Maximum-Volume Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a popular data embedding technique. Given a nonnegative data matrix $X$, it aims at finding two lower dimensional matrices, $W$ and $H$, such that $X\approx WH$, where the factors $W$ and $H$ are constrained to be element-wise nonnegative. The factor $W$ serves as a basis for the columns of $X$. In order to obtain more interpretable and unique solutions, minimum-volume NMF (MinVol NMF) minimizes the volume of $W$. In this paper, we consider the dual approach, where the volume of $H$ is maximized instead; this is referred to as maximum-volume NMF (MaxVol NMF). MaxVol NMF is identifiable under the same conditions as MinVol NMF in the noiseless case, but it behaves rather differently in the presence of noise. In practice, MaxVol NMF is much more effective to extract a sparse decomposition and does not generate rank-deficient solutions. In fact, we prove that the solutions of MaxVol NMF with the largest volume correspond to clustering the columns of $X$ in disjoint clusters, while the solutions of MinVol NMF with smallest volume are rank deficient. We propose two algorithms to solve MaxVol NMF. We also present a normalized variant of MaxVol NMF that exhibits better performance than MinVol NMF and MaxVol NMF, and can be interpreted as a continuum between standard NMF and orthogonal NMF. We illustrate our results in the context of hyperspectral unmixing.

Super-résolution non supervisée d'images hyperspectrales de télédétection utilisant un entraînement entièrement synthétique

Hyperspectral single image super-resolution (SISR) aims to enhance spatial resolution while preserving the rich spectral information of hyperspectral images. Most existing methods rely on supervised learning with high-resolution ground truth data, which is often unavailable in practice. To overcome this limitation, we propose an unsupervised learning approach based on synthetic abundance data. The hyperspectral image is first decomposed into endmembers and abundance maps through hyperspectral unmixing. A neural network is then trained to super-resolve these maps using data generated with the dead leaves model, which replicates the statistical properties of real abundances. The final super-resolution hyperspectral image is reconstructed by recombining the super-resolved abundance maps with the endmembers. Experimental results demonstrate the effectiveness of our method and the relevance of synthetic data for training.

Synthetic Abundance Maps for Unsupervised Super-Resolution of Hyperspectral Remote Sensing Images

Hyperspectral single image super-resolution (HS-SISR) aims to enhance the spatial resolution of hyperspectral images to fully exploit their spectral information. While considerable progress has been made in this field, most existing methods are supervised and require ground truth data for training-data that is often unavailable in practice. To overcome this limitation, we propose a novel unsupervised training framework for HS-SISR, based on synthetic abundance data. The approach begins by unmixing the hyperspectral image into endmembers and abundances. A neural network is then trained to perform abundance super-resolution using synthetic abundances only. These synthetic abundance maps are generated from a dead leaves model whose characteristics are inherited from the low-resolution image to be super-resolved. This trained network is subsequently used to enhance the spatial resolution of the original image's abundances, and the final super-resolution hyperspectral image is reconstructed by combining them with the endmembers. Experimental results demonstrate both the training value of the synthetic data and the effectiveness of the proposed method.

Unsupervised Super-Resolution of Hyperspectral Remote Sensing Images Using Fully Synthetic Training

Considerable work has been dedicated to hyperspectral single image super-resolution to improve the spatial resolution of hyperspectral images and fully exploit their potential. However, most of these methods are supervised and require some data with ground truth for training, which is often non-available. To overcome this problem, we propose a new unsupervised training strategy for the super-resolution of hyperspectral remote sensing images, based on the use of synthetic abundance data. Its first step decomposes the hyperspectral image into abundances and endmembers by unmixing. Then, an abundance super-resolution neural network is trained using synthetic abundances, which are generated using the dead leaves model in such a way as to faithfully mimic real abundance statistics. Next, the spatial resolution of the considered hyperspectral image abundances is increased using this trained network, and the high resolution hyperspectral image is finally obtained by recombination with the endmembers. Experimental results show the training potential of the synthetic images, and demonstrate the method effectiveness.

Deep Deterministic Nonlinear ICA via Total Correlation Minimization with Matrix-Based Entropy Functional

Blind source separation, particularly through independent component analysis (ICA), is widely utilized across various signal processing domains for disentangling underlying components from observed mixed signals, owing to its fully data-driven nature that minimizes reliance on prior assumptions. However, conventional ICA methods rely on an assumption of linear mixing, limiting their ability to capture complex nonlinear relationships and to maintain robustness in noisy environments. In this work, we present deep deterministic nonlinear independent component analysis (DDICA), a novel deep neural network-based framework designed to address these limitations. DDICA leverages a matrix-based entropy function to directly optimize the independence criterion via stochastic gradient descent, bypassing the need for variational approximations or adversarial schemes. This results in a streamlined training process and improved resilience to noise. We validated the effectiveness and generalizability of DDICA across a range of applications, including simulated signal mixtures, hyperspectral image unmixing, modeling of primary visual receptive fields, and resting-state functional magnetic resonance imaging (fMRI) data analysis. Experimental results demonstrate that DDICA effectively separates independent components with high accuracy across a range of applications. These findings suggest that DDICA offers a robust and versatile solution for blind source separation in diverse signal processing tasks.

