SDecGMCA (Spherical Deconvolution Generalized Morphological Component Analysis) is an algorithm aiming at solving joint Deconvolution and Blind Source Separation (DBSS) problems with spherical data.
The 2D adaptation of the algorithm can be found here.
Let us consider the imaging model:
,
where:
are the
multiwavelength spherical data, stacked in a matrix,
is measurement operator,
is the mixing matrix,
are the
sources, 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 on the sphere. The equation can be simplified in the spherical harmonics domain for each harmonic coefficient
and for all channels:
.
The sources are assumed to be sparse in the starlet representation . SDecGMCA 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.
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.
SDecGMCA has been developed with Python 3.7. It depends on the Python Healpy library, which is only supported on Linux and MAC OS X, not Windows.
The following Python libraries need to be installed to run the code:
- NumPy,
- Healpy (see https://healpy.readthedocs.io/en/latest/install.html for the installation procedure),
- Matplotlib [optional].
SDecGMCA 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 A and S (the harmonic projection of the sources is in addition stored in Slm).
Below is the list of the main attributes of the SDecGMCA class.
| Parameter | Type | Information | Default value |
|---|---|---|---|
X |
(m,p) float or (m,t) complex numpy.ndarray | input data in Healpix representation or in spherical harmonic domain, each row corresponds to an observation | N/A |
Hl |
(m,lmax+1) float numpy.ndarray | convolution kernels in spherical harmonics domain (lmax : maximum frequency) | N/A |
n |
int | number of sources to be estimated | N/A |
alm_in |
bool | the data X are in the spherical harmonic domain (Healpix nside=lmax/3) | False |
M |
(p,) float numpy.ndarray | mask in Healpix representation | 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 SDecGMCA 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 |
iterSH |
int | number of iterations for the spherical harmonic transforms | 3 |
Perform a DBSS on the data X with 4 sources.
sdecgmca = SDecGMCA(X, Hl, n=4, k=3, nscales=3, c_wu=numpy.array([5, 0.5]), c_ref=0.5, nStd=1e-7)
sdecgmca.run()
S = sdecgmca.S.copy() # estimated sources
A = sdecgmca.A.copy() # estimated mixing matrix- Rémi Carloni Gertosio
- Jérôme Bobin
R. Carloni Gertosio, J. Bobin, Joint deconvolution and unsupervised source separation for data on the sphere
This project is licensed under the LGPL-3.0 License.