The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schrödinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schrödinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.

The representations of the algebra of coordinates and momenta of noncommutative phase space are given. We study, as an example, the harmonic oscillator in noncommutative space of any dimension. Finally the map of Schrödinger equation from noncommutative space to commutative space is obtained.

Accurate segmentation of the optic disc (OD) and cup (OC) in fundus images from different datasets is critical for glaucoma disease screening. The cross-domain discrepancy (domain shift) hinders the generalization of deep neural networks to work on different domain datasets. In this work, we present an unsupervised domain adaptation framework, called Boundary and Entropy-driven Adversarial Learning (BEAL), to improve the OD and OC segmentation performance, especially on the ambiguous boundary regions. In particular, our proposed BEAL framework utilizes the adversarial learning to encourage the boundary prediction and mask probability entropy map (uncertainty map) of the target domain to be similar to the source ones, generating more accurate boundaries and suppressing the high uncertainty predictions of OD and OC segmentation. We evaluate the proposed BEAL framework on two public retinal fundus image datasets (Drishti-GS and RIM-ONE-r3), and the experiment results demonstrate that our method outperforms the state-of-the-art unsupervised domain adaptation methods. Our code is available at https://github.com/EmmaW8/BEAL.

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The success of deep convolutional neural networks is partially attributed to the massive amount of annotated training data. However, in practice, medical data annotations are usually expensive and time-consuming to be obtained. Considering multi-modality data with the same anatomic structures are widely available in clinic routine, in this paper, we aim to exploit the prior knowledge (e.g., shape priors) learned from one modality (aka., assistant modality) to improve the segmentation performance on another modality (aka., target modality) to make up annotation scarcity. To alleviate the learning difficulties caused by modality-specific appearance discrepancy, we first present an Image Alignment Module (IAM) to narrow the appearance gap between assistant and target modality data. We then propose a novel Mutual Knowledge Distillation (MKD) scheme to thoroughly exploit the modality-shared knowledge to facilitate the target-modality segmentation. To be specific, we formulate our framework as an integration of two individual segmentors. Each segmentor not only explicitly extracts one modality knowledge from corresponding annotations, but also implicitly explores another modality knowledge from its counterpart in mutual-guided manner. The ensemble of two segmentors would further integrate the knowledge from both modalities and generate reliable segmentation results on target modality. Experimental results on the public multi-class cardiac segmentation data, i.e., MM-WHS 2017, show that our method achieves large improvements on CT segmentation by utilizing additional MRI data and outperforms other state-of-the-art multi-modality learning methods.