26,184 research outputs found
A Fast Algorithm for a Weighted Low Rank Approximation
Matrix low rank approximation including the classical PCA and the robust PCA
(RPCA) method have been applied to solve the background modeling problem in
video analysis. Recently, it has been demonstrated that a special weighted low
rank approximation of matrices can be made robust to the outliers similar to
the -norm in RPCA method. In this work, we propose a new algorithm that
can speed up the existing algorithm for solving the special weighted low rank
approximation and demonstrate its use in background estimation problem
Weighted Low-Rank Approximation of Matrices and Background Modeling
We primarily study a special a weighted low-rank approximation of matrices
and then apply it to solve the background modeling problem. We propose two
algorithms for this purpose: one operates in the batch mode on the entire data
and the other one operates in the batch-incremental mode on the data and
naturally captures more background variations and computationally more
effective. Moreover, we propose a robust technique that learns the background
frame indices from the data and does not require any training frames. We
demonstrate through extensive experiments that by inserting a simple weight in
the Frobenius norm, it can be made robust to the outliers similar to the
norm. Our methods match or outperform several state-of-the-art online
and batch background modeling methods in virtually all quantitative and
qualitative measures.Comment: arXiv admin note: text overlap with arXiv:1707.0028
Weighted Low Rank Approximation for Background Estimation Problems
Classical principal component analysis (PCA) is not robust to the presence of
sparse outliers in the data. The use of the norm in the Robust PCA
(RPCA) method successfully eliminates the weakness of PCA in separating the
sparse outliers. In this paper, by sticking a simple weight to the Frobenius
norm, we propose a weighted low rank (WLR) method to avoid the often
computationally expensive algorithms relying on the norm. As a proof
of concept, a background estimation model has been presented and compared with
two norm minimization algorithms. We illustrate that as long as a
simple weight matrix is inferred from the data, one can use the weighted
Frobenius norm and achieve the same or better performance
On a Problem of Weighted Low-Rank Approximation of Matrices
We study a weighted low rank approximation that is inspired by a problem of
constrained low rank approximation of matrices as initiated by the work of
Golub, Hoffman, and Stewart (Linear Algebra and Its Applications, 88-89(1987),
317-327). Our results reduce to that of Golub, Hoffman, and Stewart in the
limiting cases. We also propose an algorithm based on the alternating direction
method to solve our weighted low rank approximation problem and compare it with
the state-of-art general algorithms such as the weighted total alternating
least squares and the EM algorithm
Weighted Schatten -Norm Minimization for Image Denoising and Background Subtraction
Low rank matrix approximation (LRMA), which aims to recover the underlying
low rank matrix from its degraded observation, has a wide range of applications
in computer vision. The latest LRMA methods resort to using the nuclear norm
minimization (NNM) as a convex relaxation of the nonconvex rank minimization.
However, NNM tends to over-shrink the rank components and treats the different
rank components equally, limiting its flexibility in practical applications. We
propose a more flexible model, namely the Weighted Schatten -Norm
Minimization (WSNM), to generalize the NNM to the Schatten -norm
minimization with weights assigned to different singular values. The proposed
WSNM not only gives better approximation to the original low-rank assumption,
but also considers the importance of different rank components. We analyze the
solution of WSNM and prove that, under certain weights permutation, WSNM can be
equivalently transformed into independent non-convex -norm subproblems,
whose global optimum can be efficiently solved by generalized iterated
shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g.,
image denoising and background subtraction. Extensive experimental results
show, both qualitatively and quantitatively, that the proposed WSNM can more
effectively remove noise, and model complex and dynamic scenes compared with
state-of-the-art methods.Comment: 13 pages, 11 figure
- …