An important task when processing sensor data is to distinguish relevant from irrelevant data. This paper describes a method for an iterative singular value decomposition that maintains a model of the background via singular vectors spanning a subspace of the image space, thus providing a way to determine the amount of new information contained in an incoming frame. We update the singular vectors spanning the background space in a computationally efficient manner and provide the ability to perform block-wise updates, leading to a fast and robust adaptive SVD computation. The effects of those two properties and the success of the overall method to perform a state of the art background subtraction are shown in both qualitative and quantitative evaluations.
The method from @cite_9 is outlined, in its basic form, as follows: Given: SVD of @math , and @math Aim: Compute SVD for @math @math Update: @math with [ U_ k+1 = U_k Q , ] [ V_ k+1 = (P'_k P_k)^T , ] where @math results from a QR-decomposition, @math , @math and @math result from the SVD of a @math matrix. @math and @math are permutation matrices. For details, see @cite_9 . In the original version of the iterative SVD, the matrix @math is (formally) of dimension @math . Since in image processing @math captures the amount of pixels of one image, an explicit representation of @math consumes too much memory to be efficient which suggests to represent @math in terms of Householder reflections. This ensures that the of the SVD of @math is bounded by @math , and the step @math requires @math .
@cite_9
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Abstract We consider the problem of updating the SVD when augmenting a “tall thin” matrix, i.e., a rectangular matrix A ∈ R m × n with m ≫ n . Supposing that an SVD of A is already known, and given a matrix B ∈ R m × n ′ , we derive an efficient method to compute and efficiently store the SVD of the augmented matrix [ A B ] ∈ R m × ( n + n ′ ) . This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted.