Source code for magni.imaging.dictionaries._mtx1D
"""
..
Copyright (c) 2014-2017, Magni developers.
All rights reserved.
See LICENSE.rst for further information.
Module providing 1D matrices for building 2D separable transforms.
Routine listings
----------------
get_DCT_transform_matrix(N)
Return the normalised N-by-N discrete cosine transform (DCT) matrix.
get_DFT_transform_matrix(N)
Return the normalised N-by-N discrete fourier transform (DFT) matrix.
"""
from __future__ import division
import numpy as np
import scipy.linalg
from magni.utils.validation import decorate_validation as _decorate_validation
from magni.utils.validation import validate_generic as _generic
from magni.utils.validation import validate_levels as _levels
from magni.utils.validation import validate_numeric as _numeric
[docs]def get_DCT_transform_matrix(N):
"""
Return the normalised N-by-N discrete cosine transform (DCT) matrix.
Applying the returned transform matrix to a vector x: D.dot(x) yields the
DCT of x. Applying the returned transform matrix to a matrix A: D.dot(A)
applies the DCT to the columns of A. Taking D.dot(A.dot(D.T)) applies the
DCT to both columns and rows, i.e. a full 2D separable DCT transform. The
inverse transform (the 1D IDCT) is D.T.
Parameters
----------
N : int
The size of the DCT transform matrix to return.
Returns
-------
D : ndarray
The DCT transform matrix.
Notes
-----
The returned DCT matrix normalised such that is consitutes a orthonormal
transform as given by equations (2.119) and (2.120) in [1]_.
References
----------
.. [1] A.N. Akansu, R.A. Haddad, and P.R. Haddad, *Multiresolution Signal
Decomposition: Transforms, Subbands, and Wavelets*, Academic Press,
2000.
Examples
--------
For example, get a 5-by-5 DCT matrix
>>> import numpy as np
>>> from magni.imaging.dictionaries import get_DCT_transform_matrix
>>> D = get_DCT_transform_matrix(5)
>>> np.round(np.abs(D), 4)
array([[ 0.4472, 0.4472, 0.4472, 0.4472, 0.4472],
[ 0.6015, 0.3717, 0. , 0.3717, 0.6015],
[ 0.5117, 0.1954, 0.6325, 0.1954, 0.5117],
[ 0.3717, 0.6015, 0. , 0.6015, 0.3717],
[ 0.1954, 0.5117, 0.6325, 0.5117, 0.1954]])
and apply the 2D DCT transform to a dummy image
>>> np.random.seed(6021)
>>> img = np.random.randn(5, 5)
>>> img_dct = D.dot(img.dot(D.T))
>>> np.round(img_dct, 4)
array([[-0.5247, -0.0225, 0.9098, 0.369 , -0.477 ],
[ 1.7309, -0.4142, 1.9455, -0.6726, -1.3676],
[ 0.6987, 0.5355, 0.7213, -0.8498, -0.1023],
[ 0.0078, -0.0545, 0.3649, -1.4694, 1.732 ],
[-1.5864, 0.156 , 0.8932, -0.8091, 0.5056]])
"""
@_decorate_validation
def validate_input():
_numeric('N', 'integer', range_='[1;inf)')
validate_input()
nn, rr = np.meshgrid(*map(np.arange, (N, N)))
D = np.cos((2 * nn + 1) * rr * np.pi / (2 * N))
D[0, :] /= np.sqrt(N)
D[1:, :] /= np.sqrt(N/2)
return D
[docs]def get_DFT_transform_matrix(N):
"""
Return the normalised N-by-N discrete fourier transform (DFT) matrix.
Applying the returned transform matrix to a vector x: D.dot(x) yields the
DFT of x. Applying the returned transform matrix to a matrix A: D.dot(A)
applies the DFT to the columns of A. Taking D.dot(A.dot(D.T)) applies the
DFT to both columns and rows, i.e. a full 2D separable DFT transform. The
inverse transform (the 1D IDFT) is D.T.
Parameters
----------
N : int
The size of the DFT transform matrix to return.
Returns
-------
D : ndarray
The DFT transform matrix.
See Also
--------
scipy.linalg.dft : The function used to generate the DFT transform matrix.
Notes
-----
The returned DFT matrix normalised such that is consitutes a orthonormal
transform as given by equations (2.105) and (2.109) in [2]_.
References
----------
.. [2] A.N. Akansu, R.A. Haddad, and P.R. Haddad, *Multiresolution Signal
Decomposition: Transforms, Subbands, and Wavelets*, Academic Press,
2000.
Examples
--------
For example, get a 5-by-5 DFT matrix
>>> import numpy as np, scipy.fftpack
>>> from magni.imaging.dictionaries import get_DFT_transform_matrix
>>> D = get_DFT_transform_matrix(5)
>>> np.round(D, 2)
array([[ 0.45+0.j , 0.45+0.j , 0.45+0.j , 0.45+0.j , 0.45+0.j ],
[ 0.45+0.j , 0.14-0.43j, -0.36-0.26j, -0.36+0.26j, 0.14+0.43j],
[ 0.45+0.j , -0.36-0.26j, 0.14+0.43j, 0.14-0.43j, -0.36+0.26j],
[ 0.45+0.j , -0.36+0.26j, 0.14-0.43j, 0.14+0.43j, -0.36-0.26j],
[ 0.45+0.j , 0.14+0.43j, -0.36+0.26j, -0.36-0.26j, 0.14-0.43j]])
and apply the 2D DFT transform to a dummy image
>>> np.random.seed(6021)
>>> img = np.random.randn(5, 5)
>>> img_dft = D.dot(img.dot(D.T))
>>> np.round(img_dft, 2)
array([[-0.52+0.j , 0.44+0.48j, 0.11-0.39j, 0.11+0.39j, 0.44-0.48j],
[ 1.04-0.59j, 1.32+0.13j, -1.20-0.39j, 0.35+0.66j, 0.19-0.36j],
[ 0.18-1.24j, 0.75+0.44j, -0.75+0.72j, -0.52-0.8j , 0.77+0.13j],
[ 0.18+1.24j, 0.77-0.13j, -0.52+0.8j , -0.75-0.72j, 0.75-0.44j],
[ 1.04+0.59j, 0.19+0.36j, 0.35-0.66j, -1.20+0.39j, 1.32-0.13j]])
which may be shifted to have the zero-frequency component at the center of
the spectrum
>>> np.round(scipy.fftpack.fftshift(img_dft), 2)
array([[-0.75-0.72j, 0.75-0.44j, 0.18+1.24j, 0.77-0.13j, -0.52+0.8j ],
[-1.20+0.39j, 1.32-0.13j, 1.04+0.59j, 0.19+0.36j, 0.35-0.66j],
[ 0.11+0.39j, 0.44-0.48j, -0.52+0.j , 0.44+0.48j, 0.11-0.39j],
[ 0.35+0.66j, 0.19-0.36j, 1.04-0.59j, 1.32+0.13j, -1.20-0.39j],
[-0.52-0.8j , 0.77+0.13j, 0.18-1.24j, 0.75+0.44j, -0.75+0.72j]])
"""
@_decorate_validation
def validate_input():
_numeric('N', 'integer', range_='[1;inf)')
validate_input()
D = scipy.linalg.dft(N, scale='sqrtn')
return D