Christophe Kervazo

Sparse blind source
separation: from handcrafted
to automatic methods
Christophe Kervazo
Joint works with Elsa Angelini, Jérôme Bobin, Cécile Chenot, Abdelkhalak Chetoui, Jérémy Cohen, Erwan Dereure,
Mohammad Fahes, Rassim Hadjeres, Florent Sureau, Florence Tupin, and others…

View Slide

L2S seminar
Blind Source Separation (BSS): example of applications
2
Goal: unmix some signals
Figures: courtesy from: *https://cacm.acm.org/news/190656-3d-printed-device-helps-computers-solve-cocktail-party-problem/fulltext, ** Rapin, J. (2014). Décompositions parcimonieuses pour
l'analyse avancée de données en spectrométrie pour la Santé, *** Chenot, C. (2017). Parcimonie, diversité morphologique et séparation robuste de sources.**** Courtesy of F. Acero
Cocktail party problem*

Hyperspectral unmixing in remote sensing***
Spectroscopy** Show-through removal
Astrophysics****
…
/

View Slide

L2S seminar
BSS: Linear model [Comon10]
3
Goal of BSS : estimate A* and S* from X
X : m rows observations and t samples columns (m x t)

A* : mixing function (m x n)

S* : sources (n x t) - the sources will be assumed to be sparse

N : noise and model imperfections (m x t)
X = A⇤S⇤ + N
(up to limited indeterminacies)
= + … + N
X
x
y
x
y
x
y
A⇤1
S⇤
1
+
S⇤
2
A⇤2

View Slide

L2S seminar
Sparse BSS [Zibulevsky01]
4
Ill-posed problem [1]

=> introduce prior about S:

- ICA [Comon10] : Independent Component Analysis

- NMF [Gillis12] : Non-negative Matrix Factorization

- SMF [Zibulevsky01] : Sparse Matrix Factorization
S
X = A⇤S⇤ + N
: wavelet domain
S
(S⇤
S)1
Sparsity: Most of the source coefficients are close to 0 in
S
S⇤
1
Direct domain

View Slide

L2S seminar
BSS as an optimization problem
5
Written as an optimization problem [Zibulevsky01, Bobin07]:
RS : Regularization parameters (size n x t)
How to solve sparse BSS in practice?
Data-
fi
delity Sparsity Oblique constraint
argmin
A2Rm⇥n,S2Rn⇥t
1
2
kX ASk2
F
+ RS (S T
S
)
1
+ ◆{8i2[1,n];kAik2
`2
=1}
(A)
T
S
: is a sparsifying transform
Challenges:

- Non-smooth (needs advanced optimization tools: proximal operators [Parikh14])

- Non-convex (non-unique minima)

- Difficult hyperparameter choice
=> Difficult optimization problem, requiring some difficult tuning from
the user

=> In the following, we will see how to make these choices more automatic

View Slide

L2S seminar
Outline
6
I - Optimization strategy for sparse BSS: towards automatic parameter
choices
III - New data-driven methods for sparse BSS through algorithm unrolling
II - Extension to single-observation complex component separation:
continuous target extraction in SAR images

View Slide

L2S seminar
Optimization frameworks
7
pALS
PALM [Bolte14]
Gradient based Least square based
While not converged over and do:
A S
S ←
𝒮
ηRS
(S − η (AS − X))
A ← Π∥.∥2
=1 (A − ξ (AS − X))
While not stabilized over
and :
A S
S ←
𝒮
RS
(A†X)
A ←
𝒮
RS
(XS†)
While not converged over do:
S
S ←
𝒮
ηRS
(S − η (AS − X))
A ← Π∥.∥2
=1 (A − ξ (AS − X))
While not converged over do:
A
While not converged over and do:
A S
BCD [Tseng01]
argmin
A2Rm⇥n,S2Rn⇥t
1
2
kX ASk2
F
+ RS (S T
S
)
1
+ ◆{8i2[1,n];kAik2
`2
=1}
(A)
with:

- the proximal operator of the -norm (soft-thresholding of parameters , applied element-wise)

- the projection on the unit sphere

- some gradient step-sizes
𝒮
ηRS
( . ) ℓ1
ηRS
Π∥.∥2
=1
( . ) ℓ2
η, ξ
While PALM is in theory the best algorithm, it is in practice often reported to have bad
practical results in BSS. Why?

