Area-Preserving Mappings for the Visualization of Medical Structures

Zhu, Lei; Haker, Steven; Tannenbaum, Allen

doi:10.1007/978-3-540-39903-2_35

Cited by 20 publications

(14 citation statements)

References 12 publications

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“…Zhu et al [65] applied optimal mass transport for flattening blood vessel in an area preserving mapping for medical visualization. Haker et al [25] proposed to use optimal mass transport for image registration and warping, the method is parameter free and has the unique global optimum.…”

Section: Previous Workmentioning

confidence: 99%

Optimal Mass Transport for Shape Matching and Comparison

Wang

Shi

et al. 2015

IEEE Trans. Pattern Anal. Mach. Intell.

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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(n2) 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.

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Section: Previous Workmentioning

confidence: 99%

Optimal Mass Transport for Shape Matching and Comparison

Wang

Shi

et al. 2015

IEEE Trans. Pattern Anal. Mach. Intell.

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“…The area preserving mapping is based on Optimal Mass Transport (OMT) theory, which has been applied for image registration and warping [19,26] and visualization [36]. Our method has fundamental differences from these existing methods.…”

Section: Comparisonmentioning

confidence: 99%

“…Area preserving mapping has been applied for visualizing branched vessels and intestinal tracts in [36], which combined Kantorovich's approach with conformal mapping. Optimal mass transport mapping based on Kantorovich's approach has been applied for image registration in [19].…”

Section: Related Workmentioning

confidence: 99%

Area Preserving Brain Mapping

Zeng

Wang

et al. 2013

2013 IEEE Conference on Computer Vision and Pattern Recognition

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“…In this section, we show how to use the theory of optimal mass transport to construct such an area-preserving mapping from the output of the conformal mapping algorithm. Here, we summarize the procedure; more technical details and a mathematical proof may be found in [9] and [23] and in the Appendix.…”

Section: Area-preserving Mappingsmentioning

confidence: 99%

Flattening maps for the visualization of multibranched vessels

Zhu

Haker

Tannenbaum

2005

IEEE Trans. Med. Imaging

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In this paper, we present two novel algorithms which produce flattened visualizations of branched physiological surfaces, such as vessels. The first approach is a conformal mapping algorithm based on the minimization of two Dirichlet functionals. From a triangulated representation of vessel surfaces, we show how the algorithm can be implemented using a finite element technique. The second method is an algorithm which adjusts the conformal mapping to produce a flattened representation of the original surface while preserving areas. This approach employs the theory of optimal mass transport. Furthermore, a new way of extracting center lines for vessel fly-throughs is provided.

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Area-Preserving Mappings for the Visualization of Medical Structures

Cited by 20 publications

References 12 publications

Optimal Mass Transport for Shape Matching and Comparison

Optimal Mass Transport for Shape Matching and Comparison

Area Preserving Brain Mapping

Flattening maps for the visualization of multibranched vessels

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