The Multiscale Structure of Non-Differentiable Image Manifolds

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dc.citation.bibtexNameinproceedingsen_US
dc.citation.conferenceNameProc. SPIEen_US
dc.citation.locationSan Diego, CAen_US
dc.contributor.authorWakin, Michaelen_US
dc.contributor.authorDonoho, Daviden_US
dc.contributor.authorChoi, Hyeokhoen_US
dc.contributor.authorBaraniuk, Richard G.en_US
dc.contributor.orgDigital Signal Processing (http://dsp.rice.edu/)en_US
dc.date.accessioned2007-10-31T01:09:03Z
dc.date.available2007-10-31T01:09:03Z
dc.date.issued2005-08-01en
dc.date.modified2006-06-05en_US
dc.date.note2005-07-07en_US
dc.date.submitted2005-08-01en_US
dc.descriptionConference Paperen_US
dc.description.abstractIn this paper, we study families of images generated by varying a parameter that controls the appearance of the object/scene in each image. Each image is viewed as a point in high-dimensional space; the family of images forms a low-dimensional submanifold that we call an image appearance manifold (IAM). We conduct a detailed study of some representative IAMs generated by translations/rotations of simple objects in the plane and by rotations of objects in 3-D space. Our central, somewhat surprising, finding is that IAMs generated by images with sharp edges are nowhere differentiable. Moreover, IAMs have an inherent multiscale structure in that approximate tangent planes fitted to <i>ps</i>-neighborhoods continually twist off into new dimensions as the scale parameter $\eps$ varies. We explore and explain this phenomenon. An additional, more exotic kind of local non-differentiability happens at some exceptional parameter points where occlusions cause image edges to disappear. These non-differentiabilities help to understand some key phenomena in image processing. They imply that Newton's method will not work in general for image registration, but that a multiscale Newton's method will work. Such a multiscale Newton's method is similar to existing coarse-to-fine differential estimation algorithms for image registration; the manifold perspective offers a well-founded theoretical motivation for the multiscale approach and allows quantitative study of convergence and approximation. The manifold viewpoint is also generalizable to other image understanding problems.en_US
dc.description.sponsorshipTexas Instrumentsen_US
dc.description.sponsorshipOffice of Naval Researchen_US
dc.description.sponsorshipNational Science Foundationen_US
dc.identifier.citationM. Wakin, D. Donoho, H. Choi and R. G. Baraniuk, "The Multiscale Structure of Non-Differentiable Image Manifolds," 2005.
dc.identifier.doihttp://dx.doi.org/10.1117/12.617822en_US
dc.identifier.urihttps://hdl.handle.net/1911/20432
dc.language.isoeng
dc.publisherSPIEen_US
dc.subjectImage appearance manifolds*
dc.subjectnon-differentiable manifolds*
dc.subjectangle between subspaces*
dc.subjectsampling theorems*
dc.subjectmultiscale registration*
dc.subjectpose estimation.*
dc.subject.keywordImage appearance manifoldsen_US
dc.subject.keywordnon-differentiable manifoldsen_US
dc.subject.keywordangle between subspacesen_US
dc.subject.keywordsampling theoremsen_US
dc.subject.keywordmultiscale registrationen_US
dc.subject.keywordpose estimation.en_US
dc.subject.otherMultiscale geometry processingen_US
dc.titleThe Multiscale Structure of Non-Differentiable Image Manifoldsen_US
dc.typeConference paper
dc.type.dcmiText
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