加权TV与低秩约束提升压缩感知图像重建效果

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本文主要探讨了"有效的压感通过加权总变化和加权核范数调整在压缩感知(CS)方法中的应用"。作者Mingli Zhang、Christian Desrosiers和Caiming Zhang来自加拿大的École de technologie supérieure软件与IT工程部门以及中国的山东大学计算机科学技术学院，他们共同合作，旨在解决传统压缩感知技术在图像恢复过程中存在的问题。传统的压缩感知方法，如使用总变分(Total Variation, TV)和非局部块相似性，虽然能提升性能，但往往会导致图像中重要细节的丢失或重构后的伪影。为了克服这些问题，研究人员提出了一个创新的CS方法，即通过自适应加权TV策略来增强边缘保留。这种方法认识到图像中的冗余性，并利用加权低秩正则化进一步提升重建效果。作者引入了一种基于交替方向乘子法(Alternating Direction Method of Multipliers, ADMM)的优化策略，这种策略使得图像的高效重建成为可能。实验结果显示，与现有最先进的CS方法相比，他们的新方法在处理不同类型的图像时表现更为优越，能够提供更高质量的重构结果，同时保持图像细节和边缘的完整性。这种方法对于那些对图像质量和细节敏感的应用领域，如医学成像、遥感和视频压缩等，具有显著的优势。这篇研究论文不仅提出了一个改进的压缩感知理论框架，也为实际的图像处理和信号恢复提供了实用的工具和技术。

EFFECTIVE COMPRESSIVE SENSING VIA REWEIGHTED TOTAL VARIATION

AND WEIGHTED NUCLEAR NORM REGULARIZATION

Mingli Zhang

, Christian Desrosiers

, Caiming Zhang

2,3

Software and IT Engineering department,

Ecole de technologie sup

erieure, Montreal, Canada, H3C 1K3

Shandong Provincial Key Laboratory of Digital Media Technology, Jinan 250014, China

School of Computer Science and Technology, Shandong University, Jinan 250100, China

ABSTRACT

Total variation (TV) and non-local patch similarity have been

used successfully to enhance the performance of compressive

sensing (CS) approaches. However, such techniques can of-

ten remove important details in the image or introduce recon-

struction artifacts. This paper presents a novel CS method,

which uses an adaptive reweighted TV strategy to better pre-

serve image edges. Our method also leverages the redundancy

of non-local image patches through the use of weighted low

rank regularization. An optimization strategy based on the

ADMM algorithm is used to reconstruct images efﬁciently.

Experimental results show our method to outperform state-

of-the-art CS approaches, for various sampling ratios.

Index Terms— Compressive sensing, Total variation,

Reweighted TV, Non-local self similarity, Low rank, ADMM.

1. INTRODUCTION

Compressive (or compressed) sensing (CS) has been widely

exploited in image processing, with numerous applications in

photography [1], video [2], spectral imaging [3] and medical

imaging [4, 5]. The key idea of this technique is that, if the

sampling matrix satisﬁes a condition known as the restricted

isometry property (RIP), a sparse signal can be accurately

reconstructed from undersampled measurements [6–8]. For-

mally, the model for reconstructing an image x ∈ R

from

measurements y ∈ R

can be formulated as

y = Φx + ν, (1)

where Φ ∈ R

m×n

is a known measurement operator, and ν is

additive noise (e.g., Gaussian, Rice, etc.).

In most CS methods, the task of recovering x is deﬁned

as an inverse problem:

ˆx = argmin

kΦx − yk

+ λΨ(x), (2)

This work is supported by

ETS, FRQNT (210057) and National Natural

Science Foundation of China (61272431,61373080).

where Ψ(x) is a regularization prior, and λ is a parameter con-

trolling the trade-off between data ﬁdelity and the regulariza-

tion of x. Generally, the regularization prior is based on the

principle that x is sparse under some suitable transform such

as wavelets [9]. Another popular regularization approach is

total variation (TV) [9,10], which is represented as:

TV(∇x) =

i=1

(∇

)

+ (∇

)

. (3)

Because this approach penalizes gradients uniformly, it may

result in the loss of image details like edges. To overcome this

problem, a reweighted TV model was proposed in [7], where

the gradient of a pixel i is penalized according to a weight w

i.e.

TV(∇x, w) =

i=1

(∇

)

+ (∇

)

. (4)

Weights w are updated iteratively from x in such a way that

regions with sharp gradients (e.g., edges) are less penalized

than uniform ones:

t+1

k∇x

+ 

, (5)

where  is a small positive constant.

Another strategy to improve reconstruction is to exploit

the redundancy of local patterns in the image, represented

as small patches of pixels [11–14]. In this strategy, groups

of similar patches are regularized by applying a sparsifying

transform, for instance based on wavelets [13] or dictionary

learning [14]. Similar patches can also be regularized us-

ing the fact that matrices having these patches as columns or

rows have a low-rank [12, 15]. While CS methods based on

patch similarity can signiﬁcantly improve the reconstruction

when few samples are available, they also have the tendency

to over-smooth images by “averaging” similar patches [16].

Motivated by the aforementioned observations, we present

a novel CS method based on reweighted TV and weighted

low-rank regularization of similar patches. The main contri-

butions of our work are as follows:

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