pysparse.representation package

Submodules

pysparse.representation.dcts module

pysparse.representation.dcts.dct1(x, axis=0)

1-Dimension Discrete cosine transform

The DCT of signal \(x[n], n=0, 1,\cdots, N-1\) is expressed as

(1)\[y[k] = {\rm DCT}(x[n]) = \left\{ {\begin{array}{lll} {\sqrt{\frac{2}{N}}\sum_{n=0}^{N-1}x[n]\frac{1}{\sqrt 2}, \quad k=0}\\ {\sqrt{\frac{2}{N}}\sum_{n=0}^{N-1}x[n]{\rm cos}\frac{(2n + 1)k\pi}{2N}, \quad k=1, \cdots, N-1} \end{array}} \right. \]

where, \(k=0, 1, \cdots, N-1\)

N. Ahmed, T. Natarajan, and K. R. Rao. Discrete cosine transform. IEEE Transactions on Computers, C-23(1):90–93, 1974. doi:10.1109/T-C.1974.223784

Parameters:x (numpy array) – signal vector or matrix Keyword Arguments:  axis (number) – transformation axis when x is a matrix (default: {0}, col) Returns:y – the coefficients. Return type:numpy array

pysparse.representation.dcts.dct2(X)

2-Dimension Discrete cosine transform

dct1(dct1(X, axis=0), axis=1)

Parameters:X (numpy array) – signal matrix Returns:Y – coefficients matrix Return type:numpy array

pysparse.representation.dcts.dctdict(N, isnorm=False, verbose=False)

Complete DCT dictionary

(2)\[{\bm D} = \sqrt{\frac{2}{N}}\left[\begin{array}{cccc}{1/\sqrt{2}} & {1/\sqrt{2}} & {1/\sqrt{2}} & {\cdots} & {1/\sqrt{2}} \\ {\cos \frac{\pi}{2 N}} & {\cos \frac{3 \pi}{2 N}} & {\cos \frac{5 \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1) \pi}{2 N}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\cos \frac{(N-1) \pi}{2 N}} & {\cos \frac{3(N-1) \pi}{2 N}} & {\cos \frac{5(N-1) \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1)(N-1) \pi}{2 N}}\end{array}\right] \]

Because \({\bm z} = {\bm D}{\bm x}\), \({\bm D}{\bm D}^T = {\bm I}\) So, \({\bm x} = {\bm D}^T{\bm z}\)

Parameters:N (integer) – The dictionary is of size \(N\times N\) Returns:D – DCT dictionary ( \({\bm D}^T\) ). Return type:numpy array

pysparse.representation.dcts.dctmat(N)

Discrete cosine transform matrix

(3)\[{\bm y} = {\bm D}{\bm x} \]

where, \({\bm x} = (x_n)_{N\times 1}, x_n = x[n]\), \({\bm D} = (d_{ij})_{N\times N}\) can be expressed as

\[{\bm D} = \sqrt{\frac{2}{N}}\left[\begin{array}{cccc}{1/\sqrt{2}} & {1/\sqrt{2}} & {1/\sqrt{2}} & {\cdots} & {1/\sqrt{2}} \\ {\cos \frac{\pi}{2 N}} & {\cos \frac{3 \pi}{2 N}} & {\cos \frac{5 \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1) \pi}{2 N}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\cos \frac{(N-1) \pi}{2 N}} & {\cos \frac{3(N-1) \pi}{2 N}} & {\cos \frac{5(N-1) \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1)(N-1) \pi}{2 N}}\end{array}\right] \]

Parameters:N (integer) – signal dimesion. Returns:T – DCT matrix. Return type:ndarray

pysparse.representation.dcts.idct1(y, axis=0)

1-Dimension Inverse Discrete cosine transform

(4)\[{\bm x} = {\bm D}^{-1}{\bm y} = {\bm D}^T{\bm y} \]

Parameters:y (numpy array) – coefficients Keyword Arguments:  axis (number) – IDCT along which axis (default: {0}) Returns:x – recovered signal. Return type:numpy array

pysparse.representation.dcts.idct2(X)

2-Dimension Inverse Discrete cosine transform

idct1(idct1(X, axis=0), axis=1)

