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 """ Ridge filters. Ridge filters can be used to detect continuous edges, such as vessels, neurites, wrinkles, rivers, and other tube-like structures. The present class of ridge filters relies on the eigenvalues of the Hessian matrix of image intensities to detect tube-like structures where the intensity changes perpendicular but not along the structure. """ from warnings import warn import numpy as np from ..util import img_as_float, invert from .._shared.utils import assert_nD def _divide_nonzero(array1, array2, cval=1e-10): """ Divides two arrays. Denominator is set to small value where zero to avoid ZeroDivisionError and return finite float array. Parameters ---------- array1 : (N, ..., M) ndarray Array 1 in the enumerator. array2 : (N, ..., M) ndarray Array 2 in the denominator. cval : float, optional Value used to replace zero entries in the denominator. Returns ------- array : (N, ..., M) ndarray Quotient of the array division. """ # Copy denominator denominator = np.copy(array2) # Set zero entries of denominator to small value denominator[denominator == 0] = cval # Return quotient return np.divide(array1, denominator) def _sortbyabs(array, axis=0): """ Sort array along a given axis by absolute values. Parameters ---------- array : (N, ..., M) ndarray Array with input image data. axis : int Axis along which to sort. Returns ------- array : (N, ..., M) ndarray Array sorted along a given axis by absolute values. Notes ----- Modified from: http://stackoverflow.com/a/11253931/4067734 """ # Create auxiliary array for indexing index = list(np.ix_(*[np.arange(i) for i in array.shape])) # Get indices of abs sorted array index[axis] = np.abs(array).argsort(axis) # Return abs sorted array return array[tuple(index)] def compute_hessian_eigenvalues(image, sigma, sorting='none'): """ Compute Hessian eigenvalues of nD images. For 2D images, the computation uses a more efficient, skimage-based algorithm. Parameters ---------- image : (N, ..., M) ndarray Array with input image data. sigma : float Smoothing factor of image for detection of structures at different (sigma) scales. sorting : {'val', 'abs', 'none'}, optional Sorting of eigenvalues by values ('val') or absolute values ('abs'), or without sorting ('none'). Default is 'none'. Returns ------- eigenvalues : (D, N, ..., M) ndarray Array with (sorted) eigenvalues of Hessian eigenvalues for each pixel of the input image. """ # Import has to be here due to circular import error from ..feature import hessian_matrix, hessian_matrix_eigvals # Convert image to float image = img_as_float(image) # Make nD hessian hessian_elements = hessian_matrix(image, sigma=sigma, order='rc') # Correct for scale hessian_elements = [(sigma ** 2) * e for e in hessian_elements] # Compute Hessian eigenvalues hessian_eigenvalues = np.array(hessian_matrix_eigvals(hessian_elements)) if sorting == 'abs': # Sort eigenvalues by absolute values in ascending order hessian_eigenvalues = _sortbyabs(hessian_eigenvalues, axis=0) elif sorting == 'val': # Sort eigenvalues by values in ascending order hessian_eigenvalues = np.sort(hessian_eigenvalues, axis=0) # Return Hessian eigenvalues return hessian_eigenvalues def meijering(image, sigmas=range(1, 10, 2), alpha=None, black_ridges=True): """ Filter an image with the Meijering neuriteness filter. This filter can be used to detect continuous ridges, e.g. neurites, wrinkles, rivers. It can be used to calculate the fraction of the whole image containing such objects. Calculates the eigenvectors of the Hessian to compute the similarity of an image region to neurites, according to the method described in _. Parameters ---------- image : (N, M[, ..., P]) ndarray Array with input image data. sigmas : iterable of floats, optional Sigmas used as scales of filter alpha : float, optional Frangi correction constant that adjusts the filter's sensitivity to deviation from a plate-like structure. black_ridges : boolean, optional When True (the default), the filter detects black ridges; when False, it detects white ridges. Returns ------- out : (N, M[, ..., P]) ndarray Filtered image (maximum of pixels across all scales). References ---------- ..  Meijering, E., Jacob, M., Sarria, J. C., Steiner, P., Hirling, H., Unser, M. (2004). Design and validation of a tool for neurite tracing and analysis in fluorescence microscopy images. Cytometry Part A, 58(2), 167-176. :DOI:`10.1002/cyto.a.20022` """ # Check (sigma) scales sigmas = np.asarray(sigmas) if np.any(sigmas < 0.0): raise ValueError('Sigma values less than zero are not valid') # Get image dimensions ndim = image.ndim # Set parameters if alpha is None: alpha = 1.0 / ndim # Invert image to detect bright ridges on dark background if not black_ridges: image = invert(image) # Generate empty (n+1)D arrays for storing auxiliary images filtered at # different (sigma) scales filtered_array = np.zeros(sigmas.shape + image.shape) # Filtering for all (sigma) scales for i, sigma in enumerate(sigmas): # Calculate (sorted) eigenvalues eigenvalues = compute_hessian_eigenvalues(image, sigma, sorting='val') if ndim > 1: # Set coefficients for scaling eigenvalues coefficients = [alpha] * ndim coefficients = 1 # Compute auxiliary variables l_i = e_i + sum_{j!