2DecGMCA

2DecGMCA (Deconvolution Generalized Morphological Component Analysis) is an algorithm aiming at solving joint Deconvolution and Blind Source Separation (DBSS) problems. It is a 2D adaptation of SDecGMCA, which processes spherical data.

Introduction
Procedure
Getting Started
Parameters
Example
Authors
Reference
License

Introduction

Let us consider the imaging model:

,

where:

are the multiwavelength data, flattened and stacked in a matrix,
is measurement operator,
is the mixing matrix,
are the sources, flattened and stacked in a matrix,
is the noise.

In the following, the measurement operator is assumed channel-dependant, linear and isotropic. Thus, for channel :

,

where denotes the convolution product. The equation can be simplified in Fourier domain:

The sources are assumed to be sparse in the starlet representation . DecGMCA aims at minimizing the following objective function with respect to A and S:

,

where denotes the element-wise product, are the sparsity regularization parameters and is the oblique ensemble. Moreover, non-negativity constraints on and/or may be added.

Procedure

The algorithm is built upon a sparsity-enforcing projected alternate least-square procedure, which updates iteratively the sources and the mixing matrix. In a nutshell, when either or is updated, a first least-squares estimate is computed by minimizing the data-fidelity term. This step is then followed by the application of the proximal operator of the corresponding regularization term.

In contrast to standard BSS problems, the least-square update of the sources is not necessarily stable with respect to noise. Thus, an extra Tikhonov regularization is added.

The separation is comprised of two stages. The first stage estimates a first guess of the mixing matrix and the sources (warm-up); it is required to provide robustness with respect to the initial point. The second stage refines the separation by employing a more precise Tikhonov regularization strategy (refinement). Lastly, the sources are improved during a finale step with the output mixing matrix.

During the warm-up, the Tikhonov regularization coefficients are calculated from the current estimation of the mixing matrix. During the refinement, they are rather calculated from the current estimation of the sources spectra.

Getting Started

Requirements

2DecGMCA has been developed with Python 3.7.

Prerequisites

The following Python libraries need to be installed to run the code:

NumPy,
Matplotlib [optional].

DecGMCA class

2DecGMCA is implemented in a class. The data and the parameters of the separation are provided at the initialization of the object. The separation is performed by running the method run. The results are stored in the attributes S and A.

Parameters

Below is the list of the main attributes of the DecGMCA class.

Parameter	Type	Information	Default value
`X`	(m,p) float numpy.ndarray	input data, each row corresponds to a flattened observation	N/A
`Hfft`	(m,p) float numpy.ndarray	convolution kernels in Fourier domain (flattened & with 0-frequency shifted to the center)	N/A
`n`	int	number of sources to be estimated	N/A
`fft_in`	bool	the data X are in Fourier domain	False
`M`	(p,) float numpy.ndarray	mask	None
`nnegA`	bool	non-negativity constraint on A	False
`nnegS`	bool	non-negativity constraint on S	False
`nneg`	bool	non-negativity constraint on A and S. If not None, overrides nnegA and nnegS	None
`c_wu`	float or (2,) numpy.ndarray	Tikhonov regularization hyperparameter at warm-up	0.5
`c_ref`	float	Tikhonov regularization hyperparameter at refinement	0.5
`nStd`	float	noise standard deviation (compulsory for spectrum-based regularization & analytical estimation of the noise std in source domain)	N/A
`useMad`	bool	True: noise std estimated with MAD, False: noise std estimated analytically	False
`nscales`	int	number of starlet detail scales	3
`k`	float	parameter of the k-std thresholding	3
`K_max`	float	maximal L0 norm of the sources. Being a percentage, it should be between 0 and 1	0.5
`thrEnd`	bool	perform thresholding during the finale estimation of the sources	True
`eps`	(3,) float numpy.ndarray	stopping criteria of (1) the warm-up, (2) the refinement and (3) the finale refinement of S	[1e-2, 1e-4, 1e-4]
`verb`	int	verbosity level, from 0 (mute) to 5 (most talkative)	0

Below is the list of the other attributes of the DecGMCA class, which can reasonably be equal to their default values.

Parameter	Type	Information	Default value
`AInit`	(m,n) float numpy.ndarray	initial value for the mixing matrix. If None, PCA-based initialization	None
`keepWuRegStr`	bool	keep warm-up regularization strategy during refinement (else: spectra-based coefficients)	False
`cstWuRegStr`	bool	use constant regularization coefficients during warm-up (else: mixing-matrix-based coefficients)	False
`minWuIt`	int	minimum number of iterations at warm-up	100
`cwuDec`	int	number of iterations for the decrease of c_wu (if c_wu is an array)	minWuIt/2
`L1`	bool	if False, L0 rather than L1 penalization	True
`doRw`	bool	do L1 reweighing during refinement (only if L1 penalization)	True

Example

Perform a DBSS on the data X with 4 sources.

decgmca = DecGMCA(X, Hfft, n=4, k=3, nscales=3, c_wu=numpy.array([5, 0.5]), c_ref=0.5, nStd=1e-7)
decgmca.run()
S = decgmca.S.copy()  # estimated sources
A = decgmca.A.copy()  # estimated mixing matrix

Authors

Rémi Carloni Gertosio
Jérôme Bobin

Reference

R. Carloni Gertosio, J. Bobin, Joint deconvolution and unsupervised source separation for data on the sphere

License

This project is licensed under the LGPL-3.0 License.

RCarloniGertosio/2DecGMCA