# magni.imaging.dictionaries._mtx1D module¶

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.
magni.imaging.dictionaries._mtx1D.get_DCT_transform_matrix(N)[source]

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. 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

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]])

magni.imaging.dictionaries._mtx1D.get_DFT_transform_matrix(N)[source]

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. D (ndarray) – The DFT transform matrix.

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

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]])