text: import itertools
  import math
  import numpy as np
  from scipy import ndimage as ndi
  from collections import OrderedDict
  from collections.abc import Iterable
  from ..exposure import histogram
  from .._shared.utils import check_nD, warn
  from ..transform import integral_image
  from ..util import crop, dtype_limits
  from ..filters._multiotsu import _find_threshold_multiotsu
  
  
  __all__ = ['try_all_threshold',
             'threshold_otsu',
             'threshold_yen',
             'threshold_isodata',
             'threshold_li',
             'threshold_local',
             'threshold_minimum',
             'threshold_mean',
             'threshold_niblack',
             'threshold_sauvola',
             'threshold_triangle',
             'apply_hysteresis_threshold',
             'threshold_multiotsu']
  
  
  def _try_all(image, methods=None, figsize=None, num_cols=2, verbose=True):
      """Returns a figure comparing the outputs of different methods.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Input image.
      methods : dict, optional
          Names and associated functions.
          Functions must take and return an image.
      figsize : tuple, optional
          Figure size (in inches).
      num_cols : int, optional
          Number of columns.
      verbose : bool, optional
          Print function name for each method.
  
      Returns
      -------
      fig, ax : tuple
          Matplotlib figure and axes.
      """
      from matplotlib import pyplot as plt
  
      # Handle default value
      methods = methods or {}
  
      num_rows = math.ceil((len(methods) + 1.) / num_cols)
      fig, ax = plt.subplots(num_rows, num_cols, figsize=figsize,
                             sharex=True, sharey=True)
      ax = ax.ravel()
  
      ax[0].imshow(image, cmap=plt.cm.gray)
      ax[0].set_title('Original')
  
      i = 1
      for name, func in methods.items():
          ax[i].set_title(name)
          try:
              ax[i].imshow(func(image), cmap=plt.cm.gray)
          except Exception as e:
              ax[i].text(0.5, 0.5, "%s" % type(e).__name__,
                         ha="center", va="center", transform=ax[i].transAxes)
          i += 1
          if verbose:
              print(func.__orifunc__)
  
      for a in ax:
          a.axis('off')
  
      fig.tight_layout()
      return fig, ax
  
  
  def try_all_threshold(image, figsize=(8, 5), verbose=True):
      """Returns a figure comparing the outputs of different thresholding methods.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Input image.
      figsize : tuple, optional
          Figure size (in inches).
      verbose : bool, optional
          Print function name for each method.
  
      Returns
      -------
      fig, ax : tuple
          Matplotlib figure and axes.
  
      Notes
      -----
      The following algorithms are used:
  
      * isodata
      * li
      * mean
      * minimum
      * otsu
      * triangle
      * yen
  
      Examples
      --------
      >>> from skimage.data import text
      >>> fig, ax = try_all_threshold(text(), figsize=(10, 6), verbose=False)
      """
      def thresh(func):
          """
          A wrapper function to return a thresholded image.
          """
          def wrapper(im):
              return im > func(im)
          try:
              wrapper.__orifunc__ = func.__orifunc__
          except AttributeError:
              wrapper.__orifunc__ = func.__module__ + '.' + func.__name__
          return wrapper
  
      # Global algorithms.
      methods = OrderedDict({'Isodata': thresh(threshold_isodata),
                             'Li': thresh(threshold_li),
                             'Mean': thresh(threshold_mean),
                             'Minimum': thresh(threshold_minimum),
                             'Otsu': thresh(threshold_otsu),
                             'Triangle': thresh(threshold_triangle),
                             'Yen': thresh(threshold_yen)})
  
      return _try_all(image, figsize=figsize,
                      methods=methods, verbose=verbose)
  
  
  def threshold_local(image, block_size, method='gaussian', offset=0,
                      mode='reflect', param=None, cval=0):
      """Compute a threshold mask image based on local pixel neighborhood.
  
      Also known as adaptive or dynamic thresholding. The threshold value is
      the weighted mean for the local neighborhood of a pixel subtracted by a
      constant. Alternatively the threshold can be determined dynamically by a
      given function, using the 'generic' method.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Input image.
      block_size : int
          Odd size of pixel neighborhood which is used to calculate the
          threshold value (e.g. 3, 5, 7, ..., 21, ...).
      method : {'generic', 'gaussian', 'mean', 'median'}, optional
          Method used to determine adaptive threshold for local neighbourhood in
          weighted mean image.
  
          * 'generic': use custom function (see ``param`` parameter)
          * 'gaussian': apply gaussian filter (see ``param`` parameter for custom\
                        sigma value)
          * 'mean': apply arithmetic mean filter
          * 'median': apply median rank filter
  
          By default the 'gaussian' method is used.
      offset : float, optional
          Constant subtracted from weighted mean of neighborhood to calculate
          the local threshold value. Default offset is 0.
      mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional
          The mode parameter determines how the array borders are handled, where
          cval is the value when mode is equal to 'constant'.
          Default is 'reflect'.
      param : {int, function}, optional
          Either specify sigma for 'gaussian' method or function object for
          'generic' method. This functions takes the flat array of local
          neighbourhood as a single argument and returns the calculated
          threshold for the centre pixel.
      cval : float, optional
          Value to fill past edges of input if mode is 'constant'.
  