Diffusion Posterior Sampler for Hyperspectral Unmixing with Spectral Variability Modeling

Linear spectral mixture models (LMM) provide a concise form to disentangle the constituent materials (endmembers) and their corresponding proportions (abundance) in a single pixel. The critical challenges are how to model the spectral prior distribution and spectral variability. Prior knowledge and spectral variability can be rigorously modeled under the Bayesian framework, where posterior estimation of Abundance is derived by combining observed data with endmember prior distribution. Considering the key challenges and the advantages of the Bayesian framework, a novel method using a diffusion posterior sampler for semiblind unmixing, denoted as DPS4Un, is proposed to deal with these challenges with the following features: (1) we view the pretrained conditional spectrum diffusion model as a posterior sampler, which can combine the learned endmember prior with observation to get the refined abundance distribution. (2) Instead of using the existing spectral library as prior, which may raise bias, we establish the image-based endmember bundles within superpixels, which are used to train the endmember prior learner with diffusion model. Superpixels make sure the sub-scene is more homogeneous. (3) Instead of using the image-level data consistency constraint, the superpixel-based data fidelity term is proposed. (4) The endmember is initialized as Gaussian noise for each superpixel region, DPS4Un iteratively updates the abundance and endmember, contributing to spectral variability modeling. The experimental results on three real-world benchmark datasets demonstrate that DPS4Un outperforms the state-of-the-art hyperspectral unmixing methods.

A Provably-Correct and Robust Convex Model for Smooth Separable NMF

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

SWAN: Self-supervised Wavelet Neural Network for Hyperspectral Image Unmixing

In this article, we present SWAN: a three-stage, self-supervised wavelet neural network for joint estimation of endmembers and abundances from hyperspectral imagery. The contiguous and overlapping hyperspectral band images are first expanded to Biorthogonal wavelet basis space that provides sparse, distributed, and multi-scale representations. The idea is to exploit latent symmetries from thus obtained invariant and covariant features using a self-supervised learning paradigm. The first stage, SWANencoder maps the input wavelet coefficients to a compact lower-dimensional latent space. The second stage, SWANdecoder uses the derived latent representation to reconstruct the input wavelet coefficients. Interestingly, the third stage SWANforward learns the underlying physics of the hyperspectral image. A three-stage combined loss function is formulated in the image acquisition domain that eliminates the need for ground truth and enables self-supervised training. Adam is employed for optimizing the proposed loss function, while Sigmoid with a dropout of 0.3 is incorporated to avoid possible overfitting. Kernel regularizers bound the magnitudes and preserve spatial variations in the estimated endmember coefficients. The output of SWANencoder represents estimated abundance maps during inference, while weights of SWANdecoder are retrieved to extract endmembers. Experiments are conducted on two benchmark synthetic data sets with different signal-to-noise ratios as well as on three real benchmark hyperspectral data sets while comparing the results with several state-of-the-art neural network-based unmixing methods. The qualitative, quantitative, and ablation results show performance enhancement by learning a resilient unmixing function as well as promoting self-supervision and compact network parameters for practical applications.

HAMscope: a snapshot Hyperspectral Autofluorescence Miniscope for real-time molecular imaging

We introduce HAMscope, a compact, snapshot hyperspectral autofluorescence miniscope that enables real-time, label-free molecular imaging in a wide range of biological systems. By integrating a thin polymer diffuser into a widefield miniscope, HAMscope spectrally encodes each frame and employs a probabilistic deep learning framework to reconstruct 30-channel hyperspectral stacks (452 to 703 nm) or directly infer molecular composition maps from single images. A scalable multi-pass U-Net architecture with transformer-based attention and per-pixel uncertainty estimation enables high spatio-spectral fidelity (mean absolute error ~ 0.0048) at video rates. While initially demonstrated in plant systems, including lignin, chlorophyll, and suberin imaging in intact poplar and cork tissues, the platform is readily adaptable to other applications such as neural activity mapping, metabolic profiling, and histopathology. We show that the system generalizes to out-of-distribution tissue types and supports direct molecular mapping without the need for spectral unmixing. HAMscope establishes a general framework for compact, uncertainty-aware spectral imaging that combines minimal optics with advanced deep learning, offering broad utility for real-time biochemical imaging across neuroscience, environmental monitoring, and biomedicine.

Transformer-Guided Content-Adaptive Graph Learning for Hyperspectral Unmixing

Hyperspectral unmixing (HU) targets to decompose each mixed pixel in remote sensing images into a set of endmembers and their corresponding abundances. Despite significant progress in this field using deep learning, most methods fail to simultaneously characterize global dependencies and local consistency, making it difficult to preserve both long-range interactions and boundary details. This letter proposes a novel transformer-guided content-adaptive graph unmixing framework (T-CAGU), which overcomes these challenges by employing a transformer to capture global dependencies and introducing a content-adaptive graph neural network to enhance local relationships. Unlike previous work, T-CAGU integrates multiple propagation orders to dynamically learn the graph structure, ensuring robustness against noise. Furthermore, T-CAGU leverages a graph residual mechanism to preserve global information and stabilize training. Experimental results demonstrate its superiority over the state-of-the-art methods. Our code is available at https://github.com/xianchaoxiu/T-CAGU.