slice(filtered.shape[ax]-data.shape[ax],None,None)
        filtered = filtered[tuple(sl)]
    return filtered + baseline
    
    
def downsample(data, n, axis=0, xvals='subsample'):
    """Downsample by averaging points together across axis.
    If multiple axes are specified, runs once per axis.
    If a metaArray is given, then the axis values can be either subsampled
    or downsampled to match.
    """
    ma = None
    if (hasattr(data, 'implements') and data.implements('MetaArray')):
        ma = data
        data = data.view(np.ndarray)
        
    
    if hasattr(axis, '__len__'):
        if not hasattr(n, '__len__'):
            n = [n]*len(axis)
        for i in range(len(axis)):
            data = downsample(data, n[i], axis[i])
        return data
    
    if n <= 1:
        return data
    nPts = int(data.shape[axis] / n)
    s = list(data.shape)
    s[axis] = nPts
    s.insert(axis+1, n)
    sl = [slice(None)] * data.ndim
    sl[axis] = slice(0, nPts*n)
    d1 = data[tuple(sl)]
    #print d1.shape, s
    d1.shape = tuple(s)
    d2 = d1.mean(axis+1)
    
    if ma is None:
        return d2
    else:
        info = ma.infoCopy()
        if 'values' in info[axis]:
            if xvals == 'subsample':
                info[axis]['values'] = info[axis]['values'][::n][:nPts]
            elif xvals == 'downsample':
                info[axis]['values'] = downsample(info[axis]['values'], n)
        return MetaArray(d2, info=info)


def _compute_backfill_indices(isfinite):
    # the presence of inf/nans result in an empty QPainterPath being generated
    # this behavior started in Qt 5.12.3 and was introduced in this commit
    # https://github.com/qt/qtbase/commit/c04bd30de072793faee5166cff866a4c4e0a9dd7
    # We therefore replace non-finite values

    # credit: Divakar https://stackoverflow.com/a/41191127/643629
    mask = ~isfinite
    idx = np.arange(len(isfinite))
    idx[mask] = -1
    np.maximum.accumulate(idx, out=idx)
    first = np.searchsorted(idx, 0)
    if first < len(isfinite):
        # Replace all non-finite entries from beginning of arr with the first finite one
        idx[:first] = first
        return idx
    else:
        return None


def _arrayToQPath_all(x, y, finiteCheck):
    n = x.shape[0]
    if n == 0:
        return QtGui.QPainterPath()

    backfill_idx = None
    if finiteCheck:
        isfinite = np.isfinite(x) & np.isfinite(y)
        if not np.all(isfinite):
            backfill_idx = _compute_backfill_indices(isfinite)

    chunksize = 10000
    numchunks = (n + chunksize - 1) // chunksize
    minchunks = 3

    if numchunks < minchunks:
        # too few chunks, batching would be a pessimization
        poly = create_qpolygonf(n)
        arr = ndarray_from_qpolygonf(poly)

        if backfill_idx is None:
            arr[:, 0] = x
            arr[:, 1] = y
        else:
            arr[:, 0] = x[backfill_idx]
            arr[:, 1] = y[backfill_idx]

        path = QtGui.QPainterPath()
        if hasattr(path, 'reserve'):    # Qt 5.13
            path.reserve(n)
        path.addPolygon(poly)
        return path

    # at this point, we have numchunks >= minchunks

    path = QtGui.QPainterPath()
    if hasattr(path, 'reserve'):    # Qt 5.13
        path.reserve(n)
    subpoly = QtGui.QPolygonF()
    subpath = None
    for idx in range(numchunks):
        sl = slice(idx*chunksize, min((idx+1)*chunksize, n))
        currsize = sl.stop - sl.start
        if currsize != subpoly.size():
            if hasattr(subpoly, 'resize'):
                subpoly.resize(currsize)
            else:
                subpoly.fill(QtCore.QPointF(), currsize)
        subarr = ndarray_from_qpolygonf(subpoly)
        if backfill_idx is None:
            subarr[:, 0] = x[sl]
            subarr[:, 1] = y[sl]
        else:
            bfv = backfill_idx[sl]  # view
            subarr[:, 0] = x[bfv]
            subarr[:, 1] = y[bfv]
        if subpath is None:
            subpath = QtGui.QPainterPath()
        subpath.addPolygon(subpoly)
        path.connectPath(subpath)
        if hasattr(subpath, 'clear'):   # Qt 5.13
            subpath.clear()
        else:
            subpath = None
    return path


def _arrayToQPath_finite(x, y, isfinite=None):
    n = x.shape[0]
    if n == 0:
        return QtGui.QPainterPath()

    if isfinite is None:
        isfinite = np.isfinite(x) & np.isfinite(y)

    path = QtGui.QPainterPath()
    if hasattr(path, 'reserve'):    # Qt 5.13
        path.reserve(n)

    sidx = np.nonzero(~isfinite)[0] + 1
    # note: the chunks are views
    xchunks = np.split(x, sidx)
    ychunks = np.split(y, sidx)
    chunks = list(zip(xchunks, ychunks))

    # create a single polygon able to hold the largest chunk
    maxlen = max(len(chunk) for chunk in xchunks)
    subpoly = create_qpolygonf(maxlen)
    subarr = ndarray_from_qpolygonf(subpoly)

    # resize and fill do not change the capacity
    if hasattr(subpoly, 'resize'):
        subpoly_resize = subpoly.resize
    else:
        # PyQt will be less efficient
        subpoly_resize = lambda n, v=QtCore.QPointF() : subpoly.fill(v, n)

    # notes:
    # - we backfill the non-finite in order to get the same image as the
    #   old codepath on the CI. somehow P1--P2 gets rendered differently
    #   from P1--P2--P2
    # - we do not generate MoveTo(s) that are not followed by a LineTo,
    #   thus the QPainterPath can be different from the old codepath's

    # all chunks except the last chunk have a trailing non-finite
    for xchunk, ychunk in chunks[:-1]:
        lc = len(xchunk)
        if lc <= 1:
            # len 1 means we have a string of non-finite
            continue
        subpoly_resize(lc)
        subarr[:lc, 0] = xchunk
        subarr[:lc, 1] = ychunk
        subarr[lc-1] = subarr[lc-2] # fill non-finite with its neighbour
        path.addPolygon(subpoly)

    # handle last chunk, which is either all-finite or empty
    for xchunk, ychunk in chunks[-1:]:
        lc = len(xchunk)
        if lc <= 1:
            # can't draw a line with just 1 point
            continue
        subpoly_resize(lc)
        subarr[:lc, 0] = xchunk
        subarr[:lc, 1] = ychunk
        path.addPolygon(subpoly)

    return path


def arrayToQPath(x, y, connect='all', finiteCheck=True):
    """
    Convert an array of x,y coordinates to QPainterPath as efficiently as
    possible. The *connect* argument may be 'all', indicating that each point
    should be connected to the next; 'pairs', indicating that each pair of
    points should be connected, or an array of int32 values (0 or 1) indicating
    connections.
    
