An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors

This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classic alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity

The low-rank decomposition of correlation-enhanced superpixels for video segmentation

Low-rank decomposition (LRD) is an effective scheme to explore the affinity among superpixels in the image and video segmentation. However, the superpixel feature collected based on colour, shape, and texture may be rough, incompatible, and even conflicting if multiple features extracted in various manners are vectored and stacked straight together. It poses poor correlation, inconsistence on intra-category superpixels, and similarities on inter-category superpixels. This paper proposes a correlation-enhanced superpixel for video segmentation in the framework of LRD. Our algorithm mainly consists of two steps, feature analysis to establish the initial affinity among superpixels, followed by construction of a correlation-enhanced superpixel. This work is very helpful to perform LRD effectively and find the affinity accurately and quickly. Experiments conducted on datasets validate the proposed method. Comparisons with the state-of-the-art algorithms show higher speed and more precise in video segmentation

이동 물체 감지 및 분진 영상 복원의 연구

학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2021. 2. 강명주.Robust principal component analysis(RPCA), a method used to decom- pose a matrix into the sum of a low-rank matrix and a sparse matrix, has been proven effective in modeling the static background of videos. However, because a dynamic background cannot be represented by a low-rank matrix, measures additional to the RPCA are required. In this thesis, we propose masked RPCA to process backgrounds containing moving textures. First- order Marcov random field (MRF) is used to generate a mask that roughly labels moving objects and backgrounds. To estimate the background, the rank minimization process is then applied with the mask multiplied. During the iteration, the background rank increases as the object mask expands, and the weight of the rank constraint term decreases, which increases the accuracy of the background. We compared the proposed method with state- of-art, end-to-end methods to demonstrate its advantages. Subsequently, we suggest novel dedusting method based on dust-optimized transmission map and deep image prior. This method consists of estimating atmospheric light and transmission in that order, which is similar to dark channel prior-based dehazing methods. However, existing atmospheric light estimating methods widely used in dehazing schemes give an overly bright estimation, which results in unrealistically dark dedusting results. To ad- dress this problem, we propose a segmentation-based method that gives new estimation in atmospheric light. Dark channel prior based transmission map with new atmospheric light gives unnatural intensity ordering and zero value at low transmission regions. Therefore, the transmission map is refined by scattering model based transformation and dark channel adaptive non-local total variation (NLTV) regularization. Parameter optimizing steps with deep image prior(DIP) gives the final dedusting result.강건 주성분 분석은 배경 감산을 통한 동영상의 전경 추출의 방법으로 이 용되어왔으나, 동적배경은저계수행렬로표현될수없기때문에동적배경 감산에성능적한계를가지고있었다. 우리는전경과배경을구분하는일계마 르코프연쇄를도입해정적배경을나타내는항과곱하고이것을이용한새로 운형태의강건주성분분석을제안하여동적배경감산문제를해결한다. 해당 최소화문제는반복적인교차최적화를통하여해결한다. 이어서대기중의미세 먼지에의해오염된영상을복원한다. 영상분할과암흑채널가정에기반하여 깊이지도를구하고, 비국소총변동최소화를통하여정제한다. 이후깊은영상 가정에기반한영상생성기를통하여최종적으로복원된영상을구한다. 실험을 통하여제안된방법을다른방법들과비교하고질적인측면과양적인측면모 두에서우수함을확인한다.Abstract i 1 Introduction 1 1.1 Moving Object Detection In Dynamic Backgrounds 1 1.2 Image Dedusting 2 2 Preliminaries 4 2.1 Moving Object Detection In Dynamic Backgrounds 4 2.1.1 Literature review 5 2.1.2 Robust principal component analysis(RPCA) and their application status 7 2.1.3 Graph cuts and α-expansion algorithm 14 2.2 Image Dedusting 16 2.2.1 Image dehazing methods 16 2.2.2 Dust model 18 2.2.3 Non-local total variation(NLTV) 19 3 Dynamic Background Subtraction With Masked RPCA 21 3.1 Motivation 21 3.1.1 Motivation of background modeling 21 3.1.2 Mask formulation 23 3.1.3 Model 24 3.2 Optimization 25 3.2.1 L-Subproblem 25 3.2.2 L˜-Subproblem 26 3.2.3 M-Subproblem 27 3.2.4 p-Subproblem 28 3.2.5 Adaptive parameter control 28 3.2.6 Convergence 29 3.3 Experimental results 31 3.3.1 Benchmark Algorithms And Videos 31 3.3.2 Implementation 32 3.3.3 Evaluation 32 4 Deep Image Dedusting With Dust-Optimized Transmission Map 41 4.1 Transmission estimation 41 4.1.1 Atmospheric light estimation 41 4.1.2 Transmission estimation 43 4.2 Scene radiance recovery 47 4.3 Experimental results 51 4.3.1 Implementation 51 4.3.2 Evaluation 52 5 Conclusion 58 Abstract (in Korean) 69 Acknowledgement (in Korean) 70Docto

Low rank methods for optimizing clustering

Complex optimization models and problems in machine learning often have the majority of information in a low rank subspace. By careful exploitation of these low rank structures in clustering problems, we find new optimization approaches that reduce the memory and computational cost. We discuss two cases where this arises. First, we consider the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective as a way to address overlapping and outliers in an integrated fashion. Optimizing this discrete objective is NP-hard, and even though there is a convex relaxation of the objective, straightforward convex optimization approaches are too expensive for large datasets. We utilize low rank structures in the solution matrix of the convex formulation and use a low-rank factorization of the solution matrix directly as a practical alternative. The resulting optimization problem is non-convex, but has a smaller number of solution variables, and can be locally optimized using an augmented Lagrangian method. In addition, we consider two fast multiplier methods to accelerate the convergence of the augmented Lagrangian scheme: a proximal method of multipliers and an alternating direction method of multipliers. For the proximal augmented Lagrangian, we show a convergence result for the non-convex case with bound-constrained subproblems. When the clustering performance is evaluated on real-world datasets, we show this technique is effective in finding the ground-truth clusters and cohesive overlapping communities in real-world networks. The second case is where the low-rank structure appears in the objective function. Inspired by low rank matrix completion techniques, we propose a low rank symmetric matrix completion scheme to approximate a kernel matrix. For the kernel k-means problem, we show empirically that the clustering performance with the approximation is comparable to the full kernel k-means