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# CEA-jiangming / DecGMCA Public

Joint Multichannel Deconvolution and Blind Source Separation algorithm

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# DecGMCA

DecGMCA (Deconvolution Generalized Morphological Component Analysis) is a sparsity-based algorithm aiming at solving joint multichannel Deconvolution and Blind Source Separation (DBSS) problem.

For more details, please refer to the paper Joint Multichannel Deconvolution and Blind Source Separation (https://arxiv.org/abs/1703.02650)

## Introduction

Considering a multichannel DBSS problem:

Y = H * ( AS ) + N

Y: observation of size Nc by Np for an Nc-channel and Np-pixel imaging system.

H: a linear convolution kernel of size Nc by Np (e.g. downsampling matrix, psf), often ill-conditioned in practice.

A: unknown mixing matrix of size Nc by Ns, representing Ns sources are blindly linearly mixed (entities of row i represents the weighted contribution of sources at the given channel number i).

S: sources of size Ns by Np, representing Ns sources of Np pixels (sources are aligned to row vectors).

N: additional noise of size Nc by Np, supposed to be Gaussian.

This problem can be conveniently written in Fourier space, which has a lot of interests in Fourier imaging systems such as radio interferometry, MRI, etc. In Fourier space, noticing the convolution is transformed to product and A is unchanged as its entities are actually scalar factors applied to sources, the problem is written as

\hat{Y} = \hat{H} ( A \hat{S} ) + \hat{N}

The sources are sparse in a given dictionary $\Phi$. Thus, the yielding optimization problem is written as follows:

min_{A,S} ||\hat{Y} - \hat{H} A \hat{S}||_2^2 + \lambda ||S \Phi||_p, where p-norm is 0-norm or 1-norm.

## Optimization

Challenges:

1. As the above optimization is non-convex, only critical point can be expected.
2. Convolution kernel is ill-conditioned, leading to the unstability of the solution.

The main idea of solution is based on alternating minimization but we do not directly apply alternating proximity-based algorithms due to its computational demanding. Our DecGMCA employs an alternating projected least-squares procedure to approach the critical point plus a proximity-based procedure to finally refine the solution. Thus, DecGMCA is structured as:

• Intitialization
• Alternating projected least-squares
• Refinement step

### Initialization

A simple initialization (of matrix A) can be realized by randomization. One can also have more accurate initialization which depends on the form of the convolution kernel H:

• If H is a downsampling matrix, we apply several iterations of matrix completion scheme (e.g. SVT algorithm). Then the initialization is acheived by selecting Ns eigonvectors of left singular matrix after the application of singular value decomposition (svd) on the completed data.
• If H is a convolution kernel (not a downsampling matrix), we only keep Ns eigonvectors of left singular matrix as the initialization after svd on the completed data.

### Alternating projected least-squares

The procedure is based on the alternating update of one variable with respect to the other.

#### Update S with respect to A

This update can be divided into two steps: approximation of S via least-squares and sparsity thresholding.

As for the approximation of S via least-squares, due to the ill-conditioned kernel H, a regularization parameter $\epsilon$ is involved to stablize the deconvolution. This parameter acts as a Tikhonov parameter: If $\epsilon$ is large, the system will be more regularized but the solution is less accurate. Conversely, if $\epsilon$ is small, the system will be less regularized but the solution is more accurate. The second step is sparsity thresholding to have a clean estimate of S. This step is realized by hard-thresholding with threshold $\lambda$. (Although soft-thresholding for l1-norm has more beautiful mathematical convergence proof, we argue that hard-thresholding has no biais effect and has empirical convergence according to our experiments). The consideration of $\epsilon$ and $\lambda$ is extensively studied in the paper.

#### Update A with respect to S

This update is just a simple least-squares. One should notice that columns of A should be l2 normalized.

### Refinement step

The alternating projected least-squares is efficient but does not necessarily ensure the optimal solution. This is owing to the fact that the projected least-squares has algorithmic biais compared to the exact projection realized by proximal algorithms. Thus, in order to refine the solution (often S), we resolve the problem of updating S with respect to A by using proximal algorithms such as Forward-Backward, Condat-Vu primal dual, etc.

## Python DecGMCA package

### Prerequisites

#### Basic python environment

This package has been tested with python 2.7, some python libraries are required to ensure a correct working:

• numpy
• scipy
• matplotlib
• astropy The above libraries are accessible in macport, pip or other package manager systems. For instance, via macport:
port install some-package-name


or via pip:

pip install some-package-name


One may need root permission for the above operations.

#### Accelaration of codes? Interface python with C++ and paralization

For large-scale data, one may be not satisfied python (python can be up to 50 times slower than C/C++). In this DecGMCA package, we have an option to interface python with C++ and paralize the codes. The following packages are required:

• GCC (tested with GCC 4.9)
• CMake (tested with v3.9)
• Boost (tested with v1.58)
• Cfitsio
• OMP (tested with GCC 4.9) These packages are easily installed via apt-get (Linux), homebrew (MacOS) or directly from the website. For instance, via apt-get
apt-get install some-package-name


or via homebrew

brew install some-package-name


#### Problem with MacOS?

The default C compiler of MacOS is Clang. One should set GCC as the default C compiler and compile all dependencies (Boost, Cfitsio, etc.)

