dc.contributor.author Wakin, MichaelDonoho, DavidChoi, HyeokhoBaraniuk, Richard G. Wakin, MichaelDonoho, DavidChoi, HyeokhoBaraniuk, Richard G. 2007-10-31T01:09:03Z 2007-10-31T01:09:03Z 2005-08-01 2005-08-01 M. Wakin, D. Donoho, H. Choi and R. G. Baraniuk, "The Multiscale Structure of Non-Differentiable Image Manifolds," 2005. https://hdl.handle.net/1911/20432 Conference Paper 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 ps-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. Texas Instruments Office of Naval Research National Science Foundation eng SPIE Image appearance manifoldsnon-differentiable manifoldsangle between subspacessampling theoremsmultiscale registrationpose estimation. Multiscale geometry processing The Multiscale Structure of Non-Differentiable Image Manifolds Conference paper 2005-07-07 inproceedings 2006-06-05 Digital Signal Processing (http://dsp.rice.edu/) Image appearance manifoldsnon-differentiable manifoldsangle between subspacessampling theoremsmultiscale registrationpose estimation. San Diego, CA Proc. SPIE Text Text http://dx.doi.org/10.1117/12.617822
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