from math import sqrt from numbers import Real import numpy as np from scipy import ndimage as ndi STREL_4 = np.array([[0, 1, 0], [1, 1, 1], [0, 1, 0]], dtype=np.uint8) STREL_8 = np.ones((3, 3), dtype=np.uint8) # Coefficients from # Ohser J., Nagel W., Schladitz K. (2002) The Euler Number of Discretized Sets # - On the Choice of Adjacency in Homogeneous Lattices. # In: Mecke K., Stoyan D. (eds) Morphology of Condensed Matter. Lecture Notes # in Physics, vol 600. Springer, Berlin, Heidelberg. # The value of coefficients correspond to the contributions to the Euler number # of specific voxel configurations, which are themselves encoded thanks to a # LUT. Computing the Euler number from the addition of the contributions of # local configurations is possible thanks to an integral geometry formula # (see the paper by Ohser et al. for more details). EULER_COEFS2D_4 = [0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0] EULER_COEFS2D_8 = [0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0] EULER_COEFS3D_26 = np.array( [ 0, 1, 1, 0, 1, 0, -2, -1, 1, -2, 0, -1, 0, -1, -1, 0, 1, 0, -2, -1, -2, -1, -1, -2, -6, -3, -3, -2, -3, -2, 0, -1, 1, -2, 0, -1, -6, -3, -3, -2, -2, -1, -1, -2, -3, 0, -2, -1, 0, -1, -1, 0, -3, -2, 0, -1, -3, 0, -2, -1, 0, 1, 1, 0, 1, -2, -6, -3, 0, -1, -3, -2, -2, -1, -3, 0, -1, -2, -2, -1, 0, -1, -3, -2, -1, 0, 0, -1, -3, 0, 0, 1, -2, -1, 1, 0, -2, -1, -3, 0, -3, 0, 0, 1, -1, 4, 0, 3, 0, 3, 1, 2, -1, -2, -2, -1, -2, -1, 1, 0, 0, 3, 1, 2, 1, 2, 2, 1, 1, -6, -2, -3, -2, -3, -1, 0, 0, -3, -1, -2, -1, -2, -2, -1, -2, -3, -1, 0, -1, 0, 4, 3, -3, 0, 0, 1, 0, 1, 3, 2, 0, -3, -1, -2, -3, 0, 0, 1, -1, 0, 0, -1, -2, 1, -1, 0, -1, -2, -2, -1, 0, 1, 3, 2, -2, 1, -1, 0, 1, 2, 2, 1, 0, -3, -3, 0, -1, -2, 0, 1, -1, 0, -2, 1, 0, -1, -1, 0, -1, -2, 0, 1, -2, -1, 3, 2, -2, 1, 1, 2, -1, 0, 2, 1, -1, 0, -2, 1, -2, 1, 1, 2, -2, 3, -1, 2, -1, 2, 0, 1, 0, -1, -1, 0, -1, 0, 2, 1, -1, 2, 0, 1, 0, 1, 1, 0, ] ) def euler_number(image, connectivity=None): """Calculate the Euler characteristic in binary image. For 2D objects, the Euler number is the number of objects minus the number of holes. For 3D objects, the Euler number is obtained as the number of objects plus the number of holes, minus the number of tunnels, or loops. Parameters ---------- image: (M, N[, P]) ndarray Input image. If image is not binary, all values greater than zero are considered as the object. connectivity : int, optional Maximum number of orthogonal hops to consider a pixel/voxel as a neighbor. Accepted values are ranging from 1 to input.ndim. If ``None``, a full connectivity of ``input.ndim`` is used. 4 or 8 neighborhoods are defined for 2D images (connectivity 1 and 2, respectively). 6 or 26 neighborhoods are defined for 3D images, (connectivity 1 and 3, respectively). Connectivity 2 is not defined. Returns ------- euler_number : int Euler characteristic of the set of all objects in the image. Notes ----- The Euler characteristic is an integer number that describes the topology of the set of all objects in the input image. If object is 4-connected, then background is 8-connected, and conversely. The computation of the Euler characteristic is based on an integral geometry formula in discretized space. In practice, a neighborhood configuration is constructed, and a LUT is applied for each configuration. The coefficients used are the ones of Ohser et al. It can be useful to compute the Euler characteristic for several connectivities. A large relative difference between results for different connectivities suggests that the image resolution (with respect to the size of objects and holes) is too low. References ---------- .. [1] S. Rivollier. Analyse d’image geometrique et morphometrique par diagrammes de forme et voisinages adaptatifs generaux. PhD thesis, 2010. Ecole Nationale Superieure des Mines de Saint-Etienne. https://tel.archives-ouvertes.fr/tel-00560838 .. [2] Ohser J., Nagel W., Schladitz