arXiv subjects

Xianfeng Gu

Publications and source records attributed to Xianfeng Gu.

At least 37 records · Page 2Linked to original sources

arXiv2019

Mode Collapse and Regularity of Optimal Transportation Maps

This work builds the connection between the regularity theory of optimal transportation map, Monge-Ampère equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Ampère equation, if the support of the target measure is disconnected or just non-convex, the optimal transportation mapping is discontinuous. General DNNs can only approximate continuous mappings. This intrinsic conflict leads to the convergence difficulty and mode collapse in GANs. We test our hypothesis that the supports of real data distribution are in general non-convex, therefore the discontinuity is unavoidable using an Autoencoder combined with discrete optimal transportation map (AE-OT framework) on the CelebA data set. The testing result is positive. Furthermore, we propose to approximate the continuous Brenier potential directly based on discrete Brenier theory to tackle mode collapse. Comparing with existing method, this method is more accurate and effective.

cs.LG↗

arXiv2018

Metric Based Quadrilateral Mesh Generation

This work proposes a novel metric based algorithm for quadrilateral mesh generating. Each quad-mesh induces a Riemannian metric satisfying special conditions: the metric is a flat metric with cone signualrites conformal to the original metric, the total curvature satisfies the Gauss-Bonnet condition, the holonomy group is a subgroup of the rotation group $\{e^{ikπ/2}\}$, furthermore there is cross field obtained by parallel translation which is aligned with the boundaries, and its streamlines are finite geodesics. Inversely, such kind of metric induces a quad-mesh. Based on discrete Ricci flow and conformal structure deformation, one can obtain a metric satisfying all the conditions and obtain the desired quad-mesh. This method is rigorous, simple and automatic. Our experimental results demonstrate the efficiency and efficacy of the algorithm.

cs.CG↗

arXiv2018

LMap: Shape-Preserving Local Mappings for Biomedical Visualization

Visualization of medical organs and biological structures is a challenging task because of their complex geometry and the resultant occlusions. Global spherical and planar mapping techniques simplify the complex geometry and resolve the occlusions to aid in visualization. However, while resolving the occlusions these techniques do not preserve the geometric context, making them less suitable for mission-critical biomedical visualization tasks. In this paper, we present a shape-preserving local mapping technique for resolving occlusions locally while preserving the overall geometric context. More specifically, we present a novel visualization algorithm, LMap, for conformally parameterizing and deforming a selected local region-of-interest (ROI) on an arbitrary surface. The resultant shape-preserving local mappings help to visualize complex surfaces while preserving the overall geometric context. The algorithm is based on the robust and efficient extrinsic Ricci flow technique, and uses the dynamic Ricci flow algorithm to guarantee the existence of a local map for a selected ROI on an arbitrary surface. We show the effectiveness and efficacy of our method in three challenging use cases: (1) multimodal brain visualization, (2) optimal coverage of virtual colonoscopy centerline flythrough, and (3) molecular surface visualization.

cs.CV↗

arXiv2018

Spherical Parameterization Balancing Angle and Area Distortions

This work presents a novel framework for spherical mesh parameterization. An efficient angle-preserving spherical parameterization algorithm is introduced, which is based on dynamic Yamabe flow and the conformal welding method with solid theoretic foundation. An area-preserving spherical parameterization is also discussed, which is based on discrete optimal mass transport theory. Furthermore, a spherical parameterization algorithm, which is based on the polar decomposition method, balancing angle distortion and area distortion is presented. The algorithms are tested on 3D geometric data and the experiments demonstrate the efficiency and efficacy of the proposed methods.

cs.GR↗

arXiv2018

Corresponding Supine and Prone Colon Visualization Using Eigenfunction Analysis and Fold Modeling

We present a method for registration and visualization of corresponding supine and prone virtual colonoscopy scans based on eigenfunction analysis and fold modeling. In virtual colonoscopy, CT scans are acquired with the patient in two positions, and their registration is desirable so that physicians can corroborate findings between scans. Our algorithm performs this registration efficiently through the use of Fiedler vector representation (the second eigenfunction of the Laplace-Beltrami operator). This representation is employed to first perform global registration of the two colon positions. The registration is then locally refined using the haustral folds, which are automatically segmented using the 3D level sets of the Fiedler vector. The use of Fiedler vectors and the segmented folds presents a precise way of visualizing corresponding regions across datasets and visual modalities. We present multiple methods of visualizing the results, including 2D flattened rendering and the corresponding 3D endoluminal views. The precise fold modeling is used to automatically find a suitable cut for the 2D flattening, which provides a less distorted visualization. Our approach is robust, and we demonstrate its efficiency and efficacy by showing matched views on both the 2D flattened colons and in the 3D endoluminal view. We analytically evaluate the results by measuring the distance between features on the registered colons, and we also assess our fold segmentation against 20 manually labeled datasets. We have compared our results analytically to previous methods, and have found our method to achieve superior results. We also prove the hot spots conjecture for modeling cylindrical topology using Fiedler vector representation, which allows our approach to be used for general cylindrical geometry modeling and feature extraction.

