Detection of boundary curves on the piecewise smooth boundary surface of three dimensional solids

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• 作者：Robert Houska ^{rthouska@math.uh.edu" class="auth_mail" title="E-mail the corresponding author} ; Demetrio Labate ; ^{dlabate@math.uh.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Analysis of singularities ; Continuous wavelet transform ; Edge detection ; Shearlets ; Sparse representations ; Wavelets
• 刊名：Applied and Computational Harmonic Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：40
• 期：1
• 页码：137-171
• 全文大小：840 K

文摘

Suppose that Ω   is a three-dimensional solid with boundary surface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si1.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=0592fd990c6e757b566d9813b5ca7c98" title="Click to view the MathML source">S=S₁∪⋯∪S_qclass="mathContainer hidden">class="mathCode">$S=S_1\cup \cdots \cup S_q$, where each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si2.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=abec799fd25ccae627dbee18a51f4b24" title="Click to view the MathML source">S_rclass="mathContainer hidden">class="mathCode">$S_r$ is a smooth surface with boundary curve class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si3.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=dd9d3443374a3eaf000a2cfd92e5cbc0" title="Click to view the MathML source">Γ_rclass="mathContainer hidden">class="mathCode">${\Gamma }_r$. Multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of Ω by precisely identifying the boundary set where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si5.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=01486e924756c4d0679046bdf30d3981" title="Click to view the MathML source">n_r(p)class="mathContainer hidden">class="mathCode">$n_r(p)$ denotes the normal vector to the surface class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si2.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=abec799fd25ccae627dbee18a51f4b24" title="Click to view the MathML source">S_rclass="mathContainer hidden">class="mathCode">$S_r$ at p  . This property has resulted in the successful application of multiscale directional methods in a variety of image processing problems, since edges and boundary sets are usually the most informative features in many types of multidimensional data. However, existing methods are ill-suited to capture those edge-type singularities in the three-dimensional setting resulting from the intersection of piecewise smooth boundary surfaces. In this paper, we introduce a new multiscale directional system based on a modification of the shearlet framework and prove that the associated continuous transform has the ability to precisely identify both the location and orientation of the boundary curves class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si3.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=dd9d3443374a3eaf000a2cfd92e5cbc0" title="Click to view the MathML source">Γ_rclass="mathContainer hidden">class="mathCode">${\Gamma }_r$ from the solid Ω. This paper extends a number of results appeared in the literature in recent years to the challenging problem of extracting curvilinear singularities in three-dimensional objects and is motivated by image analysis problems arising from areas including biomedical and seismic imaging and astronomy.