Minimization of the @math norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving the problem at the heart of @math norm minimization are slow, and therefore cannot scale to large problems. A new method for the minimization of the @math norm is presented here, which provides a speedup of multiple orders of magnitude for data with high dimension. This method, termed Fast @math Minimization, allows robust regression to be applied to a class of problems which were previously inaccessible. It is shown how the @math norm minimization problem can be broken up into smaller sub-problems, which can then be solved extremely efficiently. Experimental results demonstrate the radical reduction in computation time, along with robustness against large numbers of outliers in a few model-fitting problems.
Robust statistical techniques, including the aforementioned robust regression and outlier removal methods, can significantly improve the performance of their classic counterparts. However, they have rarely been applied in image analysis field, to problems such as visual recognition, due to their computational expense. The M-estimator method is utilized in @cite_24 for face recognition and achieved high accuracy even when illumination change and pixel corruption were present. In @cite_40 , the authors propose a theoretical framework combining reconstructive and discriminative subspace methods for robust classification and regression. This framework acts on subsets of pixels in images to detect outliers.
@cite_24 @cite_40
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In this paper we address the problem of illumination invariant face recognition. Using a fundamental concept that in general, patterns from a single object class lie on a linear subspace [2], we develop a linear model representing a probe image as a linear combination of class-specific galleries. In the presence of noise, the well-conditioned inverse problem is solved using the robust Huber estimation and the decision is ruled in favor of the class with the minimum reconstruction error. The proposed Robust Linear Regression Classification (RLRC) algorithm is extensively evaluated for two standard databases and has shown good performance index compared to the state-of-art robust approaches. Linear subspace methods that provide sufficient reconstruction of the data, such as PCA, offer an efficient way of dealing with missing pixels, outliers, and occlusions that often appear in the visual data. Discriminative methods, such as LDA, which, on the other hand, are better suited for classification tasks, are highly sensitive to corrupted data. We present a theoretical framework for achieving the best of both types of methods: an approach that combines the discrimination power of discriminative methods with the reconstruction property of reconstructive methods which enables one to work on subsets of pixels in images to efficiently detect and reject the outliers. The proposed approach is therefore capable of robust classification with a high-breakdown point. We also show that subspace methods, such as CCA, which are used for solving regression tasks, can be treated in a similar manner. The theoretical results are demonstrated on several computer vision tasks showing that the proposed approach significantly outperforms the standard discriminative methods in the case of missing pixels and images containing occlusions and outliers.