      Returns
      -------
      threshold : (N, M) ndarray
          Threshold image. All pixels in the input image higher than the
          corresponding pixel in the threshold image are considered foreground.
  
      References
      ----------
      .. [1] https://docs.opencv.org/modules/imgproc/doc/miscellaneous_transformations.html?highlight=threshold#adaptivethreshold
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()[:50, :50]
      >>> binary_image1 = image > threshold_local(image, 15, 'mean')
      >>> func = lambda arr: arr.mean()
      >>> binary_image2 = image > threshold_local(image, 15, 'generic',
      ...                                         param=func)
      """
      if block_size % 2 == 0:
          raise ValueError("The kwarg ``block_size`` must be odd! Given "
                           "``block_size`` {0} is even.".format(block_size))
      check_nD(image, 2)
      thresh_image = np.zeros(image.shape, 'double')
      if method == 'generic':
          ndi.generic_filter(image, param, block_size,
                             output=thresh_image, mode=mode, cval=cval)
      elif method == 'gaussian':
          if param is None:
              # automatically determine sigma which covers > 99% of distribution
              sigma = (block_size - 1) / 6.0
          else:
              sigma = param
          ndi.gaussian_filter(image, sigma, output=thresh_image, mode=mode,
                              cval=cval)
      elif method == 'mean':
          mask = 1. / block_size * np.ones((block_size,))
          # separation of filters to speedup convolution
          ndi.convolve1d(image, mask, axis=0, output=thresh_image, mode=mode,
                         cval=cval)
          ndi.convolve1d(thresh_image, mask, axis=1, output=thresh_image,
                         mode=mode, cval=cval)
      elif method == 'median':
          ndi.median_filter(image, block_size, output=thresh_image, mode=mode,
                            cval=cval)
      else:
          raise ValueError("Invalid method specified. Please use `generic`, "
                           "`gaussian`, `mean`, or `median`.")
  
      return thresh_image - offset
  
  
  def threshold_otsu(image, nbins=256):
      """Return threshold value based on Otsu's method.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Grayscale input image.
      nbins : int, optional
          Number of bins used to calculate histogram. This value is ignored for
          integer arrays.
  
      Returns
      -------
      threshold : float
          Upper threshold value. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      Raises
      ------
      ValueError
           If ``image`` only contains a single grayscale value.
  
      References
      ----------
      .. [1] Wikipedia, https://en.wikipedia.org/wiki/Otsu's_Method
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()
      >>> thresh = threshold_otsu(image)
      >>> binary = image <= thresh
  
      Notes
      -----
      The input image must be grayscale.
      """
      if len(image.shape) > 2 and image.shape[-1] in (3, 4):
          msg = "threshold_otsu is expected to work correctly only for " \
                "grayscale images; image shape {0} looks like an RGB image"
          warn(msg.format(image.shape))
  
      # Check if the image is multi-colored or not
      if image.min() == image.max():
          raise ValueError("threshold_otsu is expected to work with images "
                           "having more than one color. The input image seems "
                           "to have just one color {0}.".format(image.min()))
  
      hist, bin_centers = histogram(image.ravel(), nbins, source_range='image')
      hist = hist.astype(float)
  
      # class probabilities for all possible thresholds
      weight1 = np.cumsum(hist)
      weight2 = np.cumsum(hist[::-1])[::-1]
      # class means for all possible thresholds
      mean1 = np.cumsum(hist * bin_centers) / weight1
      mean2 = (np.cumsum((hist * bin_centers)[::-1]) / weight2[::-1])[::-1]
  
      # Clip ends to align class 1 and class 2 variables:
      # The last value of ``weight1``/``mean1`` should pair with zero values in
      # ``weight2``/``mean2``, which do not exist.
      variance12 = weight1[:-1] * weight2[1:] * (mean1[:-1] - mean2[1:]) ** 2
  
      idx = np.argmax(variance12)
      threshold = bin_centers[:-1][idx]
      return threshold
  
  
  def threshold_yen(image, nbins=256):
      """Return threshold value based on Yen's method.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Input image.
      nbins : int, optional
          Number of bins used to calculate histogram. This value is ignored for
          integer arrays.
  