View Slide

L2S seminar
First study: why is PALM difficult to use in sparse BSS?
8
Median of CA
- 3x10-4 change => 30 dB drop

- In diagonal, a 2x10-3 change => 7 dB drop
Zoom on maximum
1
2
1
2
Empirical study: study sensitivity to regularization parameters
- Simulated n = 2 sources (dynamic range = 0.6)

- Compute [Bobin15] : indeterminacy correction
Difficult choice of RS
RS =
2
6
6
4
1 1 ... 1
2 2 ... 2
... ... ... ...
n n ... n
3
7
7
5
Let us first assume one parameter per source:
CA = 10 log(mean(|P ˆ
A†A⇤ Id
|)), P
GMCA: 36 dB

View Slide

L2S seminar
Empirical study conclusions: sensitivity of PALM + grid search
9
- Low efficiency: for given X, hard to choose RS

- Low versatility: a choice of RS does not generalize to other X

- Low reliability: sensitive to initial point
Kervazo, C., Bobin, J., Chenot, C., & Sureau, F. (2020). Use of PALM for ℓ1 sparse matrix factorization: Difficulty and rationalization of a two-step
approach. Digital Signal Processing, 97, 102611.
=> Practical aspect not well discussed in the literature (grid-search, use of the
true unknown factorization…)

=> Intractable for real data

View Slide

L2S seminar 10
Comparison between sparse BSS algorithms
BCD1 pALS2 / GMCA3 PALM4
Reference [Tseng01] [Bobin07] [Bolte14]
Fast ✕
Convergence ✓ ✓
✕
Automatic parameter
choice
✕ ✕
✓
✓✓ ✓
Easy to use
Mathematically

sound
How to obtain both an easy to use and mathematically sound algorithm?
Work on PALM
Find a min. ? ✓ ✓
✕
argmin
A2Rm⇥n,S2Rn⇥t
1
2
kX ASk2
F
+ RS (S T
S
)
1
+ ◆{8i2[1,n];kAik2
`2
=1}
(A)
Robust to spurious
min.
✕ ✓ ✕
1Block Coordinate Descent, 2 projected Alternating Least-Square, 3 Generalized Morphological Component Analysis, 4 Proximal
Alternating Linearized Minimization

View Slide

L2S seminar
More about GMCA automatic hyper-parameter choice
11
- In GMCA, use of MAD of the current source estimation (new value at each iteration):
- MAD: robust STD estimator. Insensible to sparse perturbations: MAD(S⇤) ' 0
- Such a choice is motivated by a fixed point argument: assume that at a given iteration,
GMCA has perfectly estimated , then the source update is given by:
A* S
small
RS =
2
6
6
4
1 1 ... 1
2 2 ... 2
... ... ... ...
n n ... n
3
7
7
5
argmin
A2Rm⇥n,S2Rn⇥t
1
2
kX ASk2
F
+ RS (S T
S
)
1
+ ◆{8i2[1,n];kAik2
`2
=1}
(A) with
S ←
𝒮
RS
(A*†
X) =
𝒮
RS
(A*†
A*S* + A*†
N)
=
𝒮
RS
(S* + A*†
N)
- Therefore, to make , the soft-thresholding should remove the Gaussian noise

=> by thresholding with , 99% of the noise is removed

=> here, is estimated using
S ← S* A*†
N
λi
= 3 ×
𝚜
𝚝 𝚍
[A*†
N]i
𝚜 𝚝
𝚍
[A*†
N]i
𝙼 𝙰
𝙳
( ̂
A†X)
MAD(s) ' 1.48 median(|s median(s)|)
with
λi
= κ
𝙼
𝙰𝙳
[ ̂
A†X]i

View Slide

L2S seminar
Choosing hyper-parameters in PALM?
12
If we assume errors on the sources: = S* + s (in particular, s non-zero at initialization)
s ̂
S
Implement same heuristic as in GMCA:

If true A* and S* are found, similar interpretation at convergence
=> Unadapted threshold choice with the MAD due to interferences
Add dense interferences

(not present GMCA)
i
'  ⇥ MAD
⇥
A⇤T N A⇤T A⇤s
⇤
i
i
'  ⇥ MAD A⇤T N
i
Need to limit interferences :
- Good initialization (s(0) small)

