#cython: cdivision=True #cython: boundscheck=False #cython: nonecheck=False #cython: wraparound=False import numpy as np cimport numpy as cnp cimport cython from libc.math cimport cos, sin, floor, ceil, sqrt, abs, M_PI cpdef bilinear_ray_sum(cnp.double_t[:, :] image, cnp.double_t theta, cnp.double_t ray_position): """ Compute the projection of an image along a ray. Parameters ---------- image : 2D array, dtype=float Image to project. theta : float Angle of the projection ray_position : float Position of the ray within the projection Returns ------- projected_value : float Ray sum along the projection norm_of_weights : A measure of how long the ray's path through the reconstruction circle was """ theta = theta / 180. * M_PI cdef cnp.double_t radius = image.shape[0] // 2 - 1 cdef cnp.double_t projection_center = image.shape[0] // 2 cdef cnp.double_t rotation_center = image.shape[0] // 2 # (s, t) is the (x, y) system rotated by theta cdef cnp.double_t t = ray_position - projection_center # s0 is the half-length of the ray's path in the reconstruction circle cdef cnp.double_t s0 s0 = sqrt(radius**2 - t**2) if radius**2 >= t**2 else 0. cdef Py_ssize_t Ns = 2 * ( ceil(2 * s0)) # number of steps # along the ray cdef cnp.double_t ray_sum = 0. cdef cnp.double_t weight_norm = 0. cdef cnp.double_t ds, dx, dy, x0, y0, x, y, di, dj, cdef cnp.double_t index_i, index_j, weight cdef Py_ssize_t k, i, j if Ns > 0: # step length between samples ds = 2 * s0 / Ns dx = -ds * cos(theta) dy = -ds * sin(theta) # point of entry of the ray into the reconstruction circle x0 = s0 * cos(theta) - t * sin(theta) y0 = s0 * sin(theta) + t * cos(theta) for k in range(Ns+1): x = x0 + k * dx y = y0 + k * dy index_i = x + rotation_center index_j = y + rotation_center i = floor(index_i) j = floor(index_j) di = index_i - floor(index_i) dj = index_j - floor(index_j) # Use linear interpolation between values # Where values fall outside the array, assume zero if i > 0 and j > 0: weight = (1. - di) * (1. - dj) * ds ray_sum += weight * image[i, j] weight_norm += weight**2 if i > 0 and j < image.shape[1] - 1: weight = (1. - di) * dj * ds ray_sum += weight * image[i, j+1] weight_norm += weight**2 if i < image.shape[0] - 1 and j > 0: weight = di * (1 - dj) * ds ray_sum += weight * image[i+1, j] weight_norm += weight**2 if i < image.shape[0] - 1 and j < image.shape[1] - 1: weight = di * dj * ds ray_sum += weight * image[i+1, j+1] weight_norm += weight**2 return ray_sum, weight_norm cpdef bilinear_ray_update(cnp.double_t[:, :] image, cnp.double_t[:, :] image_update, cnp.double_t theta, cnp.double_t ray_position, cnp.double_t projected_value): """ Compute the update along a ray using bilinear interpolation. Parameters ---------- image : 2D array, dtype=float Current reconstruction estimate image_update : 2D array, dtype=float Array of same shape as ``image``. Updates will be added to this array. theta : float Angle of the projection ray_position : float Position of the ray within the projection projected_value : float Projected value (from the sinogram) Returns ------- deviation : Deviation before updating the image """ cdef cnp.double_t ray_sum, weight_norm, deviation ray_sum, weight_norm = bilinear_ray_sum(image, theta, ray_position) if weight_norm > 0.: deviation = -(ray_sum - projected_value) / weight_norm else: deviation = 0. theta = theta / 180. * M_PI cdef cnp.double_t radius = image.shape[0] // 2 - 1 cdef cnp.double_t projection_center = image.shape[0] // 2 cdef cnp.double_t rotation_center = image.shape[0] // 2 # (s, t) is the (x, y) system rotated by theta cdef cnp.double_t t = ray_position - projection_center # s0 is the half-length of the ray's path in the reconstruction circle cdef cnp.double_t s0 s0 = sqrt(radius*radius - t*t) if radius**2 >= t**2 else 0. cdef Py_ssize_t Ns = 2 * ( ceil(2 * s0)) cdef cnp.double_t hamming_beta = 0.46164 # beta for equiripple Hamming window cdef cnp.double_t ds, dx, dy, x0, y0, x, y, di, dj, index_i, index_j cdef cnp.double_t hamming_window cdef Py_ssize_t k, i, j if Ns > 0: # Step length between samples ds = 2 * s0 / Ns dx = -ds * cos(theta) dy = -ds * sin(theta) # Point of entry of the ray into the reconstruction circle x0 = s0 * cos(theta) - t * sin(theta) y0 = s0 * sin(theta) + t * cos(theta) for k in range(Ns+1): x = x0 + k * dx y = y0 + k * dy index_i = x + rotation_center index_j = y + rotation_center i = floor(index_i) j = floor(index_j) di = index_i - floor(index_i) dj = index_j - floor(index_j) hamming_window = ((1 - hamming_beta) - hamming_beta * cos(2 * M_PI * k / (Ns - 1))) if i > 0 and j > 0: image_update[i, j] += (deviation * (1. - di) * (1. - dj) * ds * hamming_window) if i > 0 and j < image.shape[1] - 1: image_update[i, j+1] += (deviation * (1. - di) * dj * ds * hamming_window) if i < image.shape[0] - 1 and j > 0: image_update[i+1, j] += (deviation * di * (1 - dj) * ds * hamming_window) if i < image.shape[0] - 1 and j < image.shape[1] - 1: image_update[i+1, j+1] += (deviation * di * dj * ds * hamming_window) return deviation @cython.boundscheck(True) def sart_projection_update(cnp.double_t[:, :] image not None, cnp.double_t theta, cnp.double_t[:] projection not None, cnp.double_t projection_shift=0.): """ Compute update to a reconstruction estimate from a single projection using bilinear interpolation. Parameters ---------- image : 2D array, dtype=float Current reconstruction estimate theta : float Angle of the projection projection : 1D array, dtype=float Projected values, taken from the sinogram projection_shift : float Shift the position of the projection by this many pixels before using it to compute an update to the reconstruction estimate Returns ------- image_update : 2D array, dtype=float Array of same shape as ``image`` containing updates that should be added to ``image`` to improve the reconstruction estimate """ cdef cnp.ndarray[cnp.double_t, ndim=2] image_update = np.zeros_like(image) cdef cnp.double_t ray_position cdef Py_ssize_t i for i in range(projection.shape[0]): ray_position = i + projection_shift bilinear_ray_update(image, image_update, theta, ray_position, projection[i]) return image_update