cs.CV↗

arXiv2018

Latent Space Optimal Transport for Generative Models

Variational Auto-Encoders enforce their learned intermediate latent-space data distribution to be a simple distribution, such as an isotropic Gaussian. However, this causes the posterior collapse problem and loses manifold structure which can be important for datasets such as facial images. A GAN can transform a simple distribution to a latent-space data distribution and thus preserve the manifold structure, but optimizing a GAN involves solving a Min-Max optimization problem, which is difficult and not well understood so far. Therefore, we propose a GAN-like method to transform a simple distribution to a data distribution in the latent space by solving only a minimization problem. This minimization problem comes from training a discriminator between a simple distribution and a latent-space data distribution. Then, we can explicitly formulate an Optimal Transport (OT) problem that computes the desired mapping between the two distributions. This means that we can transform a distribution without solving the difficult Min-Max optimization problem. Experimental results on an eight-Gaussian dataset show that the proposed OT can handle multi-cluster distributions. Results on the MNIST and the CelebA datasets validate the effectiveness of the proposed method.

cs.LG↗

arXiv2018

Variational Wasserstein Clustering

We propose a new clustering method based on optimal transportation. We solve optimal transportation with variational principles, and investigate the use of power diagrams as transportation plans for aggregating arbitrary domains into a fixed number of clusters. We iteratively drive centroids through target domains while maintaining the minimum clustering energy by adjusting the power diagrams. Thus, we simultaneously pursue clustering and the Wasserstein distances between the centroids and the target domains, resulting in a measure-preserving mapping. We demonstrate the use of our method in domain adaptation, remeshing, and representation learning on synthetic and real data.

cs.CV↗

arXiv2018

Robust Edge-Preserved Surface Mesh Polycube Deformation

The problem of polycube construction or deformation is an essential problem in computer graphics. In this paper, we present a robust, simple, efficient and automatic algorithm to deform the meshes of arbitrary shapes into their polycube ones. We derive a clear relationship between a mesh and its corresponding polycube shape. Our algorithm is edge-preserved, and works on surface meshes with or without boundaries. Our algorithm outperforms previous ones in speed, robustness, efficiency. Our method is simple to implement. To demonstrate the robustness and effectiveness of our method, we apply it to hundreds of models of varying complexity and topology. We demonstrat that our method compares favorably to other state-of-the-art polycube deformation methods.

cs.GR↗

arXiv2018

Conformal Mesh Parameterization Using Discrete Calabi Flow

In this paper, we introduce discrete Calabi flow to the graphics research community and present a novel conformal mesh parameterization algorithm. Calabi energy has a succinct and explicit format. Its corresponding flow is conformal and convergent under certain conditions. Our method is based on the Calabi energy and Calabi flow with solid theoretical and mathematical base. We demonstrate our approach on dozens of models and compare it with other related flow based methods, such as the well-known Ricci flow and CETM. Our experiments show that the performance of our algorithm is comparably the same with other methods. The discrete Calabi flow in our method provides another perspective on conformal flow and conformal parameterization.

cs.GR↗

arXiv2018

Classification of lung nodules in CT images based on Wasserstein distance in differential geometry

Lung nodules are commonly detected in screening for patients with a risk for lung cancer. Though the status of large nodules can be easily diagnosed by fine needle biopsy or bronchoscopy, small nodules are often difficult to classify on computed tomography (CT). Recent works have shown that shape analysis of lung nodules can be used to differentiate benign lesions from malignant ones, though existing methods are limited in their sensitivity and specificity. In this work we introduced a new 3D shape analysis within the framework of differential geometry to calculate the Wasserstein distance between benign and malignant lung nodules to derive an accurate classification scheme. The Wasserstein distance between the nodules is calculated based on our new spherical optimal mass transport, this new algorithm works directly on sphere by using spherical metric, which is much more accurate and efficient than previous methods. In the process of deformation, the area-distortion factor gives a probability measure on the unit sphere, which forms the Wasserstein space. From known cases of benign and malignant lung nodules, we can calculate a unique optimal mass transport map between their correspondingly deformed Wasserstein spaces. This transportation cost defines the Wasserstein distance between them and can be used to classify new lung nodules into either the benign or malignant class. To the best of our knowledge, this is the first work that utilizes Wasserstein distance for lung nodule classification. The advantages of Wasserstein distance are it is invariant under rigid motions and scalings, thus it intrinsically measures shape distance even when the underlying shapes are of high complexity, making it well suited to classify lung nodules as they have different sizes, orientations, and appearances.

q-bio.QM↗

arXiv2018

Brenier approach for optimal transportation between a quasi-discrete measure and a discrete measure