- Reweighted (adapt thresholds to sources)
ℓ1
PALM
λi
≃
𝙼𝙰𝙳
[A*†
N]i
GMCA

View Slide

L2S seminar
Algorithm: embed PALM within a 2-step approach
GMCA robustness
to initialization
Automatic

parameter choice

(computed on good

initialization)
Reweighting

from GMCA
PALM theoretical

background
13
RS = ⇤SG

View Slide

L2S seminar
Experiment on realistic astrophysics data
14
Experiments from simulated Chandra data

n = 3 sources

t = 128 x 128 pixels

m = 12 observations
S*
ˆ
S
S⇤ ˆ
S
2-step
Fast
Convergence ✓
Automatic Rs
Find a min. ? ✓
Reliable
✓
✓
✓
SNR (dB) 10 15 20 30 60
2 step 15.0 16.3 17.
4

19.7 20.9
PALM 11.9 13.3 13.5 14.2 14.5
GMCA 13.2 14.8 15.1 17.1 18.6
EFICA [Kodolvsky06] 8.8 10.3 14.0 18.9 19.4
RNA [Zibulevsky03] 9.8 12.6 15.6 18.3 18.4
CA (in dB)

View Slide

L2S seminar
Recent use: bioluminescence
15
- Application in biomedical imaging (real dataset)

- Modality: optical imaging based on the counting of photons emitted by biological tissues
through a bioluminescence reaction

- Goal: enable to separate multiple tumors in mice, based on their temporal
bioluminescence activity
Dereure E., Kervazo C., Seguin J., Garofalakis A., Mignet N., Angelini E., Olivo-Marin J.-C., Sparse non-negative matrix factorization
for preclinical bioluminescent imaging, Accepted to ISBI 2023 conference

View Slide

L2S seminar
Recent use: bioluminescence
16
- Non-negative extension of the above work: PALM is used for minimizing
arg min
A,S
1
2
∥X − AS∥2
F
+ ∥R ⊙ S∥1
+ i.≥0
(A) . + i.≥0
(S) + i{∀j∈[1,n],∥Aj∥ℓ2
≤1}
(A)
- A good initialization and some reweighting are used, the thresholds are based on the MAD

- In addition, a gain is used for each source to handle their imblance.
No sparsity
Fixed gain

No reweighting
Fixed gain

Reweighting
A d a p t a t i v e
gain

A d a p t a t i v e
gain

View Slide

L2S seminar
Outline
17
choices

View Slide

Target extraction
SAR imaging & objective
18
- SAR imaging is a remote sensing modality, yielding complex
images

- SAR images are the superposition

- a background

- some targets
x ∈ ℂn×n
n ∈ ℂn×n
t ∈ ℂn×n
Here, we will work on the complex images : x = n + t ∈ ℂn×n
Goal: retrieve both and from , which can be seen as a complex single-channel source
separation problem
n t
|x| |n| |t|

View Slide

Target extraction
Additional information about the problem at hand
19
- The targets are assumed to be a Dirac in the scene, which are then convoluted by the SAR
imaging system PSF, which is a cardinal sine

- If the targets were lying on the image sampling grid, they would also be Dirac on the image

- However, they have continuous positions in the scene => they ressemble cardinal sines,
making their separation harder (we cannot just extract the local maxima in the image)
- For each target, we want to estimate:

- Its amplitude

- The localization of the maximum with a subpixellic accuracy
Objective

View Slide

Target extraction
Sparsity as a separation principle
20
- and (background) follow Gaussian laws (of unknown variance)

- Only a few targets are present in the image
ℜ{n} ℑ{n}
Additional assumptions:

As finding and from is ill-posed, we resort to additional assumptions:
n t x = n + t
First (naive) approach
- The PSF knowledge could be exploited by creating a dictionary containing all
the cardinal sines shifted at all the sampling points of the grid

- To exploit sparsity, instead of using the pseudo-norm, we can relax it:
Φ ∈ ℝn×n
ℓ0
- For the sake of clarity, we will consider in the following vectors instead of
images
x ∈ ℝn
arg min
α∈ℝρN
1
2
∥x − Φα∥2
2
+ λ∥α∥1
,
with a set of pixel-wise hyperparameter
λ ∈ ℝ
- For subpixelic precisions, we can oversample the dictionary by a factor
ρ
- Most existing methods are greedy approaches, extracting the targets one at a time

View Slide

Target extraction
Remark: optimization trick
21
- In practice, computing is expensive, as it is in

