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We presented a novel radiomics approach using multimodality MRI to predict the expression of an oncogene (O6-Methylguanine-DNA methyltransferase, MGMT) and overall survival (OS) of glioblastoma (GBM) patients. Specifically, we employed an EffNetV2-T, which was down scaled and modified from EfficientNetV2, as the feature extractor. Besides, we used evidential layers based to control the distribution of prediction outputs. The evidential layers help to classify the high-dimensional radiomics features to predict the methylation status of MGMT and OS. Tests showed that our model achieved an accuracy of 0.844, making it possible to use as a clinic-enabling technique in the diagnosing and management of GBM. Comparison results indicated that our method performed better than existing work. 

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. 

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. 

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. 

Optimal transportation (OT) maps play fundamental roles in many engineering and medical fields. The computation of optimal transportation maps can be reduced to solve highly non-linear Monge-Ampere equations. In this work, we summarize the geometric variational framework to solve optimal transportation maps in Euclidean spaces. We generalize the method to solve worst transportation maps and discuss about the symmetry between the optimal and the worst transportation maps. Many algorithms from computational geometry are incorporated into the method to improve the efficiency, the accuracy and the robustness of computing optimal transportation. 

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.