Rongjie Lai

Compressed modes for variational problems in mathematics and physics

Significance Intuition suggests that many interesting phenomena in physics, chemistry, and materials science are “short-sighted”—that is, perturbation in a small spatial region only affects its immediate surroundings. In mathematical terms, near-sightedness is described by functions of finite range. As an example, the so-called Wannier functions in quantum mechanics are localized functions, which contain all the information about the properties of the system, including its spectral properties. This work’s main research objective is to develop theory and numerical methods that can systematically derive functions that span the energy spectrum of a given quantum-mechanical system and are nonzero only in a finite spatial region. These ideas hold the key for developing efficient methods for solving partial differential equations of mathematical physics. This article describes a general formalism for obtaining spatially localized (“sparse”) solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger’s equation in quantum mechanics. Sparsity is achieved by adding an regularization term to the variational principle, which is shown to yield solutions with compact support (“compressed modes”). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size.

A Splitting Method for Orthogonality Constrained Problems

Orthogonality constrained problems are widely used in science and engineering. However, it is challenging to solve these problems efficiently due to the non-convex constraints. In this paper, a splitting method based on Bregman iteration is represented to tackle the optimization problems with orthogonality constraints. With the proposed method, the constrained problems can be iteratively solved by computing the corresponding unconstrained problems and orthogonality constrained quadratic problems with analytic solutions. As applications, we demonstrate the robustness of our method in several problems including direction fields correction, noisy color image restoration and global conformal mapping for genus-0 surfaces construction. Numerical comparisons with existing methods are also conducted to illustrate the efficiency of our algorithms.

Anisotropic Laplace-Beltrami eigenmaps: Bridging Reeb graphs and skeletons

In this paper we propose a novel approach of computing skeletons of robust topology for simply connected surfaces with boundary by constructing Reeb graphs from the eigen-functions of an anisotropic Laplace-Beltrami operator. Our work brings together the idea of Reeb graphs and skeletons by incorporating a flux-based weight function into the Laplace-Beltrami operator. Based on the intrinsic geometry of the surface, the resulting Reeb graph is pose independent and captures the global profile of surface geometry. Our algorithm is very efficient and it only takes several seconds to compute on neuroanatomical structures such as the cingulate gyrus and corpus callosum. In our experiments, we show that the Reeb graphs serve well as an approximate skeleton with consistent topology while following the main body of conventional skeletons quite accurately.

Efficient Algorithm for Level Set Method Preserving Distance Function

The level set method is a popular technique for tracking moving interfaces in several disciplines, including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. First, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features, s.a. the curvature or the contour normal. Second, the level set algorithm is slow because the time step is limited by the standard Courant-Friedrichs-Lewy (CFL) condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate, and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications, s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient l1 optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods prove to be 5-6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.

Robust Surface Reconstruction via Laplace-Beltrami Eigen-Projection and Boundary Deformation

In medical shape analysis, a critical problem is reconstructing a smooth surface of correct topology from a binary mask that typically has spurious features due to segmentation artifacts. The challenge is the robust removal of these outliers without affecting the accuracy of other parts of the boundary. In this paper, we propose a novel approach for this problem based on the Laplace-Beltrami (LB) eigen-projection and properly designed boundary deformations. Using the metric distortion during the LB eigen-projection, our method automatically detects the location of outliers and feeds this information to a well-composed and topology-preserving deformation. By iterating between these two steps of outlier detection and boundary deformation, we can robustly filter out the outliers without moving the smooth part of the boundary. The final surface is the eigen-projection of the filtered mask boundary that has the correct topology, desired accuracy and smoothness. In our experiments, we illustrate the robustness of our method on different input masks of the same structure, and compare with the popular SPHARM tool and the topology preserving level set method to show that our method can reconstruct accurate surface representations without introducing artificial oscillations. We also successfully validate our method on a large data set of more than 900 hippocampal masks and demonstrate that the reconstructed surfaces retain volume information accurately.

