Xianfeng Gu

Na Lei, Zezeng Li, Zebin Xu et al.

Intelligent Mesh Generation (IMG) represents a novel and promising field of research, utilizing machine learning techniques to generate meshes. Despite its relative infancy, IMG has significantly broadened the adaptability and practicality of mesh generation techniques, delivering numerous breakthroughs and unveiling potential future pathways. However, a noticeable void exists in the contemporary literature concerning comprehensive surveys of IMG methods. This paper endeavors to fill this gap by providing a systematic and thorough survey of the current IMG landscape. With a focus on 113 preliminary IMG methods, we undertake a meticulous analysis from various angles, encompassing core algorithm techniques and their application scope, agent learning objectives, data types, targeted challenges, as well as advantages and limitations. We have curated and categorized the literature, proposing three unique taxonomies based on key techniques, output mesh unit elements, and relevant input data types. This paper also underscores several promising future research directions and challenges in IMG. To augment reader accessibility, a dedicated IMG project page is available at \url{https://github.com/xzb030/IMG_Survey}.

Zezeng Li, ShengHao Li, Zhanpeng Wang et al.

Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent progress in designing fast samplers for DPMs achieves a trade-off between sampling speed and sample quality by knowledge distillation or adjusting the variance schedule or the denoising equation. However, it can't be optimal in both aspects and often suffer from mode mixture in short steps. To tackle this problem, we innovatively regard inverse diffusion as an optimal transport (OT) problem between latents at different stages and propose the DPM-OT, a unified learning framework for fast DPMs with a direct expressway represented by OT map, which can generate high-quality samples within around 10 function evaluations. By calculating the semi-discrete optimal transport map between the data latents and the white noise, we obtain an expressway from the prior distribution to the data distribution, while significantly alleviating the problem of mode mixture. In addition, we give the error bound of the proposed method, which theoretically guarantees the stability of the algorithm. Extensive experiments validate the effectiveness and advantages of DPM-OT in terms of speed and quality (FID and mode mixture), thus representing an efficient solution for generative modeling. Source codes are available at https://github.com/cognaclee/DPM-OT

Pengfei Liu, Wenjin Deng, Hengda Li et al.

It is still an interesting and challenging problem to synthesize a vivid and realistic singing face driven by music signal. In this paper, we present a method for this task with natural motions of the lip, facial expression, head pose, and eye states. Due to the coupling of the mixed information of human voice and background music in common signals of music audio, we design a decouple-and-fuse strategy to tackle the challenge. We first decompose the input music audio into human voice stream and background music stream. Due to the implicit and complicated correlation between the two-stream input signals and the dynamics of the facial expressions, head motions and eye states, we model their relationship with an attention scheme, where the effects of the two streams are fused seamlessly. Furthermore, to improve the expressiveness of the generated results, we propose to decompose head movements generation into speed generation and direction generation, and decompose eye states generation into the short-time eye blinking generation and the long-time eye closing generation to model them separately. We also build a novel SingingFace Dataset to support the training and evaluation of this task, and to facilitate future works on this topic. Extensive experiments and user study show that our proposed method is capable of synthesizing vivid singing face, which is better than state-of-the-art methods qualitatively and quantitatively.

Alakh Aggarwal, Jikai Wang, Steven Hogue et al.

Recent research works have focused on generating human models and garments from their 2D images. However, state-of-the-art researches focus either on only a single layer of the garment on a human model or on generating multiple garment layers without any guarantee of the intersection-free geometric relationship between them. In reality, people wear multiple layers of garments in their daily life, where an inner layer of garment could be partially covered by an outer one. In this paper, we try to address this multi-layer modeling problem and propose the Layered-Garment Net (LGN) that is capable of generating intersection-free multiple layers of garments defined by implicit function fields over the body surface, given the person's near front-view image. With a special design of garment indication fields (GIF), we can enforce an implicit covering relationship between the signed distance fields (SDF) of different layers to avoid self-intersections among different garment surfaces and the human body. Experiments demonstrate the strength of our proposed LGN framework in generating multi-layer garments as compared to state-of-the-art methods. To the best of our knowledge, LGN is the first research work to generate intersection-free multiple layers of garments on the human body from a single image.

Yuxue Ren, Baowei Jiang, Wei Chen et al.

