A Low-Rank Model for Compressive Spectral Image Classification

Low rank approximation of matrices is an important tool in machine learning. Given a data matrix, low rank approximation helps to find factors, patterns, and provides concise representations for the data. Research on low rank approximation usually focuses on real matrices. However, in many applications data are binary (categorical) rather than continuous. This leads to the problem of low rank approximation of binary matrices. Here we are given a d x n binary matrix A and a small integer k < d. The goal is to find two binary matrices U and V of sizes d x k and k x n respectively, so that the Frobenius norm of A - U V is minimized. There are two models of this problem, depending on the definition of the dot product of binary vectors: The GF(2) model and the Boolean semiring model. Unlike low rank approximation of a real matrix which can be efficiently solved by Singular Value Decomposition, we show that approximation of a binary matrix is NP-hard, even for k=1.\nIn this paper, our main concern is the problem of Column Subset Selection (CSS), in which the low rank matrix U must be formed by k columns of the data matrix, and we are interested in the approximation ratio achievable by CSS for binary matrices. For the GF(2) model, we show that CSS has approximation ratio bounded by k/2+1+k/(2(2^k-1)) and this is asymptotically tight. For the Boolean model, it turns out that CSS is no longer sufficient to obtain a bound. We then develop a Generalized CSS (GCSS) procedure in which the columns of U are generated from Boolean formulas operating bitwise on selected columns of the data matrix. We show that the approximation ratio achieved by GCSS is bounded by 2^(k-1)+1, and argue that an exponential dependency on k is seems inherent.

Least Squares Support Vector Machine (LS-SVM) has been recently applied to non-parametric identification of Linear Parameter Varying (LPV) systems, described by the AutoRegressive with eXogenous input (ARX). However, the online application of LPV-ARX system in the LS-SVM setting requires high computational time, related to the number of training data used to compute the coefficients of the identified model, limiting the possibility to use the method to real-time applications. In this paper, the authors propose the Low-Rank (LR) matrix approximation and a pruning based approach to compute a sparse solution. In particular, the pruning algorithm is considered to compute off-line a sparse solution of Lagrangian multipliers and then speed up the testing stage, whereas the LR matrix approximation allows to speed up the training stage. The proposed approach has been tested by identifying a subsystem of a vehicle powertrain model by the input/output data collected from the simulation model. The proposed approach has been compared with respect to the standard approach based on LS-SVM. The methods are tested on the considered real-world problem and the proposed approach permits to reduce the execution time of about 77% on average in the considered identification problem, corresponding to a degradation of the identification result less than 0.2% with respect to the standard solution.

The problem of matrix approximation appears ubiquitously in recommendation systems, computer vision and text mining. The prevailing assumption is that the partially observed matrix has a low-rank or can be well approximated by a low-rank matrix. However, this assumption is strictly that the partially observed matrix is globally low rank. In this paper, we propose a local sensitive formulation of matrix approximation which relaxes the global low-rank assumption, leading to a representation of the observed matrix as a weighted sum of low-rank matrices. We solve the problem by nonconvex optimization which exhibits superior performance of low rank matrix estimation when compared with convex relaxation. Our experiments show improvements in prediction accuracy over classical approaches for recommendation tasks.

Several bandwise total variation (TV) regularized low-rank (LR)-based models have been proposed to remove mixed noise in hyperspectral images (HSIs). Conventionally, the rank of LR matrix is approximated using nuclear norm (NN). The NN is defined by adding all singular values together, which is essentially a $L_{1}$ -norm of the singular values. It results in non-negligible approximation errors and thus the resulting matrix estimator can be significantly biased. Moreover, these bandwise TV-based methods exploit the spatial information in a separate manner. To cope with these problems, we propose a spatial–spectral TV (SSTV) regularized non-convex local LR matrix approximation (NonLLRTV) method to remove mixed noise in HSIs. From one aspect, local LR of HSIs is formulated using a non-convex $L_{\gamma}$ -norm, which provides a closer approximation to the matrix rank than the traditional NN. From another aspect, HSIs are assumed to be piecewisely smooth in the global spatial domain. The TV regularization is effective in preserving the smoothness and removing Gaussian noise. These facts inspire the integration of the NonLLR with TV regularization. To address the limitations of bandwise TV, we use the SSTV regularization to simultaneously consider global spatial structure and spectral correlation of neighboring bands. Experiment results indicate that the use of local non-convex penalty and global SSTV can boost the preserving of spatial piecewise smoothness and overall structural information.

