Learning Task-Aware Sampling with Shared Saliency through Density-Equalizing Mappings

In image and surface-based learning tasks, convolutional features are typically extracted using receptive fields that are sampled uniformly across the entire domain. However, informative structures are rarely distributed uniformly in practice and are often concentrated in localized regions. Such phenomena are particularly common in medical imaging, where pathological changes are spatially confined. Consequently, uniform convolution allocates equal computational effort to both informative and uninformative regions, resulting in inefficient feature extraction and suboptimal utilization of model capacity. To address this issue, we propose a framework for task-adaptive sampling that dynamically redistributes computational attention according to the spatial importance of the data. Specifically, we introduce the Density-Equalizing Convolutional Neural Network (DECNN), which employs density-equalizing mappings to guide convolution through a learned density function. The density function encodes the relative importance of different regions and induces a transformation that enlarges informative areas while compressing less relevant ones. As a result, convolutional receptive fields are redistributed non-uniformly over the domain, enabling denser sampling in task-relevant regions. By coupling this importance-driven transformation with convolution, DECNN performs adaptive feature extraction that focuses computational resources on informative structures. This leads to more efficient use of model capacity, yielding a lightweight yet expressive architecture while simultaneously producing an interpretable saliency map. Experiments on image classification and craniofacial surface analysis demonstrate that DECNN achieves competitive or superior performance with fewer parameters, accurately identifies task-relevant regions, and remains robust under complex geometric variations.

Quasi-Conformal Convolution : A Learnable Convolution for Deep Learning on Riemann Surfaces

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 Riemann 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 Riemann 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 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.