Research statement - UCLA Department of Mathematics

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Geometric image segmentation with Harmonic Active Contours In the meantime, I also worked on using the Beltrami energy as single term in a variational framework for image segmentation. As a more fancy extension of the Beltrami framework, we propose a segmentation method based on the geometric representation of images as surfaces embedded in a higher dimensional space, enabling us to naturally work with multichannel images. The segmentation is based on an active contour, described as the zero level set of the level set function, which is embedded in the image manifold along with a set of image features. Hence, both the data-fidelity and regularity terms of the active contour are jointly optimized by minimizing a single, unweighted Beltrami energy representing the hyper-surface of this embedded manifold. This approach is purely geometrical and does not require additional weighting of the energy functional to drive the segmentation task to the image contours. Indeed, the joint embedding of the image features and the level set both regularizes the level set and couples it to the image features. We have implemented both gradient-based and region-based criteria. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, enabling the use of the proposed technique in scale-space methods, volumetric data or catadioptric camera images. This constitutes a more general framework for image segmentation by defining the optimal segmentation function as the harmonic map minimizing the hyper-surface of the manifold. The proposed technique is therefore called Harmonic Active Contour (HAC). Efficient algorithm for the level set method The level set method, as also used in the aforementioned HAC framework, is a popular technique for tracking moving interfaces in several disciplines including fluid dynamics and computer vision. However, despite its high flexibility, the level set method is limited by two important numerical issues. Firstly, the level set algorithm is slow because the time step is limited by the standard CFL condition, which is also essential to the numerical stability of the iterative scheme. Secondly, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to ensure a good estimation of the contour normal and the curvature during the evolution process. Therefore we propose a new algorithm which overcomes these two fundamental problems in the level set method. The algorithm is based on recent advances of ℓ1 optimization techniques, some of which have already been employed in the fast algorithm for the optimization of the weighted Beltrami energy, above. The main idea is to split the original hard problem into sub-optimization problems which are wellknown and easy to solve, and to combine them together using an augmented Lagrangian approach to guarantee the equivalence with the original problem. This idea is borrowed from recent efficient ℓ1 minimization methods originally applied to compressed sensing. Current Postdoctoral Research Currently, I am researcher at the Department of Mathematics at UCLA, on a fellowship of the Swiss National Science Foundation (SNF). This fellowship gives me great independence in my research. A Unifying Geometric Framework for Image Processing My main focus, the topic for which my current fellowship was granted, is research on a “unifying geometric framework” for image processing. We believe that the Beltrami framework has great potential as generalized regularizing functional, in particular if equipped with a weighting function that allows multiplicative coupling of the data term with the regularization term of the specific inverse problem. Indeed, the resulting energy functional has very interesting fundamental properties, such as geometric regularization interpolating between Gaussian and TV regularization, re-parametrization invariance, applicability to any Riemannian manifold and intrinsic automatic data-dependent modulation of the local regularization strength. 4
Research in related fields, s.a. compressed sensing and optimization theory, has yielded very efficient optimization algorithms, which can be applied to the weighted Beltrami framework as well. We postulate that the weighted Beltrami framework represents an important step towards a unifying variational framework for geometric image processing, with a high degree of generality and a multitude of beneficial properties. Therefore we research, if and how other inverse problems in computer vision, image processing and other related domains can be generalized by the weighted Beltrami framework, and to develop robust and fast numerical schemes to optimize them. Beyond, I have developed two different generalizations of the Beltrami energy, that make the functional applicable to more interesting and more powerful regularization problems. Beltrami diffusion in the space of patches First, I am working on weighted Beltrami regularization in the space of patches. Recently, the use of patches has significantly gained in importance in various image processing applications. Indeed, the individual intensity or color information contained in a single local pixel is often not sufficient to completely characterize this pixel. Neighborhood information is required in order to better differentiate between textural features and noise, and diffusion on the space of patches was proposed mainly by Tschumperle. Patch-based embeddings of the “Beltrami-kind” were proposed as texture-aware edgeindicators and for denoising. Only very recently, a computationally more interesting minimization scheme was proposed. Here, I propose a novel model for image restoration, based on anisotropic diffusion on the space of patches, using the Beltrami embedding. We derive a local multiplicative coupling from the standard additive scheme and show how this automatically introduces an edge-aware pre-conditioner for diffusion. We propose a splitting scheme that naturally allows dealing with the patch-overlap and different non-linearities in a very elegant and efficient way. Graph-based and non-local Beltrami energy Beyond patch-diffusion, (patch-based) non-local regularization currently produces very promising results. For example, the current denoising state-of-the art is achieved by sparsification in patch-group transform-domain (BM3D). Currently, I am working on rendering the Beltrami-energy fully nonlocal, by extending its definition to non-local operators as introduced e.g., by Osher and Gilboa. This extension makes the benefits of Beltrami regularization, such as the intrinsic inter-channel coupling in vectorial or color images, readily available for data defined on graphs. This applies to non-local regularization where the graph-edge-weights are defined by patch-distances, but we equally see important usage in color-image processing, where node-distances are defined by various color-distances instead. Beyond, the anisotropic, inherently multichannel Beltrami-regularization thereby becomes available for any graph-based inverse problem, such as clustering or segmentation, with immediate applications in machine learning. Non-local Retinex Retinex is a theory on the human visual perception, introduced in 1971 by Edwin Land. It was an attempt to explain how the human visual system, as a combination of processes both in the retina and the cortex, is capable of adaptively coping with illumination spatially varying both in intensity and color. In image processing, the retinex theory has been implemented in various different flavors, each particularly adapted to specific tasks, including color balancing, contrast enhancement, dynamic range compression and shadow removal in consumer electronics and imaging, bias field correction in medical imaging or even illumination normalization, e.g. for face detection. In this project, I develop a unifying framework for retinex that is able to reproduce many of the existing retinex implementations within a single model, including all gradient-fidelity based models, variational models, and kernel-based models. The fundamental assumption, as shared with many 5
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Research statement - UCLA Department of Mathematics

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