    Parameters
    ----------
    x : (N,) ndarray
        x-values to be plotted
    y : (N,) ndarray
        y-values to be plotted, must be same length as `x`
    connect : {'all', 'pairs', 'finite', (N,) ndarray}, optional
        Argument detailing how to connect the points in the path. `all` will 
        have sequential points being connected.  `pairs` generates lines
        between every other point.  `finite` only connects points that are
        finite.  If an ndarray is passed, containing int32 values of 0 or 1,
        only values with 1 will connect to the previous point.  Def
    finiteCheck : bool, default Ture
        When false, the check for finite values will be skipped, which can
        improve performance. If nonfinite values are present in `x` or `y`,
        an empty QPainterPath will be generated.
    
    Returns
    -------
    QPainterPath
        QPainterPath object to be drawn
    
    Raises
    ------
    ValueError
        Raised when the connect argument has an invalid value placed within.

    Notes
    -----
    A QPainterPath is generated through one of two ways.  When the connect
    parameter is 'all', a QPolygonF object is created, and
    ``QPainterPath.addPolygon()`` is called.  For other connect parameters
    a ``QDataStream`` object is created and the QDataStream >> QPainterPath
    operator is used to pass the data.  The memory format is as follows

    numVerts(i4)
    0(i4)   x(f8)   y(f8)    <-- 0 means this vertex does not connect
    1(i4)   x(f8)   y(f8)    <-- 1 means this vertex connects to the previous vertex
    ...
    cStart(i4)   fillRule(i4)
    
    see: https://github.com/qt/qtbase/blob/dev/src/gui/painting/qpainterpath.cpp

    All values are big endian--pack using struct.pack('>d') or struct.pack('>i')
    This binary format may change in future versions of Qt
    """

    n = x.shape[0]
    if n == 0:
        return QtGui.QPainterPath()

    connect_array = None
    if isinstance(connect, np.ndarray):
        # make connect argument contain only str type
        connect_array, connect = connect, 'array'

    isfinite = None

    if connect == 'finite':
        if not finiteCheck:
            # if user specified to skip finite check, then we skip the heuristic
            return _arrayToQPath_finite(x, y)

        # otherwise use a heuristic
        # if non-finite aren't that many, then use_qpolyponf
        isfinite = np.isfinite(x) & np.isfinite(y)
        nonfinite_cnt = n - np.sum(isfinite)
        all_isfinite = nonfinite_cnt == 0
        if all_isfinite:
            # delegate to connect='all'
            connect = 'all'
            finiteCheck = False
        elif nonfinite_cnt / n < 2 / 100:
            return _arrayToQPath_finite(x, y, isfinite)
        else:
            # delegate to connect=ndarray
            # finiteCheck=True, all_isfinite=False
            connect = 'array'
            connect_array = isfinite

    if connect == 'all':
        return _arrayToQPath_all(x, y, finiteCheck)

    backstore = QtCore.QByteArray()
    backstore.resize(4 + n*20 + 8)      # contents uninitialized
    backstore.replace(0, 4, struct.pack('>i', n))
    # cStart, fillRule (Qt.FillRule.OddEvenFill)
    backstore.replace(4+n*20, 8, struct.pack('>ii', 0, 0))
    arr = np.frombuffer(backstore, dtype=[('c', '>i4'), ('x', '>f8'), ('y', '>f8')],
        count=n, offset=4)

    backfill_idx = None
    if finiteCheck:
        if isfinite is None:
            isfinite = np.isfinite(x) & np.isfinite(y)
            all_isfinite = np.all(isfinite)
        if not all_isfinite:
            backfill_idx = _compute_backfill_indices(isfinite)

    if backfill_idx is None:
        arr['x'] = x
        arr['y'] = y
    else:
        arr['x'] = x[backfill_idx]
        arr['y'] = y[backfill_idx]

    # decide which points are connected by lines
    if connect == 'pairs':
        arr['c'][0::2] = 0
        arr['c'][1::2] = 1  # connect every 2nd point to every 1st one
    elif connect == 'array':
        # Let's call a point with either x or y being nan is an invalid point.
        # A point will anyway not connect to an invalid point regardless of the
        # 'c' value of the invalid point. Therefore, we should set 'c' to 0 for
        # the next point of an invalid point.
        arr['c'][:1] = 0  # the first vertex has no previous vertex to connect
        arr['c'][1:] = connect_array[:-1]
    else:
        raise ValueError('connect argument must be "all", "pairs", "finite", or array')

    path = QtGui.QPainterPath()
    if hasattr(path, 'reserve'):    # Qt 5.13
        path.reserve(n)

    ds = QtCore.QDataStream(backstore)
    ds >> path
    return path

def ndarray_from_qpolygonf(polyline):
    nbytes = 2 * len(polyline) * 8
    if QT_LIB.startswith('PyQt'):
        buffer = polyline.data()
        if buffer is None:
            buffer = Qt.sip.voidptr(0)
        buffer.setsize(nbytes)
    else:
        ptr = polyline.data()
        if ptr is None:
            ptr = 0
        buffer = Qt.shiboken.VoidPtr(ptr, nbytes, True)
    memory = np.frombuffer(buffer, np.double).reshape((-1, 2))
    return memory

def create_qpolygonf(size):
    polyline = QtGui.QPolygonF()
    if QT_LIB.startswith('PyQt'):
        polyline.fill(QtCore.QPointF(), size)
    else:
        polyline.resize(size)
    return polyline

def arrayToQPolygonF(x, y):
    """
    Utility function to convert two 1D-NumPy arrays representing curve data
    (X-axis, Y-axis data) into a single open polygon (QtGui.PolygonF) object.
    
    Thanks to PythonQwt for making this code available
    
    License/copyright: MIT License © Pierre Raybaut 2020.

