The multiscale structure of non-differentiable image manifolds

Michael B. Wakin, David L. Donoho, Hyeokho Choi, Richard G. Baraniuk

Research output: Chapter in Book/Report/Conference proceedingConference contribution

35 Scopus citations

Abstract

In 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 ε-neighborhoods continually twist off into new dimensions as the scale parameter ε 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.

Original languageEnglish (US)
Title of host publicationProceedings of SPIE - The International Society for Optical Engineering
EditorsM. Papadakis, A.F. Laine, M.A. Unser
Pages1-17
Number of pages17
Volume5914
DOIs
StatePublished - 2005
EventWavelets XI - San Diego, CA, United States
Duration: Jul 31 2005Aug 3 2005

Other

OtherWavelets XI
CountryUnited States
CitySan Diego, CA
Period7/31/058/3/05

Keywords

  • Angle between subspaces
  • Image appearance manifolds
  • Multiscale registration
  • Non-differentiable manifolds
  • Pose estimation
  • Sampling theorems

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Condensed Matter Physics

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