Algorithmic issues in modeling motion
ACM Computing Surveys (CSUR)
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In digital image processing, images are treated as two dimensional signals and are processed using signal processing techniques. This thesis focuses on a signal decomposition technique that partitions an input image into components with different geometric scales. This multiscale decomposition technique is based on the minimization of an energy that is a weighted sum of a total variation and an L1 distance (TV-L 1). Although the minimization of the L1-fidelity term penalizes the intensity difference between the input and an output images, the decomposition of the input image by the TV-L1 model is solely determined by the geometric scales of the features (homogeneous parts) in the image, independent of the feature intensity. This important property was previously observed by Alliney, Nikolova, Chan and Esedoglu. Here we present in Chapter 2 a rigorous proof of this property that is based on a newly established equivalence between the minimizations of the TV-L1 energy and a sequence of geometric energies. Furthermore, we use this property to obtain other properties, including geometric and morphological invariance, of the TV-L 1 model and to discuss new computational methods. To reveal the differences between the TV-L1 model and other TV-based models for image decomposition, we present the related ROF, Meyer, and Vese-Osher models and formulate all four TV-based models as second-order cone programs (SOCPs). We used an SOCP interior-point method, combined with nested dissection to reduce computational complexity, to numerically solve the SOCPs corresponding to these models and obtained solutions for both 1D and 2D examples. Our tests show that the decomposition by the TV -L1 model is qualitatively different from those produced by the other TV based models and, therefore, is very unique. Because of its special properties shown both analytically and numerically, the TV-L1 model quickly finds applications in medical imaging and computer vision. In the first application, cDNA microarray images with inhomogeneous background are processed by the TV-L 1 model to have the large-scale inhomogeneous background separated from the useful foreground cDNA spots. In the second application, a multiplicative version of the TV-L1 model is applied to face images with varying illumination conditions. The resulting quotient images are successfully used for face recognition.