Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture industries. Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. This work conquers this problem by changing the Riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. The computational algorithms are based on Ricci flow and nonlinear heat diffusion methods. The approach is general and robust. We employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards.

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.

Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be preloaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. The paper describes algorithmic innovations as well as implementations on a real testbed.

Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration and shape space. The computation of the optimal mass transport map is based on Monge-Brenier theory, in comparison to the conventional method based on Monge-Kantorovich theory, this method significantly improves the efficiency by reducing computational complexity from O(n(2)) to O(n) . For surface registration problem, one commonly used approach is to use conformal map to convert the shapes into some canonical space. Although conformal mappings have small angle distortions, they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis. This work proposes to compose the conformal map with the optimal mass transport map to get the unique area-preserving map, which is intrinsic to the Riemannian metric, unique, and diffeomorphic. For shape space study, this work introduces a novel Riemannian framework, Conformal Wasserstein Shape Space, by combing conformal geometry and optimal mass transport theory. In our work, all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping, which pushes the area element on the surface to a probability measure on the disk. The optimal mass transport provides a map from the shape space of all topological disks with metrics to the Wasserstein space of the disk and the pullback Wasserstein metric equips the shape space with a Riemannian metric. We validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach.

This work proposes a novel framework for optimization in the constrained diffeomorphism space for deformable surface registration. First the diffeomorphism space is modeled as a special complex functional space on the source surface, the Beltrami coefficient space. The physically plausible constraints, in terms of feature landmarks and deformation types, define subspaces in the Beltrami coefficient space. Then the harmonic energy of the registration is minimized in the constrained subspaces. The minimization is achieved by alternating two steps: 1) optimization - diffuse the Beltrami coefficient, and 2) projection - first deform the conformal structure by the current Beltrami coefficient and then compose with a harmonic map from the deformed conformal structure to the target. The registration result is diffeomorphic, satisfies the physical landmark and deformation constraints, and minimizes the conformality distortion. Experiments on human facial surfaces demonstrate the efficiency and efficacy of the proposed registration framework.

Shape analysis is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial but difficult. It usually involves an efficient representation of 2D shape space with a metric, so that its mathematical structure can be used for further analysis. Although the study of 2D simply-connected shapes has been subject to a corpus of literatures, the analysis of multiply-connected shapes is comparatively less studied. In this work, we propose a representation for general 2D multiply-connected domains with arbitrary topologies using conformal welding. A metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains (unit disk with several inner disks removed), using holomorphic 1-forms. A set of diffeomorphisms of the unit circle S(1) can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme.

Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture. Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. This work conquer this problem by changing the Riemannian metric on the target surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. The computational algorithms are based on the Ricci flow method and the method is general and robust. We apply our algorithm to study constrained human brain surface registration problem. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are always diffeomorphic, and achieve relative high performance when evaluated with some popular cortical surface registration evaluation standards.

Brain mapping transforms the brain cortical surface to canonical planar domains, which plays a fundamental role in morphological study. Most existing brain mapping methods are based on angle preserving maps, which may introduce large area distortions. This work proposes an area preserving brain mapping method based on Monge-Brenier theory. The brain mapping is intrinsic to the Riemannian metric, unique, and diffeomorphic. The computation is equivalent to convex energy minimization and power Voronoi diagram construction. Comparing to the existing approaches based on Monge-Kantorovich theory, the proposed one greatly reduces the complexity (from n2unknowns to n ), and improves the simplicity and efficiency. Experimental results on caudate nucleus surface mapping and cortical surface mapping demonstrate the efficacy and efficiency of the proposed method. Conventional methods for caudate nucleus surface mapping may suffer from numerical instability, in contrast, current method produces diffeomorpic mappings stably. In the study of cortical surface classification for recognition of Alzheimer's Disease, the proposed method outperforms some other morphometry features.

Surface registration plays a fundamental role in many applications in computer vision and aims at finding a one-to-one correspondence between surfaces. Conformal mapping based surface registration methods conformally map 2D/3D surfaces onto 2D canonical domains and perform the matching on the 2D plane. This registration framework reduces dimensionality, and the result is intrinsic to Riemannian metric and invariant under isometric deformation. However, conformal mapping will be affected by inconsistent boundaries and non-isometric deformations of surfaces. In this work, we quantify the effects of boundary variation and non-isometric deformation to conformal mappings, and give the theoretical upper bounds for the distortions of conformal mappings under these two factors. Besides giving the thorough theoretical proofs of the theorems, we verified them by concrete experiments using 3D human facial scans with dynamic expressions and varying boundaries. Furthermore, we used the distortion estimates for reducing search range in feature matching of surface registration applications. The experimental results are consistent with the theoretical predictions and also demonstrate the performance improvements in feature tracking.

In this paper we propose an algorithm to construct a “space filling” curve for a sensor network with holes. Mathematically, for a given multi-hole domain R, we generate a path P that is provably aperiodic (i.e., any point is covered at most a constant number of times) and dense (i.e., any point of R is arbitrarily close to P). In a discrete setting as in a sensor network, the path visits the nodes with progressive density, which can adapt to the budget of the path length. Given a higher budget, the path covers the network with higher density. With a lower budget the path becomes proportional sparser. We show how this density-adaptive space filling curve can be useful for applications such as serial data fusion, motion planning for data mules, sensor node indexing, and double ruling type in-network data storage and retrieval. We show by simulation results the superior performance of using our algorithm vs standard space filling curves and random walks.

Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, location errors and link dynamics, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We follow the approach taken by Sarkar et. al. [24] to find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our new contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be pre-loaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. Being extremely light-weight, greedy routing on the virtual coordinates is shown to be very robust to mobility, link dynamics and non-unit disk graph connectivity models.

We propose a new colon flattening algorithm that is efficient, shape-preserving, and robust to topological noise. Unlike previous approaches, which require a mandatory topological denoising to remove fake handles, our algorithm directly flattens the colon surface without any denoising. In our method, we replace the original Euclidean metric of the colon surface with a heat diffusion metric that is insensitive to topological noise. Using this heat diffusion metric, we then solve a Laplacian equation followed by an integration step to compute the final flattening. We demonstrate that our method is shape-preserving and the shape of the polyps are well preserved. The flattened colon also provides an efficient way to enhance the navigation and inspection in virtual colonoscopy. We further show how the existing colon registration pipeline is made more robust by using our colon flattening. We have tested our method on several colon wall surfaces and the experimental results demonstrate the robustness and the efficiency of our method.

We present the conformal magnifier, a novel interactive focus+context visualization technique that magnifies a region of interest (ROI) using conformal mapping. Our framework supports the arbitrary shape design of magnifiers for the user to enlarge the ROI while globally deforming the context region without any cropping. By using the mathematically well-defined conformal mapping theory and algorithm, the ROI is magnified with local shape preservation (angle distortion minimization), while the transition area between the focus and context regions is deformed smoothly and continuously. After the selection of a specified magnifier shape, our system can automatically magnify the ROI in real time with full resolution even for large volumetric data sets. These properties are important for many visualization applications, especially for the computer aided detection and diagnosis (CAD). Our framework is suitable for diverse applications, including the map visualization, and volumetric visualization. Experimental results demonstrate the effectiveness, robustness, and efficiency of our framework.

A novel method for registering 3D surfaces with large deformations is presented, which is based on quasi-conformal geometry. A general diffeomorphism distorts the conformal structure of the surface, which is represented as the Beltrami coefficient. Inversely, the diffeomorphism can be determined by the Beltrami coefficient in an essentially unique way. Our registration method first extracts the features on the surfaces, then estimates the Beltrami coefficient, and finally uniquely determines the registration mapping by solving Beltrami equations using curvature flow. The method is 1) general, it can search the desired registration in the whole space of diffeomorphisms, which includes the conventional searching spaces, such as rigid motions, isometric transformations or conformal mappings; 2) global optimal, the global optimum is determined by the method unique up to a 3 dimensional transformation group; 3) robust, it handles large surfaces with complicated topologies; 4) rigorous, it has solid theoretic foundation. Experiments on the real surfaces with large deformations and complicated topologies demonstrate the efficiency, robustness of the proposed method.

In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the Koebe-Andreev-Thurston Theorem for any 3-connected planar graphs. In our paper we use discrete Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differs from one another by a Möbius transformation. We employ an optimization routine to look for the Möbius transformation such that the nodes are spread on the polyhedron as uniformly as possible. We evaluated the load balancing property of this greedy routing scheme and showed favorable comparison with previous schemes.

When visualizing tubular 3D structures, external representations are often used for guidance and display, and such views in 2D can often contain occlusions. Virtual dissection methods have been proposed where the entire 3D structure can be mapped to the 2D plane, though these will lose context by straightening curved sections. We present a new method of creating maps of 3D tubular structures that yield a succinct view while preserving the overall geometric structure. Given a dominant view plane for the structure, its curve skeleton is first projected to a 2D skeleton. This 2D skeleton is adjusted to account for distortions in length, modified to remove intersections, and optimized to preserve the shape of the original 3D skeleton. Based on this shaped 2D skeleton, a boundary for the map of the object is obtained based on a slicing path through the structure and the radius around the skeleton. The sliced structure is conformally mapped to a rectangle and then deformed via harmonic mapping to match the boundary placement. This flattened map preserves the general geometric context of a 3D object in a 2D display, and rendering of this flattened map can be accomplished using volumetric ray casting. We have evaluated our method on real datasets of human colon models.

In this paper, we introduce a new approach for partial 3D face recognition, which makes use of shape decomposition over the rigid part of a face. To explore the descriptiveness of shape dissimilarity over an isometric part of a face, which has lower probability to be influenced by expression, we transform a 3D shape to a 2D domain using conformal mapping and use shape decomposition as a similarity measurement. In our work we investigate several classifiers as well as several shape descriptors for recognition purposes. Recognition tests on a subset of the FRGC data set show approximately 80% rank-one recognition rate using only the eyes and nose part of the face.

We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms.