We present a novel, effective method for global point cloud registration problems by geometric topology. Based on many point cloud pairwise registration methods (e.g ICP), we focus on the problem of accumulated error for the composition of transformations along any loops. The major technical contribution of this paper is a linear method for the elimination of errors, using only solving a Poisson equation. We demonstrate the consistency of our method from Hodge-Helmhotz decomposition theorem and experiments on multiple RGBD datasets of real-world scenes. The experimental results also demonstrate that our global registration method runs quickly and provides accurate reconstructions.

Zezeng Li, Weimin Wang, Yuming Zhao et al.

Recent CLIP-guided 3D generation methods have achieved promising results but struggle with generating faithful 3D shapes that conform with input text due to the gap between text and image embeddings. To this end, this paper proposes HOTS3D which makes the first attempt to effectively bridge this gap by aligning text features to the image features with spherical optimal transport(SOT). However, in high-dimensional situations, solving the SOT remains a challenge. To obtain the SOT map for high-dimensional features obtained from CLIP encoding of two modalities, we mathematically formulate and derive the solution based on Villani's theorem, which can directly align two hyper-sphere distributions without manifold exponential maps. Furthermore, we implement it by leveraging input convex neural networks (ICNNs) for the optimal Kantorovich potential. With the optimally mapped features, a diffusion-based generator is utilized to decode them into 3D shapes. Extensive quantitative and qualitative comparisons with state-of-the-art methods demonstrate the superiority of HOTS3D for text-to-3D generation, especially in the consistency with text semantics.

Rui Xu, Longdu Liu, Ningna Wang et al.

In mesh simplification, common requirements like accuracy, triangle quality, and feature alignment are often considered as a trade-off. Existing algorithms concentrate on just one or a few specific aspects of these requirements. For example, the well-known Quadric Error Metrics (QEM) approach prioritizes accuracy and can preserve strong feature lines/points as well but falls short in ensuring high triangle quality and may degrade weak features that are not as distinctive as strong ones. In this paper, we propose a smooth functional that simultaneously considers all of these requirements. The functional comprises a normal anisotropy term and a Centroidal Voronoi Tessellation (CVT) energy term, with the variables being a set of movable points lying on the surface. The former inherits the spirit of QEM but operates in a continuous setting, while the latter encourages even point distribution, allowing various surface metrics. We further introduce a decaying weight to automatically balance the two terms. We selected 100 CAD models from the ABC dataset, along with 21 organic models, to compare the existing mesh simplification algorithms with ours. Experimental results reveal an important observation: the introduction of a decaying weight effectively reduces the conflict between the two terms and enables the alignment of weak features. This distinctive feature sets our approach apart from most existing mesh simplification methods and demonstrates significant potential in shape understanding.

Weixing Xie, Xiao Dong, Yong Yang et al.

With the popularity of monocular videos generated by video sharing and live broadcasting applications, reconstructing and editing dynamic scenes in stationary monocular cameras has become a special but anticipated technology. In contrast to scene reconstructions that exploit multi-view observations, the problem of modeling a dynamic scene from a single view is significantly more under-constrained and ill-posed. Inspired by recent progress in neural rendering, we present a novel framework to tackle 4D decomposition problem for dynamic scenes in monocular cameras. Our framework utilizes decomposed static and dynamic feature planes to represent 4D scenes and emphasizes the learning of dynamic regions through dense ray casting. Inadequate 3D clues from a single-view and occlusion are also particular challenges in scene reconstruction. To overcome these difficulties, we propose deep supervised optimization and ray casting strategies. With experiments on various videos, our method generates higher-fidelity results than existing methods for single-view dynamic scene representation.

David Gu, Peter Belcak, Roger Wattenhofer

We challenge the prevailing assumption that LLMs must rely fully on sub-word tokens for high-quality text generation. To this end, we propose the "Generative Pretrained Thoughtformer" (GPTHF), a hierarchical transformer language model capable of text generation by compressing text into sentence embeddings and employing a sentence attention mechanism. GPTHF retains GPT's architecture, modifying only token interactions via dynamic sparse attention masks. Our experiments show that GPTHF achieves an up to an order of magnitude improvement in FLOPs efficiency and a threefold increase in runtime speed compared to equally-sized GPT models in the low-size regime. This is achieved through a unique generation method that caches and reuses sentence embeddings, allowing significant portions of the input to bypass large parts of the network.

Daoxi Cao, Hangbei Cheng, Yijin Li et al.