=i} alpha * e_j auxiliary = [np.sum([eigenvalues[i] * np.roll(coefficients, j)[i] for j in range(ndim)], axis=0) for i in range(ndim)] # Compute maximum over auxiliary variables auxiliary = np.max(auxiliary, axis=0) # Rescale image intensity filtered = np.abs(auxiliary) / np.abs(np.max(auxiliary)) # Remove background filtered = np.where(filtered > 0, filtered, 0) # Store results in (n+1)D matrices filtered_array[i] = filtered # Return for every pixel the maximum value over all (sigma) scales return np.max(filtered_array, axis=0) def sato(image, sigmas=range(1, 10, 2), black_ridges=True): """ Filter an image with the Sato tubeness filter. This filter can be used to detect continuous ridges, e.g. tubes, wrinkles, rivers. It can be used to calculate the fraction of the whole image containing such objects. Defined only for 2-D and 3-D images. Calculates the eigenvectors of the Hessian to compute the similarity of an image region to tubes, according to the method described in _. Parameters ---------- image : (N, M[, P]) ndarray Array with input image data. sigmas : iterable of floats, optional Sigmas used as scales of filter. black_ridges : boolean, optional When True (the default), the filter detects black ridges; when False, it detects white ridges. Returns ------- out : (N, M[, P]) ndarray Filtered image (maximum of pixels across all scales). References ---------- ..  Sato, Y., Nakajima, S., Shiraga, N., Atsumi, H., Yoshida, S., Koller, T., ..., Kikinis, R. (1998). Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. Medical image analysis, 2(2), 143-168. :DOI:`10.1016/S1361-8415(98)80009-1` """ # Check image dimensions assert_nD(image, [2, 3]) # Check (sigma) scales sigmas = np.asarray(sigmas) if np.any(sigmas < 0.0): raise ValueError('Sigma values less than zero are not valid') # Invert image to detect bright ridges on dark background if not black_ridges: image = invert(image) # Generate empty (n+1)D arrays for storing auxiliary images filtered # at different (sigma) scales filtered_array = np.zeros(sigmas.shape + image.shape) # Filtering for all (sigma) scales for i, sigma in enumerate(sigmas): # Calculate (sorted) eigenvalues lamba1, *lambdas = compute_hessian_eigenvalues(image, sigma, sorting='val') # Compute tubeness, see equation (9) in reference _. # np.abs(lambda2) in 2D, np.sqrt(np.abs(lambda2 * lambda3)) in 3D filtered = np.abs(np.multiply.reduce(lambdas)) ** (1/len(lambdas)) # Remove background and store results in (n+1)D matrices filtered_array[i] = np.where(lambdas[-1] > 0, filtered, 0) # Return for every pixel the maximum value over all (sigma) scales return np.max(filtered_array, axis=0) def frangi(image, sigmas=range(1, 10, 2), scale_range=None, scale_step=None, beta1=None, beta2=None, alpha=0.5, beta=0.5, gamma=15, black_ridges=True): """ Filter an image with the Frangi vesselness filter. This filter can be used to detect continuous ridges, e.g. vessels, wrinkles, rivers. It can be used to calculate the fraction of the whole image containing such objects. Defined only for 2-D and 3-D images. Calculates the eigenvectors of the Hessian to compute the similarity of an image region to vessels, according to the method described in _. Parameters ---------- image : (N, M[, P]) ndarray Array with input image data. sigmas : iterable of floats, optional Sigmas used as scales of filter, i.e., np.arange(scale_range, scale_range, scale_step) scale_range : 2-tuple of floats, optional The range of sigmas used. scale_step : float, optional Step size between sigmas. alpha : float, optional Frangi correction constant that adjusts the filter's sensitivity to deviation from a plate-like structure. beta = beta1 : float, optional Frangi correction constant that adjusts the filter's sensitivity to deviation from a blob-like structure. gamma = beta2 : float, optional Frangi correction constant that adjusts the filter's sensitivity to areas of high variance/texture/structure. black_ridges : boolean, optional When True (the default), the filter detects black ridges; when False, it detects white ridges. Returns ------- out : (N, M[, P]) ndarray Filtered image (maximum of pixels across all scales). Notes ----- Written by Marc Schrijver, November 2001 Re-Written by D. J. Kroon, University of Twente, May 2009, _ Adoption of 3D version from D. G. Ellis, Januar 20017, _ References ---------- ..  