For in-network storage schemes, one maps data, indexed in a logical space, to the distributed sensor locations. When the physical sensor network has an irregular shape and possibly holes, the mapping of data to sensors often creates unbalanced storage load with high data concentration on nodes near network boundaries. In this paper we propose to map data to a covering space, which is a tiling of the plane with copies of the sensor network, such that the sensors receive uniform storage load and traffic. We propose distributed algorithms to construct the covering space with Ricci flow and Möbius transforms. The use of the covering space improves the performance of many in-network storage and retrieval schemes such as geographical hash tables (GHTs) or the double rulings (quorum based schemes), and provides better load balanced routing.

Ricci flow is a powerful curvature flow method, which is invariant to rigid motion, scaling, isometric, and conformal deformations. We present the first application of surface Ricci flow in computer vision. Previous methods based on conformal geometry, which only handle 3D shapes with simple topology, are subsumed by the Ricci flow-based method, which handles surfaces with arbitrary topology. We present a general framework for the computation of Ricci flow, which can design any Riemannian metric by user-defined curvature. The solution to Ricci flow is unique and robust to noise. We provide implementation details for Ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry. Our Ricci flow-based method can convert all 3D problems into 2D domains and offers a general framework for 3D shape analysis. We demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration, and shape indexing. Surfaces with large nonrigid anisotropic deformations can be registered using Ricci flow with constraints of feature points and curves. We show how conformal equivalence can be used to index shapes in a 3D surface shape space with the use of Teichmüller space coordinates. Experimental results are shown on 3D face data sets with large expression deformations and on dynamic heart data.

In virtual colonoscopy, CT scans are typically acquired with the patient in both supine (facing up) and prone (facing down) positions. The registration of these two scans is desirable so that the user can clarify situations or confirm polyp findings at a location in one scan with the same location in the other, thereby improving polyp detection rates and reducing false positives. However, this supine-prone registration is challenging because of the substantial distortions in the colon shape due to the patient's change in position. We present an efficient algorithm and framework for performing this registration through the use of conformal geometry to guarantee that the registration is a diffeomorphism (a one-to-one and onto mapping). The taeniae coli and colon flexures are automatically extracted for each supine and prone surface, employing the colon geometry. The two colon surfaces are then divided into several segments using the flexures, and each segment is cut along a taenia coli and conformally flattened to the rectangular domain using holomorphic differentials. The mean curvature is color encoded as texture images, from which feature points are automatically detected using graph cut segmentation, mathematic morphological operations, and principal component analysis. Corresponding feature points are found between supine and prone and are used to adjust the conformal flattening to be quasi-conformal, such that the features become aligned. We present multiple methods of visualizing our results, including 2D flattened rendering, corresponding 3D endoluminal views, and rendering of distortion measurements. We demonstrate the efficiency and efficacy of our registration method by illustrating matched views on both the 2D flattened colon images and in the 3D volume rendered colon endoluminal view. We analytically evaluate the correctness of the results by measuring the distance between features on the registered colons.

Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.

Surface parameterization refers to the process of mapping the surface to canonical planar domains, which plays crucial roles in texture mapping and shape analysis purposes. Most existing techniques focus on simply connected surfaces. It is a challenging problem for multiply connected genus zero surfaces. This work generalizes conventional Koebe's method for multiply connected planar domains. According to Koebe's uniformization theory, all genus zero multiply connected surfaces can be mapped to a planar disk with multiply circular holes. Furthermore, this kind of mappings are angle preserving and differ by Möbius transformations. We introduce a practical algorithm to explicitly construct such a circular conformal mapping. Our algorithm pipeline is as follows: suppose the input surface has n boundaries, first we choose 2 boundaries, and fill the other n -- 2 boundaries to get a topological annulus; then we apply discrete Yamabe flow method to conformally map the topological annulus to a planar annulus; then we remove the filled patches to get a planar multiply connected domain. We repeat this step for the planar domain iteratively. The two chosen boundaries differ from step to step. The iterative construction leads to the desired conformal mapping, such that all the boundaries are mapped to circles. In theory, this method converges quadratically faster than conventional Koebe's method. We give theoretic proof and estimation for the converging rate. In practice, it is much more robust and efficient than conventional non-linear methods based on curvature flow. Experimental results demonstrate the robustness and efficiency of the method.

Teichmuller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmuller space, only if they can be conformally mapped to each other. Teichmuller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichmuller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to decompose the surface to pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichmuller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmuller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the potential value of the method for geometric database indexing, shape comparison and classification.

Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an infinite number of loops. Finding a canonical representative within a homotopy class will simplify many computational tasks in computational topology, such as loop homotopy detection, pants decomposition. Furthermore, the canonical representative can be used as the shape descriptor. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class. The main strategy is to use hyperbolic structure, such that each homotopy class has a unique closed geodesic, which is the representative. The following is the algorithm pipeline: for a given surface with negative Euler number, we apply hyperbolic Yamabe curvature flow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. Theories and algorithms are explained thoroughly in details. Experimental results are reported to show the efficiency and efficacy of the algorithm. The unique homotopy class representative can be applied for homotopy detection and shape comparison.

Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmuller theory. The Teichmuller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, a curve represents a deformation process from one class to the other. We apply Teichmuller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic; invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmuller coordinates is practical, stable and efficient. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with -1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmuller space.