Parameters:X (numpy array) – signal matrix Returns:Y – coefficients matrix Return type:numpy array

pysparse.representation.dcts.odctdict(dictshape, isnorm=False, verbose=False)

Overcomplete 1D-DCT dictionary

(5)\[{\bm D} = \sqrt{\frac{2}{N}}\left[\begin{array}{cccc}{1/\sqrt{2}} & {1/\sqrt{2}} & {1/\sqrt{2}} & {\cdots} & {1/\sqrt{2}} \\ {\cos \frac{\pi}{2 N}} & {\cos \frac{3 \pi}{2 N}} & {\cos \frac{5 \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1) \pi}{2 N}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\cos \frac{(M-1) \pi}{2 N}} & {\cos \frac{3(M-1) \pi}{2 N}} & {\cos \frac{5(M-1) \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1)(M-1) \pi}{2 N}}\end{array}\right] \]

(6)\[{\bm D} = \left[\frac{{\bm d}_0}{\|{\bm d}_0\|_2}, \frac{{\bm d}_1}{\|{\bm d}_1\|_2},\cdots, \frac{{\bm d}_{N-1}}{\|{\bm d}_{N-1}\|_2}\right] \]

Parameters:

dictshape (tuple) – dictionary shape

Keyword Arguments:  

• isnorm (bool) – normlize atoms (default: {False})
• verbose (bool) – display log (default: {True})

pysparse.representation.dcts.odctndict(dictshape, axis=-1, isnorm=False, verbose=False)

generates Overcomplete nD-DCT dictionary

(7)\[{\bm D}_{nd} = {\bm D}_{(n-1)d} \otimes {\bm D}_{(n-1)d}. \]

Parameters:

dictshape (list or tuple) – shape of DCT dict [M, N]

Keyword Arguments:  

• axis (number) – Axis along which the dct is computed. If -1 then the transform is multidimensional(default=-1) (default: {-1})
• isnorm (bool) – normlize atoms (default: {False})
• verbose (bool) – display log info (default: {True})

Returns:

OD – Overcomplete nD-DCT dictionary

Return type:

ndarray

pysparse.representation.dfts module

pysparse.representation.dfts.dft1(x, axis=0)

1-Dimension Discrete cosine transform

The DFT of signal \(x[n], n=0, 1,\cdots, N-1\) is expressed as

(8)\[y[k] = {\rm DFT}(x[n]) = \left\{ {\begin{array}{lll} {\sqrt{\frac{2}{N}}\sum_{n=0}^{N-1}x[n]\frac{1}{\sqrt 2}, \quad k=0}\\ {\sqrt{\frac{2}{N}}\sum_{n=0}^{N-1}x[n]{\rm cos}\frac{(2n + 1)k\pi}{2N}, \quad k=1, \cdots, N-1} \end{array}} \right. \]

where, \(k=0, 1, \cdots, N-1\)

N. Ahmed, T. Natarajan, and K. R. Rao. Discrete cosine transform. IEEE Transactions on Computers, C-23(1):90–93, 1974. doi:10.1109/T-C.1974.223784

Parameters:x (numpy array) – signal vector or matrix Keyword Arguments:  axis (number) – transformation axis when x is a matrix (default: {0}, col) Returns:y – the coefficients. Return type:numpy array

pysparse.representation.dfts.dft2(X)

2-Dimension Discrete cosine transform

dft1(dft1(X, axis=0), axis=1)

Parameters:X (numpy array) – signal matrix Returns:Y – coefficients matrix Return type:numpy array

pysparse.representation.dfts.dftdict(N, isnorm=False, verbose=False)

Complete DFT dictionary

(9)\[{\bm D} = \sqrt{\frac{2}{N}}\left[\begin{array}{cccc}{1/\sqrt{2}} & {1/\sqrt{2}} & {1/\sqrt{2}} & {\cdots} & {1/\sqrt{2}} \\ {\cos \frac{\pi}{2 N}} & {\cos \frac{3 \pi}{2 N}} & {\cos \frac{5 \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1) \pi}{2 N}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\cos \frac{(N-1) \pi}{2 N}} & {\cos \frac{3(N-1) \pi}{2 N}} & {\cos \frac{5(N-1) \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1)(N-1) \pi}{2 N}}\end{array}\right] \]