K. (2002) The Euler Number of Discretized Sets - On the Choice of Adjacency in Homogeneous Lattices. In: Mecke K., Stoyan D. (eds) Morphology of Condensed Matter. Lecture Notes in Physics, vol 600. Springer, Berlin, Heidelberg. Examples -------- >>> import numpy as np >>> SAMPLE = np.zeros((100,100,100)); >>> SAMPLE[40:60, 40:60, 40:60]=1 >>> euler_number(SAMPLE) # doctest: +ELLIPSIS 1... >>> SAMPLE[45:55,45:55,45:55] = 0; >>> euler_number(SAMPLE) # doctest: +ELLIPSIS 2... >>> SAMPLE = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0], ... [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0], ... [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], ... [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], ... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0], ... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0], ... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0], ... [1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0], ... [0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1], ... [0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1]]) >>> euler_number(SAMPLE) # doctest: 0 >>> euler_number(SAMPLE, connectivity=1) # doctest: 2 """ # as image can be a label image, transform it to binary image = (image > 0).astype(int) image = np.pad(image, pad_width=1, mode='constant') # check connectivity if connectivity is None: connectivity = image.ndim # config variable is an adjacency configuration. A coefficient given by # variable coefs is attributed to each configuration in order to get # the Euler characteristic. if image.ndim == 2: config = np.array([[0, 0, 0], [0, 1, 4], [0, 2, 8]]) if connectivity == 1: coefs = EULER_COEFS2D_4 else: coefs = EULER_COEFS2D_8 bins = 16 else: # 3D images if connectivity == 2: raise NotImplementedError( 'For 3D images, Euler number is implemented ' 'for connectivities 1 and 3 only' ) config = np.array( [ [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 1, 4], [0, 2, 8]], [[0, 0, 0], [0, 16, 64], [0, 32, 128]], ] ) if connectivity == 1: coefs = EULER_COEFS3D_26[::-1] else: coefs = EULER_COEFS3D_26 bins = 256 # XF has values in the 0-255 range in 3D, and in the 0-15 range in 2D, # with one unique value for each binary configuration of the # 27-voxel cube in 3D / 8-pixel square in 2D, up to symmetries XF = ndi.convolve(image, config, mode='constant', cval=0) h = np.bincount(XF.ravel(), minlength=bins) if image.ndim == 2: return coefs @ h else: return int(0.125 * coefs @ h) def perimeter(image, neighborhood=4): """Calculate total perimeter of all objects in binary image. Parameters ---------- image : (M, N) ndarray Binary input image. neighborhood : 4 or 8, optional Neighborhood connectivity for border pixel determination. It is used to compute the contour. A higher neighborhood widens the border on which the perimeter is computed. Returns ------- perimeter : float Total perimeter of all objects in binary image. References ---------- .. [1] K. Benkrid, D. Crookes. Design and FPGA Implementation of a Perimeter Estimator. The Queen's University of Belfast. http://www.cs.qub.ac.uk/~d.crookes/webpubs/papers/perimeter.doc Examples -------- >>> from skimage import data, util >>> from skimage.measure import label >>> # coins image (binary) >>> img_coins = data.coins() > 110 >>> # total perimeter of all objects in the image >>> perimeter(img_coins, neighborhood=4) # doctest: +ELLIPSIS 7796.867... >>> perimeter(img_coins, neighborhood=8) # doctest: +ELLIPSIS 8806.268... """ if image.ndim != 2: raise NotImplementedError('`perimeter` supports 2D images only') if neighborhood == 4: strel = STREL_4 else: strel = STREL_8 image = image.astype(np.uint8) eroded_image = ndi.binary_erosion(image, strel, border_value=0) border_image = image - eroded_image perimeter_weights = np.zeros(50, dtype=np.float64) perimeter_weights[[5, 7, 15, 17, 25, 27]] = 1 perimeter_weights[[21, 33]] = sqrt(2) perimeter_weights[[13, 