- Fortunately, can be seen as a circulant matrix and the cost function can then be rewritten
using a convolution
Φ ℝn×ρn
Φ
- All the convolutions can then be computed in the Fourier domain. As such, we never need
to compute explicitly !
Φ
- The minimization can then be performed by using for instance the ISTA/FISTA algorithm
arg min
α∈ℝρN
1
2
∥x − Φα∥2
2
+ λ∥α∥1
,
arg min
α∈ℝρN
1
2
∥x − sinc * α∥2
2
+ λ∥α∥1
,

View Slide

Target extraction
Naive approach: limitations
22
- Exemple of results on a simulation
|x| |α|
- The targets are split in the neighboring pixels

- Too much targets are found compared to the ground truth
, zoom
|α|
- Two reasons:

- The dictionary might be ill-conditioned (especially, when is large)

- Difficulty to set the hyperparameters
Φ ρ
λ ∈ ℝ

View Slide

Target extraction
Better approach: Continuous Basis Pursuit
23
- To bypass the fact that the dictionary might be ill-conditioned, we propose to use a CBP
approach.

- Since the positions of the target are continuous, CBP is based on a Taylor expansion of the
sinc function around the grid points.
Φ
∑
i
αi
sinc(π(t − iΔ + li
)) ≃
|li
|≤Δ
∑
i
αi
sinc(π(t − iΔ)) +
αi
li
2
sinc′

(π(t − iΔ))
Positions
on the grid
Displacement compared to
the grid (continuous)
= ∑
i
αi
sinc(π(t − iΔ)) + δi
sinc′

(π(t − iΔ)), with δi
=
αi
li
2

View Slide

Target extraction
CBP cost function
24
- Using the above Taylor expansion, we can derive a new cost function:
arg min
α∈ℂρn, δ∈ℝρn
1
2
∥x − Φα − Ψδ ⊙ α∥2
2
+ λ∥α∥1
+ ι.≤ Δ
2
(δ)
- Where:

- The component-wise product is denoted as

- The columns of the dictionary are shifted derivatives of the sinc

- The is related to the shift compared to the regular grid and is a continuous variable
⊙
Ψ
δ
- Once and are found, it is easy to find the displacement over the regular grid by shifting
the grid coordinates of
α δ
Δ
2
δ

View Slide

Target extraction
CBP: optimization framework
25
- In the real case, the CBP cost function
arg min
α∈ℂsN, δ∈ℝsN
1
2
∥x − Φα − Ψδ ⊙ α∥2
2
+ λ∥α∥1
+ ι.≤ Δ
2
(δ)
can be rewritten in a convex way [Ekanadham2011].

- However, in the complex-case, it is non-convex.

- Still, it is multi-convex and we can therefore use the PALM algorithm in order to try
to minimize it.
Initialize with FISTA, and will zero shift

While not converged

return
α δ
α =
𝒮
λη (α + η [Φ* + diag(δ)Ψ*] (x − Φα − Ψα ⊙ δ))
δ ← ΠΔ
2
(δ + ξ[diag( ¯
α)Ψ*](x − Φα − Ψα ⊙ δ))
α, δ
All the products with are done using convolutions for higher efficiency
Φ, Φ*, Ψ, Ψ*
PALM for CBP

View Slide

Target extraction
Extension to images
26
- In the case of images (2-D signals) , the methodology easily extends:
x ∈ ℂsN×sN
arg min
α∈ℂsN, δh∈ℝsN, δv∈ℝsN
1
2
∥x − Φα − Ψhδh ⊙ α − Ψvδv ⊙ α∥2
2
+ λ∥α∥1
+ ι.≤ Δ
2
(δh) + + ι.≤ Δ
2
(δv)
where:

- contain the horizontal and vertical displacement, respectively

-
The columns of are and the columns of are
δh, δv
Φh
∂sinc(x, y)
∂x
Φv
∂sinc(x, y)
∂y

View Slide

Target extraction
Regularization parameter setting
27
- Choice of in:
λ
arg min
α∈ℂρn, δ∈ℝρn
1
2
∥x − Φα − Ψδ ⊙ α∥2
2
+ λ∥α∥1
+ ι.≤ Δ
2
(δ)
In principle:

- should not be constant for the whole image, as it is related to the background level

=> we can rather use a -map, with a value for each pixel

- Similarly to what we did in the previous part, we use a fixed-point argument :
λ
Λ λ (Λ ∈ ℝρn)
=> if the target are well separated, , which follows a Rayleigh law,
should be canceled by the proximal operator of the -norm
γ [ΦT + diag(δ)ΨT] n
ℓ1
α* + γ[ΦT + diag(δ)ΨT](x − Φα* − Ψα* ⊙ δ) ≃ α* + γ [ΦT + diag(δ)ΨT] n
=> standard deviation is estimated using the MAD.
γ [ΦT + diag(δ)ΨT] n

