"""watershed.py - watershed algorithm This module implements a watershed algorithm that apportions pixels into marked basins. The algorithm uses a priority queue to hold the pixels with the metric for the priority queue being pixel value, then the time of entry into the queue - this settles ties in favor of the closest marker. Some ideas taken from Soille, "Automated Basin Delineation from Digital Elevation Models Using Mathematical Morphology", Signal Processing 20 (1990) 171-182. The most important insight in the paper is that entry time onto the queue solves two problems: a pixel should be assigned to the neighbor with the largest gradient or, if there is no gradient, pixels on a plateau should be split between markers on opposite sides. Originally part of CellProfiler, code licensed under both GPL and BSD licenses. Website: http://www.cellprofiler.org Copyright (c) 2003-2009 Massachusetts Institute of Technology Copyright (c) 2009-2011 Broad Institute All rights reserved. Original author: Lee Kamentsky """ from _heapq import heappush, heappop import numpy as np import scipy.ndimage from ..filter import rank_order from .._shared.utils import deprecated from . import _watershed def watershed(image, markers, connectivity=None, offset=None, mask=None): """ Return a matrix labeled using the watershed segmentation algorithm Parameters ---------- image: ndarray (2-D, 3-D, ...) of integers Data array where the lowest value points are labeled first. markers: ndarray of the same shape as `image` An array marking the basins with the values to be assigned in the label matrix. Zero means not a marker. This array should be of an integer type. connectivity: ndarray, optional An array with the same number of dimensions as `image` whose non-zero elements indicate neighbors for connection. Following the scipy convention, default is a one-connected array of the dimension of the image. offset: array_like of shape image.ndim, optional offset of the connectivity (one offset per dimension) mask: ndarray of bools or 0s and 1s, optional Array of same shape as `image`. Only points at which mask == True will be labeled. Returns ------- out: ndarray A labeled matrix of the same type and shape as markers See also -------- skimage.segmentation.random_walker: random walker segmentation A segmentation algorithm based on anisotropic diffusion, usually slower than the watershed but with good results on noisy data and boundaries with holes. Notes ----- This function implements a watershed algorithm [1]_that apportions pixels into marked basins. The algorithm uses a priority queue to hold the pixels with the metric for the priority queue being pixel value, then the time of entry into the queue - this settles ties in favor of the closest marker. Some ideas taken from Soille, "Automated Basin Delineation from Digital Elevation Models Using Mathematical Morphology", Signal Processing 20 (1990) 171-182 The most important insight in the paper is that entry time onto the queue solves two problems: a pixel should be assigned to the neighbor with the largest gradient or, if there is no gradient, pixels on a plateau should be split between markers on opposite sides. This implementation converts all arguments to specific, lowest common denominator types, then passes these to a C algorithm. Markers can be determined manually, or automatically using for example the local minima of the gradient of the image, or the local maxima of the distance function to the background for separating overlapping objects (see example). References ---------- .. [1] http://en.wikipedia.org/wiki/Watershed_%28image_processing%29 .. [2] http://cmm.ensmp.fr/~beucher/wtshed.html Examples -------- The watershed algorithm is very useful to separate overlapping objects >>> # Generate an initial image with two overlapping circles >>> x, y = np.indices((80, 80)) >>> x1, y1, x2, y2 = 28, 28, 44, 52 >>> r1, r2 = 16, 20 >>> mask_circle1 = (x - x1)**2 + (y - y1)**2 < r1**2 >>> mask_circle2 = (x - x2)**2 + (y - y2)**2 < r2**2 >>> image = np.logical_or(mask_circle1, mask_circle2) >>> # Now we want to separate the two objects in image >>> # Generate the markers as local maxima of the distance >>> # to the background >>> from scipy import ndimage >>> distance = ndimage.distance_transform_edt(image) >>> from skimage.feature import peak_local_max >>> local_maxi = peak_local_max(distance, labels=image, ... footprint=np.ones((3, 3)), ... indices=False) >>> markers = ndimage.label(local_maxi)[0] >>> labels = watershed(-distance, markers, mask=image) The algorithm works also for 3-D images, and can be used for example to separate overlapping spheres. """ if connectivity is None: c_connectivity = scipy.ndimage.generate_binary_structure(image.ndim, 1) else: c_connectivity = np.array(connectivity, bool) if c_connectivity.ndim != image.ndim: raise ValueError("Connectivity dimension must be same as image") if offset is None: if any([x % 2 == 0 for x in c_connectivity.shape]): raise ValueError("Connectivity array must have an unambiguous " "center") # # offset to center of connectivity array # offset = np.array(c_connectivity.shape) // 2 # pad the image, markers, and mask so that we can use the mask to # keep from running off the edges pads = offset def pad(im): new_im = np.zeros([i + 2 * p for i, p in zip(im.shape, pads)], im.dtype) new_im[[slice(p, -p, None) for p in pads]] = im return new_im if mask is not None: mask = pad(mask) else: mask = pad(np.ones(image.shape, bool)) image = pad(image) markers = pad(markers) c_image = rank_order(image)[0].astype(np.int32) c_markers = np.ascontiguousarray(markers, dtype=np.int32) if