Zhiyuan Lyu

Zhiyuan Lyu, Lok Ming Lui, Gary P. T. Choi

Surface parameterization is a fundamental task in geometry processing and plays an important role in many science and engineering applications. In recent years, the density-equalizing map, a shape deformation technique based on the physical principle of density diffusion, has been utilized for the parameterization of simply connected and multiply connected open surfaces. More recently, a spherical density-equalizing mapping method has been developed for the parameterization of genus-0 closed surfaces. However, for genus-0 closed surfaces with extreme geometry, using a spherical domain for the parameterization may induce large geometric distortion. In this work, we develop a novel method for computing density-equalizing maps of genus-0 closed surfaces onto an ellipsoidal domain. This allows us to achieve ellipsoidal area-preserving parameterizations and ellipsoidal parameterizations with controlled area change. We further propose an energy minimization approach that combines density-equalizing maps and quasi-conformal maps, which allows us to produce ellipsoidal density-equalizing quasi-conformal maps for achieving a balance between density-equalization and quasi-conformality. Using our proposed methods, we can significantly improve the performance of surface remeshing for genus-0 closed surfaces. Experimental results on a large variety of genus-0 closed surfaces are presented to demonstrate the effectiveness of our proposed methods.

Zhiyuan Lyu, Gary P. T. Choi

Over the past several decades, geometric mapping methods have been extensively developed and utilized for many practical problems in science and engineering. To assess the quality of geometric mappings, one common consideration is their conformality. In particular, it is well-known that conformal mappings preserve angles and hence the local geometry, which is beneficial in many applications. Therefore, many existing works have focused on the angular distortion as a measure of the conformality of mappings. More recently, quasi-conformal theory has attracted increasing attention in the development of geometric mapping methods, in which the Beltrami coefficient has also been considered as a representation of the conformal distortion. However, the precise connection between these two concepts has not been analyzed. In this work, we study the connection between the two concepts and establish a series of theoretical results. In particular, we discover a simple relationship between the norm of the Beltrami coefficient of a mapping and the absolute angular distortion of triangle elements under the mapping. We can further estimate the maximal angular distortion using a simple formula in terms of the Beltrami coefficient. We verify the developed theoretical results and estimates using numerical experiments on multiple geometric mapping methods, covering conformal mapping, quasi-conformal mapping, and area-preserving mapping algorithms, for a variety of surface meshes in biology and engineering. Altogether, by establishing the theoretical foundation for the relationship between the angular distortion and Beltrami coefficient, our work opens up new avenues for the quantification and analysis of surface mapping algorithms.

Feijiang Han, Hengtao Cui, Licheng Guo et al.

Large Language Models (LLMs) often fail on complex reasoning tasks due to flawed question comprehension, not just flawed logic. This paper presents a systematic investigation into these comprehension failures. Our work yields three key insights: (1) the step-by-step principle, effective for calculation, can be migrated to the reading process to enhance comprehension; (2) increasing the proportion of question-related tokens (e.g., via repetition) succeeds by refocusing attention, a mechanism that can be explicitly controlled; and (3) backward dependencies represent a core bottleneck for decoder-only models that persists even with strong methods like Chain-of-Thought. Based on these findings, we introduce the Step-by-Step Reading (SSR) family of prompts. This multi-stage approach culminates in SSR++, a method specifically engineered to deepen model comprehension by guiding it to parse questions with finer granularity, focus attention on critical tokens, and resolve backward dependencies through iterative re-contextualization. SSR++ sets a new state-of-the-art on multiple reasoning benchmarks, and our analysis confirms it works by directly mitigating semantic misunderstanding. These results demonstrate that guiding how a model reads is a powerful and efficient method for improving its reasoning ability.