      Returns
      -------
      threshold : float
          Upper threshold value. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      References
      ----------
      .. [1] Yen J.C., Chang F.J., and Chang S. (1995) "A New Criterion
             for Automatic Multilevel Thresholding" IEEE Trans. on Image
             Processing, 4(3): 370-378. :DOI:`10.1109/83.366472`
      .. [2] Sezgin M. and Sankur B. (2004) "Survey over Image Thresholding
             Techniques and Quantitative Performance Evaluation" Journal of
             Electronic Imaging, 13(1): 146-165, :DOI:`10.1117/1.1631315`
             http://www.busim.ee.boun.edu.tr/~sankur/SankurFolder/Threshold_survey.pdf
      .. [3] ImageJ AutoThresholder code, http://fiji.sc/wiki/index.php/Auto_Threshold
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()
      >>> thresh = threshold_yen(image)
      >>> binary = image <= thresh
      """
      hist, bin_centers = histogram(image.ravel(), nbins, source_range='image')
      # On blank images (e.g. filled with 0) with int dtype, `histogram()`
      # returns ``bin_centers`` containing only one value. Speed up with it.
      if bin_centers.size == 1:
          return bin_centers[0]
  
      # Calculate probability mass function
      pmf = hist.astype(np.float32) / hist.sum()
      P1 = np.cumsum(pmf)  # Cumulative normalized histogram
      P1_sq = np.cumsum(pmf ** 2)
      # Get cumsum calculated from end of squared array:
      P2_sq = np.cumsum(pmf[::-1] ** 2)[::-1]
      # P2_sq indexes is shifted +1. I assume, with P1[:-1] it's help avoid
      # '-inf' in crit. ImageJ Yen implementation replaces those values by zero.
      crit = np.log(((P1_sq[:-1] * P2_sq[1:]) ** -1) *
                    (P1[:-1] * (1.0 - P1[:-1])) ** 2)
      return bin_centers[crit.argmax()]
  
  
  def threshold_isodata(image, nbins=256, return_all=False):
      """Return threshold value(s) based on ISODATA method.
  
      Histogram-based threshold, known as Ridler-Calvard method or inter-means.
      Threshold values returned satisfy the following equality::
  
          threshold = (image[image <= threshold].mean() +
                       image[image > threshold].mean()) / 2.0
  
      That is, returned thresholds are intensities that separate the image into
      two groups of pixels, where the threshold intensity is midway between the
      mean intensities of these groups.
  
      For integer images, the above equality holds to within one; for floating-
      point images, the equality holds to within the histogram bin-width.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Input image.
      nbins : int, optional
          Number of bins used to calculate histogram. This value is ignored for
          integer arrays.
      return_all: bool, optional
          If False (default), return only the lowest threshold that satisfies
          the above equality. If True, return all valid thresholds.
  
      Returns
      -------
      threshold : float or int or array
          Threshold value(s).
  
      References
      ----------
      .. [1] Ridler, TW & Calvard, S (1978), "Picture thresholding using an
             iterative selection method"
             IEEE Transactions on Systems, Man and Cybernetics 8: 630-632,
             :DOI:`10.1109/TSMC.1978.4310039`
      .. [2] Sezgin M. and Sankur B. (2004) "Survey over Image Thresholding
             Techniques and Quantitative Performance Evaluation" Journal of
             Electronic Imaging, 13(1): 146-165,
             http://www.busim.ee.boun.edu.tr/~sankur/SankurFolder/Threshold_survey.pdf
             :DOI:`10.1117/1.1631315`
      .. [3] ImageJ AutoThresholder code,
             http://fiji.sc/wiki/index.php/Auto_Threshold
  
      Examples
      --------
      >>> from skimage.data import coins
      >>> image = coins()
      >>> thresh = threshold_isodata(image)
      >>> binary = image > thresh
      """
      hist, bin_centers = histogram(image.ravel(), nbins, source_range='image')
  
      # image only contains one unique value
      if len(bin_centers) == 1:
          if return_all:
              return bin_centers
          else:
              return bin_centers[0]
  
      hist = hist.astype(np.float32)
  
      # csuml and csumh contain the count of pixels in that bin or lower, and
      # in all bins strictly higher than that bin, respectively
      csuml = np.cumsum(hist)
      csumh = csuml[-1] - csuml
  
      # intensity_sum contains the total pixel intensity from each bin
      intensity_sum = hist * bin_centers
  
      # l and h contain average value of all pixels in that bin or lower, and
      # in all bins strictly higher than that bin, respectively.
      # Note that since exp.histogram does not include empty bins at the low or
      # high end of the range, csuml and csumh are strictly > 0, except in the
      # last bin of csumh, which is zero by construction.
      # So no worries about division by zero in the following lines, except
      # for the last bin, but we can ignore that because no valid threshold
      # can be in the top bin.
      # To avoid the division by zero, we simply skip over the last element in
      # all future computation.
      csum_intensity = np.cumsum(intensity_sum)
      lower = csum_intensity[:-1] / csuml[:-1]
      higher = (csum_intensity[-1] - csum_intensity[:-1]) / csumh[:-1]
  
      # isodata finds threshold values that meet the criterion t = (l + m)/2
      # where l is the mean of all pixels <= t and h is the mean of all pixels
      # > t, as calculated above. So we are looking for places where
      # (l + m) / 2 equals the intensity value for which those l and m figures
      # were calculated -- which is, of course, the histogram bin centers.
      # We only require this equality to be within the precision of the bin
      # width, of course.
      all_mean = (lower + higher) / 2.0
      bin_width = bin_centers[1] - bin_centers[0]
  