    Parameters
    ----------
    x : np.array
        x-axis coordinates for data to be plotted, must have have ndim of 1
    y : np.array
        y-axis coordinates for data to be plotted, must have ndim of 1 and 
        be the same length as x
    
    Returns
    -------
    QPolygonF
        Open QPolygonF object that represents the path looking to be plotted
    
    Raises
    ------
    ValueError
        When xdata or ydata does not meet the required criteria
    """
    if not (
        x.size == y.size == x.shape[0] == y.shape[0]
    ):
        raise ValueError("Arguments must be 1D and the same size")
    size = x.size
    polyline = create_qpolygonf(size)
    memory = ndarray_from_qpolygonf(polyline)
    memory[:, 0] = x
    memory[:, 1] = y
    return polyline

#def isosurface(data, level):
    #"""
    #Generate isosurface from volumetric data using marching tetrahedra algorithm.
    #See Paul Bourke, "Polygonising a Scalar Field Using Tetrahedrons"  (http://local.wasp.uwa.edu.au/~pbourke/geometry/polygonise/)
    
    #*data*   3D numpy array of scalar values
    #*level*  The level at which to generate an isosurface
    #"""
    
    #facets = []
    
    ### mark everything below the isosurface level
    #mask = data < level
    
    #### make eight sub-fields 
    #fields = np.empty((2,2,2), dtype=object)
    #slices = [slice(0,-1), slice(1,None)]
    #for i in [0,1]:
        #for j in [0,1]:
            #for k in [0,1]:
                #fields[i,j,k] = mask[slices[i], slices[j], slices[k]]
    
    
    
    ### split each cell into 6 tetrahedra
    ### these all have the same 'orienation'; points 1,2,3 circle 
    ### clockwise around point 0
    #tetrahedra = [
        #[(0,1,0), (1,1,1), (0,1,1), (1,0,1)],
        #[(0,1,0), (0,1,1), (0,0,1), (1,0,1)],
        #[(0,1,0), (0,0,1), (0,0,0), (1,0,1)],
        #[(0,1,0), (0,0,0), (1,0,0), (1,0,1)],
        #[(0,1,0), (1,0,0), (1,1,0), (1,0,1)],
        #[(0,1,0), (1,1,0), (1,1,1), (1,0,1)]
    #]
    
    ### each tetrahedron will be assigned an index
    ### which determines how to generate its facets.
    ### this structure is: 
    ###    facets[index][facet1, facet2, ...]
    ### where each facet is triangular and its points are each 
    ### interpolated between two points on the tetrahedron
    ###    facet = [(p1a, p1b), (p2a, p2b), (p3a, p3b)]
    ### facet points always circle clockwise if you are looking 
    ### at them from below the isosurface.
    #indexFacets = [
        #[],  ## all above
        #[[(0,1), (0,2), (0,3)]],  # 0 below
        #[[(1,0), (1,3), (1,2)]],   # 1 below
        #[[(0,2), (1,3), (1,2)], [(0,2), (0,3), (1,3)]],   # 0,1 below
        #[[(2,0), (2,1), (2,3)]],   # 2 below
        #[[(0,3), (1,2), (2,3)], [(0,3), (0,1), (1,2)]],   # 0,2 below
        #[[(1,0), (2,3), (2,0)], [(1,0), (1,3), (2,3)]],   # 1,2 below
        #[[(3,0), (3,1), (3,2)]],   # 3 above
        #[[(3,0), (3,2), (3,1)]],   # 3 below
        #[[(1,0), (2,0), (2,3)], [(1,0), (2,3), (1,3)]],   # 0,3 below
        #[[(0,3), (2,3), (1,2)], [(0,3), (1,2), (0,1)]],   # 1,3 below
        #[[(2,0), (2,3), (2,1)]], # 0,1,3 below
        #[[(0,2), (1,2), (1,3)], [(0,2), (1,3), (0,3)]],   # 2,3 below
        #[[(1,0), (1,2), (1,3)]], # 0,2,3 below
        #[[(0,1), (0,3), (0,2)]], # 1,2,3 below
        #[]  ## all below
    #]
    
    #for tet in tetrahedra:
        
        ### get the 4 fields for this tetrahedron
        #tetFields = [fields[c] for c in tet]
        
        ### generate an index for each grid cell
        #index = tetFields[0] + tetFields[1]*2 + tetFields[2]*4 + tetFields[3]*8
        
        ### add facets
        #for i in range(index.shape[0]):                 # data x-axis
            #for j in range(index.shape[1]):             # data y-axis
                #for k in range(index.shape[2]):         # data z-axis
                    #for f in indexFacets[index[i,j,k]]:  # faces to generate for this tet
                        #pts = []
                        #for l in [0,1,2]:      # points in this face
                            #p1 = tet[f[l][0]]  # tet corner 1
                            #p2 = tet[f[l][1]]  # tet corner 2
                            #pts.append([(p1[x]+p2[x])*0.5+[i,j,k][x]+0.5 for x in [0,1,2]]) ## interpolate between tet corners
                        #facets.append(pts)

    #return facets
    

def isocurve(data, level, connected=False, extendToEdge=False, path=False):
    """
    Generate isocurve from 2D data using marching squares algorithm.
    
    ============== =========================================================
    **Arguments:**
    data           2D numpy array of scalar values
    level          The level at which to generate an isosurface
    connected      If False, return a single long list of point pairs
                   If True, return multiple long lists of connected point 
                   locations. (This is slower but better for drawing 
                   continuous lines)
    extendToEdge   If True, extend the curves to reach the exact edges of 
                   the data. 
    path           if True, return a QPainterPath rather than a list of 
                   vertex coordinates. This forces connected=True.
    ============== =========================================================
    
    This function is SLOW; plenty of room for optimization here.
    """    
    
    if path is True:
        connected = True
    
    if extendToEdge:
        d2 = np.empty((data.shape[0]+2, data.shape[1]+2), dtype=data.dtype)
        d2[1:-1, 1:-1] = data
        d2[0, 1:-1] = data[0]
        d2[-1, 1:-1] = data[-1]
        d2[1:-1, 0] = data[:, 0]
        d2[1:-1, -1] = data[:, -1]
        d2[0,0] = d2[0,1]
        d2[0,-1] = d2[1,-1]
        d2[-1,0] = d2[-1,1]
        d2[-1,-1] = d2[-1,-2]
        data = d2
    
    sideTable = [
        [],
        [0,1],
        [1,2],
        [0,2],
        [0,3],
        [1,3],
        [0,1,2,3],
        [2,3],
        [2,3],
        [0,1,2,3],
        [1,3],
        [0,3],
        [0,2],
        [1,2],
        [0,1],
        []
        ]
    
    edgeKey=[
        [(0,1), (0,0)],
        [(0,0), (1,0)],
        [(1,0), (1,1)],
        [(1,1), (0,1)]
        ]
    
    
    lines = []
    
    ## mark everything below the isosurface level
    mask = data < level
    
    ### make four sub-fields and compute indexes for grid cells
    index = np.zeros([x-1 for x in data.shape], dtype=np.ubyte)
    fields = np.empty((2,2), dtype=object)
    slices = [slice(0,-1), slice(1,None)]
    for i in [0,1]:
        for j in [0,1]:
            fields[i,j] = mask[slices[i], slices[j]]
            #vertIndex = i - 2*j*i + 3*j + 4*k  ## this is just to match Bourk's vertex numbering scheme
            vertIndex = i+2*j
            #print i,j,k," : ", fields[i,j,k], 2**vertIndex
            np.add(index, fields[i,j] * 2**vertIndex, out=index, casting='unsafe')
            #print index
    #print index
    