The hyperspectral images of airplanes and flying birds are obtained by a xiSpec snapshot mosaic hyperspectral cameras, supported by the Interuniversity Microelectronics Centre (IMEC). The single frame Infrared Patch-Image (IPI) model is used to detect the small targets of airplanes and flying birds under complex cloud background in the hyperspectral images. Based on the non-local autocorrelation property of the background image, the method assumes that the target image is a sparse matrix and the background image is a low-rank matrix. The small target detection is transformed into an optimization problem of recovering the low-rank and sparse matrix. Using stable principal component tracking solution, the decomposed background and target are obtained. The results show that this method can detect bright and dark small targets in complex background at the same time, and the hyperspectral image can effectively improve the detection rate. More importantly, the detection ability is closely related to the intensity difference of the target against the background. Thus, the optimal waveband of different targets can be given by combining the target detection results and the intensity difference curves. This has a guiding significance for the design of specific point target detection payloads.

Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely, the weighted Schatten p-norm minimization (WSNM), to generalize the NNM to the Schatten p-norm minimization with weights assigned to different singular values. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers the importance of different rank components. We analyze the solution of WSNM and prove that, under certain weights permutation, WSNM can be equivalently transformed into independent non-convex lp-norm subproblems, whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g., image denoising and background subtraction. Extensive experimental results show, both qualitatively and quantitatively, that the proposed WSNM can more effectively remove noise, and model the complex and dynamic scenes compared with state-of-the-art methods.

A hyperspectral image (HSI) can be described as a set of images with spatial information across different spectral bands. Compressive spectral imaging techniques (CSI) permit to capture a 3-dimensional hyperspectral scene using 2 dimensional coded and multiplexed projections. Recovering the original scene from a very few projections can be valuable in applications such as remote sensing, video surveillance and biomedical imaging. Typically, HSI exhibit high correlations both, in the spatial and spectral dimensions. Thus, exploiting these correlations allows to accurately recover the original scene from compressed measurements. Traditional approaches exploit the sparsity of the scene when represented in a proper basis. For this purpose, an optimization problem that seeks to minimize a joint e 2 − e 1 norm is solved to obtain the original scene. However, there exist some HSI with an important feature which does not have been widely exploited; HSI are commonly low rank, thus only a few number of spectral signatures are presented in the image. Therefore, this paper proposes an approach to recover a simultaneous sparse and low rank hyperspectral image by exploiting both features at the same time. The proposed approach solves an optimization problem that seeks to minimize the e 2 -norm, penalized by the e 1 -norm, to force the solution to be sparse, and penalized by the nuclear norm to force the solution to be low rank. Theoretical analysis along with a set of simulations over different data sets show that simultaneously exploiting low rank and sparse structures enhances the performance of the recovery algorithm and the quality of the recovered image with an average improvement of around 3 dB in terms of the peak-signal to noise ratio (PSNR).

Hyperspectral image (HSI) is often corrupted by various kinds of noises. This letter proposes an innovative HSI denoising approach by leveraging the graph signal processing (GSP) theory and the low-rank (LR) matrix recovery model. With GSP, the piecewise smoothness (PWS) property of the HSI can be efficiently characterized, leading to a new regularization for HSI denoising, termed the adaptive weight graph total variation (AWGTV) regularization. Then, the denoising problem is formulated into a constrained optimization problem that incorporates the AWGTV and the LR property of the HSI. An augmented Lagrange multiplier method is adopted to solve the problem. Numerical experiments conducted on synthetic and real-world datasets and comparisons with existing methods demonstrate the effectiveness of the proposed denoising algorithm.

We describe a local low rank approximation method based on super-rays for light field compression. Super-rays can be seen as a set of super-pixels that are coherent across all light field views. A super-ray based disparity estimation method is proposed using a low rank prior, in order to be able to align all the super-pixels forming each super-ray. A dedicated super-ray construction method is described that constrains the super-pixels forming a given super-ray to be all of the same shape and size, dealing with occlusions. This constraint is needed so that the super-rays can be used as a support of angular dimensionality reduction based on low rank matrix approximation. A low rank matrix approximation is then computed on the disparity compensated super-rays using a singular value decomposition (SVD). A coding algorithm is then described for the different components of the resulting low rank approximation. Experimental results show performance gains compared with two reference light field coding schemes (HEVC-based scheme and JPEG-Pleno VM 1.1).

