Low-Rank Recovery from Linear Observations

It is critical to classify the landing terrain from aerial images when an unmanned aerial vehicle lands at an unprepared site autonomously by using a vision sensor. Owing to the interference of illumination variations and noises, different terrains may show a similar image feature and the same terrain may have a different image feature, which brings great difficulties to image classification. To address this issue, a terrain classification method based on low-rank recovery and sparse representation is proposed. Color moments and Gabor texture feature are extracted and fused to construct a discriminative dictionary. Then, we perform low-rank matrix recovery for the dictionary by using augmented Lagrange multipliers and classify the test samples by sparse-representation-based classification. Experimental results on an aerial image database that we prepared by using the DJI Phantom 3 Advanced UAV verify the classification accuracy and robustness of the proposed method.

Fabric defect detection plays an important role in improving the quality of fabric product. In this paper, a novel fabric defect detection method based on visual saliency using deep feature and low-rank recovery was proposed. First, unsupervised training is carried out by the initial network parameters based on MNIST large datasets. The supervised fine-tuning of fabric image library based on Convolutional Neural Networks (CNNs) is implemented, and then more accurate deep neural network model is generated. Second, the fabric images are uniformly divided into the image block with the same size, then we extract their multi-layer deep features using the trained deep network. Thereafter, all the extracted features are concentrated into a feature matrix. Third, low-rank matrix recovery is adopted to divide the feature matrix into the low-rank matrix which indicates the background and the sparse matrix which indicates the salient defect. In the end, the iterative optimal threshold segmentation algorithm is utilized to segment the saliency maps generated by the sparse matrix to locate the fabric defect area. Experimental results demonstrate that the feature extracted by CNN is more suitable for characterizing the fabric texture than the traditional LBP, HOG and other hand-crafted features extraction method, and the proposed method can accurately detect the defect regions of various fabric defects, even for the image with complex texture.

Automatic classification of terrain surfaces from an aerial image is essential for an autonomous unmanned aerial vehicle (UAV) landing at an unprepared site by using vision. Diverse terrain surfaces may show similar spectral properties due to the illumination and noise that easily cause poor classification performance. To address this issue, a multi-stage classification algorithm based on low-rank recovery and multi-feature fusion sparse representation is proposed. First, color moments and Gabor texture feature are extracted from training data and stacked as column vectors of a dictionary. Then we perform low-rank matrix recovery for the dictionary by using augmented Lagrange multipliers and construct a multi-stage terrain classifier. Experimental results on an aerial map database that we prepared verify the classification accuracy and robustness of the proposed method.

In this paper, we propose a framework of non-convex low-rank recovery and alignment for arbitrary tensor data. Specially, by using Schatten-p (\(0<p<1\), the same below) norm and \(\ell _p\) norm to relax the rank function and \(\ell _0\) norm respectively, the model requires much weaker incoherence conditions to guarantee a successful recovery than the common used nuclear norm and \(\ell _1\) norm. At the same time, we adopt a set of transformations which acts on the images of the tensor data to compensate the possible misalignments of images. By solving the optimal transformations, the strict alignments of the images are achieved in the low-rank recovery process. Furthermore, we propose an efficient algorithm based on the method of Alternating Direction Method of Multipliers (ADMM) for the non-convex optimization problem. The extensive experiments on the artificial data sets and real image data sets show the superiority of our method in image alignment and denoising.

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional subspace H, spanned by rank 1 tensors, such that any non-zero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimal-size hitting sets for tensors of degree 2 (matrices), and obtain quasi-polynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit). We also show connections to the task of performing low-rank recovery of matrices, which is studied in the field of compressed sensing. Low-rank recovery asks (say, over the reals) to recover a matrix M from few measurements, under the promise that M is rank <=r. We also give a formal connection between low-rank recovery and the task of sparse (vector) recovery: any sparse-recovery algorithm that exactly recovers vectors of length n and sparsity 2r, using m non-adaptive measurements, yields a low-rank recovery scheme for exactly recovering nxn matrices of rank <=r, making 2nm non-adaptive measurements. Furthermore, if the sparse-recovery algorithm runs in time \tau, then the low-rank recovery algorithm runs in time O(rn^2+n\tau). We obtain this reduction using linear-algebraic techniques, and not using convex optimization, which is more commonly seen in compressed sensing algorithms. By using a dual Reed-Solomon code, we are able to (deterministically) construct low-rank recovery schemes taking 4nr measurements over the reals, such that the measurements can be all rank-1 matrices, or all sparse matrices.