      # Look only at thresholds that are below the actual all_mean value,
      # for consistency with the threshold being included in the lower pixel
      # group. Otherwise can get thresholds that are not actually fixed-points
      # of the isodata algorithm. For float images, this matters less, since
      # there really can't be any guarantees anymore anyway.
      distances = all_mean - bin_centers[:-1]
      thresholds = bin_centers[:-1][(distances >= 0) & (distances < bin_width)]
  
      if return_all:
          return thresholds
      else:
          return thresholds[0]
  
  
  # Computing a histogram using np.histogram on a uint8 image with bins=256
  # doesn't work and results in aliasing problems. We use a fully specified set
  # of bins to ensure that each uint8 value false into its own bin.
  _DEFAULT_ENTROPY_BINS = tuple(np.arange(-0.5, 255.51, 1))
  
  
  def _cross_entropy(image, threshold, bins=_DEFAULT_ENTROPY_BINS):
      """Compute cross-entropy between distributions above and below a threshold.
  
      Parameters
      ----------
      image : array
          The input array of values.
      threshold : float
          The value dividing the foreground and background in ``image``.
      bins : int or array of float, optional
          The number of bins or the bin edges. (Any valid value to the ``bins``
          argument of ``np.histogram`` will work here.) For an exact calculation,
          each unique value should have its own bin. The default value for bins
          ensures exact handling of uint8 images: ``bins=256`` results in
          aliasing problems due to bin width not being equal to 1.
  
      Returns
      -------
      nu : float
          The cross-entropy target value as defined in [1]_.
  
      Notes
      -----
      See Li and Lee, 1993 [1]_; this is the objective function ``threshold_li``
      minimizes. This function can be improved but this implementation most
      closely matches equation 8 in [1]_ and equations 1-3 in [2]_.
  
      References
      ----------
      .. [1] Li C.H. and Lee C.K. (1993) "Minimum Cross Entropy Thresholding"
             Pattern Recognition, 26(4): 617-625
             :DOI:`10.1016/0031-3203(93)90115-D`
      .. [2] Li C.H. and Tam P.K.S. (1998) "An Iterative Algorithm for Minimum
             Cross Entropy Thresholding" Pattern Recognition Letters, 18(8): 771-776
             :DOI:`10.1016/S0167-8655(98)00057-9`
      """
      histogram, bin_edges = np.histogram(image, bins=bins, density=True)
      bin_centers = np.convolve(bin_edges, [0.5, 0.5], mode='valid')
      t = np.flatnonzero(bin_centers > threshold)[0]
      m0a = np.sum(histogram[:t])  # 0th moment, background
      m0b = np.sum(histogram[t:])
      m1a = np.sum(histogram[:t] * bin_centers[:t])  # 1st moment, background
      m1b = np.sum(histogram[t:] * bin_centers[t:])
      mua = m1a / m0a  # mean value, background
      mub = m1b / m0b
      nu = -m1a * np.log(mua) - m1b * np.log(mub)
      return nu
  
  
  def threshold_li(image, *, tolerance=None, initial_guess=None,
                   iter_callback=None):
      """Compute threshold value by Li's iterative Minimum Cross Entropy method.
  
      Parameters
      ----------
      image : ndarray
          Input image.
  
      tolerance : float, optional
          Finish the computation when the change in the threshold in an iteration
          is less than this value. By default, this is half the smallest
          difference between intensity values in ``image``.
  
      initial_guess : float or Callable[[array[float]], float], optional
          Li's iterative method uses gradient descent to find the optimal
          threshold. If the image intensity histogram contains more than two
          modes (peaks), the gradient descent could get stuck in a local optimum.
          An initial guess for the iteration can help the algorithm find the
          globally-optimal threshold. A float value defines a specific start
          point, while a callable should take in an array of image intensities
          and return a float value. Example valid callables include
          ``numpy.mean`` (default), ``lambda arr: numpy.quantile(arr, 0.95)``,
          or even :func:`skimage.filters.threshold_otsu`.
  
      iter_callback : Callable[[float], Any], optional
          A function that will be called on the threshold at every iteration of
          the algorithm.
  
      Returns
      -------
      threshold : float
          Upper threshold value. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      References
      ----------
      .. [1] Li C.H. and Lee C.K. (1993) "Minimum Cross Entropy Thresholding"
             Pattern Recognition, 26(4): 617-625
             :DOI:`10.1016/0031-3203(93)90115-D`
      .. [2] Li C.H. and Tam P.K.S. (1998) "An Iterative Algorithm for Minimum
             Cross Entropy Thresholding" Pattern Recognition Letters, 18(8): 771-776
             :DOI:`10.1016/S0167-8655(98)00057-9`
      .. [3] Sezgin M. and Sankur B. (2004) "Survey over Image Thresholding
             Techniques and Quantitative Performance Evaluation" Journal of
             Electronic Imaging, 13(1): 146-165
             :DOI:`10.1117/1.1631315`
      .. [4] ImageJ AutoThresholder code, http://fiji.sc/wiki/index.php/Auto_Threshold
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()
      >>> thresh = threshold_li(image)
      >>> binary = image > thresh
      """
      # Remove nan:
      image = image[~np.isnan(image)]
      if image.size == 0:
          return np.nan
  
      # Make sure image has more than one value; otherwise, return that value
      # This works even for np.inf
      if np.all(image == image.flat[0]):
          return image.flat[0]
  
      # At this point, the image only contains np.inf, -np.inf, or valid numbers
      image = image[np.isfinite(image)]
      # if there are no finite values in the image, return 0. This is because
      # at this point we *know* that there are *both* inf and -inf values,
      # because inf == inf evaluates to True. We might as well separate them.
      if image.size == 0:
          return 0.
  