Latent fingerprint segmentation with adaptive total variation model

Image segmentation is an important step in automatic fingerprint identification systems. While tremendous progress has been made in rolled and plain fingerprint segmentation, the segmentation of latent fingerprints is still a difficult problem. Features used for rolled and plain fingerprint images fail to work properly on latent images due to the poor quality in ridge information and the presence of multiple types of strong structured noise. In this work, we present an adaptive total variation (TV) model to achieve effective latent fingerprint segmentation. The proposed solution can remove various types of structured noise existing in a single latent image and automatically locate the region of interest (ROI), which contains primarily the latent fingerprint. Then, the following tasks such as fingerprint feature extraction and matching can be conducted in the ROI only. In the proposed TV-based image model, one can adaptively adjust the weight coefficient of the fidelity term in L1-norm depending on the background noise level, which is estimated via TV-based texture analysis. We apply the proposed TV-based segmentation algorithm to the NIST SD27 latent fingerprint database to demonstrate its superior performance.

A framework for intrinsic image processing on surfaces

After many years of study, the subject of image processing on the plane, or more generally in Euclidean space is well developed. However, more and more practical problems in different areas, such as computer vision, computer graphics, geometric modeling, and medical imaging, inspire us to consider imaging on surfaces beyond imaging on Euclidean domains. Several approaches, such as implicit representation approaches and parameterization approaches, are investigated about image processing on surfaces. Most of these methods require certain preprocessing to convert image problems on surfaces to image problems in Euclidean spaces. In this work, we use differential geometry techniques to directly study image problems on surfaces. By using our approach, all plane image variation models and their algorithms can be naturally adapted to study image problems on surfaces. As examples, we show how to generalize Rudin-Osher-Fatemi (ROF) denoising model [1] and convexified Chan-Vese (CV) [2] segmentation model on surfaces, and then demonstrate how to adapt popular algorithms to solve the total variation related problems on surfaces. This intrinsic approach provides us a robust and efficient method to directly study image processing, in particular, total variation problems on surfaces without requiring any preprocessing.

Laplace-Beltrami nodal counts: A new signature for 3D shape analysis

In this paper we develop a new approach of analyzing 3D shapes based on the eigen-system of the Laplace-Beltrami operator. While the eigenvalues of the Laplace-Beltrami operator have been used previously in shape analysis, they are unable to differentiate isospectral shapes. To overcome this limitation, we propose here a new signature based on nodal counts of the eigenfunctions. This signature provides a compact representation of the geometric information that is missing in the eigenvalues. In our experiments, we demonstrate that the proposed signature can successfully classify anatomical shapes with similar eigenvalues.

Harmonic Surface Mapping with Laplace-Beltrami Eigenmaps

In this paper we propose a novel approach for the mapping of 3D surfaces. With the Reeb graph of Laplace-Beltrami eigenmaps, our method automatically detects stable landmark features intrinsic to the surface geometry and use them as boundary conditions to compute harmonic maps to the unit sphere. The resulting maps are diffeomorphic, robust to natural pose variations, and establish correspondences for geometric features shared across population. In the experiments, we demonstrate our method on three subcortical structures: the hippocampus, putamen, and caudate nucleus. A group study is also performed to generate a statistically significant map of local volume losses in the hippocampus of patients with secondary progressive multiple sclerosis.

Geometric understanding of point clouds using Laplace-Beltrami operator

In this paper, we propose a general framework for approximating differential operator directly on point clouds and use it for geometric understanding on them. The discrete approximation of differential operator on the underlying manifold represented by point clouds is based only on local approximation using nearest neighbors, which is simple, efficient and accurate. This allows us to extract the complete local geometry, solve partial differential equations and perform intrinsic calculations on surfaces. Since no mesh or parametrization is needed, our method can work with point clouds in any dimensions or co-dimensions or even with variable dimensions. The computation complexity scaled well with the number of points and the intrinsic dimensions (rather than the embedded dimensions). We use this method to define the Laplace-Beltrami (LB) operator on point clouds, which links local and global information together. With this operator, we propose a few key applications essential to geometric understanding for point clouds, including the computation of LB eigenvalues and eigenfunctions, the extraction of skeletons from point clouds, and the extraction of conformal structures from point clouds.