View Slide

Target extraction
Target localization: results
28
- Comparison of :

- Basis Pursuit with a threshold based on the true noise level

- Continuous Basis Pursuit with a threshold based on the true noise level

- Continuous Basis Pursuit with a threshold based on the MAD rule
Λ*
Λ*
Median target localization error
Blue: true targets; red: estimated targets
CBP, thresholds
Λ* CBP, MAD-based thresholds

View Slide

Target extraction
Target localization: results
29
- Proposed a new method for target extraction in SAR signals

- Based on convex relaxation, contrary to most other methods

- Leveraged Continuous Basis Pursuit to obtain a higher subpixellic accuracy

- Used the PALM algorithm to minimize the corresponding cost function

- An automatic threshold choice is used
Remaining work:

- Further tests on real datasets

- New ways of choosing the thresholds (Plug-and-Play methods / Unrolling ?)
Kervazo, C., & Ladjal, S. (2022). Extraction des positions continues des cibles dans les signaux RSO,
GRETSI conference.

View Slide

L2S seminar
Outline
30
choices

View Slide

GDR MIA Plug-and-Play workshop
Limitations of tackling sparse BSS using PALM
31
• If we have access to a data base with examples of mixtures and the corresponding
factors A* and S*, can we make PALM better work by introducing some learnt
components?

• The method we use here is algorithm unrolling

=> enables to bypass the cumbersome hyper-parameter choice

=> much more computationally efficient than PALM

=> yield interpretable neural networks

Contribution
• We highlighted the difficulty of tuning BSS methods [Kervazo20]

• We used an heuristic based on the use of the MAD for finding some decent hyperparameters

• But it is still handcrafted, and the MAD might not be the best noise STD estimator.

• In addition, PALM might require in any case several thousands of iterations to converge,
reducing its applicability

View Slide

Algorithm unrolling: setting
32
Sparse Blind Source Separation: minimization of the cost function
• Let us consider sparse BSS, in which (for the moment) we only want to estimate from a
(when is estimated, can be quite well estimated by least squares)
S*
X S* A*
• Let us further assume that :

• We have training datasets such that:
atrain
1X, 2X, . . , atrainX
X = A* S* + N
,
1X = 1A* 1S* +1 N
2X = 2A* 2S* +2 N
. . .
atrainX = atrainA* atrainS* +atrain N
• For each training dataset , we have access to the corresponding source
iX iS*
with « of the same kind as » (same for and
)
1A*, 2A*, . . , atrainA* A* 1S*, 2S*, . . , atrainS*
S*
Algorithm unrolling is then a principled method to construct neural network architectures
enabling to recover S*

View Slide

Algorithm unrolling: methodology
33
• (Classical) iterative method to estimate :
S*
with the algorithm parameters
(gradient step sizes…)
θ
• Algorithm unrolling truncates this scheme to rewrite it in the form of a neural network with a
small number of layers (iterations):
• The algorithms parameters becomes trainable on a training set (i.e. they becomes the
weights of the neural network)

• The number of iterations is usually much smaller than in the original algorithm
θ(k)
L
S ←
𝒮
λ
LS
(
S −
1
LS
AT(A*S − X)
)
It can be sketched as: X S
fθ
(A)
update
A
while not converged do:
Update A
X S
f(1)
θ1
(A) f(2)
θ2
(A) f(3)
θ3
(A) f(L)
θL
(A)
…

View Slide

Being more specific: LISTA algorithm
34
• [Gregor, Lecun 10] rewrote the gradient-proximal update in the form of a neural network

• It is based on a reparametrization of the update
S ←
𝒮
λ
L
(
S −
1
L
AT(AS − X))
S ←
𝒮
λ
LS
(W1
S +W2
X)
• This update is a typical NN update: linearity followed by non-linearity

•
If there was no learning, and (but note that this is not
exploited in LISTA, since and are learnt independently)
W1
= I −
1
LS
ATA W2
=
1
LS
AT
W1
W2
Θ = {λ/LS
, W1
, W2
}
LISTA update
S ←
𝒮
λ
L ((
I −
1
L
ATA
)
S +
1
L
ATX
)
⇔
learning some parts of the update