c_markers.ndim != c_image.ndim: raise ValueError("markers (ndim=%d) must have same # of dimensions " "as image (ndim=%d)" % (c_markers.ndim, c_image.ndim)) if c_markers.shape != c_image.shape: raise ValueError("image and markers must have the same shape") if mask is not None: c_mask = np.ascontiguousarray(mask, dtype=bool) if c_mask.ndim != c_markers.ndim: raise ValueError("mask must have same # of dimensions as image") if c_markers.shape != c_mask.shape: raise ValueError("mask must have same shape as image") c_markers[np.logical_not(mask)] = 0 else: c_mask = None c_output = c_markers.copy() # # We pass a connectivity array that pre-calculates the stride for each # neighbor. # # The result of this bit of code is an array with one row per # point to be considered. The first column is the pre-computed stride # and the second through last are the x,y...whatever offsets # (to do bounds checking). c = [] image_stride = np.array(image.strides) // image.itemsize for i in range(np.product(c_connectivity.shape)): multiplier = 1 offs = [] indexes = [] ignore = True for j in range(len(c_connectivity.shape)): idx = (i // multiplier) % c_connectivity.shape[j] off = idx - offset[j] if off: ignore = False offs.append(off) indexes.append(idx) multiplier *= c_connectivity.shape[j] if (not ignore) and c_connectivity.__getitem__(tuple(indexes)): stride = np.dot(image_stride, np.array(offs)) offs.insert(0, stride) c.append(offs) c = np.array(c, dtype=np.int32) pq, age = __heapify_markers(c_markers, c_image) pq = np.ascontiguousarray(pq, dtype=np.int32) if np.product(pq.shape) > 0: # If nothing is labeled, the output is empty and we don't have to # do anything c_output = c_output.flatten() if c_mask is None: c_mask = np.ones(c_image.shape, np.int8).flatten() else: c_mask = c_mask.astype(np.int8).flatten() _watershed.watershed(c_image.flatten(), pq, age, c, c_mask, c_output) c_output = c_output.reshape(c_image.shape)[[slice(1, -1, None)] * image.ndim] try: return c_output.astype(markers.dtype) except: return c_output # ---------------------- deprecated ------------------------------ # Deprecate slower pure-Python code, that we keep only for # pedagogical purposes def __heapify_markers(markers, image): """Create a priority queue heap with the markers on it""" stride = np.array(image.strides) // image.itemsize coords = np.argwhere(markers != 0) ncoords = coords.shape[0] if ncoords > 0: pixels = image[markers != 0] age = np.arange(ncoords) offset = np.zeros(coords.shape[0], int) for i in range(image.ndim): offset = offset + stride[i] * coords[:, i] pq = np.column_stack((pixels, age, offset, coords)) # pixels = top priority, age=second ordering = np.lexsort((age, pixels)) pq = pq[ordering, :] else: pq = np.zeros((0, markers.ndim + 3), int) return (pq, ncoords) def _slow_watershed(image, markers, connectivity=8, mask=None): """Return a matrix labeled using the watershed algorithm Use the `watershed` function for a faster execution. This pure Python function is solely for pedagogical purposes. Parameters ---------- image: 2-d ndarray of integers a two-dimensional matrix where the lowest value points are labeled first. markers: 2-d ndarray of integers a two-dimensional matrix marking the basins with the values to be assigned in the label matrix. Zero means not a marker. connectivity: {4, 8}, optional either 4 for four-connected or 8 (default) for eight-connected mask: 2-d ndarray of bools, optional don't label points in the mask Returns ------- out: ndarray A labeled matrix of the same type and shape as markers Notes ----- This function implements a watershed algorithm [1]_that apportions pixels into marked basins. The algorithm uses a priority queue to hold the pixels with the metric for the priority queue being pixel value, then the time of entry into the queue - this settles ties in favor of the closest marker. Some ideas taken from Soille, "Automated Basin Delineation from Digital Elevation Models Using Mathematical Morphology", Signal Processing 20 (1990) 171-182 The most important insight in the paper is that entry time onto the queue solves two problems: a pixel should be assigned to the neighbor with the largest gradient or, if there is no gradient, pixels on a plateau should be split between markers on opposite sides. This implementation converts all arguments to specific, lowest common denominator types, then passes these to a C algorithm. Markers can be determined manually, or automatically using for example the local minima of the gradient of the image, or the local maxima of the distance function to the background for separating overlapping objects. """ if connectivity not in (4, 8): raise ValueError("Connectivity was %d: it should be either \ four or eight" % (connectivity)) image = np.array(image) markers = np.array(markers) labels = markers.copy() max_x = markers.shape[0] max_y = markers.shape[1] if connectivity == 4: connect_increments = ((1, 0), (0, 1), (-1, 0), (0, -1)) else: connect_increments = ((1, 0), (1, 1), (0, 1), (-1, 1), (-1, 0), (-1, -1), (0, -1), (1, -1)) pq, age = __heapify_markers(markers, image) pq = pq.tolist() # # The second step pops a value off of the queue, then labels and pushes # the neighbors # while len(pq): pix_value, pix_age, ignore, pix_x, pix_y = heappop(pq) pix_label = labels[pix_x, pix_y] for xi, yi in connect_increments: x = pix_x + xi y = pix_y + yi if x < 0 or y < 0 or x >= max_x or y >= max_y: continue if labels[x, y]: continue if mask is not None and not mask[x, y]: continue # label the pixel labels[x, y] = pix_label # put the pixel onto the queue heappush(pq, [image[x, y], age, 0, x, y]) age += 1 return labels