Frangi, A. F., Niessen, W. J., Vincken, K. L., & Viergever, M. A. (1998,). Multiscale vessel enhancement filtering. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 130-137). Springer Berlin Heidelberg. :DOI:`10.1007/BFb0056195` ..  Kroon, D. J.: Hessian based Frangi vesselness filter. ..  Ellis, D. G.: https://github.com/ellisdg/frangi3d/tree/master/frangi """ # Check deprecated keyword parameters if beta1: warn('Use keyword parameter `beta` instead of `beta1` which ' 'will be removed in version 0.17.') beta = beta1 if beta2: warn('Use keyword parameter `gamma` instead of `beta2` which ' 'will be removed in version 0.17.') gamma = beta2 if scale_range and scale_step: warn('Use keyword parameter `sigmas` instead of `scale_range` and ' '`scale_range` which will be removed in version 0.17.') sigmas = np.arange(scale_range, scale_range, scale_step) # Check image dimensions assert_nD(image, [2, 3]) # Check (sigma) scales sigmas = np.asarray(sigmas) if np.any(sigmas < 0.0): raise ValueError('Sigma values less than zero are not valid') # Rescale filter parameters alpha_sq = 2 * alpha ** 2 beta_sq = 2 * beta ** 2 gamma_sq = 2 * gamma ** 2 # Get image dimensions ndim = image.ndim # Invert image to detect dark ridges on light background if black_ridges: image = invert(image) # Generate empty (n+1)D arrays for storing auxiliary images filtered # at different (sigma) scales filtered_array = np.zeros(sigmas.shape + image.shape) lambdas_array = np.zeros(sigmas.shape + image.shape) # Filtering for all (sigma) scales for i, sigma in enumerate(sigmas): # Calculate (abs sorted) eigenvalues lambda1, *lambdas = compute_hessian_eigenvalues(image, sigma, sorting='abs') # Compute sensitivity to deviation from a plate-like structure # see equations (11) and (15) in reference _ r_a = np.inf if ndim == 2 else _divide_nonzero(*lambdas) ** 2 # Compute sensitivity to deviation from a blob-like structure, # see equations (10) and (15) in reference _, # np.abs(lambda2) in 2D, np.sqrt(np.abs(lambda2 * lambda3)) in 3D filtered_raw = np.abs(np.multiply.reduce(lambdas)) ** (1/len(lambdas)) r_b = _divide_nonzero(lambda1, filtered_raw) ** 2 # Compute sensitivity to areas of high variance/texture/structure, # see equation (12)in reference _ r_g = sum([lambda1 ** 2] + [lambdai ** 2 for lambdai in lambdas]) # Compute output image for given (sigma) scale and store results in # (n+1)D matrices, see equations (13) and (15) in reference _ filtered_array[i] = ((1 - np.exp(-r_a / alpha_sq)) * np.exp(-r_b / beta_sq) * (1 - np.exp(-r_g / gamma_sq))) lambdas_array[i] = np.max(lambdas, axis=0) # Remove background filtered_array[lambdas_array > 0] = 0 # Return for every pixel the maximum value over all (sigma) scales return np.max(filtered_array, axis=0) def hessian(image, sigmas=range(1, 10, 2), scale_range=None, scale_step=None, beta1=None, beta2=None, alpha=0.5, beta=0.5, gamma=15, black_ridges=True): """Filter an image with the Hybrid Hessian filter. This filter can be used to detect continuous edges, e.g. vessels, wrinkles, rivers. It can be used to calculate the fraction of the whole image containing such objects. Defined only for 2-D and 3-D images. Almost equal to Frangi filter, but uses alternative method of smoothing. Refer to _ to find the differences between Frangi and Hessian filters. Parameters ---------- image : (N, M[, P]) ndarray Array with input image data. sigmas : iterable of floats, optional Sigmas used as scales of filter, i.e., np.arange(scale_range, scale_range, scale_step) scale_range : 2-tuple of floats, optional The range of sigmas used. scale_step : float, optional Step size between sigmas. beta = beta1 : float, optional Frangi correction constant that adjusts the filter's sensitivity to deviation from a blob-like structure. gamma = beta2 : float, optional Frangi correction constant that adjusts the filter's sensitivity to areas of high variance/texture/structure. black_ridges : boolean, optional When True (the default), the filter detects black ridges; when False, it detects white ridges. Returns ------- out : (N, M[, P]) ndarray Filtered image (maximum of pixels across all scales). Notes ----- Written by Marc Schrijver (November 2001) Re-Written by D. J. Kroon University of Twente (May 2009) _ References ---------- ..  Ng, C. C., Yap, M. H., Costen, N., & Li, B. (2014,). Automatic wrinkle detection using hybrid Hessian filter. In Asian Conference on Computer Vision (pp. 609-622). Springer International Publishing. :DOI:`10.1007/978-3-319-16811-1_40` ..  Kroon, D. J.: Hessian based Frangi vesselness filter. """ filtered = frangi(image, sigmas=sigmas, scale_range=scale_range, scale_step=scale_step, beta1=beta1, beta2=beta2, alpha=alpha, beta=beta, gamma=gamma, black_ridges=black_ridges) filtered[filtered <= 0] = 1 return filtered
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