    ## add lines
    for i in range(index.shape[0]):                 # data x-axis
        for j in range(index.shape[1]):             # data y-axis     
            sides = sideTable[index[i,j]]
            for l in range(0, len(sides), 2):     ## faces for this grid cell
                edges = sides[l:l+2]
                pts = []
                for m in [0,1]:      # points in this face
                    p1 = edgeKey[edges[m]][0] # p1, p2 are points at either side of an edge
                    p2 = edgeKey[edges[m]][1]
                    v1 = data[i+p1[0], j+p1[1]] # v1 and v2 are the values at p1 and p2
                    v2 = data[i+p2[0], j+p2[1]]
                    f = (level-v1) / (v2-v1)
                    fi = 1.0 - f
                    p = (    ## interpolate between corners
                        p1[0]*fi + p2[0]*f + i + 0.5, 
                        p1[1]*fi + p2[1]*f + j + 0.5
                        )
                    if extendToEdge:
                        ## check bounds
                        p = (
                            min(data.shape[0]-2, max(0, p[0]-1)),
                            min(data.shape[1]-2, max(0, p[1]-1)),                        
                        )
                    if connected:
                        gridKey = i + (1 if edges[m]==2 else 0), j + (1 if edges[m]==3 else 0), edges[m]%2
                        pts.append((p, gridKey))  ## give the actual position and a key identifying the grid location (for connecting segments)
                    else:
                        pts.append(p)
                
                lines.append(pts)

    if not connected:
        return lines
                
    ## turn disjoint list of segments into continuous lines

    #lines = [[2,5], [5,4], [3,4], [1,3], [6,7], [7,8], [8,6], [11,12], [12,15], [11,13], [13,14]]
    #lines = [[(float(a), a), (float(b), b)] for a,b in lines]
    points = {}  ## maps each point to its connections
    for a,b in lines:
        if a[1] not in points:
            points[a[1]] = []
        points[a[1]].append([a,b])
        if b[1] not in points:
            points[b[1]] = []
        points[b[1]].append([b,a])

    ## rearrange into chains
    for k in list(points.keys()):
        try:
            chains = points[k]
        except KeyError:   ## already used this point elsewhere
            continue
        #print "===========", k
        for chain in chains:
            #print "  chain:", chain
            x = None
            while True:
                if x == chain[-1][1]:
                    break ## nothing left to do on this chain
                    
                x = chain[-1][1]
                if x == k:  
                    break ## chain has looped; we're done and can ignore the opposite chain
                y = chain[-2][1]
                connects = points[x]
                for conn in connects[:]:
                    if conn[1][1] != y:
                        #print "    ext:", conn
                        chain.extend(conn[1:])
                #print "    del:", x
                del points[x]
            if chain[0][1] == chain[-1][1]:  # looped chain; no need to continue the other direction
                chains.pop()
                break
                

    ## extract point locations 
    lines = []
    for chain in points.values():
        if len(chain) == 2:
            chain = chain[1][1:][::-1] + chain[0]  # join together ends of chain
        else:
            chain = chain[0]
        lines.append([p[0] for p in chain])
    
    if not path:
        return lines ## a list of pairs of points
    
    path = QtGui.QPainterPath()
    for line in lines:
        path.moveTo(*line[0])
        for p in line[1:]:
            path.lineTo(*p)
    
    return path
    
    
def traceImage(image, values, smooth=0.5):
    """
    Convert an image to a set of QPainterPath curves.
    One curve will be generated for each item in *values*; each curve outlines the area
    of the image that is closer to its value than to any others.
    
    If image is RGB or RGBA, then the shape of values should be (nvals, 3/4)
    The parameter *smooth* is expressed in pixels.
    """
    if values.ndim == 2:
        values = values.T
    values = values[np.newaxis, np.newaxis, ...].astype(float)
    image = image[..., np.newaxis].astype(float)
    diff = np.abs(image-values)
    if values.ndim == 4:
        diff = diff.sum(axis=2)
        
    labels = np.argmin(diff, axis=2)
    
    paths = []
    for i in range(diff.shape[-1]):    
        d = (labels==i).astype(float)
        d = gaussianFilter(d, (smooth, smooth))
        lines = isocurve(d, 0.5, connected=True, extendToEdge=True)
        path = QtGui.QPainterPath()
        for line in lines:
            path.moveTo(*line[0])
            for p in line[1:]:
                path.lineTo(*p)
        
        paths.append(path)
    return paths
    
    
    
IsosurfaceDataCache = None
def isosurface(data, level):
    """
    Generate isosurface from volumetric data using marching cubes algorithm.
    See Paul Bourke, "Polygonising a Scalar Field"  
    (http://paulbourke.net/geometry/polygonise/)
    
    *data*   3D numpy array of scalar values. Must be contiguous.
    *level*  The level at which to generate an isosurface
    
    Returns an array of vertex coordinates (Nv, 3) and an array of 
    per-face vertex indexes (Nf, 3)    
    """
    ## For improvement, see:
    ## 
    ## Efficient implementation of Marching Cubes' cases with topological guarantees.
    ## Thomas Lewiner, Helio Lopes, Antonio Wilson Vieira and Geovan Tavares.
    ## Journal of Graphics Tools 8(2): pp. 1-15 (december 2003)
    
