__all__ = ['threshold_adaptive', 'threshold_otsu', 'threshold_yen', 'threshold_isodata'] import numpy as np import scipy.ndimage from skimage.exposure import histogram 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 Uneven 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) """ thresh_image = np.zeros(image.shape, 'double') if method == 'generic': scipy.ndimage.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 scipy.ndimage.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 scipy.ndimage.convolve1d(image, mask, axis=0, output=thresh_image, mode=mode) scipy.ndimage.convolve1d(thresh_image, mask, axis=1, output=thresh_image, mode=mode) elif method == 'median': scipy.ndimage.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 : array 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 intensities that less or equal of this value assumed as foreground. 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 """ hist, bin_centers = histogram(image, 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 : array 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 intensities that less or equal of this value assumed as 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 .. [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 .. [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, 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 : array 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" .. [2] IEEE Transactions on Systems, Man and Cybernetics 8: 630-632, http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4310039 .. [3] 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 .. [4] 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, 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]