23]] = (1 + sqrt(2)) / 2 perimeter_image = ndi.convolve( border_image, np.array([[10, 2, 10], [2, 1, 2], [10, 2, 10]]), mode='constant', cval=0, ) # You can also write # return perimeter_weights[perimeter_image].sum() # but that was measured as taking much longer than bincount + np.dot (5x # as much time) perimeter_histogram = np.bincount(perimeter_image.ravel(), minlength=50) total_perimeter = perimeter_histogram @ perimeter_weights return total_perimeter def perimeter_crofton(image, directions=4): """Calculate total Crofton perimeter of all objects in binary image. Parameters ---------- image : (M, N) ndarray Input image. If image is not binary, all values greater than zero are considered as the object. directions : 2 or 4, optional Number of directions used to approximate the Crofton perimeter. By default, 4 is used: it should be more accurate than 2. Computation time is the same in both cases. Returns ------- perimeter : float Total perimeter of all objects in binary image. Notes ----- This measure is based on Crofton formula [1], which is a measure from integral geometry. It is defined for general curve length evaluation via a double integral along all directions. In a discrete space, 2 or 4 directions give a quite good approximation, 4 being more accurate than 2 for more complex shapes. Similar to :func:`~.measure.perimeter`, this function returns an approximation of the perimeter in continuous space. References ---------- .. [1] https://en.wikipedia.org/wiki/Crofton_formula .. [2] S. Rivollier. Analyse d’image geometrique et morphometrique par diagrammes de forme et voisinages adaptatifs generaux. PhD thesis, 2010. Ecole Nationale Superieure des Mines de Saint-Etienne. https://tel.archives-ouvertes.fr/tel-00560838 Examples -------- >>> from skimage import data, util >>> from skimage.measure import label >>> # coins image (binary) >>> img_coins = data.coins() > 110 >>> # total perimeter of all objects in the image >>> perimeter_crofton(img_coins, directions=2) # doctest: +ELLIPSIS 8144.578... >>> perimeter_crofton(img_coins, directions=4) # doctest: +ELLIPSIS 7837.077... """ if image.ndim != 2: raise NotImplementedError('`perimeter_crofton` supports 2D images only') # as image could be a label image, transform it to binary image image = (image > 0).astype(np.uint8) image = np.pad(image, pad_width=1, mode='constant') XF = ndi.convolve( image, np.array([[0, 0, 0], [0, 1, 4], [0, 2, 8]]), mode='constant', cval=0 ) h = np.bincount(XF.ravel(), minlength=16) # definition of the LUT if directions == 2: coefs = [ 0, np.pi / 2, 0, 0, 0, np.pi / 2, 0, 0, np.pi / 2, np.pi, 0, 0, np.pi / 2, np.pi, 0, 0, ] else: coefs = [ 0, np.pi / 4 * (1 + 1 / (np.sqrt(2))), np.pi / (4 * np.sqrt(2)), np.pi / (2 * np.sqrt(2)), 0, np.pi / 4 * (1 + 1 / (np.sqrt(2))), 0, np.pi / (4 * np.sqrt(2)), np.pi / 4, np.pi / 2, np.pi / (4 * np.sqrt(2)), np.pi / (4 * np.sqrt(2)), np.pi / 4, np.pi / 2, 0, 0, ] total_perimeter = coefs @ h return total_perimeter def _normalize_spacing(spacing, ndims): """Normalize spacing parameter. The `spacing` parameter should be a sequence of numbers matching the image dimensions. If `spacing` is a scalar, assume equal spacing along all dimensions. Parameters --------- spacing : Any User-provided `spacing` keyword. ndims : int Number of image dimensions. Returns ------- spacing : array Corrected spacing. Raises ------ ValueError If `spacing` is invalid. """ spacing = np.array(spacing) if spacing.shape == (): spacing = np.broadcast_to(spacing, shape=(ndims,)) elif spacing.shape != (ndims,): raise ValueError( f"spacing isn't a scalar nor a sequence of shape {(ndims,)}, got {spacing}." ) if not all(isinstance(s, Real) for s in spacing): raise TypeError( f"Element of spacing isn't float or integer type, got {spacing}." ) if not all(np.isfinite(spacing)): raise ValueError( f"Invalid spacing parameter. All elements must be finite, got {spacing}." ) return spacing