    ## Precompute lookup tables on the first run
    global IsosurfaceDataCache
    if IsosurfaceDataCache is None:
        ## map from grid cell index to edge index.
        ## grid cell index tells us which corners are below the isosurface,
        ## edge index tells us which edges are cut by the isosurface.
        ## (Data stolen from Bourk; see above.)
        edgeTable = np.array([
            0x0  , 0x109, 0x203, 0x30a, 0x406, 0x50f, 0x605, 0x70c,
            0x80c, 0x905, 0xa0f, 0xb06, 0xc0a, 0xd03, 0xe09, 0xf00,
            0x190, 0x99 , 0x393, 0x29a, 0x596, 0x49f, 0x795, 0x69c,
            0x99c, 0x895, 0xb9f, 0xa96, 0xd9a, 0xc93, 0xf99, 0xe90,
            0x230, 0x339, 0x33 , 0x13a, 0x636, 0x73f, 0x435, 0x53c,
            0xa3c, 0xb35, 0x83f, 0x936, 0xe3a, 0xf33, 0xc39, 0xd30,
            0x3a0, 0x2a9, 0x1a3, 0xaa , 0x7a6, 0x6af, 0x5a5, 0x4ac,
            0xbac, 0xaa5, 0x9af, 0x8a6, 0xfaa, 0xea3, 0xda9, 0xca0,
            0x460, 0x569, 0x663, 0x76a, 0x66 , 0x16f, 0x265, 0x36c,
            0xc6c, 0xd65, 0xe6f, 0xf66, 0x86a, 0x963, 0xa69, 0xb60,
            0x5f0, 0x4f9, 0x7f3, 0x6fa, 0x1f6, 0xff , 0x3f5, 0x2fc,
            0xdfc, 0xcf5, 0xfff, 0xef6, 0x9fa, 0x8f3, 0xbf9, 0xaf0,
            0x650, 0x759, 0x453, 0x55a, 0x256, 0x35f, 0x55 , 0x15c,
            0xe5c, 0xf55, 0xc5f, 0xd56, 0xa5a, 0xb53, 0x859, 0x950,
            0x7c0, 0x6c9, 0x5c3, 0x4ca, 0x3c6, 0x2cf, 0x1c5, 0xcc ,
            0xfcc, 0xec5, 0xdcf, 0xcc6, 0xbca, 0xac3, 0x9c9, 0x8c0,
            0x8c0, 0x9c9, 0xac3, 0xbca, 0xcc6, 0xdcf, 0xec5, 0xfcc,
            0xcc , 0x1c5, 0x2cf, 0x3c6, 0x4ca, 0x5c3, 0x6c9, 0x7c0,
            0x950, 0x859, 0xb53, 0xa5a, 0xd56, 0xc5f, 0xf55, 0xe5c,
            0x15c, 0x55 , 0x35f, 0x256, 0x55a, 0x453, 0x759, 0x650,
            0xaf0, 0xbf9, 0x8f3, 0x9fa, 0xef6, 0xfff, 0xcf5, 0xdfc,
            0x2fc, 0x3f5, 0xff , 0x1f6, 0x6fa, 0x7f3, 0x4f9, 0x5f0,
            0xb60, 0xa69, 0x963, 0x86a, 0xf66, 0xe6f, 0xd65, 0xc6c,
            0x36c, 0x265, 0x16f, 0x66 , 0x76a, 0x663, 0x569, 0x460,
            0xca0, 0xda9, 0xea3, 0xfaa, 0x8a6, 0x9af, 0xaa5, 0xbac,
            0x4ac, 0x5a5, 0x6af, 0x7a6, 0xaa , 0x1a3, 0x2a9, 0x3a0,
            0xd30, 0xc39, 0xf33, 0xe3a, 0x936, 0x83f, 0xb35, 0xa3c,
            0x53c, 0x435, 0x73f, 0x636, 0x13a, 0x33 , 0x339, 0x230,
            0xe90, 0xf99, 0xc93, 0xd9a, 0xa96, 0xb9f, 0x895, 0x99c,
            0x69c, 0x795, 0x49f, 0x596, 0x29a, 0x393, 0x99 , 0x190,
            0xf00, 0xe09, 0xd03, 0xc0a, 0xb06, 0xa0f, 0x905, 0x80c,
            0x70c, 0x605, 0x50f, 0x406, 0x30a, 0x203, 0x109, 0x0   
            ], dtype=np.uint16)
        
