Abstract. Deep learning on non-Euclidean domains is important for analyzing complex geometric data that lacks common coordinate systems and familiar Euclidean properties. A central challenge in this field is to define convolution on domains, which inherently possess irregular and non-Euclidean structures. In this work, we introduce quasi-conformal convolution (QCC), a novel framework for defining convolution on simply-connected open surfaces using quasi-conformal theories. Each QCC operator is linked to a specific quasi-conformal mapping, enabling the adjustment of the convolution operation through manipulation of this mapping. By utilizing trainable estimator modules that produce quasi-conformal mappings, QCC facilitates adaptive and learnable convolution operators that can be dynamically adjusted according to the underlying data structured on the surfaces. QCC unifies a broad range of spatially defined convolutions, facilitating the learning of tailored convolution operators on each underlying surface optimized for specific tasks. Building on this foundation, we develop the quasi-conformal convolutional neural network (QCCNN) to address a variety of tasks related to geometric data. We validate the efficacy of QCCNN through the classification of images defined on curvilinear simply-connected open Riemann surfaces, demonstrating superior performance in this context. Additionally, we explore its potential in medical applications, including craniofacial analysis using 3D facial data and lesion segmentation on 3D human faces, achieving enhanced accuracy and reliability.

Abstract. Interactive segmentation allows users to provide meaningful input to guide the segmentation process. However, an important problem in interactive segmentation lies in determining how to incorporate minimal yet meaningful user guidance into the segmentation model. In this paper, we propose the quasi-conformal interactive segmentation (QIS) model, which incorporates user input in the form of positive and negative clicks. Users mark a few pixels belonging to the object region as positive clicks, indicating that the segmentation model should include a region around these clicks. Conversely, negative clicks are provided on pixels belonging to the background, instructing the model to exclude the region near these clicks from the segmentation mask. By solving our proposed theoretical supported model, the segmentation mask is obtained by deforming a template mask with the same topology as the object of interest using a quasi-conformal mapping. This approach makes each user input effectively used and helps to avoid topological errors in the segmentation results. We provide a thorough theoretical analysis of the proposed model for its ability to include or exclude regions of interest or disinterest based on the user’s indication. To evaluate the performance of QIS, we conduct experiments on synthesized images, medical images, and natural images. The results demonstrate the efficacy of our proposed method.

We propose the Topology-Preserving Segmentation Network, a deformation-based model that can extract objects in an image while maintaining their topological properties. This network generates segmentation masks that have the same topology as the template mask, even when trained with limited data. The network consists of two components: the Deformation Estimation Network, which produces a deformation map that warps the template mask to enclose the region of interest, and the Beltrami Adjustment Module, which ensures the bijectivity of the deformation map by truncating the associated Beltrami coefficient based on Quasiconformal theories. The proposed network can also be trained in an unsupervised manner, eliminating the need for labeled training data. This is achieved by incorporating an unsupervised segmentation loss. Our experimental results on various image datasets show that TPSN achieves better segmentation accuracy than state-of-the-art models with correct topology. Furthermore, we demonstrate TPSN’s ability to handle multiple object segmentation.

Deformation analysis is crucial in many applications, especially in medical image analysis. Analyzing the deformation pattern of anatomical structures provides important information for disease analysis. With degraded images or uncertainties, getting a deterministic solution of the deformation is challenging. In some cases, there may also be multiple solutions with different probabilities. As such, it is important to analyze the probability distribution of deformations, given data information with uncertainty. In this work, we propose to use computational Quasiconformal (QC) Teichmuller theories to parameterize the space of deformations. A distribution over the space of special features, called the QC features, can be computed and applied for deformation analysis. Extensive experiments are carried out on both synthetic data and real medical images, which demonstrate the efficacy of the proposed framework.

We introduce the first path-oriented drone trajectory planning algorithm, which performs continuous (i.e., dense ) image acquisition along an aerial path and explicitly factors path quality into an optimization along with scene reconstruction quality. Specifically, our method takes as input a rough 3D scene proxy and produces a drone trajectory and image capturing setup, which efficiently yields a high-quality reconstruction of the 3D scene based on three optimization objectives: one to maximize the amount of 3D scene information that can be acquired along the entirety of the trajectory, another to optimize the scene capturing efficiency by maximizing the scene information that can be acquired per unit length along the aerial path, and the last one to minimize the total turning angles along the aerial path, so as to reduce the number of sharp turns. Our search scheme is based on the rapidly-exploring random tree framework, resulting in a final trajectory as a single path through the search tree. Unlike state-of-the-art works, our joint optimization for view selection and path planning is performed in a single step. We comprehensively evaluate our method not only on benchmark virtual datasets as in existing works but also on several large-scale real urban scenes. We demonstrate that the continuous paths optimized by our method can effectively reduce onsite acquisition cost using drones, while achieving high-fidelity 3D reconstruction, compared to existing planning methods and oblique photography, a mature and popular industry solution.