"""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. """ import numpy as np from scipy import ndimage as ndi from . import _watershed_cy from ..morphology.extrema import local_minima from ..morphology._util import (_validate_connectivity, _offsets_to_raveled_neighbors) from ..util import crop, regular_seeds def _validate_inputs(image, markers, mask, connectivity): """Ensure that all inputs to watershed have matching shapes and types. Parameters ---------- image : array The input image. markers : int or array of int The marker image. mask : array, or None A boolean mask, True where we want to compute the watershed. connectivity : int in {1, ..., image.ndim} The connectivity of the neighborhood of a pixel. Returns ------- image, markers, mask : arrays The validated and formatted arrays. Image will have dtype float64, markers int32, and mask int8. If ``None`` was given for the mask, it is a volume of all 1s. Raises ------ ValueError If the shapes of the given arrays don't match. """ n_pixels = image.size if mask is None: # Use a complete `True` mask if none is provided mask = np.ones(image.shape, bool) else: mask = np.asanyarray(mask, dtype=bool) n_pixels = np.sum(mask) if mask.shape != image.shape: message = (f'`mask` (shape {mask.shape}) must have same shape ' f'as `image` (shape {image.shape})') raise ValueError(message) if markers is None: markers_bool = local_minima(image, connectivity=connectivity) * mask footprint = ndi.generate_binary_structure(markers_bool.ndim, connectivity) markers = ndi.label(markers_bool, structure=footprint)[0] elif not isinstance(markers, (np.ndarray, list, tuple)): # not array-like, assume int # given int, assume that number of markers *within mask*. markers = regular_seeds(image.shape, int(markers / (n_pixels / image.size))) markers *= mask else: markers = np.asanyarray(markers) * mask if markers.shape != image.shape: message = (f'`markers` (shape {markers.shape}) must have same ' f'shape as `image` (shape {image.shape})') raise ValueError(message) return (image.astype(np.float64), markers, mask.astype(np.int8)) def watershed(image, markers=None, connectivity=1, offset=None, mask=None, compactness=0, watershed_line=False): """Find watershed basins in `image` flooded from given `markers`. Parameters ---------- image : ndarray (2-D, 3-D, ...) Data array where the lowest value points are labeled first. markers : int, or ndarray of int, same shape as `image`, optional The desired number of markers, or an array marking the basins with the values to be assigned in the label matrix. Zero means not a marker. If ``None`` (no markers given), the local minima of the image are used as markers. 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. compactness : float, optional Use compact watershed [3]_ with given compactness parameter. Higher values result in more regularly-shaped watershed basins. watershed_line : bool, optional If watershed_line is True, a one-pixel wide line separates the regions obtained by the watershed algorithm. The line has the label 0. Note that the method used for adding this line expects that marker regions are not adjacent; the watershed line may not catch borders between adjacent marker regions. 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]_ [2]_ 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] https://en.wikipedia.org/wiki/Watershed_%28image_processing%29 .. [2] http://cmm.ensmp.fr/~beucher/wtshed.html .. [3] Peer Neubert & Peter Protzel (2014). Compact Watershed and Preemptive SLIC: On Improving Trade-offs of Superpixel Segmentation Algorithms. ICPR 2014, pp 996-1001. :DOI:`10.1109/ICPR.2014.181` https://www.tu-chemnitz.de/etit/proaut/publications/cws_pSLIC_ICPR.pdf Examples -------- The watershed algorithm is useful to separate overlapping objects. We first 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) Next, we want to separate the two circles. We generate markers at the maxima of the distance to the background: >>> from scipy import ndimage as ndi >>> distance = ndi.distance_transform_edt(image) >>> from skimage.feature import peak_local_max >>> max_coords = peak_local_max(distance, labels=image, ... footprint=np.ones((3, 3))) >>> local_maxima = np.zeros_like(image, dtype=bool) >>> local_maxima[tuple(max_coords.T)] = True >>> markers = ndi.label(local_maxima)[0] Finally, we run the watershed on the image and markers: >>> labels = watershed(-distance, markers, mask=image) The algorithm works also for 3-D images, and can be used for example to separate overlapping spheres. """ image, markers, mask = _validate_inputs(image, markers, mask, connectivity) connectivity, offset = _validate_connectivity(image.ndim, connectivity, offset) # pad the image, markers, and mask so that we can use the mask to # keep from running off the edges pad_width = [(p, p) for p in offset] image = np.pad(image, pad_width, mode='constant') mask = np.pad(mask, pad_width, mode='constant').ravel() output = np.pad(markers, pad_width, mode='constant') flat_neighborhood = _offsets_to_raveled_neighbors( image.shape, connectivity, center=offset) marker_locations = np.flatnonzero(output) image_strides = np.array(image.strides, dtype=np.intp) // image.itemsize _watershed_cy.watershed_raveled(image.ravel(), marker_locations, flat_neighborhood, mask, image_strides, compactness, output.ravel(), watershed_line) output = crop(output, pad_width, copy=True) return output