        ## Table of triangles to use for filling each grid cell.
        ## Each set of three integers tells us which three edges to
        ## draw a triangle between.
        ## (Data stolen from Bourk; see above.)
        triTable = [
            [],
            [0, 8, 3],
            [0, 1, 9],
            [1, 8, 3, 9, 8, 1],
            [1, 2, 10],
            [0, 8, 3, 1, 2, 10],
            [9, 2, 10, 0, 2, 9],
            [2, 8, 3, 2, 10, 8, 10, 9, 8],
            [3, 11, 2],
            [0, 11, 2, 8, 11, 0],
            [1, 9, 0, 2, 3, 11],
            [1, 11, 2, 1, 9, 11, 9, 8, 11],
            [3, 10, 1, 11, 10, 3],
            [0, 10, 1, 0, 8, 10, 8, 11, 10],
            [3, 9, 0, 3, 11, 9, 11, 10, 9],
            [9, 8, 10, 10, 8, 11],
            [4, 7, 8],
            [4, 3, 0, 7, 3, 4],
            [0, 1, 9, 8, 4, 7],
            [4, 1, 9, 4, 7, 1, 7, 3, 1],
            [1, 2, 10, 8, 4, 7],
            [3, 4, 7, 3, 0, 4, 1, 2, 10],
            [9, 2, 10, 9, 0, 2, 8, 4, 7],
            [2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4],
            [8, 4, 7, 3, 11, 2],
            [11, 4, 7, 11, 2, 4, 2, 0, 4],
            [9, 0, 1, 8, 4, 7, 2, 3, 11],
            [4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1],
            [3, 10, 1, 3, 11, 10, 7, 8, 4],
            [1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4],
            [4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3],
            [4, 7, 11, 4, 11, 9, 9, 11, 10],
            [9, 5, 4],
            [9, 5, 4, 0, 8, 3],
            [0, 5, 4, 1, 5, 0],
            [8, 5, 4, 8, 3, 5, 3, 1, 5],
            [1, 2, 10, 9, 5, 4],
            [3, 0, 8, 1, 2, 10, 4, 9, 5],
            [5, 2, 10, 5, 4, 2, 4, 0, 2],
            [2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8],
            [9, 5, 4, 2, 3, 11],
            [0, 11, 2, 0, 8, 11, 4, 9, 5],
            [0, 5, 4, 0, 1, 5, 2, 3, 11],
            [2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5],
            [10, 3, 11, 10, 1, 3, 9, 5, 4],
            [4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10],
            [5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3],
            [5, 4, 8, 5, 8, 10, 10, 8, 11],
            [9, 7, 8, 5, 7, 9],
            [9, 3, 0, 9, 5, 3, 5, 7, 3],
            [0, 7, 8, 0, 1, 7, 1, 5, 7],
            [1, 5, 3, 3, 5, 7],
            [9, 7, 8, 9, 5, 7, 10, 1, 2],
            [10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3],
            [8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2],
            [2, 10, 5, 2, 5, 3, 3, 5, 7],
            [7, 9, 5, 7, 8, 9, 3, 11, 2],
            [9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11],
            [2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7],
            [11, 2, 1, 11, 1, 7, 7, 1, 5],
            [9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11],
            [5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0],
            [11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0],
            [11, 10, 5, 7, 11, 5],
            [10, 6, 5],
            [0, 8, 3, 5, 10, 6],
            [9, 0, 1, 5, 10, 6],
            [1, 8, 3, 1, 9, 8, 5, 10, 6],
            [1, 6, 5, 2, 6, 1],
            [1, 6, 5, 1, 2, 6, 3, 0, 8],
            [9, 6, 5, 9, 0, 6, 0, 2, 6],
            [5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8],
            [2, 3, 11, 10, 6, 5],
            [11, 0, 8, 11, 2, 0, 10, 6, 5],
            [0, 1, 9, 2, 3, 11, 5, 10, 6],
            [5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11],
            [6, 3, 11, 6, 5, 3, 5, 1, 3],
            [0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6],
            [3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9],
            [6, 5, 9, 6, 9, 11, 11, 9, 8],
            [5, 10, 6, 4, 7, 8],
            [4, 3, 0, 4, 7, 3, 6, 5, 10],
            [1, 9, 0, 5, 10, 6, 8, 4, 7],
            [10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4],
            [6, 1, 2, 6, 5, 1, 4, 7, 8],
            [1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7],
            [8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6],
            [7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9],
            [3, 11, 2, 7, 8, 4, 10, 6, 5],
            [5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11],
            [0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6],
            [9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6],
            [8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6],
            [5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11],
            [0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7],
            [6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9],
            [10, 4, 9, 6, 4, 10],
            [4, 10, 6, 4, 9, 10, 0, 8, 3],
            [10, 0, 1, 10, 6, 0, 6, 4, 0],
            [8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10],
            [1, 4, 9, 1, 2, 4, 2, 6, 4],
            [3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4],
            [0, 2, 4, 4, 2, 6],
            [8, 3, 2, 8, 2, 4, 4, 2, 6],
            [10, 4, 9, 10, 6, 4, 11, 2, 3],
            [0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6],
            [3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10],
            [6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1],
            [9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3],
            [8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1],
            [3, 11, 6, 3, 6, 0, 0, 6, 4],
            [6, 4, 8, 11, 6, 8],
            [7, 10, 6, 7, 8, 10, 8, 9, 10],
            [0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10],
            [10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0],
            [10, 6, 7, 10, 7, 1, 1, 7, 3],
            [1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7],
            [2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9],
            [7, 8, 0, 7, 0, 6, 6, 0, 2],
            [7, 3, 2, 6, 7, 2],
            [2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7],
            [2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7],
            [1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11],
            [11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1],
            [8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6],
            [0, 9, 1, 11, 6, 7],
            [7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0],
            [7, 11, 6],
            [7, 6, 11],
            [3, 0, 8, 11, 7, 6],
            [0, 1, 9, 11, 7, 6],
            [8, 1, 9, 8, 3, 1, 11, 7, 6],
            [10, 1, 2, 6, 11, 7],
            [1, 2, 10, 3, 0, 8, 6, 11, 7],
            [2, 9, 0, 2, 10, 9, 6, 11, 7],
            [6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8],
            [7, 2, 3, 6, 2, 7],
            [7, 0, 8, 7, 6, 0, 6, 2, 0],
            [2, 7, 6, 2, 3, 7, 0, 1, 9],
            [1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6],
            [10, 7, 6, 10, 1, 7, 1, 3, 7],
            [10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8],
            [0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7],
            [7, 6, 10, 7, 10, 8, 8, 10, 9],
            [6, 8, 4, 11, 8, 6],
            [3, 6, 11, 3, 0, 6, 0, 4, 6],
            [8, 6, 11, 8, 4, 6, 9, 0, 1],
            [9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6],
            [6, 8, 4, 6, 11, 8, 2, 10, 1],
            [1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6],
            [4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9],
            [10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3],
            [8, 2, 3, 8, 4, 2, 4, 6, 2],
            [0, 4, 2, 4, 6, 2],
            [1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8],
            [1, 9, 4, 1, 4, 2, 2, 4, 6],
            [8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1],
            [10, 1, 0, 10, 0, 6, 6, 0, 4],
            [4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3],
            [10, 9, 4, 6, 10, 4],
            [4, 9, 5, 7, 6, 11],
            [0, 8, 3, 4, 9, 5, 11, 7, 6],
            [5, 0, 1, 5, 4, 0, 7, 6, 11],
            [11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5],
            [9, 5, 4, 10, 1, 2, 7, 6, 11],
            [6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5],
            [7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2],
            [3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6],
            [7, 2, 3, 7, 6, 2, 5, 4, 9],
            [9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7],
            [3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0],
            [6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8],
            [9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7],
            [1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4],
            [4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10],
            [7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10],
            [6, 9, 5, 6, 11, 9, 11, 8, 9],
            [3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5],
            [0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11],
            [6, 11, 3, 6, 3, 5, 5, 3, 1],
            [1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6],
            [0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10],
            [11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5],
            [6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3],
            [5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2],
            [9, 5, 6, 9, 6, 0, 0, 6, 2],
            [1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8],
            [1, 5, 6, 2, 1, 6],
            [1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6],
            [10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0],
            [0, 3, 8, 5, 6, 10],
            [10, 5, 6],
            [11, 5, 10, 7, 5, 11],
            [11, 5, 10, 11, 7, 5, 8, 3, 0],
            [5, 11, 7, 5, 10, 11, 1, 9, 0],
            [10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1],
            [11, 1, 2, 11, 7, 1, 7, 5, 1],
            [0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11],
            [9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7],
            [7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2],
            [2, 5, 10, 2, 3, 5, 3, 7, 5],
            [8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5],
            [9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2],
            [9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2],
            [1, 3, 5, 3, 7, 5],
            [0, 8, 7, 0, 7, 1, 1, 7, 5],
            [9, 0, 3, 9, 3, 5, 5, 3, 7],
            [9, 8, 7, 5, 9, 7],
            [5, 8, 4, 5, 10, 8, 10, 11, 8],
            [5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0],
            [0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5],
            [10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4],
            [2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8],
            [0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11],
            [0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5],
            [9, 4, 5, 2, 11, 3],
            [2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4],
            [5, 10, 2, 5, 2, 4, 4, 2, 0],
            [3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9],
            [5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2],
            [8, 4, 5, 8, 5, 3, 3, 5, 1],
            [0, 4, 5, 1, 0, 5],
            [8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5],
            [9, 4, 5],
            [4, 11, 7, 4, 9, 11, 9, 10, 11],
            [0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11],
            [1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11],
            [3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4],
            [4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2],
            [9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3],
            [11, 7, 4, 11, 4, 2, 2, 4, 0],
            [11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4],
            [2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9],
            [9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7],
            [3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10],
            [1, 10, 2, 8, 7, 4],
            [4, 9, 1, 4, 1, 7, 7, 1, 3],
            [4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1],
            [4, 0, 3, 7, 4, 3],
            [4, 8, 7],
            [9, 10, 8, 10, 11, 8],
            [3, 0, 9, 3, 9, 11, 11, 9, 10],
            [0, 1, 10, 0, 10, 8, 8, 10, 11],
            [3, 1, 10, 11, 3, 10],
            [1, 2, 11, 1, 11, 9, 9, 11, 8],
            [3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9],
            [0, 2, 11, 8, 0, 11],
            [3, 2, 11],
            [2, 3, 8, 2, 8, 10, 10, 8, 9],
            [9, 10, 2, 0, 9, 2],
            [2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8],
            [1, 10, 2],
            [1, 3, 8, 9, 1, 8],
            [0, 9, 1],
            [0, 3, 8],
            []
        ]    
        edgeShifts = np.array([  ## maps edge ID (0-11) to (x,y,z) cell offset and edge ID (0-2)
            [0, 0, 0, 0],   
            [1, 0, 0, 1],
            [0, 1, 0, 0],
            [0, 0, 0, 1],
            [0, 0, 1, 0],
            [1, 0, 1, 1],
            [0, 1, 1, 0],
            [0, 0, 1, 1],
            [0, 0, 0, 2],
            [1, 0, 0, 2],
            [1, 1, 0, 2],
            [0, 1, 0, 2],
            #[9, 9, 9, 9]  ## fake
        ], dtype=np.uint16) # don't use ubyte here! This value gets added to cell index later; will need the extra precision.
        nTableFaces = np.array([len(f)/3 for f in triTable], dtype=np.ubyte)
        faceShiftTables = [None]
        for i in range(1,6):
            ## compute lookup table of index: vertexes mapping
            faceTableI = np.zeros((len(triTable), i*3), dtype=np.ubyte)
            faceTableInds = np.argwhere(nTableFaces == i)
            faceTableI[faceTableInds[:,0]] = np.array([triTable[j[0]] for j in faceTableInds])
            faceTableI = faceTableI.reshape((len(triTable), i, 3))
            faceShiftTables.append(edgeShifts[faceTableI])
            
