论文简介：
毕业论文-压缩传感与多尺度几何分析的研究，共62页，20718字
中文摘要
压缩传感是一种新型的数字信号采样方式。自 2004 年其理论提出以来，压缩
传感迅速成为研究热点。本文主要讨论压缩传感作用在二维图像信号上的情形。
对于未知的信号采样，压缩传感对信号采样首先要假设信号在某一个基底上能够
被稀疏的表示。对于二维图像信号，目前一般假设图像在二维正交小波基上有稀
疏表示。然而，这样的假设是不符合实际的。二维正交小波的构成决定了其能比
较好的表示图像中水平、垂直和对角线方向的边缘信息，对于图像中丰富的其他
方向的边缘信息，二维正交小波没能达到对应一维小波的表示能力。
与此同时，除了二维正交小波，人们也发展起多种更能稀疏表示图像中信息
的变换方法。与二维正交小波变换比较类似的有双树小波。在多尺度集合分析这
一领域中，Curvelet 变换、Contourlet 变换等方法相比于基于小波的图像变换具有
更强的稀疏表示能力。
本文讨论当信号假设在不同基底下稀疏时，压缩传感方法的效果。在双树小
波变换假设下对图像信号进行了仿真实验。仿真说明双树小波要比二维正交小波
效果好，同时文章给出双树小波实际应用中的方法。另外，文章分析了在其他变
换假设下做压缩传感会遇到的问题，并给出了理论上在最佳变换假设下做压缩传
感的效果的分析。
关键词：压缩传感；多尺度几何分析；稀疏性；变换
ABSTRACT
Compressed sensing is a new framework of digital signal acquisition. Since the
theory was first published in 2004, compressed sensing has quickly grown into a hot
area of research. In this paper, we discuss its performance on digital images. To acquire
an unknown signal, the compressed sensing approach is based on the assumption that
the signal can be sparsely represented on a certain basis. For most image signals, we
most commonly assume the basis to be the 2D orthogonal wavelet basis. However, this
assumption is not without flaw. The 2D orthogonal wavelet basis is effective in
representing edges in the horizontal, perpendicular and diagonal directions. For images
with abundant edges, the 2D orthogonal wavelet is not as effective as its 1D
counterpart.
Meanwhile, besides the 2D orthogonal wavelet, other transforms that can better
capture image information are developed. Along the wavelet line, we have the
dual-tree wavelet transform. In the area of multiscale geometric analysis, the curvelet
transform, contourlet transform etc have better performance than wavelets.
In this paper, we discuss the performance of compressed sensing when different
bases are assumed for the images. In the dual-tree wavelet case, experiments show its
improvement over the 2D orthogonal wavelet basis in the image compressed sensing.
Also, we give the setup for dual-tree wavelet in actual implementation. Additionally,
we discuss the difficulties encountered when compressed sensing is used under other
transform assumptions. We also give the analysis of CS performance under optimal
basis assumption.
Keywords：compressed sensing; multiscale geometry analysis; sparsity; transform
目录
第 1 章 引 言 ............................................................................................5
1.1 研究背景 ...........................................................................................5
1.1.1 压缩传感 .......................................................................................5
1.1.2 多尺度几何分析 ...........................................................................6
1.2 研究问题 ...........................................................................................6
1.3 文章结构 ...........................................................................................7
第 2 章 相关理论知识 ...............................................................................8
2.1 压缩传感的基本理论 .........................................................................8
2.2 小波的问题与多尺度几何分析 ........................................................10
2.2.1 二维正交小波基对图像的稀疏表示能力 ....................................11
2.2.2 Curvelet 变换与 Contourlet 变换基 ..............................................12
2.2.3 二维双树小波变换 ......................................................................19
第 3 章 理想最优基假设下压缩传感的效果 ...........................................22
第 4 章 在二维双树小波基下的压缩传感 ...............................................24
4.1 等效实验方法 ..................................................................................24
4.2 实验仿真及结果 ..............................................................................25
4.3 实际方法 .........................................................................................39
第 5 章 Curvelet 与 Contourlet 变换在压缩传感中的适应性 ................42
5.1 计算复杂度 ......................................................................................42
5.2 恢复效果的分析 ..............................................................................43
第 6 章 总结 ............................................................................................44
插图索引 ...................................................................................................45
表格索引 ...................................................................................................46
参考文献 ...................................................................................................47
致 谢 .........................................................................................................49
声 明 ...........................................................................错误！未定义书签。
附录 A 外文资料的调研阅读报告（或书面翻译） ..................................51

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• 毕业论文-压缩传感与多尺度几何分析的研究
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