The multiscale structure of non-differentiable image manifolds Export

@inproceedings{wakin05mult, abstract = {In this paper, we study families of images generated by varying aparameter 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 -neighborhoods continually twist off into new dimensions as the scale parameter 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.}, author = {Wakin, Michael B. and Donoho, David L. and Choi, Hyeokho and Baraniuk, Richard G.}, citeulike-article-id = {4754117}, citeulike-linkout-0 = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PSISDG00591400000159141B000001&idtype=cvips&gifs=yes}, citeulike-linkout-1 = {http://link.aip.org/link/?PSI/5914/59141B}, editor = {Papadakis, Manos and Laine, Andrew F. and Unser, Michael A.}, journal = {Wavelets XI}, keywords = {donoho, image, manifold, onur2009}, location = {San Diego, CA, USA}, number = {1}, pages = {59141B+}, posted-at = {2009-06-05 17:56:17}, priority = {2}, publisher = {SPIE}, title = {The multiscale structure of non-differentiable image manifolds}, url = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PSISDG00591400000159141B000001&idtype=cvips&gifs=yes}, volume = {5914}, year = {2005} }

TY - CONF ID - wakin05mult L3 - citeulike-article-id:4754117 N2 - In this paper, we study families of images generated by varying aparameter 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 -neighborhoods continually twist off into new dimensions as the scale parameter 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. CY - San Diego, CA, USA IS - 1 JF - Wavelets XI TI - The multiscale structure of non-differentiable image manifolds PB - SPIE VL - 5914 SP - 59141B KW - donoho KW - image KW - manifold KW - onur2009 AU - Wakin, Michael AU - Donoho, David AU - Choi, Hyeokho AU - Baraniuk, Richard A2 - Papadakis, Manos A2 - Laine, Andrew A2 - Unser, Michael PY - 2005/// UR - http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PSISDG00591400000159141B000001&idtype=cvips&gifs=yes ER -