Guaranteed Best Sparse Solutions for Spectral Unmixing

Fri, 17 Oct 2025 00:00:00 +0000

Ensemble des solutions parcimonieuses exactes en démélange spectral : algorithme garanti et analyse des solutions

Mon, 30 Jun 2025 00:00:00 +0000

Branch-and-bound algorithm for exact ℓ0-norm sparse spectral unmixing

Sun, 15 Jun 2025 00:00:00 +0000

MIMOSA UNMIX

Tue, 15 Apr 2025 00:00:00 +0000

Exact resolution of the sparse spectral unmixing problem

Thu, 23 Sep 2021 00:00:00 +0000

$$\min_{\boldsymbol{x}\in [0,1]^{Q}}\quad \frac{1}{2}\big|\big| \boldsymbol{y}-\mathbf{H}\boldsymbol{x}\big|\big|_{2}^{2}\quad \textrm{st. } \big|\big|\boldsymbol{x}\big|\big| _{0} \leq K, \quad\boldsymbol{1} _{Q} ^{\intercal} \boldsymbol{x} =1$$

Abstract: Hyperspectral imaging involves the simultaneous acquisition of images across a wide range of wavelengths. A recurring challenge in processing this data is to solve a source separation problem: due to the low spatial resolution of the instruments, the light reflectance spectrum measured at a given position results from the superposition of elementary spectra, i.e. a mixture for which the proportions must be determined.

In so-called supervised unmixture, the mixture is sought within a known dictionary of reference spectra. A physical constraint, known as the parsimony constraint, then stipulates that a small number of components in the mixture is sufficient to describe each observed spectrum. Mathematically, this involves fitting a linear model using the least squares method, subject to the constraint that the vector of sought-after coefficients has few non-zero components, i.e. a small “norm” $\ell_0$. This is an NP-hard problem that essentially falls within the field of combinatorial optimisation.

Whilst most work in this field considers relaxed approaches, prioritising low computational time, we propose an exact solution method that guarantees the optimality of the solutions obtained.

During his PhD research, Ramzi Ben Mhenni designed branch-and-bound algorithms for exact sparse optimization. Here, we propose a specific version of these algorithms to solve the problem of mixture decomposition. Indeed, the mixture coefficients are positive and sum to unity, constraints that require a dedicated algorithmic strategy.

When the mixture is composed of a limited number of spectra, which is generally the case when these spectra are learned from a set of observations, we show that this approach remains computationally efficient and yields better solutions compared to methods in the existing literature. When the dictionary size increases, eg. for a dictionary composed of a large number of laboratory-measured spectra, the obtained solutions remain of higher quality, but with a much higher computational cost.

A free solver, implemented in C++, is available to the scientific community.

MimosaUnmix_1inst_meth $(pwd) 2 bb_homotopy_fcls eq
L2L0_ASC_ANC_BB_Rhom_fcls
Instance : SA_SNR40_K4_instance30
K = 2 Q = 50 N = 224 ASC form : equality
-----------------------------------------------------------------------
#(it) = 10 #(node) = 19 T = 0.083769 card(L) = 2
xUB* = 1' 34'
UB* = 0.312598 sum(xUB*) = 1
-----------------------------------------------------------------------
[...]
-----------------------------------------------------------------------
#(it) = 100 #(node) = 199 T = 0.319659 card(L) = 1
xUB* = 24 39'
UB* = 0.170461 sum(xUB*) = 1
-----------------------------------------------------------------------
SA_SNR40_K4_instance30 - bb_homotopy_fcls
T = 0.322625 (s)
#(node) = 211 / best(node) = 221
UB* = 0.170461
xUB* :
x(24) = 0.234965
x(39)' = 0.765035
sum(xUB*) = 1

Branch-and-bound | Mehdi LATIF

Guaranteed Best Sparse Solutions for Spectral Unmixing

Ensemble des solutions parcimonieuses exactes en démélange spectral : algorithme garanti et analyse des solutions

Branch-and-bound algorithm for exact ℓ0-norm sparse spectral unmixing

MIMOSA UNMIX

Exact resolution of the sparse spectral unmixing problem