This paper considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as [Formula: see text] norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the [Formula: see text]-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this [Formula: see text] norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated [Formula: see text] norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated [Formula: see text] norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.

Abstract. The automatic classification of land cover types from hyperspectral images is a challenging problem due to (among others) the large amount of spectral bands and their high spatial and spectral correlation. The extraction of meaningful features, that enables a subsequent classifier to distinguish between different land cover classes, is often limited to a subset of all available data dimensions which is found by band selection techniques or other methods of dimensionality reduction. This work applies Projection-Based Random Forests to hyperspectral images, which not only overcome the need of an explicit feature extraction, but also provide mechanisms to automatically select spectral bands that contain original (i.e. non-redundant) as well as highly meaningful information for the given classification task. The proposed method is applied to four challenging hyperspectral datasets and it is shown that the effective number of spectral bands can be considerably limited without loosing too much of classification performance, e.g. a loss of 1 % accuracy if roughly 13 % of all available bands are used.

Hyperspectral (HS) images (HSI) provide a vast amount of spatial and spectral information based on the high dimensionality of the pixels in a wide range of wavelengths. A HS image usually requires massive storage capacity, which demands high compression rates to save space with preservation of data integrity. HS image can be deemed as three dimensional data cube where different wavelengths (W) form the third dimension along with X and Y dimensions. To get a better compression result, spatial redundancy of HS images can be exploited using different coders along X, Y, or W direction. This article focuses on taking maximum advantage of HS images redundancy by rearranging HS image into different 3D data cubes and proposes a directionlet based compression scheme constituted the optimal compression plane (OCP) for adaptive best approximation of geometric matrix. The OCP, calculated by the spectral correlation, is used to the prediction and determination of which reconstructed plane can reach higher compression rates while minimizing data loss of hyperspectral data. Moreover, we also rearrange the 3D data cube into different 2D image planes and investigate the compression ratio using different coders. The schema can be used for both lossless and lossy compression. Our experimental results show that the new framework optimizes the performance of the compression using a number of coding methods (inclusive of lossless/lossy HEVC, motion JPEG, JPG2K, and JPEG) for HSIs with different visual content.KeywordsHyperspectral imageslossless compressionoptimal compression plane (OCP)

In recent years, low rank matrix theory becomes one of the research hotspots in the image processing field. In this report, we construct a linear equation via born approximation for the electromagnetic inverse scattering problem, then solve the linear equation with Low Rank Constraint. The low rank matrix representation is applied to quickly and precisely reconstruct the low rank data matrix in GPR imaging. Simulation results and performance analysis show that Low Rank Reconstruction helps to reduce the memory space and computational complexity in GPR imaging.

Machine-learning interatomic potentials (MLIPs) have become a mainstay in computationally guided materials science, surpassing traditional force fields due to their flexible functional form and superior accuracy in reproducing physical properties of materials. This flexibility is achieved through mathematically rigorous basis sets that describe interatomic interactions within a local atomic environment. The number of parameters in these basis sets influences both the size of the training dataset required and the computational speed of the MLIP. Consequently, compressing MLIPs by reducing the number of parameters is a promising route to more efficient simulations. In this work, we use low-rank matrix and tensor factorizations under fixed-rank constraints to achieve this compression. In addition, we demonstrate that an algorithm with automatic rank augmentation helps to find a deeper local minimum of the fitted potential. The methodology is mainly verified using the Moment Tensor Potential (MTP) model and benchmarked on multi-component systems: a Mo-Nb-Ta-W medium-entropy alloy, molten LiF-NaF-KF, and a glycine molecular crystal. The proposed approach achieves up to 50% compression without any loss of MTP accuracy. We also demonstrate that the developed methodology is universal and can be applied to compress other MLIPs on the example of atomic cluster expansion.

We propose new approximate alternating projection methods, based on randomized sketching, for the low-rank nonnegative matrix approximation problem: find a low-rank approximation of a nonnegative matrix that is nonnegative, but whose factors can be arbitrary. We calculate the computational complexities of the proposed methods and evaluate their performance in numerical experiments. The comparison with the known deterministic alternating projection methods shows that the randomized approaches are faster and exhibit similar convergence properties.