Pipeline transportation is the most economic and reasonable transport means of petroleum and natural gas. Influenced by complex corrosion environment, it is challenging to detect anomalies accurately with precise boundaries. Inspired by the idea of low-rank (LR) recovery, a novel anomaly detection model with a spatial–temporal regularization (STLR) is proposed to distinguish anomalies and background from magnetic flux leakage (MFL) measurements. By adding the STLR, the model enlarges the differences between the two parts and improves the effectiveness of anomaly detection. Then, an anomaly detection framework is proposed, where the raw MFL measurements are first transformed including profile transformation and filter transformation to prepare the measurements with high saliency. Second, a measurement fusion method is introduced to divide the transformed measurements into multiple blocks and fuse all detection results together. In experiments, the simulated MFL measurements generated by the magnetic dipole model are used to verify the method and obtain optimal parameters, and the real MFL measurements collected from experimental pipelines and in-service pipelines are evaluated for comparisons with other baseline methods.

In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based sparse and low-rank recovery is of considerable interest and it in fact is a main driver of the recent progress in nonconvex and nonsmooth optimization. This paper gives an overview of this topic in various fields in signal processing, statistics and machine learning, including compressive sensing (CS), sparse regression and variable selection, sparse signals separation, sparse principal component analysis (PCA), large covariance and inverse covariance matrices estimation, matrix completion, and robust PCA. We present recent developments of nonconvex regularization based sparse and low-rank recovery in these fields, addressing the issues of penalty selection, applications and the convergence of nonconvex algorithms. Code is available at https://github.com/FWen/ncreg.git.

In this article, we consider the problem of simultaneous low-rank recovery and sparse projection. More specifically, a new Robust Principal Component Analysis (RPCA)-based framework called Sparse Projection and Low-Rank Recovery (SPLRR) is proposed for handwriting representation and salient stroke feature extraction. In addition to achieving a low-rank component encoding principal features and identify errors or missing values from a given data matrix as RPCA, SPLRR also learns a similarity-preserving sparse projection for extracting salient stroke features and embedding new inputs for classification. These properties make SPLRR applicable for handwriting recognition and stroke correction and enable online computation. A cosine-similarity-style regularization term is incorporated into the SPLRR formulation for encoding the similarities of local handwriting features. The sparse projection and low-rank recovery are calculated from a convex minimization problem that can be efficiently solved in polynomial time. Besides, the supervised extension of SPLRR is also elaborated. The effectiveness of our SPLRR is examined by extensive handwritten digital repairing, stroke correction, and recognition based on benchmark problems. Compared with other related techniques, SPLRR delivers strong generalization capability and state-of-the-art performance for handwriting representation and recognition.

In order to accurately detect the patterned fabric defects, a novel patterned fabric detection algorithm based on Gabor-HOG (GHOG) and low-rank recovery is proposed. Firstly, Gabor filter preprocess the pattern fabric image to generate the Gabor maps, and then HOG feature is extracted from the blocks of Gabor maps with size of 16×16. Secondly, the feature vectors GHOG of all blocks is stacked into a feature matrix, each column represents an image block. Thirdly, an efficient low rank recovery model is built for the feature matrix, and is decomposed into a low-rank matrix (background information) and a sparse matrix (defect information) by the alternative direction multiplier method (ADMM). Finally, the saliency map generated by sparse matrix is segmented by the improved optimal threshold algorithm, to locate the defect regions. Experiment results show that recovery the proposed method can effectively detect patterned fabric defect, and outperforms the state-of-the-art methods.

In order to study how to better suppress image noise, improve image resolution, and enhance the visual effect of images, three-dimensional joint filtering – block matching (BM3D) algorithm is used to explore image denoising and image detail retention, and in image denoising, sparse expression and low rank recovery theory are introduced. The results show that BM3D algorithm has a good effect in removing Gaussian white noise, but it is lacking in suppressing impulse noise and mixed noise. The non-local BM3D algorithm is superior to sparse expression and low rank recovery in removing Gaussian white noise, but sparse expression and low rank recovery has a good effect in removing impulse noise.

