Recent Research Projects

My research mainly focuses on computational quasi-conformal geometry (CQC) and its applications to medical imaging, computer vision and computer graphics.

Modal Uncertainty Estimation via Discrete Latent Representation: Many important problems in the real world don't have unique solutions. It is thus important for machine learning models to be capable of proposing different plausible solutions with meaningful probability measures. In this work we introduce such a deep learning framework that learns the one-to-many relationships between the inputs and outputs. The key of our approach is the use of a discrete latent space, where each item represents a latent mode hypothesis for a particular type of input-output relationship. The discrete latent representations and the uncertainty associated to any input are learned jointly during training. We thus call our framework modal uncertainty estimation. We extensively validate our framework on both real and synthetic datasets.
Shape Analysis of Partial Surfaces via Partial Surface Mapping Technique: We develop a framework for shape analysis of partial surfaces using inconsistent surface mapping technqiue. Traditional landmark-based geometric morphometrics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid surface mapping methods rely on a strong assumption of the global consistency of two surfaces. From a practical point of view, given two anatomical surfaces with prominent feature landmarks, it is more desirable to have a method that automatically detects the most relevant parts of the two surfaces and finds the optimal landmark-matching alignment between those parts, without assuming any global 1-1 correspondence between the two partial surfaces. Our method is capable of solving this problem using an inconsistent surface mapping technique based on quasi-conformal theory. It further enables us to quantify the dissimilarity of two shapes using quasi-conformal distortion and differences in mean and Gaussian curvatures, thereby providing a natural way for shape classification. Experiments on Platyrrhine molars demonstrate the effectiveness of our method and shed light on the interplay between function and shape in nature. Project Page
Inconsistent Shape Registration via Optimization of Mapping Distortions: We address the problem of registering two surfaces, of which a natural bijection between them does not exist. More precisely, only a partial subset of the source domain is assumed to be in correspondence with a subset of the target domain. We call such a problem an inconsistent surface registration problem. This problem is challenging as the corresponding regions on each surfaces and a meaningful bijection between them have to be simultaneously determined. In this paper, we propose a variational model to solve the inconsistent surface registration problem by minimizing mapping distortions. Mapping distortions are described by the Beltrami coefficient as well as the differential of the mapping. Registration is then guided by feature landmarks and/or intensities, such as curvatures, defined on each surfaces. The key idea of the approach is to control angle and scale distortions via quasiconformal theory as well as minimizing landmark and/or intensity mismatch. A splitting method is proposed to iteratively search for the optimal corresponding regions as well as the optimal bijection between them. Bijectivity of the mapping is easily enforced by a thresholding of the Beltrami coefficient. We test the proposed method on both synthetic and real examples. Experimental results demonstrate the efficacy of our proposed model.
Detection of Abnormal Deformations from Normal Motions: Detection of abnormal deformations from normal motions is crucial in image analysis, especially for medical image analysis. For instance, accurate extraction of abnormal cardiac motions is a necessary procedure for cardiac disease analysis. The combination of nomral motion and abnormal deformations bring challenges to extract abnormalities. In this work, we propose a novel method to extract abnormal deformation from normal (periodic) motions by using the Beltrami coefficients (BC). BCs are used to represent a sequence of deformations over a sequence of images capturing an object of interest in motion. RPCA is then performed on the BCs to decompose the overall motion into nomral motion and abnormal deformation. Experiments have been carried out on both synthetic and real medical image sequence, which demonstrate the efficacy of our proposed method.
Shape-prior Image Segmentation using Discrete Conformality Structure: Image segmentation aims to partition an image into meaningful regions and extract important objects therein. It is an important and yet ambiguous problem in computer visions. Common methods usually impose various constraints on the extracted regions as regularization to achieve the image segmentation goal. In practical situations, it is desirable to incorporate shape prior information about the objects into the segmentation model. In this work, we present a novel shape prior image segmentation model based on discrete conformality structure. The discrete conformality structure captures the angle structures of the meshes representing the segmented objects. Shape prior information can be prescribed into the segmentation model by imposing constraints on the conformality structures. Segmentation results with prescribed local or global shape priors can be easily obtained. We illustrate our idea on various shape prior segmentation problems such as the convexity prior segmentation problem. Experimental results demonstrate the efficacy of our proposed method.
Parametrising Flat-foldable surfaces with incomplete data: We propose a novel way of computing surface folding maps via solving a linear PDE. This framework is a generalization to the existing quasiconformal methods and allows manipulation of the geometry of folding. Moreover, the crucial quantity that characterizes the geometry occurs as the coefficient of the equation, namely the Beltrami coefficient. This allows us to solve an inverse problem of parametrizing the folded surface given only partial data but with known folding topology. Various interesting applications such as fold sculpting on 3D models and self-occlusion reasoning are demonstrated to show the effectiveness of our method.
A Parallel Algorithm for Global Conformal Parameterization using Welding Maps: We propose an efficient parallel algorithm for computing the global conformal parameterization of a simply-connected open or closed surface embedded in R3. The method is specially designed for large-scale problems, of which surface meshes contain a large number of vertices. The main idea is based on the observation that a global conformal parameterization can be retrieved from local parameterizations of arbitrary patches using the welding maps. A numerical algorithm for the global conformal parameterization of a high-resolution surface mesh can then be developed by conformally glueing the local conformal parameterizations of various patches. Each local conformal parameterization can be computed in parallel. Hence, our proposed algorithm is parallelizable. The procedure conformal glueing can be efficiently carried out through a partial zipper algorithm, which involves only the boundary vertices. Extensive experiments on different simply-connected open and closed surface meshes demonstrate the efficacy of our proposed method.
Restoration of turbulence-distorted images via RPCA and Quasi-conformal Maps: We address the problem of restoring a high-quality image from an observed image sequence strongly distorted by atmospheric turbulence. A novel algorithm is proposed in this paper to reduce geometric distortion as well as space and time-varying blur due to turbulence. By considering an optimization problem, our algorithm first obtain a sharp reference image and a sub-sampled image sequence containing sharp and mildly distorted image frames with respect to the reference image. The sub-sampled image sequence is then stabilized by applying the Robust Principal Component Analysis (RPCA) on the deformation fields between image frames and warping the image frames by a quasiconformal map associated to the low-rank part of the deformation matrix. After image frames are registered to the reference image, the low-rank part of them are deblurred via a blind deconvolution, and the deblurred frmaes are then fused with the enhanced sparse part. Experiments have been carried out on both synthetic and real turbulence-distorted video. Results demonstrate our method is effecitve in alleviating distortions and blur, restoring image details and enhancing visual quality.
Image retargeting via Beltrami representation: Image retargeting aims to resize an image to one with prescribed aspect ratio. Simple scaling inevitably introduces unnatural geometric distortions on the important contents of the image. In this paper, we propose a simple and yet effective method to resize an image, which preserves the geometry of the important content, using quasiconformal theories. Our algorithm allows users to interactively identify content regions as well as line structures. Image resizing can then be acheived by warping the image by an orientation-preserving homeomorphism with controlled distortion. The warping map is represented by its Beltrami representation, which captures the local geometric distortion of the map. By carefully setting the values of the Beltrami representation, we propose three methods to resize an image for different situations. Our method is simple and does not require solving any optimization problems throughtout the process. This results in a simple and efficient algorithm to solve the image retargeting problem. Experiment results demonstrate the efficacy of our proposed method.