#cython: cdivision=True #cython: boundscheck=True #cython: nonecheck=True #cython: wraparound=False import numpy as np from numpy import pi cimport numpy as cnp cimport cython from libc.math cimport cos, sin, floor, ceil, sqrt, abs cpdef bilinear_ray_sum(cnp.ndarray[cnp.double_t, ndim=2] image, double theta, double ray_position): '''Compute the projection of an image along a ray. Parameters ---------- image : 2D array, dtype=float Image to project. :param theta: Angle of the projection. :param ray_position: 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. * pi cdef double radius = image.shape[0] // 2 - 1 cdef double projection_center = image.shape[0] // 2 - 1 cdef double rotation_center = image.shape[0] // 2 # (s, t) is the (x, y) system rotated by theta cdef double t = ray_position - projection_center # s0 is the half-length of the ray's path in the reconstruction circle cdef double s0 s0 = sqrt(radius**2 - t**2) if radius**2 >= t**2 else 0. cdef Py_ssize_t Ns = 2 * int(ceil(2 * s0)) # number of steps along the ray cdef double ray_sum = 0. cdef double weight_norm = 0. cdef double ds, dx, dy, x0, y0, x, y, di, dj, index_i, index_j 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: ray_sum += (1. - di) * (1. - dj) * image[i, j] * ds weight_norm += ((1 - di) * (1 - dj) * ds)**2 if i > 0 and j < image.shape[1] - 1: ray_sum += (1. - di) * dj * image[i, j+1] * ds weight_norm += ((1 - di) * dj * ds)**2 if i < image.shape[0] - 1 and j > 0: ray_sum += di * (1 - dj) * image[i+1, j] * ds weight_norm += (di * (1 - dj) * ds)**2 if i < image.shape[0] - 1 and j < image.shape[1] - 1: ray_sum += di * dj * image[i+1, j+1] * ds weight_norm += (di * dj * ds)**2 return ray_sum, weight_norm cpdef bilinear_ray_update(cnp.ndarray[cnp.double_t, ndim=2] image, cnp.ndarray[cnp.double_t, ndim=2] image_update, double theta, double ray_position, double projected_value): """Compute the update along a ray using bilinear interpolation. Parameters ---------- image : Current reconstruction estimate image_update : Array of same shape as ``image``. Updates will be added to this array. theta : Angle of the projection ray_position : Position of the ray within the projection projected_value : Projected value (from the sinogram) Returns ------- deviation : Deviation before updating the image """ cdef double 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. * pi cdef double radius = image.shape[0] // 2 - 1 cdef double projection_center = image.shape[0] // 2 - 1 cdef double rotation_center = image.shape[0] // 2 # (s, t) is the (x, y) system rotated by theta cdef double t = ray_position - projection_center # s0 is the half-length of the ray's path in the reconstruction circle cdef double s0 s0 = sqrt(radius*radius - t*t) if radius**2 >= t**2 else 0. cdef unsigned int Ns = 2 * int(ceil(2 * s0)) cdef double hamming_beta = 0.46164 cdef double ds, dx, dy, x0, y0, x, y, di, dj, index_i, index_j cdef double hamming_window cdef unsigned int 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*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 def sart_projection_update(cnp.ndarray[cnp.double_t, ndim=2] image, \ double theta, \ cnp.ndarray[cnp.double_t, ndim=1] projection): cdef cnp.ndarray[cnp.double_t, ndim=2] image_update = np.zeros_like(image) cdef unsigned int ray_position cdef Py_ssize_t i for i in range(projection.shape[0]): # TODO: # ip may differ from i in the future (for alignment of projections) ray_position = i bilinear_ray_update(image, image_update, theta, ray_position, projection[i]) return image_update