Applications of low-rank matrix recovery methods in computer vision

The ubiquitous availability of high-dimensional data such as images and videos has generated a lot of interest in high-dimensional data analysis. One of the key issues that needs to be addressed in real applications is the presence of large-magnitude non-Gaussian errors. For image data, the problem of deformations or domain transformations also poses interesting challenges. In this thesis, we harness recent advances in low-rank matrix recovery via convex optimization techniques to solve real problems in computer vision. This thesis also provides some theoretical analysis that extends existing results to new observation models. Low-rank matrix approximations are a popular tool in data analysis. The well-known Principal Component Analysis (PCA) algorithm is a good example. Recently, it was shown that low-rank matrices can be recovered exactly from grossly corrupted measurements via convex optimization. This framework, called Principal Component Pursuit (PCP), constitutes a powerful tool that allows us to handle corrupted measurements and even missing entries in a principled way. In this thesis, we extend existing theoretical results to the case when a large majority of the entries of the matrix are badly corrupted. On the application side, we first briefly look at the image formation model that naturally gives rise to a low-rank matrix structure, and see how PCP can be used effectively in the photometric stereo problem. We then extend the existing PCP framework in a non-trivial fashion to effectively handle domain transformations in images. The proposed ideas are used to align multiple images simultaneously with one another, as well as to represent structured and symmetric textures in a novel way that is invariant to deformations. In addition to achieving excellent performance on real data, these methods are potentially very useful for other vision tasks like 3D structure recovery for urban images, automatic camera calibration, etc. Finally, we provide some theoretical guarantees for the new observation model encountered in the aforementioned applications. In particular, we show that under some conditions it is possible to recover most low-rank matrices even when a small linear fraction of their entries has been badly corrupted and, furthermore, when only linear measurements of the corrupted matrix are available. Besides being one of the first theoretical results for this case, this dissertation opens up many exciting avenues for future research in this direction.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical & Computer Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2012

Author dc:creator

• Balasubramanian, Arvind

Contributors dc:contributor

• Ma, Yi
• Huang, Thomas S.
• Milenkovic, Olgica
• Meyn, Sean P.

• Image Alignment
• Texture Rectification
• Low-Rank Matrix Recovery
• Convex Optimization
• Photometric Stereo
• Principal Component Pursuit

Statement dc:rights

• Copyright 2012 Arvind Balasubramanian

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/31929

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/31929