import math import numpy as np from scipy import ndimage as ndi from scipy.ndimage import filters as ndif from collections import OrderedDict from ..exposure import histogram from .._shared.utils import assert_nD, warn __all__ = ['try_all_threshold', 'threshold_adaptive', 'threshold_otsu', 'threshold_yen', 'threshold_isodata', 'threshold_li', 'threshold_minimum', 'threshold_mean', 'threshold_triangle'] 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 num_rows = math.ceil((len(methods) + 1.) / num_cols) num_rows = int(num_rows) # Python 2.7 support fig, ax = plt.subplots(num_rows, num_cols, figsize=figsize, sharex=True, sharey=True, subplot_kw={'adjustable': 'box-forced'}) 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].imshow(func(image), cmap=plt.cm.gray) ax[i].set_title(name) 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_adaptive(image, block_size, method='gaussian', offset=0, mode='reflect', param=None): """Applies an adaptive threshold to an array. Also known as local or dynamic thresholding where 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 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. Returns ------- threshold : (N, M) ndarray Thresholded binary image References ---------- .. [1] http://docs.opencv.org/modules/imgproc/doc/miscellaneous_transformations.html?highlight=threshold#adaptivethreshold Examples -------- >>> from skimage.data import camera >>> image = camera()[:50, :50] >>> binary_image1 = threshold_adaptive(image, 15, 'mean') >>> func = lambda arr: arr.mean() >>> binary_image2 = threshold_adaptive(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)) assert_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) 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) 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) ndi.convolve1d(thresh_image, mask, axis=1, output=thresh_image, mode=mode) elif method == 'median': ndi.median_filter(image, block_size, output=thresh_image, mode=mode) return image > (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, http://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 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) 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) # 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) # 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 = np.cumsum(hist[::-1])[::-1] - hist # 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. So we just patch up csumh[-1] to not cause 0/0 # errors. csumh[-1] = 1 l = np.cumsum(intensity_sum) / csuml h = (np.cumsum(intensity_sum[::-1])[::-1] - intensity_sum) / csumh # 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 = (l + h) / 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 thresholds = bin_centers[(distances >= 0) & (distances < bin_width)] if return_all: return thresholds else: return thresholds[0] def threshold_li(image): """Return threshold value based on adaptation of Li's Minimum Cross Entropy method. Parameters ---------- image : (N, M) ndarray Input image. 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 """ # Copy to ensure input image is not modified image = image.copy() # Requires positive image (because of log(mean)) immin = np.min(image) image -= immin imrange = np.max(image) tolerance = 0.5 * imrange / 256 # Calculate the mean gray-level mean = np.mean(image) # Initial estimate new_thresh = mean old_thresh = new_thresh + 2 * tolerance # Stop the iterations when the difference between the # new and old threshold values is less than the tolerance while abs(new_thresh - old_thresh) > tolerance: old_thresh = new_thresh threshold = old_thresh + tolerance # range # Calculate the means of background and object pixels mean_back = image[image <= threshold].mean() mean_obj = image[image > threshold].mean() temp = (mean_back - mean_obj) / (np.log(mean_back) - np.log(mean_obj)) if temp < 0: new_thresh = temp - tolerance else: new_thresh = temp + tolerance return threshold + immin def threshold_minimum(image, nbins=256, bias='min', 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. bias : {'min', 'mid', 'max'}, optional 'min', 'mid', 'max' return lowest, middle, or highest pixel value with minimum histogram value. 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] 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(hist): # We can't use scipy.signal.argrelmax # as it fails on plateaus maximums = list() direction = 1 for i in range(hist.shape[0] - 1): if direction > 0: if hist[i + 1] < hist[i]: direction = -1 maximums.append(i) else: if hist[i + 1] > hist[i]: direction = 1 return maximums if bias not in ('min', 'mid', 'max'): raise ValueError("Unknown bias: {0}".format(bias)) hist, bin_centers = histogram(image.ravel(), nbins) smooth_hist = np.copy(hist) for counter in range(max_iter): smooth_hist = ndif.uniform_filter1d(smooth_hist, 3) maximums = find_local_maxima(smooth_hist) if len(maximums) < 3: break if len(maximums) != 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, biased to the low end (min) minimum = smooth_hist[maximums[0]] threshold = maximums[0] for i in range(maximums[0], maximums[1]+1): if smooth_hist[i] < minimum: minimum = smooth_hist[i] threshold = i if bias == 'min': return bin_centers[threshold] else: upper_bound = threshold while smooth_hist[upper_bound] == smooth_hist[threshold]: upper_bound += 1 upper_bound -= 1 if bias == 'max': return bin_centers[upper_bound] elif bias == 'mid': return bin_centers[(threshold + upper_bound) // 2] 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) 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]