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.
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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. 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]])
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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. 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]])