        ## Let's try something different:
        #faceTable = np.empty((256, 5, 3, 4), dtype=np.ubyte)   # (grid cell index, faces, vertexes, edge lookup)
        #for i,f in enumerate(triTable):
            #f = np.array(f + [12] * (15-len(f))).reshape(5,3)
            #faceTable[i] = edgeShifts[f]
        
        
        IsosurfaceDataCache = (faceShiftTables, edgeShifts, edgeTable, nTableFaces)
    else:
        faceShiftTables, edgeShifts, edgeTable, nTableFaces = IsosurfaceDataCache

    # We use strides below, which means we need contiguous array input.
    # Ideally we can fix this just by removing the dependency on strides.
    if not data.flags['C_CONTIGUOUS']:
        raise TypeError("isosurface input data must be c-contiguous.")
    
    ## mark everything below the isosurface level
    mask = data < level
    
    ### make eight sub-fields and compute indexes for grid cells
    index = np.zeros([x-1 for x in data.shape], dtype=np.ubyte)
    fields = np.empty((2,2,2), dtype=object)
    slices = [slice(0,-1), slice(1,None)]
    for i in [0,1]:
        for j in [0,1]:
            for k in [0,1]:
                fields[i,j,k] = mask[slices[i], slices[j], slices[k]]
                vertIndex = i - 2*j*i + 3*j + 4*k  ## this is just to match Bourk's vertex numbering scheme
                np.add(index, fields[i,j,k] * 2**vertIndex, out=index, casting='unsafe')
    
    ### Generate table of edges that have been cut
    cutEdges = np.zeros([x+1 for x in index.shape]+[3], dtype=np.uint32)
    edges = edgeTable[index]
    for i, shift in enumerate(edgeShifts[:12]):        
        slices = [slice(shift[j],cutEdges.shape[j]+(shift[j]-1)) for j in range(3)]
        cutEdges[slices[0], slices[1], slices[2], shift[3]] += edges & 2**i
    
    ## for each cut edge, interpolate to see where exactly the edge is cut and generate vertex positions
    m = cutEdges > 0
    vertexInds = np.argwhere(m)   ## argwhere is slow!
    vertexes = vertexInds[:,:3].astype(np.float32)
    dataFlat = data.reshape(data.shape[0]*data.shape[1]*data.shape[2])
    
    ## re-use the cutEdges array as a lookup table for vertex IDs
    cutEdges[vertexInds[:,0], vertexInds[:,1], vertexInds[:,2], vertexInds[:,3]] = np.arange(vertexInds.shape[0])
    
    for i in [0,1,2]:
        vim = vertexInds[:,3] == i
        vi = vertexInds[vim, :3]
        viFlat = (vi * (np.array(data.strides[:3]) // data.itemsize)[np.newaxis,:]).sum(axis=1)
        v1 = dataFlat[viFlat]
        v2 = dataFlat[viFlat + data.strides[i]//data.itemsize]
        vertexes[vim,i] += (level-v1) / (v2-v1)
    
    ### compute the set of vertex indexes for each face. 
    
    ## This works, but runs a bit slower.
    #cells = np.argwhere((index != 0) & (index != 255))  ## all cells with at least one face
    #cellInds = index[cells[:,0], cells[:,1], cells[:,2]]
    #verts = faceTable[cellInds]
    #mask = verts[...,0,0] != 9
    #verts[...,:3] += cells[:,np.newaxis,np.newaxis,:]  ## we now have indexes into cutEdges
    #verts = verts[mask]
    #faces = cutEdges[verts[...,0], verts[...,1], verts[...,2], verts[...,3]]  ## and these are the vertex indexes we want.
    
    
    ## To allow this to be vectorized efficiently, we count the number of faces in each 
    ## grid cell and handle each group of cells with the same number together.
    ## determine how many faces to assign to each grid cell
    nFaces = nTableFaces[index]
    totFaces = nFaces.sum()
    faces = np.empty((totFaces, 3), dtype=np.uint32)
    ptr = 0
    #import debug
    #p = debug.Profiler()
    
    ## this helps speed up an indexing operation later on
    cs = np.array(cutEdges.strides)//cutEdges.itemsize
    cutEdges = cutEdges.flatten()

    ## this, strangely, does not seem to help.
    #ins = np.array(index.strides)/index.itemsize
    #index = index.flatten()

    for i in range(1,6):
        ### expensive:
        #profiler()
        cells = np.argwhere(nFaces == i)  ## all cells which require i faces  (argwhere is expensive)
        #profiler()
        if cells.shape[0] == 0:
            continue
        cellInds = index[cells[:,0], cells[:,1], cells[:,2]]   ## index values of cells to process for this round
        #profiler()
        
        ### expensive:
        verts = faceShiftTables[i][cellInds]
        #profiler()
        np.add(verts[...,:3], cells[:,np.newaxis,np.newaxis,:], out=verts[...,:3], casting='unsafe')  ## we now have indexes into cutEdges
        verts = verts.reshape((verts.shape[0]*i,)+verts.shape[2:])
        #profiler()
        
        ### expensive:
        verts = (verts * cs[np.newaxis, np.newaxis, :]).sum(axis=2)
        vertInds = cutEdges[verts]
        #profiler()
        nv = vertInds.shape[0]
        #profiler()
        faces[ptr:ptr+nv] = vertInds #.reshape((nv, 3))
        #profiler()
        ptr += nv
        
    return vertexes, faces

    
def _pinv_fallback(tr):
    arr = np.array([tr.m11(), tr.m12(), tr.m13(),
                    tr.m21(), tr.m22(), tr.m23(),
                    tr.m31(), tr.m32(), tr.m33()])
    arr.shape = (3, 3)
    pinv = np.linalg.pinv(arr)
    return QtGui.QTransform(*pinv.ravel().tolist())


def invertQTransform(tr):
    """Return a QTransform that is the inverse of *tr*.
    A pseudo-inverse is returned if tr is not invertible.
    
