Robust Algorithms for Low-Rank and Sparse Matrix Models

Data in statistical signal processing problems is often inherently matrix-valued, and a natural first step in working with such data is to impose a model with structure that captures the distinctive features of the underlying data. Under the right model, one can design algorithms that can reliably tease weak signals out of highly corrupted data. In this thesis, we study two important classes of matrix structure: low-rankness and sparsity. In particular, we focus on robust principal component analysis (PCA) models that decompose data into the sum of low-rank and sparse (in an appropriate sense) components. Robust PCA models are popular because they are useful models for data in practice and because efficient algorithms exist for solving them. This thesis focuses on developing new robust PCA algorithms that advance the state-of-the-art in several key respects. First, we develop a theoretical understanding of the effect of outliers on PCA and the extent to which one can reliably reject outliers from corrupted data using thresholding schemes. We apply these insights and other recent results from low-rank matrix estimation to design robust PCA algorithms with improved low-rank models that are well-suited for processing highly corrupted data. On the sparse modeling front, we use sparse signal models like spatial continuity and dictionary learning to develop new methods with important adaptive representational capabilities. We also propose efficient algorithms for implementing our methods, including an extension of our dictionary learning algorithms to the online or sequential data setting. The underlying theme of our work is to combine ideas from low-rank and sparse modeling in novel ways to design robust algorithms that produce accurate reconstructions from highly undersampled or corrupted data. We consider a variety of application domains for our methods, including foreground-background separation, photometric stereo, and inverse problems such as video inpainting and dynamic magnetic resonance imaging.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/143925/1/brimoor_1.pd

Radiometric Scene Decomposition: Estimating Complex Re ectance and Natural Illumination from Images

The phrase, "a picture is worth a thousand words," is often used to emphasize the wealth of information encoded into an image. While much of this information (e.g., the identities of people in an image, the type and number of objects in an image, etc.) is readily inferred by humans, fully understanding an image is still extremely difficult for computers. One important set of information encoded into images are radiometric scene properties---the properties of a scene related to light. Each pixel in an image indicates the amount of light received by the camera after being reflected, transmitted, or emitted by objects in a scene. It follows that we can learn about the objects of the scene and the scene itself through the image by thinking about the interaction between light and geometry in a scene. The appearance of objects in an image is primarily due to three factors: the geometry of the scene, the reflectance of the surfaces, and the incident illumination of the scene. Recovering these hidden properties of scenes can give us a deep understanding of a scene. For example, the reflectance of a surface can give a hint at the material properties of that surface. In this thesis, we address the question of how to recover complex, spatially-varying reflectance functions and natural illumination in real scenes from one or more images with known or approximately-known geometry. Recovering latent radiometric properties from images is difficult because of the severe underdetermined nature of the problem (i.e., there are many potential combinations of reflectance, light, and geometry that would produce identical input images) combined with the overwhelming dimensionality of the problem. In the real world, reflectance functions are complex, requiring many parameters to accurately model. An important aspect of solving this problem is to create a compact mathematical model to express a wide range of surface reflectance. We must also carefully model scene illumination, which typically exhibits complex behavior as well. Prior work has often simply assumed the light incident to a scene is made up of one or more infinitely-distant point lights. This assumption, however, rarely holds up in practice as not only are scenes illuminated by every possible direction, they are also illuminated by other objects interreflecting one another. To accurately infer reflectance and illumination of real-world scenes, we must account for the real-world behavior of reflectance and illumination. In this work, we develop a mathematical framework for the inference of complex, spatially-varying reflectance and natural illumination in real-world scenes. We use a Bayesian approach, where the radiometric properties (i.e., reflectance and illumination) to be inferred are modeled as random variables. We can then apply statistical priors to model how reflectance and illumination often exist in the real world to help combat the ambiguities created through the image formation process. We use our framework to infer the reflectance and illumination in a variety of scenes, ultimately using it in unrestricted real-world scenes. We show that the framework is capable of recovering complex reflectance and natural illumination in the real world.Ph.D., Computer Science -- Drexel University, 201

制約付き回帰に基づく照度差ステレオ

学位の種別: 課程博士審査委員会委員 : (主査)東京大学准教授 山﨑 俊彦, 東京大学教授, 相澤 清晴, 東京大学教授 池内 克史, 東京大学教授 佐藤 真一, 東京大学教授 佐藤 洋一, 東京大学教授 苗村 健University of Tokyo(東京大学