      # Li's algorithm requires positive image (because of log(mean))
      image_min = np.min(image)
      image -= image_min
      tolerance = tolerance or np.min(np.diff(np.unique(image))) / 2
  
      # Initial estimate for iteration. See "initial_guess" in the parameter list
      if initial_guess is None:
          t_next = np.mean(image)
      elif callable(initial_guess):
          t_next = initial_guess(image)
      elif np.isscalar(initial_guess):  # convert to new, positive image range
          t_next = initial_guess - image_min
          image_max = np.max(image) + image_min
          if not 0 < t_next < np.max(image):
              msg = ('The initial guess for threshold_li must be within the '
                     'range of the image. Got {} for image min {} and max {} '
                     .format(initial_guess, image_min, image_max))
              raise ValueError(msg)
      else:
          raise TypeError('Incorrect type for `initial_guess`; should be '
                          'a floating point value, or a function mapping an '
                          'array to a floating point value.')
  
      # initial value for t_curr must be different from t_next by at
      # least the tolerance. Since the image is positive, we ensure this
      # by setting to a large-enough negative number
      t_curr = -2 * tolerance
  
      # Callback on initial iterations
      if iter_callback is not None:
          iter_callback(t_next + image_min)
  
      # Stop the iterations when the difference between the
      # new and old threshold values is less than the tolerance
      while abs(t_next - t_curr) > tolerance:
          t_curr = t_next
          foreground = (image > t_curr)
          mean_fore = np.mean(image[foreground])
          mean_back = np.mean(image[~foreground])
  
          t_next = ((mean_back - mean_fore) /
                    (np.log(mean_back) - np.log(mean_fore)))
  
          if iter_callback is not None:
              iter_callback(t_next + image_min)
  
      threshold = t_next + image_min
      return threshold
  
  
  def threshold_minimum(image, nbins=256, max_iter=10000):
      """Return threshold value based on minimum method.
  
      The histogram of the input ``image`` is computed and smoothed until
      there are only two maxima. Then the minimum in between is the threshold
      value.
  
      Parameters
      ----------
      image : (M, N) ndarray
          Input image.
      nbins : int, optional
          Number of bins used to calculate histogram. This value is ignored for
          integer arrays.
      max_iter: int, optional
          Maximum number of iterations to smooth the histogram.
  
      Returns
      -------
      threshold : float
          Upper threshold value. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      Raises
      ------
      RuntimeError
          If unable to find two local maxima in the histogram or if the
          smoothing takes more than 1e4 iterations.
  
      References
      ----------
      .. [1] C. A. Glasbey, "An analysis of histogram-based thresholding
             algorithms," CVGIP: Graphical Models and Image Processing,
             vol. 55, pp. 532-537, 1993.
      .. [2] Prewitt, JMS & Mendelsohn, ML (1966), "The analysis of cell
             images", Annals of the New York Academy of Sciences 128: 1035-1053
             :DOI:`10.1111/j.1749-6632.1965.tb11715.x`
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()
      >>> thresh = threshold_minimum(image)
      >>> binary = image > thresh
      """
  
      def find_local_maxima_idx(hist):
          # We can't use scipy.signal.argrelmax
          # as it fails on plateaus
          maximum_idxs = list()
          direction = 1
  
          for i in range(hist.shape[0] - 1):
              if direction > 0:
                  if hist[i + 1] < hist[i]:
                      direction = -1
                      maximum_idxs.append(i)
              else:
                  if hist[i + 1] > hist[i]:
                      direction = 1
  
          return maximum_idxs
  
      hist, bin_centers = histogram(image.ravel(), nbins, source_range='image')
  
      smooth_hist = hist.astype(np.float64, copy=False)
  
      for counter in range(max_iter):
          smooth_hist = ndi.uniform_filter1d(smooth_hist, 3)
          maximum_idxs = find_local_maxima_idx(smooth_hist)
          if len(maximum_idxs) < 3:
              break
  
      if len(maximum_idxs) != 2:
          raise RuntimeError('Unable to find two maxima in histogram')
      elif counter == max_iter - 1:
          raise RuntimeError('Maximum iteration reached for histogram'
                             'smoothing')
  
      # Find lowest point between the maxima
      threshold_idx = np.argmin(smooth_hist[maximum_idxs[0]:maximum_idxs[1] + 1])
  
      return bin_centers[maximum_idxs[0] + threshold_idx]
  
  
  def threshold_mean(image):
      """Return threshold value based on the mean of grayscale values.
  
