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: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):
# Roll the pth row of ai left by p places
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):
# Rolls the pth row of ai right by p places.
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
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