Correctly estimating the discrepancy between two data distributions has always been an important task in Machine Learning. Recently, Cuturi proposed the Sinkhorn distance which makes use of an approximate Optimal Transport cost between two distributions as a distance to describe distribution discrepancy. Although it has been successfully adopted in various machine learning applications (e.g. in Natural Language Processing and Computer Vision) since then, the Sinkhorn distance also suffers from two unnegligible limitations. The first one is that the Sinkhorn distance only gives an approximation of the real Wasserstein distance, the second one is the `divide by zero' problem which often occurs during matrix scaling when setting the entropy regularization coefficient to a small value. In this paper, we introduce a new Brenier approach for calculating a more accurate Wasserstein distance between two discrete distributions, this approach successfully avoids the two limitations shown above for Sinkhorn distance and gives an alternative way for estimating distribution discrepancy.

cs.CV↗

arXiv2017

Decentralized Trajectory Tracking Using Homology and Hodge Decomposition in Sensor Networks

With the recent development of localization and tracking systems for both indoor and outdoor settings, we consider the problem of sensing, representing and analyzing human movement trajectories that we expect to gather in the near future. In this paper, we propose to use the topological representation, which records how a target moves around the natural obstacles in the underlying environment. We demonstrate that the topological information can be sufficiently descriptive for many applications and efficient enough for storing, comparing and classifying these natural human trajectories. We pre-process the sensor network with a purely decentralized algorithm such that certain edges are given numerical weights. Then we can perform trajectory classification by simply summing up the edge weights along the trajectory. Our method supports real-time classification of trajectories with minimum communication cost. We test the effectiveness of our approach by showing how to classify randomly generated trajectories in a multi-level arts museum layout as well as how to distinguish real world taxi trajectories in a large city.

cs.NI↗

arXiv2016

Registration of Volumetric Prostate Scans using Curvature Flow

Radiological imaging of the prostate is becoming more popular among researchers and clinicians in searching for diseases, primarily cancer. Scans might be acquired with different equipment or at different times for prognosis monitoring, with patient movement between scans, resulting in multiple datasets that need to be registered. For these cases, we introduce a method for volumetric registration using curvature flow. Multiple prostate datasets are mapped to canonical solid spheres, which are in turn aligned and registered through the use of identified landmarks on or within the gland. Theoretical proof and experimental results show that our method produces homeomorphisms with feature constraints. We provide thorough validation of our method by registering prostate scans of the same patient in different orientations, from different days and using different modes of MRI. Our method also provides the foundation for a general group-wise registration using a standard reference, defined on the complex plane, for any input. In the present context, this can be used for registering as many scans as needed for a single patient or different patients on the basis of age, weight or even malignant and non-malignant attributes to study the differences in general population. Though we present this technique with a specific application to the prostate, it is generally applicable for volumetric registration problems.

cs.GR↗

arXiv2014

Discrete Conformal Deformation: Algorithm and Experiments

In this paper, we introduce a definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, that is to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature. We explicitly construct a discrete conformal map between the input triangulated surface and the deformed triangulated surface. Our algorithm can handle the surface with any topology with or without boundary, and can find a deformed metric for any prescribed curvature satisfying the Gauss-Bonnet formula. In addition, we present the numerical examples to show the convergence of our discrete conformality and to demonstrate the efficiency and the robustness of our algorithm.

cs.CG↗

arXiv2014

Kernel Estimation from Salient Structure for Robust Motion Deblurring

Blind image deblurring algorithms have been improving steadily in the past years. Most state-of-the-art algorithms, however, still cannot perform perfectly in challenging cases, especially in large blur setting. In this paper, we focus on how to estimate a good kernel estimate from a single blurred image based on the image structure. We found that image details caused by blurring could adversely affect the kernel estimation, especially when the blur kernel is large. One effective way to eliminate these details is to apply image denoising model based on the Total Variation (TV). First, we developed a novel method for computing image structures based on TV model, such that the structures undermining the kernel estimation will be removed. Second, to mitigate the possible adverse effect of salient edges and improve the robustness of kernel estimation, we applied a gradient selection method. Third, we proposed a novel kernel estimation method, which is capable of preserving the continuity and sparsity of the kernel and reducing the noises. Finally, we developed an adaptive weighted spatial prior, for the purpose of preserving sharp edges in latent image restoration. The effectiveness of our method is demonstrated by experiments on various kinds of challenging examples.

cs.CV↗

arXiv2014

Simplicial Ricci Flow: An Example of a Neck Pinch Singularity in 3D

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The 2-sphere cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice well-centered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model.

math.DG↗

arXiv2014

The Unified Surface Ricci Flow

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete Surface Ricci Flow, including all common schemes: Thurston's Circle Packing, Tangential Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe. Furthermore, this work also introduces a novel scheme, virtual radius circle packing, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energy for all the schemes, and Hessian of the discrete Ricci energy for schemes with Euclidean back ground geometry. The unified frame work deepen our understanding to the the discrete surface Ricci flow theory, and inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the debugging efficiency. Experimental results shows the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and effective for solving real problems.

math.GT↗

arXiv2014

Computing Teichmüller Maps between Polygons

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichmüller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $ε$ of the dilatation of the extremal map, our method uses $O(1/ε^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.

math.DG↗