Geometry-Aware Neighborhood Search for Learning Local Models for Image Reconstruction

Local learning of sparse image models has proven to be very effective to solve inverse problems in many computer vision applications. To learn such models, the data samples are often clustered using the K-means algorithm with the Euclidean distance as a dissimilarity metric. However, the Euclidean distance may not always be a good dissimilarity measure for comparing data samples lying on a manifold. In this paper, we propose two algorithms for determining a local subset of training samples from which a good local model can be computed for reconstructing a given input test sample, where we take into account the underlying geometry of the data. The first algorithm, called Adaptive Geometry-driven Nearest Neighbor search (AGNN), is an adaptive scheme which can be seen as an out-of-sample extension of the replicator graph clustering method for local model learning. The second method, called Geometry-driven Overlapping Clusters (GOC), is a less complex nonadaptive alternative for training subset selection. The proposed AGNN and GOC methods are evaluated in image super-resolution, deblurring and denoising applications and shown to outperform spectral clustering, soft clustering, and geodesic distance based subset selection in most settings.Comment: 15 pages, 10 figures and 5 table

3D Point Cloud Denoising via Deep Neural Network based Local Surface Estimation

We present a neural-network-based architecture for 3D point cloud denoising called neural projection denoising (NPD). In our previous work, we proposed a two-stage denoising algorithm, which first estimates reference planes and follows by projecting noisy points to estimated reference planes. Since the estimated reference planes are inevitably noisy, multi-projection is applied to stabilize the denoising performance. NPD algorithm uses a neural network to estimate reference planes for points in noisy point clouds. With more accurate estimations of reference planes, we are able to achieve better denoising performances with only one-time projection. To the best of our knowledge, NPD is the first work to denoise 3D point clouds with deep learning techniques. To conduct the experiments, we sample 40000 point clouds from the 3D data in ShapeNet to train a network and sample 350 point clouds from the 3D data in ModelNet10 to test. Experimental results show that our algorithm can estimate normal vectors of points in noisy point clouds. Comparing to five competitive methods, the proposed algorithm achieves better denoising performance and produces much smaller variances

Riemannian Optimization via Frank-Wolfe Methods

We study projection-free methods for constrained Riemannian optimization. In particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize Rfw to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian "linear oracle" required by RFW admits a closed-form solution; this result may be of independent interest. We further specialize RFW to the special orthogonal group and show that here too, the Riemannian "linear oracle" can be solved in closed form. Here, we describe an application to the synchronization of data matrices (Procrustes problem). We complement our theoretical results with an empirical comparison of Rfw against state-of-the-art Riemannian optimization methods and observe that RFW performs competitively on the task of computing Riemannian centroids.Comment: Under Review. Largely revised version, including an extended experimental section and an application to the special orthogonal group and the Procrustes proble