Browsing by Author "Donoho, David"

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    High-Resolution Navigation on Non-Differentiable Image Manifolds
    (2005-03-01) Wakin, Michael; Donoho, David; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
    The images generated by varying the underlying articulation parameters of an object (pose, attitude, light source position, and so on) can be viewed as points on a low-dimensional image parameter articulation manifold (IPAM) in a high-dimensional ambient space. In this paper, we develop theory and methods for the inverse problem of estimating, from a given image on or near an IPAM, the underlying parameters that produced it. Our approach is centered on the observation that, while typical image manifolds are not differentiable, they have an intrinsic multiscale geometric structure. In fact, each IPAM has a family of approximate tangent spaces, each one good at a certain resolution. Putting this structural aspect to work, we develop a new algorithm for high-accuracy parameter estimation based on a coarse-to-fine Newton iteration through the family of approximate tangent spaces. We test the algorithm in several idealized registration and pose estimation problems.
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    The Multiscale Structure of Non-Differentiable Image Manifolds
    (SPIE, 2005-08-01) Wakin, Michael; Donoho, David; Choi, Hyeokho; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
    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.