      Parameters
      ----------
      image : (N, M[, ..., P]) ndarray
          Grayscale input image.
  
      Returns
      -------
      threshold : float
          Upper threshold value. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      References
      ----------
      .. [1] C. A. Glasbey, "An analysis of histogram-based thresholding
          algorithms," CVGIP: Graphical Models and Image Processing,
          vol. 55, pp. 532-537, 1993.
          :DOI:`10.1006/cgip.1993.1040`
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()
      >>> thresh = threshold_mean(image)
      >>> binary = image > thresh
      """
      return np.mean(image)
  
  
  def threshold_triangle(image, nbins=256):
      """Return threshold value based on the triangle algorithm.
  
      Parameters
      ----------
      image : (N, M[, ..., P]) ndarray
          Grayscale input image.
      nbins : int, optional
          Number of bins used to calculate histogram. This value is ignored for
          integer arrays.
  
      Returns
      -------
      threshold : float
          Upper threshold value. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      References
      ----------
      .. [1] Zack, G. W., Rogers, W. E. and Latt, S. A., 1977,
         Automatic Measurement of Sister Chromatid Exchange Frequency,
         Journal of Histochemistry and Cytochemistry 25 (7), pp. 741-753
         :DOI:`10.1177/25.7.70454`
      .. [2] ImageJ AutoThresholder code,
         http://fiji.sc/wiki/index.php/Auto_Threshold
  
      Examples
      --------
      >>> from skimage.data import camera
      >>> image = camera()
      >>> thresh = threshold_triangle(image)
      >>> binary = image > thresh
      """
      # nbins is ignored for integer arrays
      # so, we recalculate the effective nbins.
      hist, bin_centers = histogram(image.ravel(), nbins, source_range='image')
      nbins = len(hist)
  
      # Find peak, lowest and highest gray levels.
      arg_peak_height = np.argmax(hist)
      peak_height = hist[arg_peak_height]
      arg_low_level, arg_high_level = np.where(hist > 0)[0][[0, -1]]
  
      # Flip is True if left tail is shorter.
      flip = arg_peak_height - arg_low_level < arg_high_level - arg_peak_height
      if flip:
          hist = hist[::-1]
          arg_low_level = nbins - arg_high_level - 1
          arg_peak_height = nbins - arg_peak_height - 1
  
      # If flip == True, arg_high_level becomes incorrect
      # but we don't need it anymore.
      del(arg_high_level)
  
      # Set up the coordinate system.
      width = arg_peak_height - arg_low_level
      x1 = np.arange(width)
      y1 = hist[x1 + arg_low_level]
  
      # Normalize.
      norm = np.sqrt(peak_height**2 + width**2)
      peak_height /= norm
      width /= norm
  
      # Maximize the length.
      # The ImageJ implementation includes an additional constant when calculating
      # the length, but here we omit it as it does not affect the location of the
      # minimum.
      length = peak_height * x1 - width * y1
      arg_level = np.argmax(length) + arg_low_level
  
      if flip:
          arg_level = nbins - arg_level - 1
  
      return bin_centers[arg_level]
  
  
  def _validate_window_size(axis_sizes):
      """Ensure all sizes in ``axis_sizes`` are odd.
  
      Parameters
      ----------
      axis_sizes : iterable of int
  
      Raises
      ------
      ValueError
          If any given axis size is even.
      """
      for axis_size in axis_sizes:
          if axis_size % 2 == 0:
              msg = ('Window size for `threshold_sauvola` or '
                     '`threshold_niblack` must not be even on any dimension. '
                     'Got {}'.format(axis_sizes))
              raise ValueError(msg)
  
  
  def _mean_std(image, w):
      """Return local mean and standard deviation of each pixel using a
      neighborhood defined by a rectangular window size ``w``.
      The algorithm uses integral images to speedup computation. This is
      used by :func:`threshold_niblack` and :func:`threshold_sauvola`.
  
      Parameters
      ----------
      image : ndarray
          Input image.
      w : int, or iterable of int
          Window size specified as a single odd integer (3, 5, 7, …),
          or an iterable of length ``image.ndim`` containing only odd
          integers (e.g. ``(1, 5, 5)``).
  
      Returns
      -------
      m : ndarray of float, same shape as ``image``
          Local mean of the image.
      s : ndarray of float, same shape as ``image``
          Local standard deviation of the image.
  