Because \({\bm z} = {\bm D}{\bm x}\), \({\bm D}{\bm D}^T = {\bm I}\) So, \({\bm x} = {\bm D}^T{\bm z}\)

Parameters:N (integer) – The dictionary is of size \(N\times N\) Returns:D – DFT dictionary ( \({\bm D}^T\) ). Return type:numpy array

pysparse.representation.dfts.dftmat(N)

Discrete Fourier transform matrix

(10)\[{\bm y} = {\bm D}{\bm x} \]

where, \({\bm x} = (x_n)_{N\times 1}, x_n = x[n]\), \({\bm D} = (d_{ij})_{N\times N}\) can be expressed as

\[{\bm D} = \sqrt{\frac{2}{N}}\left[\begin{array}{cccc}{1/\sqrt{2}} & {1/\sqrt{2}} & {1/\sqrt{2}} & {\cdots} & {1/\sqrt{2}} \\ {\cos \frac{\pi}{2 N}} & {\cos \frac{3 \pi}{2 N}} & {\cos \frac{5 \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1) \pi}{2 N}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\cos \frac{(N-1) \pi}{2 N}} & {\cos \frac{3(N-1) \pi}{2 N}} & {\cos \frac{5(N-1) \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1)(N-1) \pi}{2 N}}\end{array}\right] \]

Parameters:N (integer) – signal dimesion. Returns:T – DFT matrix. Return type:ndarray

pysparse.representation.dfts.idft1(y, axis=0)

1-Dimension Inverse Discrete cosine transform

(11)\[{\bm x} = {\bm D}^{-1}{\bm y} = {\bm D}^T{\bm y} \]

Parameters:y (numpy array) – coefficients Keyword Arguments:  axis (number) – IDFT along which axis (default: {0}) Returns:x – recovered signal. Return type:numpy array

pysparse.representation.dfts.idft2(X)

2-Dimension Inverse Discrete cosine transform

idft1(idft1(X, axis=0), axis=1)

Parameters:X (numpy array) – signal matrix Returns:Y – coefficients matrix Return type:numpy array

pysparse.representation.dfts.odftdict(dictshape, isnorm=False, verbose=False)

Overcomplete 1D-DFT dictionary

(12)\[{\bm D} = \sqrt{\frac{2}{N}}\left[\begin{array}{cccc}{1/\sqrt{2}} & {1/\sqrt{2}} & {1/\sqrt{2}} & {\cdots} & {1/\sqrt{2}} \\ {\cos \frac{\pi}{2 N}} & {\cos \frac{3 \pi}{2 N}} & {\cos \frac{5 \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1) \pi}{2 N}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\cos \frac{(M-1) \pi}{2 N}} & {\cos \frac{3(M-1) \pi}{2 N}} & {\cos \frac{5(M-1) \pi}{2 N}} & {\cdots} & {\cos \frac{(2 N-1)(M-1) \pi}{2 N}}\end{array}\right] \]

(13)\[{\bm D} = \left[\frac{{\bm d}_0}{\|{\bm d}_0\|_2}, \frac{{\bm d}_1}{\|{\bm d}_1\|_2},\cdots, \frac{{\bm d}_{N-1}}{\|{\bm d}_{N-1}\|_2}\right] \]

Parameters:

dictshape (tuple) – dictionary shape

Keyword Arguments:  

• isnorm (bool) – normlize atoms (default: {False})
• verbose (bool) – display log (default: {True})

pysparse.representation.dfts.odftndict(dictshape, axis=-1, isnorm=False, verbose=False)

generates Overcomplete nD-DFT dictionary

(14)\[{\bm D}_{nd} = {\bm D}_{(n-1)d} \otimes {\bm D}_{(n-1)d}. \]

Parameters:

dictshape (list or tuple) – shape of DFT dict [M, N]

Keyword Arguments:  

• axis (number) – Axis along which the dft is computed. If -1 then the transform is multidimensional(default=-1) (default: {-1})
• isnorm (bool) – normlize atoms (default: {False})
• verbose (bool) – display log info (default: {True})

Returns:

OD – Overcomplete nD-DFT dictionary

Return type:

ndarray

Module contents