| """ |
| :author: Gary Ruben, 2009 |
| :license: modified BSD |
| """ |
|
|
| __all__ = ["frt2", "ifrt2"] |
|
|
| import numpy as np |
| from numpy import roll, newaxis |
|
|
|
|
| def frt2(a): |
| """Compute the 2-dimensional finite Radon transform (FRT) for the input array. |
| |
| Parameters |
| ---------- |
| a : ndarray of int, shape (M, M) |
| Input array. |
| |
| Returns |
| ------- |
| FRT : ndarray of int, shape (M+1, M) |
| Finite Radon Transform array of coefficients. |
| |
| See Also |
| -------- |
| ifrt2 : The two-dimensional inverse FRT. |
| |
| Notes |
| ----- |
| The FRT has a unique inverse if and only if M is prime. [FRT] |
| The idea for this algorithm is due to Vlad Negnevitski. |
| |
| Examples |
| -------- |
| |
| Generate a test image: |
| Use a prime number for the array dimensions |
| |
| >>> SIZE = 59 |
| >>> img = np.tri(SIZE, dtype=np.int32) |
| |
| Apply the Finite Radon Transform: |
| |
| >>> f = frt2(img) |
| |
| References |
| ---------- |
| .. [FRT] A. Kingston and I. Svalbe, "Projective transforms on periodic |
| discrete image arrays," in P. Hawkes (Ed), Advances in Imaging |
| and Electron Physics, 139 (2006) |
| |
| """ |
| if a.ndim != 2 or a.shape[0] != a.shape[1]: |
| raise ValueError("Input must be a square, 2-D array") |
|
|
| ai = a.copy() |
| n = ai.shape[0] |
| f = np.empty((n + 1, n), np.uint32) |
| f[0] = ai.sum(axis=0) |
| for m in range(1, n): |
| |
| for row in range(1, n): |
| ai[row] = roll(ai[row], -row) |
| f[m] = ai.sum(axis=0) |
| f[n] = ai.sum(axis=1) |
| return f |
|
|
|
|
| def ifrt2(a): |
| """Compute the 2-dimensional inverse finite Radon transform (iFRT) for the input array. |
| |
| Parameters |
| ---------- |
| a : ndarray of int, shape (M+1, M) |
| Input array. |
| |
| Returns |
| ------- |
| iFRT : ndarray of int, shape (M, M) |
| Inverse Finite Radon Transform coefficients. |
| |
| See Also |
| -------- |
| frt2 : The two-dimensional FRT |
| |
| Notes |
| ----- |
| The FRT has a unique inverse if and only if M is prime. |
| See [1]_ for an overview. |
| The idea for this algorithm is due to Vlad Negnevitski. |
| |
| Examples |
| -------- |
| |
| >>> SIZE = 59 |
| >>> img = np.tri(SIZE, dtype=np.int32) |
| |
| Apply the Finite Radon Transform: |
| |
| >>> f = frt2(img) |
| |
| Apply the Inverse Finite Radon Transform to recover the input |
| |
| >>> fi = ifrt2(f) |
| |
| Check that it's identical to the original |
| |
| >>> assert len(np.nonzero(img-fi)[0]) == 0 |
| |
| References |
| ---------- |
| .. [1] A. Kingston and I. Svalbe, "Projective transforms on periodic |
| discrete image arrays," in P. Hawkes (Ed), Advances in Imaging |
| and Electron Physics, 139 (2006) |
| |
| """ |
| if a.ndim != 2 or a.shape[0] != a.shape[1] + 1: |
| raise ValueError("Input must be an (n+1) row x n column, 2-D array") |
|
|
| ai = a.copy()[:-1] |
| n = ai.shape[1] |
| f = np.empty((n, n), np.uint32) |
| f[0] = ai.sum(axis=0) |
| for m in range(1, n): |
| |
| for row in range(1, ai.shape[0]): |
| ai[row] = roll(ai[row], row) |
| f[m] = ai.sum(axis=0) |
| f += a[-1][newaxis].T |
| f = (f - ai[0].sum()) / n |
| return f |
|
|
