Research statement - UCLA Department of Mathematics

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Recent Graduate Research During my PhD thesis at EPFL, I have been working on a number of image processing problems. I would like to summarize the most important ones. Geometric image registration In this project, I first developed a novel geometric framework called geodesic active fields (GAF) for image registration in general. In image registration, one looks for the underlying deformation field that best maps one image onto another, see figure 1. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, instead, I proposed a multiplicative coupling between the registration term and the regularization term, the Beltrami energy of the embedded deformation field, see figure 2. This approach turns out to be equivalent to embedding the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including non-flat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Secondly, this method is, to the best of our knowledge, the first re-parametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization. Then, I provide an efficient numerical scheme that uses a splitting approach: data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. This fast scheme is called FastGAF. Registration of cortical feature maps and automatic parcellation Much like a tree, the human brain exhibits a highly convoluted and irregular structure, with lots of complexity and variability: sulci and gyri vary a lot between subjects. On the other hand, high level structures of the brain – the “big picture” – are highly conserved, such as the two hemispheres, the lobes and main folds. A hierarchical representation of these structures is important for example in the context of intersubject registration: Considering the complexity of the cortical surface, directly involving local small-scale features would mislead the registration to be trapped in local minima. A robust method needs to rely on large-scale features, describing the main landmarks of the cortex, such as the main gyri or sulci. Subsequently, features are to be iteratively refined to drive the registration more locally and reach the desired precision. Here, I proposed a scale-space that lends itself for a meaningful hierarchical representation of structures on cortical feature maps on the sphere. The scale-space is expected to produce “generic” brain images at coarse scale, adding more and more individual details at finer scales, as shown in figure 3. Finally, I provided an implementation of the FastGAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, to automatically delineate the corresponding regions on a subject brain given its feature map (figure 4). 2
Figure 1: Image Registration. The human skull is registered to chimpanzee and baboon by finding the deformation fields u(x), s.t. human features match chimpanzee/baboon at x + u(x). Figure 3: Scale-Space of cortical feature maps. (a)– (b) The equalized spherical map and projection on the partially inflated surface, at scales k. (c) Median thresholding illustrates the simplification of structures. Main structures, e.g. the central sulcus (red), were well preserved in coarser scales. (d)–(e) Magnitude and orientation of the gradient of the map. Figure 2: The Swiss view of the Beltrami framework. A flat map corresponds to a higher-dimensional reality. Here: the intensity information (false-color) of the rectangular topographic map of the Matterhorn region translates directly into a three-dimensional terrain model. The surface of the Matterhorn as measured by the Beltrami energy of its map, is a direct measure of its “regularity”. (geodata c○swisstopo) 3 Ground Truth Automatic Error on GT Error on Feature Figure 4: Automatic parcellation. After registering a subject to the brains in the atlas, we perform automatic labelling. First and second row show manual (“ground truth”) and automatic parcellation results on the pial and inflated cerebro-cortical surface. The incorrect regions are marked in the third row.
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Research statement - UCLA Department of Mathematics

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