skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Cortical Morphometry Analysis based on Worst Transportation Theory
Biomarkers play an important role in early detection and intervention in Alzheimer’s disease (AD). However, obtaining effective biomarkers for AD is still a big challenge. In this work, we propose to use the worst transportation cost as a univariate biomarker to index cortical morphometry for tracking AD progression. The worst transportation (WT) aims to find the least economical way to transport one measure to the other, which contrasts to the optimal transportation (OT) that finds the most economical way between measures. To compute the WT cost, we generalize the Brenier theorem for the OT map to the WT map, and show that the WT map is the gradient of a concave function satisfying the Monge-Ampere equation. We also develop an efficient algorithm to compute the WT map based on computational geometry. We apply the algorithm to analyze cortical shape difference between dementia due to AD and normal aging individuals. The experimental results reveal the effectiveness of our proposed method which yields better statistical performance than other competiting methods including the OT.  more » « less
Award ID(s):
1762287
PAR ID:
10291690
Author(s) / Creator(s):
; ; ; ; ; ; ;
Date Published:
Journal Name:
IPMI 2021: Information Processing in Medical Imaging
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Proceedings of International Conference on Computer Vision (ICCV))
    null (Ed.)
    An optimal transportation map finds the most economical way to transport one probability measure to the other. It has been applied in a broad range of applications in vision, deep learning and medical images. By Brenier theory, computing the optimal transport map is equivalent to solving a Monge-Amp\`ere equation. Due to the highly non-linear nature, the computation of optimal transportation maps in large scale is very challenging. This work proposes a simple but powerful method, the FFT-OT algorithm, to tackle this difficulty based on three key ideas. First, solving Monge-Amp\`ere equation is converted to a fixed point problem; Second, the obliqueness property of optimal transportation maps are reformulated as Neumann boundary conditions on rectangular domains; Third, FFT is applied in each iteration to solve a Poisson equation in order to improve the efficiency. Experiments on surfaces captured from 3D scanning and reconstructed from medical imaging are conducted, and compared with other existing methods. Our experimental results show that the proposed FFT-OT algorithm is simple, general and scalable with high efficiency and accuracy. 
    more » « less
  2. ; ; ; ; ; ; (, Proceedings of International Meshing Roundtable)
    null (Ed.)
    Optimal transportation (OT) finds the most economical way to transport one measure to another and plays an important role in geometric modeling and processing. Based on the Brenier theorem, the OT problem is equivalent to the Alexandrov problem, which is the dual to the Pogorelov problem. Although solving the Alexandrov/Pogorelov problem are both equivalent to solving the Monge-Amp\`{e}re equation, the former requires second type boundary condition and the latter requires much simpler Dirichlet boundary condition. Hence, we propose to use the Pogorelov map to approximate the OT map. The Pogorelov problem can be solved by a convex geometric optimization framework, in which we need to ensure the searching inside the admissible space. In this work, we prove the discrete Alexandrov maximum principle, which gives an apriori estimate of the searching. Our experimental results demonstrate that the Pogorelov map does approximate the OT map well with much more efficient computation. 
    more » « less
  3. ; ; ; ; ; ; ; (, Proceedings of International Conference on Computer Vision (ICCV))
    null (Ed.)
    Shape analysis has been playing an important role in early diagnosis and prognosis of neurodegenerative diseases such as Alzheimer's diseases (AD). However, obtaining effective shape representations remains challenging. This paper proposes to use the Alexandrov polyhedra as surface-based shape signatures for cortical morphometry analysis. Given a closed genus-0 surface, its Alexandrov polyhedron is a convex representation that encodes its intrinsic geometry information. We propose to compute the polyhedra via a novel spherical optimal transport (OT) computation. In our experiments, we observe that the Alexandrov polyhedra of cortical surfaces between pathology-confirmed AD and cognitively unimpaired individuals are significantly different. Moreover, we propose a visualization method by comparing local geometry differences across cortical surfaces. We show that the proposed method is effective in pinpointing regional cortical structural changes impacted by AD. 
    more » « less
  4. ; ; ; ; ; (, ICLR 2020)
    Generative adversarial networks (GANs) have attracted huge attention due to its capability to generate visual realistic images. However, most of the existing models suffer from the mode collapse or mode mixture problems. In this work, we give a theoretic explanation of the both problems by Figalli’s regularity theory of optimal transportation maps. Basically, the generator compute the transportation maps between the white noise distributions and the data distributions, which are in general discontinuous. However, DNNs can only represent continuous maps. This intrinsic conflict induces mode collapse and mode mixture. In order to tackle the both problems, we explicitly separate the manifold embedding and the optimal transportation; the first part is carried out using an autoencoder to map the images onto the latent space; the second part is accomplished using a GPU-based convex optimization to find the discontinuous transportation maps. Composing the extended OT map and the decoder, we can finally generate new images from the white noise. This AE-OT model avoids representing discontinuous maps by DNNs, therefore effectively prevents mode collapse and mode mixture. 
    more » « less
  5. ; ; ; ; (, Proceedings of International Meshing Roundtable)
    null (Ed.)
    Optimal transportation finds the most economical way to transport one probability measure to another, and it plays an important role in geometric modeling and processing. In this paper, we propose a moving mesh method to generate adaptive meshes by optimal transport. Given an initial mesh and a scalar density function defined on the mesh domain, our method will redistribute the mesh nodes such that they are adapted to the density function. Based on the Brenier theorem, solving an optimal transportation problem is reduced to solving a Monge-Amp\`ere equation, which is difficult to compute due to the high non-linearity. On the other hand, the optimal transportation problem is equivalent to the Alexandrov problem, which can finally induce an effective variational algorithm. Experiments show that our proposed method finds the adaptive mesh quickly and efficiently. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email