View Slide

Difficulties of applying LISTA for sparse BSS
35
• But in the original LISTA work, the A* operator is the same over all the , which is not
the case in BSS:

iX
iX = iA* iS* + iN
iA*
1
, i = 1..50 iA*
2
, i = 1..50 iA*
3
, i = 1..50 iA*
4
, i = 1..50
=> LISTA numerically obtains bad results for sparse BSS
S ←
𝒮
λ
LS
(W1
S +W2
X)
Illustration on the introductory dataset:

View Slide

LISTA-CP
36
• In [chen18], the authors have used the theoretical coupling of
• Therefore, it is important to have an update taking into account the variability of the over
the different samples
iA*
iX
W1
= I −
1
L
ATA W2
=
1
L
AT
and
to propose the LISTA-CP algorithm, leading to the update:
S ←
𝒮
λ
LS
(S − WT(AS − X))
• To handle large variations of the operator between the datasets , we propose to use a
LISTA-CP like update for update, within an alternating structure enabling to also
estimate

• This corresponds to unroll PALM in a way which is specifically Taylored for blind inverse
problems
iA* iX
iS
iA*
LISTA-CP is better suited than LISTA when all are not the same (in LISTA,
and are the same for all the samples , giving no flexibility for the update)
iA* W1
W2
iX
☺
• In LISTA-CP, the matrix appears in the update :
A
☹︎
LISTA-CP requires an estimate of iA*
S ←
𝒮
λ
LS
(W1
S +W2
X)
(LISTA update) (LISTA-CP update)

View Slide

Solving the BLIND source separation problem
37
• The way to unroll PALM was chosen according to the previous remarks and experimental
trials :
S ←
𝒮
γ (S − WT(AS − X))
A ← Π∥.∥≤1 (
A +
1
LA
(X − AS)ST
)
(LISTA-CP update)
(learning step-size)
• The loss function is chosen as
NMSE(A, A*) + NMSE(S, S*)
for k from 1 to L do :
end for
return A, S
initialize A and S with a very generic initialization
over the training set.
Learned-PALM (LPALM)

View Slide

Numerical experiments: datasets
38
• Test set:
with:
X = A*S* + N
A*
• A* coming from realistic simulations
• S* being real supernovae maps (reshaped into vectors)
• N generated Gaussian noise
S*
• Train set:
750 samples of with:
iX = iA*iS* + iN
• coming from realistic simulations (quite different from the test
set)
iA*
• generated using a Bernouilli Generalized-Gaussian
distribution
iS*
• generated Gaussian noise
iN
for . Each column
is represented with a different
color
iA* i = 1..50
LPALM is tested on astrophysical simulations of the Cassiopea A supernovae remnant as
observed by the X-ray telescope Chandra. There are emissions: synchrotron, thermal
and 2 red-shifted irons
n = 4

View Slide

Numerical results: comparison with PALM
39
Blue, plain and dashed lines: median number of iterations for PALM
and LPALM, respectively
Red, plain and dashed lines: median NMSE for PALM and LPALM,
respectively
LPALM is compared with PALM, by optimizing PALM parameters over the train set:
LPALM largely outperforms PALM, both:

- in terms of separation quality

- in terms of number of iterations

View Slide

Numerical results: comparison with other unrolled methods
40
S*
S*
LPALM largely outperforms its competitors:

- LISTA lacks of flexibility to handle varying matrices

- DNMF suffers from a training using the reconstruction error only: , which is
well-known to lead to spurious solutions
iA*
∥iX − iAiS∥2
2

View Slide

Conclusion
41
Further research paths which are currently considered:

• Perform unrolling on other kind of BSS algorithms

• Extend LPALM to remote-sensing datasets in which spectral variabilities can occur
even within a single dataset

• Study semi / unsupervised unrolled methods

• Explore new methods for SAR target extraction (based on unrolling and/or PnP)
X
• We started by showing the practical difficulty of choosing hyperparameters in
optimization-based BSS methods

• We have empirically shown that using a MAD-heuristic for the hyperparameter choice,
based on a fixed point argument, can lead to good results

• This extends to complex data in the context of target continuous position extraction in
SAR imaging

• To go a step further, and if a training set with ground-truth is available, we have shown
the interest of using algorithm unrolling to propose neural network architectures enabling
to perform BSS by mimicking the structure of PALM.

View Slide

Christophe Kervazo

Christophe Kervazo

S³ Seminar

More Decks by S³ Seminar

Other Decks in Research

Featured

Transcript