Abstract
The recovery of the underlying low-rank structure of clean data corrupted with sparse noise/outliers is attracting increasing interest. However, in many low-level vision problems, the exact target rank of the underlying structure and the particular locations and values of the sparse outliers are not known. Thus, the conventional methods cannot separate the low-rank and sparse components completely, especially in the case of gross outliers or deficient observations. Therefore, in this study, we employ the minimum description length (MDL) principle and atomic norm for low-rank matrix recovery to overcome these limitations. First, we employ the atomic norm to find all the candidate atoms of low-rank and sparse terms, and then we minimize the description length of the model in order to select the appropriate atoms of low-rank and the sparse matrices, respectively. Our experimental analyses show that the proposed approach can obtain a higher success rate than the state-of-the-art methods, even when the number of observations is limited or the corruption ratio is high. Experimental results utilizing synthetic data and real sensing applications (high dynamic range imaging, background modeling, removing noise and shadows) demonstrate the effectiveness, robustness and efficiency of the proposed method.
Highlights
Low-rank matrix recovery is important in many fields, such as image processing and computer vision [1,2,3], pattern recognition and machine learning [4,5,6] and many other applications [7,8,9]
The atomic norm induced by the convex hull of all unit-norm one-sparse vectors is the l1 -norm, and the nuclear norm is induced by taking the convex hull of an atomic set, in which the elements are all unit rank matrices [24,25,26]. To address issues such as the limited number of observations, the rank of X and the regularizing parameter γ, we propose a low-rank model based on the minimum description length (MDL) principle within the devised atomic norm (MDLAN), which is an expanded version of our published conference paper [27]
We propose an algorithm based on MDL and the atomic norm to overcome these difficulties; i.e., an unknown target rank r, the regularizing parameter λ and deficient observations or gross outliers
Summary
Low-rank matrix recovery is important in many fields, such as image processing and computer vision [1,2,3], pattern recognition and machine learning [4,5,6] and many other applications [7,8,9]. Due to the sensor or environmental reasons, the observations used in these fields are readily corrupted by noise or outliers, and so the given data matrix Y can be decomposed into low-rank and sparse components. Principal components analysis (PCA) [10] has been used widely to search for the best approximation of the underlying structure (unknown low-rank matrix X) of the given data. We adopt the F-measure as the quantitative metric for the performance evaluation of the background modeling. The F-measure, which combines precision and recall, is calculated as follows:. It is reasonable to assume that these background variations are low-rank, while the moving objects in the foreground are large in magnitude and sparse in the spatial domain. Background estimation is complex due to the presence of foreground activity such as moving people and variations in illumination
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