      References
      ----------
      .. [1] F. Shafait, D. Keysers, and T. M. Breuel, "Efficient
             implementation of local adaptive thresholding techniques
             using integral images." in Document Recognition and
             Retrieval XV, (San Jose, USA), Jan. 2008.
             :DOI:`10.1117/12.767755`
      """
      if not isinstance(w, Iterable):
          w = (w,) * image.ndim
      _validate_window_size(w)
  
      pad_width = tuple((k // 2 + 1, k // 2) for k in w)
      padded = np.pad(image.astype('float'), pad_width,
                      mode='reflect')
      padded_sq = padded * padded
  
      integral = integral_image(padded)
      integral_sq = integral_image(padded_sq)
  
      kern = np.zeros(tuple(k + 1 for k in w))
      for indices in itertools.product(*([[0, -1]] * image.ndim)):
          kern[indices] = (-1) ** (image.ndim % 2 != np.sum(indices) % 2)
  
      total_window_size = np.prod(w)
      sum_full = ndi.correlate(integral, kern, mode='constant')
      m = crop(sum_full, pad_width) / total_window_size
      sum_sq_full = ndi.correlate(integral_sq, kern, mode='constant')
      g2 = crop(sum_sq_full, pad_width) / total_window_size
      # Note: we use np.clip because g2 is not guaranteed to be greater than
      # m*m when floating point error is considered
      s = np.sqrt(np.clip(g2 - m * m, 0, None))
      return m, s
  
  
  def threshold_niblack(image, window_size=15, k=0.2):
      """Applies Niblack local threshold to an array.
  
      A threshold T is calculated for every pixel in the image using the
      following formula::
  
          T = m(x,y) - k * s(x,y)
  
      where m(x,y) and s(x,y) are the mean and standard deviation of
      pixel (x,y) neighborhood defined by a rectangular window with size w
      times w centered around the pixel. k is a configurable parameter
      that weights the effect of standard deviation.
  
      Parameters
      ----------
      image: ndarray
          Input image.
      window_size : int, or iterable of int, optional
          Window size specified as a single odd integer (3, 5, 7, …),
          or an iterable of length ``image.ndim`` containing only odd
          integers (e.g. ``(1, 5, 5)``).
      k : float, optional
          Value of parameter k in threshold formula.
  
      Returns
      -------
      threshold : (N, M) ndarray
          Threshold mask. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      Notes
      -----
      This algorithm is originally designed for text recognition.
  
      The Bradley threshold is a particular case of the Niblack
      one, being equivalent to
  
      >>> from skimage import data
      >>> image = data.page()
      >>> q = 1
      >>> threshold_image = threshold_niblack(image, k=0) * q
  
      for some value ``q``. By default, Bradley and Roth use ``q=1``.
  
  
      References
      ----------
      .. [1] W. Niblack, An introduction to Digital Image Processing,
             Prentice-Hall, 1986.
      .. [2] D. Bradley and G. Roth, "Adaptive thresholding using Integral
             Image", Journal of Graphics Tools 12(2), pp. 13-21, 2007.
             :DOI:`10.1080/2151237X.2007.10129236`
  
      Examples
      --------
      >>> from skimage import data
      >>> image = data.page()
      >>> threshold_image = threshold_niblack(image, window_size=7, k=0.1)
      """
      m, s = _mean_std(image, window_size)
      return m - k * s
  
  
  def threshold_sauvola(image, window_size=15, k=0.2, r=None):
      """Applies Sauvola local threshold to an array. Sauvola is a
      modification of Niblack technique.
  
      In the original method a threshold T is calculated for every pixel
      in the image using the following formula::
  
          T = m(x,y) * (1 + k * ((s(x,y) / R) - 1))
  
      where m(x,y) and s(x,y) are the mean and standard deviation of
      pixel (x,y) neighborhood defined by a rectangular window with size w
      times w centered around the pixel. k is a configurable parameter
      that weights the effect of standard deviation.
      R is the maximum standard deviation of a greyscale image.
  
      Parameters
      ----------
      image: ndarray
          Input image.
      window_size : int, or iterable of int, optional
          Window size specified as a single odd integer (3, 5, 7, …),
          or an iterable of length ``image.ndim`` containing only odd
          integers (e.g. ``(1, 5, 5)``).
      k : float, optional
          Value of the positive parameter k.
      r : float, optional
          Value of R, the dynamic range of standard deviation.
          If None, set to the half of the image dtype range.
  
      Returns
      -------
      threshold : (N, M) ndarray
          Threshold mask. All pixels with an intensity higher than
          this value are assumed to be foreground.
  
      Notes
      -----
      This algorithm is originally designed for text recognition.
  
      References
      ----------
      .. [1] J. Sauvola and M. Pietikainen, "Adaptive document image
             binarization," Pattern Recognition 33(2),
             pp. 225-236, 2000.
             :DOI:`10.1016/S0031-3203(99)00055-2`
  
      Examples
      --------
      >>> from skimage import data
      >>> image = data.page()
      >>> t_sauvola = threshold_sauvola(image, window_size=15, k=0.2)
      >>> binary_image = image > t_sauvola
      """
      if r is None:
          imin, imax = dtype_limits(image, clip_negative=False)
          r = 0.5 * (imax - imin)
      m, s = _mean_std(image, window_size)
      return m * (1 + k * ((s / r) - 1))
  
  
  def apply_hysteresis_threshold(image, low, high):
      """Apply hysteresis thresholding to ``image``.
  
      This algorithm finds regions where ``image`` is greater than ``high``
      OR ``image`` is greater than ``low`` *and* that region is connected to
      a region greater than ``high``.
  
      Parameters
      ----------
      image : array, shape (M,[ N, ..., P])
          Grayscale input image.
      low : float, or array of same shape as ``image``
          Lower threshold.
      high : float, or array of same shape as ``image``
          Higher threshold.
  
