679 research outputs found
From Rank Estimation to Rank Approximation: Rank Residual Constraint for Image Restoration
In this paper, we propose a novel approach to the rank minimization problem,
termed rank residual constraint (RRC) model. Different from existing low-rank
based approaches, such as the well-known nuclear norm minimization (NNM) and
the weighted nuclear norm minimization (WNNM), which estimate the underlying
low-rank matrix directly from the corrupted observations, we progressively
approximate the underlying low-rank matrix via minimizing the rank residual.
Through integrating the image nonlocal self-similarity (NSS) prior with the
proposed RRC model, we apply it to image restoration tasks, including image
denoising and image compression artifacts reduction. Towards this end, we first
obtain a good reference of the original image groups by using the image NSS
prior, and then the rank residual of the image groups between this reference
and the degraded image is minimized to achieve a better estimate to the desired
image. In this manner, both the reference and the estimated image are updated
gradually and jointly in each iteration. Based on the group-based sparse
representation model, we further provide a theoretical analysis on the
feasibility of the proposed RRC model. Experimental results demonstrate that
the proposed RRC model outperforms many state-of-the-art schemes in both the
objective and perceptual quality
Tensor Robust PCA with Nonconvex and Nonlocal Regularization
Tensor robust principal component analysis (TRPCA) is a promising way for
low-rank tensor recovery, which minimizes the convex surrogate of tensor rank
by shrinking each tensor singular values equally. However, for real-world
visual data, large singular values represent more signifiant information than
small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA)
model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our
N-TRPCA can adaptively shrink small singular values more and shrink large
singular values less. In addition, TRPCA assumes that the whole data tensor is
of low rank. This assumption is hardly satisfied in practice for natural visual
data, restricting the capability of TRPCA to recover the edges and texture
details from noisy images and videos. To this end, we integrate nonlocal
self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal
TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a
tensor and then each group tensor is recovered by our N-TRPCA. Since the
patches in one group are highly correlated, all group tensors have strong
low-rank property, leading to an improvement of recovery performance.
Experimental results demonstrate that the proposed NN-TRPCA outperforms some
existing TRPCA methods in visual data recovery. The demo code is available at
https://github.com/qguo2010/NN-TRPCA.Comment: 19 pages, 7 figure
Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications
Robust Principal Component Analysis (RPCA) via rank minimization is a
powerful tool for recovering underlying low-rank structure of clean data
corrupted with sparse noise/outliers. In many low-level vision problems, not
only it is known that the underlying structure of clean data is low-rank, but
the exact rank of clean data is also known. Yet, when applying conventional
rank minimization for those problems, the objective function is formulated in a
way that does not fully utilize a priori target rank information about the
problems. This observation motivates us to investigate whether there is a
better alternative solution when using rank minimization. In this paper,
instead of minimizing the nuclear norm, we propose to minimize the partial sum
of singular values, which implicitly encourages the target rank constraint. Our
experimental analyses show that, when the number of samples is deficient, our
approach leads to a higher success rate than conventional rank minimization,
while the solutions obtained by the two approaches are almost identical when
the number of samples is more than sufficient. We apply our approach to various
low-level vision problems, e.g. high dynamic range imaging, motion edge
detection, photometric stereo, image alignment and recovery, and show that our
results outperform those obtained by the conventional nuclear norm rank
minimization method.Comment: Accepted in Transactions on Pattern Analysis and Machine Intelligence
(TPAMI). To appea
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