Mathematics

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    Novel integro-differential schemes for multiscale image representation
    (2009) Athavale, Prashant Vinayak; Tadmor, Eitan; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Multiscale representation of a given image is the problem of constructing a family of images, where each image in this family represents a scaled version of the given image. This finds its motivation from biological vision studies. Using the hierarchical multiscale image representation proposed by Tadmor et. al. [32], an image is decomposed into sums of simpler `slices', which extract more refined information from the previous scales. This approach motivates us to propose a novel integro-differential equation (IDE), for a multiscale image representation. We examine various properties of this IDE. The advantage of formulating the IDE this way is that, although this IDE is motivated by variational approach, we no longer need to be associated with any minimization problem and can modify the IDE, suitable to our image processing needs. For example, we may need to find different scales in the image, while retaining or enhancing prominent edges, which may define boundaries of objects. We propose some edge preserving modifications to our IDE. One of the important problems in image processing is deblurring a blurred image. Images get blurred due to various reasons, such as unfocused camera lens, relative motion between the camera and the object pictured, etc. The blurring can be modeled with a continuous, linear operator. Recovering a clean image from a blurry image, is an ill-posed problem, which is solved using Tikhonov-like regularization. We propose a different IDE to solve the deblurring problem. We propose hierarchical multiscale scheme based on (BV; L1) decomposition, proposed by Chan, Esedoglu, Nikolova and Alliney [12, 25, 3]. We finally propose another hierarchical multiscale representation based on a novel weighted (BV;L1) decomposition.
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    A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV,L2) DECOMPOSITIONS
    (Copyright: Society for Industrial and Applied Mathematics, 2004) TADMOR, EITAN; NEZZAR, SUZANNE; VESE, LUMINITA
    We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 + v0, where [u0, v0] is the minimizer of a J-functional, J(f, λ0; X, Y ) = infu+v=f u X + λ0 v p Y . Such minimizers are standard tools for image manipulations (e.g., denoising, deblurring, compression); see, for example, [M. Mumford and J. Shah, Proceedings of the IEEE Computer Vision Pattern Recognition Conference, San Francisco, CA, 1985] and [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259–268]. Here, u0 should capture “essential features” of f which are to be separated from the spurious components absorbed by v0, and λ0 is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [uj+1, vj+1] = arginf J(vj, λ02j ), leading to the hierarchical decomposition, f = k j=0 uj + vk. We focus our attention on the particular case of (X, Y) = (BV,L2) decomposition. The resulting hierarchical decomposition, f ∼ j uj , is essentially nonlinear. The questions of convergence, energy decomposition, localization, and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler–Lagrange equations. Numerical results illustrate applications of the new decomposition to synthetic and real images. Both greyscale and color images are considered.