Whole-slide images (WSIs) are critical for cancer diagnosis due to their ultra-high resolution and rich semantic content. However, their massive size and the limited availability of fine-grained annotations pose substantial challenges for conventional supervised learning. We propose DSAGL (Dual-Stream Attention-Guided Learning), a novel weakly supervised classification framework that combines a teacher-student architecture with a dual-stream design. DSAGL explicitly addresses instance-level ambiguity and bag-level semantic consistency by generating multi-scale attention-based pseudo labels and guiding instance-level learning. A shared lightweight encoder (VSSMamba) enables efficient long-range dependency modeling, while a fusion-attentive module (FASA) enhances focus on sparse but diagnostically relevant regions. We further introduce a hybrid loss to enforce mutual consistency between the two streams. Experiments on CIFAR-10, NCT-CRC, and TCGA-Lung datasets demonstrate that DSAGL consistently outperforms state-of-the-art MIL baselines, achieving superior discriminative performance and robustness under weak supervision.

Xinmu Wang, Xiang Gao, Xiyun Song et al.

Animating 3D head meshes using audio inputs has significant applications in AR/VR, gaming, and entertainment through 3D avatars. However, bridging the modality gap between speech signals and facial dynamics remains a challenge, often resulting in incorrect lip syncing and unnatural facial movements. To address this, we propose OT-Talk, the first approach to leverage optimal transportation to optimize the learning model in talking head animation. Building on existing learning frameworks, we utilize a pre-trained Hubert model to extract audio features and a transformer model to process temporal sequences. Unlike previous methods that focus solely on vertex coordinates or displacements, we introduce Chebyshev Graph Convolution to extract geometric features from triangulated meshes. To measure mesh dissimilarities, we go beyond traditional mesh reconstruction errors and velocity differences between adjacent frames. Instead, we represent meshes as probability measures and approximate their surfaces. This allows us to leverage the sliced Wasserstein distance for modeling mesh variations. This approach facilitates the learning of smooth and accurate facial motions, resulting in coherent and natural facial animations. Our experiments on two public audio-mesh datasets demonstrate that our method outperforms state-of-the-art techniques both quantitatively and qualitatively in terms of mesh reconstruction accuracy and temporal alignment. In addition, we conducted a user perception study with 20 volunteers to further assess the effectiveness of our approach.

Wei Chen, Yuxue Ren, Na Lei et al.

Texture mapping is a common technology in the area of computer graphics, it maps the 3D surface space onto the 2D texture space. However, the loose texture space will reduce the efficiency of data storage and GPU memory addressing in the rendering process. Many of the existing methods focus on repacking given textures, but they still suffer from high computational cost and hardly produce a wholly tight texture space. In this paper, we propose a method to optimize the texture space and produce a new texture mapping which is compact based on global parameterization. The proposed method is computationally robust and efficient. Experiments show the effectiveness of the proposed method and the potency in improving the storage and rendering efficiency.

Hongwei Zhang, Xiaoyin Xu, Dongsheng An et al.

Backdoor attacks become a significant security concern for deep neural networks in recent years. An image classification model can be compromised if malicious backdoors are injected into it. This corruption will cause the model to function normally on clean images but predict a specific target label when triggers are present. Previous research can be categorized into two genres: poisoning a portion of the dataset with triggered images for users to train the model from scratch, or training a backdoored model alongside a triggered image generator. Both approaches require significant amount of attackable parameters for optimization to establish a connection between the trigger and the target label, which may raise suspicions as more people become aware of the existence of backdoor attacks. In this paper, we propose a backdoor attack paradigm that only requires minimal alterations (specifically, the output layer) to a clean model in order to inject the backdoor under the guise of fine-tuning. To achieve this, we leverage mode mixture samples, which are located between different modes in latent space, and introduce a novel method for conducting backdoor attacks. We evaluate the effectiveness of our method on four popular benchmark datasets: MNIST, CIFAR-10, GTSRB, and TinyImageNet.

Yingjie Feng, Jun Wang, Xianfeng Gu et al.

In diagnosing challenging conditions such as Alzheimer's disease (AD), imaging is an important reference. Non-imaging patient data such as patient information, genetic data, medication information, cognitive and memory tests also play a very important role in diagnosis. Effect. However, limited by the ability of artificial intelligence models to mine such information, most of the existing models only use multi-modal image data, and cannot make full use of non-image data. We use a currently very popular pre-trained large language model (LLM) to enhance the model's ability to utilize non-image data, and achieved SOTA results on the ADNI dataset.

