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∪⋯∪Sqclass="mathContainer hidden">class="mathCode">, 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">Srclass="mathContainer hidden">class="mathCode"> 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">Γrclass="mathContainer hidden">class="mathCode">. Multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of
Ω by precisely identifying the boundary set
class="formula" id="fm0010">
class="mathml">
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1063520315000160&_mathId=si4.gif&_user=111111111&_pii=S1063520315000160&_rdoc=1&_issn=10635203&md5=06bbad2796e21c77add05cb22fdfdb7b" title="Click to view the MathML source">N={(p,nr(p)):p∈Sr,r=1,…,q},class="mathContainer hidden">class="mathCode">class="temp" src="/sd/blank.gif">
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">nr(p)class="mathContainer hidden">class="mathCode"> 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">Srclass="mathContainer hidden">class="mathCode"> 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">Γrclass="mathContainer hidden">class="mathCode"> 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.