Characterization and analysis of edges in piecewise smooth functions

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• 作者：Kanghui Guo^a ; ^{KanghuiGuo@MissouriState.edu" class="auth_mail" title="E-mail the corresponding author} ; Demetrio Labate^b ; ^{dlabate@math.uh.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Analysis of singularities ; Continuous wavelets ; Curvelets ; Directional wavelets ; Edge detection ; Shearlets ; Wavelets
• 刊名：Applied and Computational Harmonic Analysis
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：41
• 期：1
• 页码：139-163
• 全文大小：595 K

文摘

The analysis and detection of edges is a central problem in applied mathematics and image processing.

A number of results in recent years have shown that directional multiscale methods such as continuous curvelet and shearlet transforms offer a powerful theoretical framework to capture the geometry of edge singularities, going far beyond the capabilities of the conventional wavelet transform. The continuous shearlet transform, in particular, provides a precise geometric characterization of edges in piecewise constant functions in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315001487&_mathId=si1.gif&_user=111111111&_pii=S1063520315001487&_rdoc=1&_issn=10635203&md5=d10643855d2a08fc3f7159fa0d27fcec" title="Click to view the MathML source">R²class="mathContainer hidden">class="mathCode">${\mathbb{R}}^2$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315001487&_mathId=si2.gif&_user=111111111&_pii=S1063520315001487&_rdoc=1&_issn=10635203&md5=99fd8aaa8772e4cafb572f6e62793c7d" title="Click to view the MathML source">R³class="mathContainer hidden">class="mathCode">${\mathbb{R}}^3$, including corner points. However, a question has been raised frequently: What happens if the function is piecewise smooth and not just piecewise constant? Clearly, a piecewise smooth function is a much more realistic model of images with edges.

In this paper, we extend the characterization results previously known and show that, also in the case of piecewise smooth functions, the continuous shearlet transform can detect the location and orientation of edge points, including corner points, through its asymptotic decay at fine scales. The new proof introduces innovative technical constructions to deal with the more challenging problem. The new results set the theoretical groundwork for the application of the shearlet framework to a wider class of problems from image processing.