    Note that this function is preferred over QTransform.inverted() due to
    bugs in that method. (specifically, Qt has floating-point precision issues
    when determining whether a matrix is invertible)
    """
    try:
        det = tr.determinant()
        detr = 1.0 / det    # let singular matrices raise ZeroDivisionError
        inv = tr.adjoint()
        inv *= detr
        return inv
    except ZeroDivisionError:
        return _pinv_fallback(tr)
    

def pseudoScatter(data, spacing=None, shuffle=True, bidir=False, method='exact'):
    """Return an array of position values needed to make beeswarm or column scatter plots.
    
    Used for examining the distribution of values in an array.
    
    Given an array of x-values, construct an array of y-values such that an x,y scatter-plot
    will not have overlapping points (it will look similar to a histogram).
    """
    if method == 'exact':
        return _pseudoScatterExact(data, spacing=spacing, shuffle=shuffle, bidir=bidir)
    elif method == 'histogram':
        return _pseudoScatterHistogram(data, spacing=spacing, shuffle=shuffle, bidir=bidir)


def _pseudoScatterHistogram(data, spacing=None, shuffle=True, bidir=False):
    """Works by binning points into a histogram and spreading them out to fill the bin.
    
    Faster method, but can produce blocky results.
    """
    inds = np.arange(len(data))
    if shuffle:
        np.random.shuffle(inds)
        
    data = data[inds]
    
    if spacing is None:
        spacing = 2.*np.std(data)/len(data)**0.5

    yvals = np.empty(len(data))
    
    dmin = data.min()
    dmax = data.max()
    nbins = int((dmax-dmin) / spacing) + 1
    bins = np.linspace(dmin, dmax, nbins)
    dx = bins[1] - bins[0]
    dbins = ((data - bins[0]) / dx).astype(int)
    binCounts = {}
        
    for i,j in enumerate(dbins):
        c = binCounts.get(j, -1) + 1
        binCounts[j] = c
        yvals[i] = c

    if bidir is True:
        for i in range(nbins):
            yvals[dbins==i] -= binCounts.get(i, 0) * 0.5

    return yvals[np.argsort(inds)]  ## un-shuffle values before returning


def _pseudoScatterExact(data, spacing=None, shuffle=True, bidir=False):
    """Works by stacking points up one at a time, searching for the lowest position available at each point.
    
    This method produces nice, smooth results but can be prohibitively slow for large datasets.
    """
    inds = np.arange(len(data))
    if shuffle:
        np.random.shuffle(inds)
        
    data = data[inds]
    
    if spacing is None:
        spacing = 2.*np.std(data)/len(data)**0.5
    s2 = spacing**2
    
    yvals = np.empty(len(data))
    if len(data) == 0:
        return yvals
    yvals[0] = 0
    for i in range(1,len(data)):
        x = data[i]     # current x value to be placed
        x0 = data[:i]   # all x values already placed
        y0 = yvals[:i]  # all y values already placed
        y = 0
        
        dx = (x0-x)**2  # x-distance to each previous point
        xmask = dx < s2  # exclude anything too far away
        
        if xmask.sum() > 0:
            if bidir:
                dirs = [-1, 1]
            else:
                dirs = [1]
            yopts = []
            for direction in dirs:
                y = 0
                dx2 = dx[xmask]
                dy = (s2 - dx2)**0.5   
                limits = np.empty((2,len(dy)))  # ranges of y-values to exclude
                limits[0] = y0[xmask] - dy
                limits[1] = y0[xmask] + dy    
                while True:
                    # ignore anything below this y-value
                    if direction > 0:
                        mask = limits[1] >= y
                    else:
                        mask = limits[0] <= y
                        
                    limits2 = limits[:,mask]
                    
                    # are we inside an excluded region?
                    mask = (limits2[0] < y) & (limits2[1] > y)
                    if mask.sum() == 0:
                        break
                        
                    if direction > 0:
                        y = limits2[:,mask].max()
                    else:
                        y = limits2[:,mask].min()
                yopts.append(y)
            if bidir:
                y = yopts[0] if -yopts[0] < yopts[1] else yopts[1]
            else:
                y = yopts[0]
        yvals[i] = y
    
    return yvals[np.argsort(inds)]  ## un-shuffle values before returning



def toposort(deps, nodes=None, seen=None, stack=None, depth=0):
    """Topological sort. Arguments are:
      deps    dictionary describing dependencies where a:[b,c] means "a depends on b and c"
      nodes   optional, specifies list of starting nodes (these should be the nodes 
              which are not depended on by any other nodes). Other candidate starting
              nodes will be ignored.
              
    Example::

        # Sort the following graph:
        # 
        #   B ──┬─────> C <── D
        #       │       │       
        #   E <─┴─> A <─┘
        #     
        deps = {'a': ['b', 'c'], 'c': ['b', 'd'], 'e': ['b']}
        toposort(deps)
         => ['b', 'd', 'c', 'a', 'e']
    """
    # fill in empty dep lists
    deps = deps.copy()
    for k,v in list(deps.items()):
        for k in v:
            if k not in deps:
                deps[k] = []
    
    if nodes is None:
        ## run through deps to find nodes that are not depended upon
        rem = set()
        for dep in deps.values():
            rem |= set(dep)
        nodes = set(deps.keys()) - rem
    if seen is None:
        seen = set()
        stack = []
    sorted = []
    for n in nodes:
        if n in stack:
            raise Exception("Cyclic dependency detected", stack + [n])
        if n in seen:
            continue
        seen.add(n)
        sorted.extend( toposort(deps, deps[n], seen, stack+[n], depth=depth+1))
        sorted.append(n)
    return sorted


def disconnect(signal, slot):
    """Disconnect a Qt signal from a slot.

    This method augments Qt's Signal.disconnect():

      * Return bool indicating whether disconnection was successful, rather than
        raising an exception
      * Attempt to disconnect prior versions of the slot when using pg.reload
    """
    while True:
        try:
            signal.disconnect(slot)
            return True
        except (TypeError, RuntimeError):
            slot = reload.getPreviousVersion(slot)
            if slot is None:
                return False


class SignalBlock(object):
    """Class used to temporarily block a Qt signal connection::

        with SignalBlock(signal, slot):
            # do something that emits a signal; it will
            # not be delivered to slot
    """
    def __init__(self, signal, slot):
        self.signal = signal
        self.slot = slot

    def __enter__(self):
        self.reconnect = disconnect(self.signal, self.slot)
        return self

    def __exit__(self, *args):
        if self.reconnect:
            self.signal.connect(self.slot)