In this paper, we propose a new radio frequency interference (RFI) detection method by the non-convex low rank concept to achieve higher resolution for RFI source localization problem in MIRAS. First, based on the low rank property of RFI signals, we reformulate the RFI localization model as a low rank recovery problem to mitigate the impact of rank deterioration caused by the nuclear norm. Then, we apply a low rank recovery algorithm to improve the spatial resolution by solving the non-convex low rank recovery problem. Last, we employ the MUSIC algorithm to localize RFI sources.

We propose a novel approach for rain/snow removal from videos using low-rank recovery. Rain/snowdistorted video frames are treated as a distorted 3D signal. The main goal is to separate the distortion, which is the rain or snow additive signal, from the original rain-free signal. Inter-frame information is exploited to put the problem in a convex optimization form. Then, the exact augmented Lagrangian multipliers (EALM) method is used to solve the model for the low-rank terms, which represent the rain/snow-free frames. The proposed approach has several advantages over the existing approaches. It is model-independent, i.e. it does not require shape, appearance, or speed models. Also, it does not need prior information about the acquisition environment. Three different sets of data were used for evaluation: synthetic data for simulation experiments to provide quantitative results, real static videos, and real dynamic videos. The evaluation results proved the effectiveness of the proposed approach when compared to the existing approaches.

Hyperspectral image (HSI) is often contaminated by mixed noise, which severely affects the visual quality and subsequent applications of the data. In this paper, HSI restoration based on low-rank recovery with a local neighborhood weighted spectral–spatial total variation (TV) model is proposed, which focuses on the preservation of spatial structure and spectral fidelity. The low-rank matrix model is adopted to exploit the spectral and spatial correlation information, and the $l_{1}$ -norm is used as a prior to remove the sparse noise. Furthermore, a local spatial neighborhood weighted spectral–spatial TV is utilized to jointly model the spectral–spatial prior information; specifically, the spectral and spatial differences are both considered in the TV term, and the weight is computed by considering the local neighborhood information in the spatial domain. Alternating direction method of multipliers optimization procedure is extended to solve the presented model. Experimental results demonstrate that the proposed method can remove the mixed noise, enhance the structural information simultaneously, and offer the best performance compared with several state-of-the-art HSI restoration methods.

Exemplar-based image denoising algorithms have shown great potential for image restoration with a multitude of existing models. In this paper, we interpret nonlocal similar patch-based denoising as a problem of low-rank recovery. This offers a physically plausible model and unifies several existing techniques in a single low-rank recovery framework. The framework can handle complex noise models, such as zero-mean Gaussian noise, impulse noise, and any other noise that can be approximated by mixing these two kinds of noise. Moreover, we introduce a new nonconvex surrogate for the $l_{0}$ -norm and find the optimal solution of the optimization problems when the new norm is applied to low-rank recovery. The experimental results with different kinds of noise confirm the effectiveness of the proposed low-rank recovery framework and the new norm.

Software effort estimation (SEE) is a crucial step in software development. Effort data missing usually occurs in real-world data collection. Focusing on the missing data problem, existing SEE methods employ the deletion, ignoring, or imputation strategy to address the problem, where the imputation strategy was found to be more helpful for improving the estimation performance. Current imputation methods in SEE use classical imputation techniques for missing data imputation, yet these imputation techniques have their respective disadvantages and might not be appropriate for effort data. In this paper, we aim to provide an effective solution for the effort data missing problem. Incompletion includes the drive factor missing case and effort label missing case. We introduce the low-rank recovery technique for addressing the drive factor missing case. And we employ the semi-supervised regression technique to perform imputation in the case of effort label missing. We then propose a novel effort data imputation approach, named low-rank recovery and semi-supervised regression imputation (LRSRI). Experiments on 7 widely used software effort datasets indicate that: (1) the proposed approach can obtain better effort data imputation effects than other methods; (2) the imputed data using our approach can apply to multiple estimators well.