Dongsheng An, Na Lei, Xianfeng Gu

Optimal transport (OT) plays an essential role in various areas like machine learning and deep learning. However, computing discrete optimal transport plan for large scale problems with adequate accuracy and efficiency is still highly challenging. Recently, methods based on the Sinkhorn algorithm add an entropy regularizer to the prime problem and get a trade off between efficiency and accuracy. In this paper, we propose a novel algorithm to further improve the efficiency and accuracy based on Nesterov's smoothing technique. Basically, the non-smooth c-transform of the Kantorovich potential is approximated by the smooth Log-Sum-Exp function, which finally smooths the original non-smooth Kantorovich dual functional (energy). The smooth Kantorovich functional can be optimized by the fast proximal gradient algorithm (FISTA) efficiently. Theoretically, the computational complexity of the proposed method is given by $O(n^{\frac{5}{2}} \sqrt{\log n} /ε)$, which is lower than that of the Sinkhorn algorithm. Empirically, compared with the Sinkhorn algorithm, our experimental results demonstrate that the proposed method achieves faster convergence and better accuracy with the same parameter.

Dongsheng An, Yang Guo, Min Zhang et al.

Though generative adversarial networks (GANs) areprominent models to generate realistic and crisp images,they often encounter the mode collapse problems and arehard to train, which comes from approximating the intrinsicdiscontinuous distribution transform map with continuousDNNs. The recently proposed AE-OT model addresses thisproblem by explicitly computing the discontinuous distribu-tion transform map through solving a semi-discrete optimaltransport (OT) map in the latent space of the autoencoder.However the generated images are blurry. In this paper, wepropose the AE-OT-GAN model to utilize the advantages ofthe both models: generate high quality images and at thesame time overcome the mode collapse/mixture problems.Specifically, we first faithfully embed the low dimensionalimage manifold into the latent space by training an autoen-coder (AE). Then we compute the optimal transport (OT)map that pushes forward the uniform distribution to the la-tent distribution supported on the latent manifold. Finally,our GAN model is trained to generate high quality imagesfrom the latent distribution, the distribution transform mapfrom which to the empirical data distribution will be con-tinuous. The paired data between the latent code and thereal images gives us further constriction about the generator.Experiments on simple MNIST dataset and complex datasetslike Cifar-10 and CelebA show the efficacy and efficiency ofour proposed method.

Na Lei, Yang Guo, Dongsheng An et al.

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.

Saad Nadeem, Joseph Marino, Xianfeng Gu et al.

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.

Saad Nadeem, Xianfeng Gu, Arie Kaufman

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.

Huidong Liu, Yang Guo, Na Lei et al.

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.

Min Zhang, Qianli Ma, Chengfeng Wen et al.

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.

Liang Mi, Wen Zhang, Xianfeng Gu et al.

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.

Na Lei, Zhongxuan Luo, Shing-Tung Yau et al.

Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great successes. Unfortunately, the understanding on how it works remains unclear. It has the central importance to lay down the theoretic foundation for deep learning. In this work, we give a geometric view to understand deep learning: we show that the fundamental principle attributing to the success is the manifold structure in data, namely natural high dimensional data concentrates close to a low-dimensional manifold, deep learning learns the manifold and the probability distribution on it. We further introduce the concepts of rectified linear complexity for deep neural network measuring its learning capability, rectified linear complexity of an embedding manifold describing the difficulty to be learned. Then we show for any deep neural network with fixed architecture, there exists a manifold that cannot be learned by the network. Finally, we propose to apply optimal mass transportation theory to control the probability distribution in the latent space.

Na Lei, Kehua Su, Li Cui et al.

In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models. By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Preliminary experimental results show the geometric method outperforms WGAN for approximating probability measures with multiple clusters in low dimensional space.

Yu-Yao Lin, Chien-Chun Ni, Na Lei et al.

Robot Coverage Path planning (i.e., provide full coverage of a given domain by one or multiple robots) is a classical problem in the field of robotics and motion planning. The goal is to provide nearly full coverage while also minimize duplicately visited area. In this paper we focus on the scenario of path planning on general surfaces including planar domains with complex topology, complex terrain or general surface in 3D space. The main idea is to adopt a natural, intrinsic and global parametrization of the surface for robot path planning, namely the holomorphic quadratic differentials. Except for a small number of zero points (singularities), each point on the surface is given a uv-coordinates naturally represented by a complex number. We show that natural, efficient robot paths can be obtained by using such coordinate systems. The method is based on intrinsic geometry and thus can be adapted to general surface exploration in 3D.

Lok Ming Lui, Ka Chun Lam, Tsz Wai Wong et al.

Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed using the Linear Beltrami Solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage memory can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we proposed to apply the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improve the compression ratio.