# 2D-Shape Analysis Using Conformal Mapping

@inproceedings{Sharon20042DShapeAU, title={2D-Shape Analysis Using Conformal Mapping}, author={Eitan Sharon and David Mumford}, booktitle={CVPR}, year={2004} }

The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this…

## Figures from this paper

## 172 Citations

Shape Analysis of Planar Objects with Arbitrary Topologies Using Conformal Geometry

- Computer ScienceECCV
- 2010

It is proved mathematically that the proposed signature uniquely represents shapes with arbitrary topologies using conformal geometry, and the efficacy of the proposed algorithm as a stable shape representation scheme is shown.

Shape Analysis of Planar Multiply-Connected Objects Using Conformal Welding

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2014

It is theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization, and a reconstruction algorithm to obtain shapes from their signatures is introduced.

Similarity Metric for Curved Shapes in Euclidean Space

- Mathematics2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
- 2016

The proposed method represents a given curve as a point in the deformation space, the direct product of rigid transformation matrices, such that the successive action of the matrices on a fixed starting point reconstructs the full curve.

Registration for 3D surfaces with large deformations using quasi-conformal curvature flow

- MathematicsCVPR 2011
- 2011

A novel method for registering 3D surfaces with large deformations is presented, which is based on quasi-conformal geometry and uniquely determines the registration mapping by solving Beltrami equations using curvature flow.

Two-dimensional shapes and lemniscates

- Mathematics
- 2010

A shape in the plane is an equivalence class of sufficiently smooth Jordan curves, where two curves are equivalent if one can be obtained from the other by a translation and a scaling. The…

Conformal Geometry and Its Applications on 3D Shape Matching, Recognition, and Stitching

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2007

A novel and computationally efficient shape matching framework by using least-squares conformal maps is proposed by comparing the resulting 2D parametric maps, which are stable, insensitive to resolution changes and robust to occlusion, and noise.

Computing Metrics on Riemannian Shape Manifolds: Geometric shape analysis made practical

- Mathematics, Computer Science
- 2009

Shape analysis and recognition is a field ripe with creative solutions and innovative algorithms. We give a quick introduction to several different approaches, before basing our work on a…

Harmonic Beltrami Signature: A Novel 2D Shape Representation for Object Classification

- Mathematics
- 2021

There is a one-to-one correspondence between the quotient space of HBS and the space of 2D simply-connected shapes up to a translation, rotation and scaling and the HBS is thus an effective fingerprint to represent a 2D shape.

Optimal Mass Transport for Shape Matching and Comparison

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2015

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, and is validated by numerous experiments and comparisons with prior approaches.

Shape analysis in shape space

- Mathematics
- 2012

This study aims to classify different deformations based on the shape space concept. A shape space is a quotient space in which each point corresponds to a class of shapes. The shapes of each class…

## References

SHOWING 1-10 OF 18 REFERENCES

Recognition of Shapes by Editing Shock Graphs

- Computer ScienceICCV
- 2001

The effectiveness of the proposed technique in the presence of a variety of visual transformations including occlusion, articulation and deformation of parts, shadow and highlights, viewpoint variation, and boundary perturbations is demonstrated.

Order Structure, Correspondence, and Shape Based Categories

- MathematicsShape, Contour and Grouping in Computer Vision
- 1999

A general method for finding pointwise correspondence between 2-D shapes based on the concept of order structure and using geometric hashing is proposed, which can be defined for arbitrary geometric configurations such as points lines and curves.

Flexible Syntactic Matching of Curves and Its Application to Automatic Hierarchical Classification of Silhouettes

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 1999

This paper presents extensive experiments where the flexible algorithm to match curves under substantial deformations and arbitrary large scaling and rigid transformations, and defines a dissimilarity measure which is used in order to organize the image database into shape categories.

Modal Matching for Correspondence and Recognition

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 1995

Improved formulation of modal matching utilizes a new type of finite element formulation that allows for an object's eigenmodes to be computed directly from available image information, and is applicable to data of any dimensionality.

Structural Image Restoration through Deformable Templates

- Mathematics
- 1991

Abstract Prior knowledge on the space of possible images is given in the form of a function or template in some domain. The set of all possible true images is assumed to be formed by a composition of…

Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping

- Computer ScienceTOMS
- 1996

The Schwarz- Christoffel Toolbox for MATLAB is a new implementation of Schwarz-Christoffel formulas for maps from the disk, half-plane, strip, and rectangle domains to polygon interiors, and from thedisk to polyagon exteriors.

Mathematical theories of shape: do they model perception?

- Computer ScienceOptics & Photonics
- 1991

The mathematics of shape has a long history in the fields of differential geometry and topology. But does this theory of shape address the central problem of vision: finding the best data structure…

Human image understanding: Recent research and a theory

- Computer ScienceComput. Vis. Graph. Image Process.
- 1985

Dynamic Programming for Detecting, Tracking, and Matching Deformable Contours

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 1995

The information provided by the user's selected points is explored and an optimal method to detect contours which allows a segmentation of the image is applied, based on dynamic programming (DP), and applies to a wide variety of shapes.