Among all packages, the Boost package is the most troublesome on MacOS. Here is a rapid solution:

tar -xzvf boost_1_58_0.tar.gz

• Inside the boost directory run the bootstrap.sh script specifying a path to where the libraries are to be installed as follows:
./bootstrap.sh --prefix=/opt/local/ --with-toolset=gcc

• Run the b2 script as follows:
./b2 install


## Installing

If all of the prerequisites are installed properly, one only needs to download the whole repository on his local machine.

The package includes:

• pyDecGMCA: main DecGMCA package
• mathTools.py: some useful mathematical operations
• pyUtils.py: useful routines such as two stages of update
• algoDecG.py: python DecGMCA algorithm
• pyProx.py: python proximal algorithms (used for the refinement step)
• boost_Prox.py: partially accelerated proximal algorithms
• boost_algoDecG.py: accelerated DecGMCA algorithm
• boost_utils.py: accelerated routines such as two stages of update. One needs to compile DecGMCA_utils previously.
• pyWavelet: python wavelet tools
• waveTools.py: some operations for wavelet coefficients
• wav1d.py: 1D wavelet
• wav2d.py: 2D wavelet
• starlet_utils.py: accelerated starlet transform. One needs to compile pystarlet previously.
• simulationsTools: used to generate sources for tests
• MakeExperiment.py
• simu_CS_deconv: used for running tests
• param.py: global parameters for tests
• test_CS.py: test script for compressed sensing test
• test_deconv.py: test script for deconvolution test
• evaluation: used to evaluate results, such as criteria A and S.
• evaluation.py

The following packages are needed to interface python with C++

• DecGMCA_utils
• pystarlet

Instructions for compilation (e.g. DecGMCA_utils):

• Inside the DecGMCA_utils directory create a build dossier
mkdir build

• Inside the build directory and run
cmake ..
make


Then a shared object decG.so is created. Please move this object to the directory pyDecGMCA. The same compilation goes with pystarlet as well. After the compilation, a shared object starlet.so should be moved to the directory pyWavelet.

## Execution

Given correct parameters, one only needs to run the function DecGMCA located in pyDecGMCA.

### Example

Assume noised multichannel 1D data (in Fourier space, Fourier plane option (FTPlane) should be set True) is V_N, linear operator is Mask which downsamples the data points, the number of sources is 5, the size of each source is 1 by 4096. The number of iterations is set 500, the initial epsilon for regularization is 10^{-1} and the final epsilon is 10^{-4}, the wavelet option is True with starlet wavelet and 4 scales of decomposition, the thresholding strategy is 2 with threshold 3. To avoid that the low frequency data affect the sources separation quality, a high-pass filter is applied before the update of mixing matrix. The cut-off frequency of this high-pass filter is set 1./16 without logistic smoothness (False). Both of the positivity constraints are False. The refinement step (postProc) is True with max iterations 50 and parameter (Ksig) 3.

(S_est,A_est) = DecGMCA(V_N,Mask,5,1,4096,500,1e-1,1e-4,Ndim=1,wavelet=True,scale=4,mask=True,deconv=False,wname='starlet',thresStrtg=2,FTPlane=True,fc=1./16,logistic=False,postProc=True,postProcImax=50,Kend=3.0,Ksig=3.0,positivityS=False,positivityA=False)


### Run simulations

This package consists of reproducible simulations presented in the paper. One needs to enter the directory simu_CS_deconv and run corresponding simulation scripts.

#### Multichannel compressed sensing and blind source separation simulation

This simulation corresponds to the case where H is a downsampling matrix.

In the paper, we studied the performance of DecGMCA in terms of the sub-sampling effect, the number of sources, the SNR. One can change these parameters in the file param.py. To run the simulation, only need to run the script test_CS.py.

• Remarks of parameters (in param.py):

• pcArr: array of different ratios of present data (used for compressed sensing simulation)
• ratioArr: array of different resolution ratios (used for deconvolution simulation)
• nArr: array of number of sources
• dbArr: array of different SNRs
• bdArr: array of number of bands (channels)
• numTests: number of Monte-Carlo tests
• Comparison with other methods The DecGMCA method is compared with other methods:

• Matrix completion + BSS (GMCA): controled by the option MC_GMCA_flag in the script.

The results (mixing matrix A and source matrix S) of the simulation will be all saved in the same directory. In order to evaluate the quality of the results, one needs to go to the directory evaluation and run the script script_CS.py.

#### Multichannel deconvolution and blind source separation simulation

This simulation corresponds to the case where H is an ill-conditioned linear kernel (e.g. PSF).

The design of this simulation has the same structure as the above one. In this part, we studied the performance of DecGMCA in terms of the resolution ratio (corresponding to the condition number of H), the number of sources, the SNR. Similarly, one can play with these parameters in the same file param.py. To run the simulation, only need to run the script test_deconv.py.

• Comparison with other methods The DecGMCA method is compared with other methods:
• BSS alone (GMCA): controled by the option GMCA_flag in the script.
• Sequential deconvolution (ForWaRD) and BSS (GMCA): controled by the option ForWaRD_GMCA_flag in the script.

The results (mixing matrix A and source matrix S) of the simulation will be all saved in the same directory. In order to evaluate the quality of the results, one needs to go to the directory evaluation and run the script script_kernel.py.

## Acknowledgments

This work is supported by the CEA DRF impulsion project COSMIC and the European Community through the grants PHySIS (contract no. 60174), DEDALE (contract no. 665044) and LENA (contract no. 678282) within the H2020 Framework Programe.

Joint Multichannel Deconvolution and Blind Source Separation algorithm

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