      Returns
      -------
      thresholded : array of bool, same shape as ``image``
          Array in which ``True`` indicates the locations where ``image``
          was above the hysteresis threshold.
  
      Examples
      --------
      >>> image = np.array([1, 2, 3, 2, 1, 2, 1, 3, 2])
      >>> apply_hysteresis_threshold(image, 1.5, 2.5).astype(int)
      array([0, 1, 1, 1, 0, 0, 0, 1, 1])
  
      References
      ----------
      .. [1] J. Canny. A computational approach to edge detection.
             IEEE Transactions on Pattern Analysis and Machine Intelligence.
             1986; vol. 8, pp.679-698.
             :DOI:`10.1109/TPAMI.1986.4767851`
      """
      low = np.clip(low, a_min=None, a_max=high)  # ensure low always below high
      mask_low = image > low
      mask_high = image > high
      # Connected components of mask_low
      labels_low, num_labels = ndi.label(mask_low)
      # Check which connected components contain pixels from mask_high
      sums = ndi.sum(mask_high, labels_low, np.arange(num_labels + 1))
      connected_to_high = sums > 0
      thresholded = connected_to_high[labels_low]
      return thresholded
  
  
  def threshold_multiotsu(image, classes=3, nbins=256):
      r"""Generate `classes`-1 threshold values to divide gray levels in `image`.
  
      The threshold values are chosen to maximize the total sum of pairwise
      variances between the thresholded graylevel classes. See Notes and [1]_
      for more details.
  
      Parameters
      ----------
      image : (N, M) ndarray
          Grayscale input image.
      classes : int, optional
          Number of classes to be thresholded, i.e. the number of resulting
          regions.
      nbins : int, optional
          Number of bins used to calculate the histogram. This value is ignored
          for integer arrays.
  
      Returns
      -------
      idx_thresh : array
          Array containing the threshold values for the desired classes.
  
      Notes
      -----
      This implementation relies on a Cython function whose complexity
      is :math:`O\left(\frac{Ch^{C-1}}{(C-1)!}\right)`, where :math:`h`
      is the number of histogram bins and :math:`C` is the number of
      classes desired.
  
      References
      ----------
      .. [1] Liao, P-S., Chen, T-S. and Chung, P-C., "A fast algorithm for
             multilevel thresholding", Journal of Information Science and
             Engineering 17 (5): 713-727, 2001. Available at:
             <http://ftp.iis.sinica.edu.tw/JISE/2001/200109_01.pdf>
      .. [2] Tosa, Y., "Multi-Otsu Threshold", a java plugin for ImageJ.
             Available at:
             <http://imagej.net/plugins/download/Multi_OtsuThreshold.java>
  
      Examples
      --------
      >>> from skimage.color import label2rgb
      >>> from skimage import data
      >>> image = data.camera()
      >>> thresholds = threshold_multiotsu(image)
      >>> regions = np.digitize(image, bins=thresholds)
      >>> regions_colorized = label2rgb(regions)
      """
      # calculating the histogram and the probability of each gray level.
      hist, bin_centers = histogram(image.ravel(),
                                    nbins=nbins,
                                    source_range='image')
      prob = hist / image.size
      # histogram ignores nbins for integer arrays.
      nbins = len(bin_centers)
  
      # Compute the zeroth (momP, cumulative probability) and first
      # (momS, mean) moments
  
      # building the lookup tables.
      # step 1: calculating the diagonal.
      prob[0] = 0
      momP = np.diagflat(prob)
      momS = np.diagflat(np.arange(nbins) * prob)
  
      # step 2: calculating the first row.
      momP[1, 2:] = prob[1] + np.cumsum(prob[2:])
      momS[1, 2:] = prob[1] + np.cumsum(np.arange(2, nbins) * prob[2:])
  
      # step 3: the other rows are recursively computed as:
      # A[i, j] = A[1, j] - A[1, i-1] for A in {momP, momS};  i > 1 and j > i.
      upper_tri = np.triu_indices_from(momP[2:, 2:], 1)
  
      momP[2:, 2:][upper_tri] = (momP[1, 2:][upper_tri[1]]
                                 - np.repeat(momP[1, 1:-2],
                                             np.arange(nbins - 3, 0, -1)))
      momS[2:, 2:][upper_tri] = (momS[1, 2:][upper_tri[1]]
                                 - np.repeat(momS[1, 1:-2],
                                             np.arange(nbins - 3, 0, -1)))
  
      # step 4: calculating the between class variance.
      var_btwcls = np.zeros_like(momP)
      idx = momP > 0
      var_btwcls[idx] = momS[idx] ** 2 / momP[idx]
  
      # finding max threshold candidates, depending on classes.
      # number of thresholds is equal to number of classes - 1.
      aux_thresh = _find_threshold_multiotsu(var_btwcls, classes, nbins)
  
      # correcting threshold values.
      idx_thresh = bin_centers[:-1][aux_thresh.astype('int')]
  
      return idx_thresh
  