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Hu BY, Ye C, Su JP, Liu L. Manifold-Constrained Geometric Optimization via Local Parameterizations. IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 2023; 29:1318-1329. [PMID: 34529566 DOI: 10.1109/tvcg.2021.3112896] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Many geometric optimization problems contain manifold constraints that restrict the optimized vertices on some specified manifold surface. The constraints are highly nonlinear and non-convex, therefore existing methods usually suffer from a breach of condition or low optimization quality. In this article, we present a novel divide-and-conquer methodology for manifold-constrained geometric optimization problems. Central to our methodology is to use local parameterizations to decouple the optimization with hard constraints, which transforms nonlinear constraints into linear constraints. We decompose the input mesh into a set of developable or nearly-developable overlapping patches with disc topology, then flatten each patch into the planar domain with very low isometric distortion, optimize vertices with linear constraints and recover the patch. Finally, we project it onto the constrained manifold surface. We demonstrate the applicability and robustness of our methodology through a variety of geometric optimization tasks. Experimental results show that our method performs much better than existing methods.
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Choi GPT, Qiu D, Lui LM. Shape analysis via inconsistent surface registration. Proc Math Phys Eng Sci 2020; 476:20200147. [PMID: 33223928 DOI: 10.1098/rspa.2020.0147] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2020] [Accepted: 09/09/2020] [Indexed: 11/12/2022] Open
Abstract
In this work, we develop a framework for shape analysis using inconsistent surface mapping. Traditional landmark-based geometric morphometr- ics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid surface mapping methods rely on a strong assumption of the global consistency of two surfaces. From a practical point of view, given two anatomical surfaces with prominent feature landmarks, it is more desirable to have a method that automatically detects the most relevant parts of the two surfaces and finds the optimal landmark-matching alignment between these parts, without assuming any global 1-1 correspondence between the two surfaces. Our method is capable of solving this problem using inconsistent surface registration based on quasi-conformal theory. It further enables us to quantify the dissimilarity of two shapes using quasi-conformal distortion and differences in mean and Gaussian curvatures, thereby providing a natural way for shape classification. Experiments on Platyrrhine molars demonstrate the effectiveness of our method and shed light on the interplay between function and shape in nature.
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Affiliation(s)
- Gary P T Choi
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA.,Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Di Qiu
- Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, People's Republic of China
| | - Lok Ming Lui
- Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, People's Republic of China
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Duan L, Luo X, Ruan L, Gu M. Novel method of boundary-free mesh parameterization. PLoS One 2019; 14:e0217537. [PMID: 31170192 PMCID: PMC6553849 DOI: 10.1371/journal.pone.0217537] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2018] [Accepted: 05/14/2019] [Indexed: 11/18/2022] Open
Abstract
Unless the targeted mesh is developable, metric distortion is inevitable during the process of surface mesh parameterization, thus one important objective of all involved parametric studies is to reduce the metric distortion. In order to further reduce area and angle distortion, a novel method of boundary-free mesh parameterization is presented in the paper. Firstly, the initial boundary-fixed conformal parameterization from 3D surface mesh patch to a plane is performed in the method. Then, based on the initial parameterization, the iterations of boundary-free quasi-harmonic parameterization are developed, where the tensor field is updated in each iterative step and the principal curvature direction is utilized to terminate the iteration. The solution of the novel method is convenient to calculate since it involves a series of linear systems. In our novel parameterization method, lower metric distortion and considerable efficiency have been obtained in experiments.
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Affiliation(s)
- Liming Duan
- ICT Research Center, Key Laboratory of Optoelectronic Technology and System of the Education Ministry of China, Chongqing University, Chongqing, China
- College of Mechanical Engineering, Chongqing University, Chongqing, China
- Engineering Research Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing, China
| | - Xueqing Luo
- ICT Research Center, Key Laboratory of Optoelectronic Technology and System of the Education Ministry of China, Chongqing University, Chongqing, China
- College of Mechanical Engineering, Chongqing University, Chongqing, China
- Engineering Research Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing, China
| | - Lang Ruan
- ICT Research Center, Key Laboratory of Optoelectronic Technology and System of the Education Ministry of China, Chongqing University, Chongqing, China
- College of Mechanical Engineering, Chongqing University, Chongqing, China
- Engineering Research Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing, China
| | - Minghui Gu
- ICT Research Center, Key Laboratory of Optoelectronic Technology and System of the Education Ministry of China, Chongqing University, Chongqing, China
- College of Mechanical Engineering, Chongqing University, Chongqing, China
- Engineering Research Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing, China
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Reyneke CJF, Luthi M, Burdin V, Douglas TS, Vetter T, Mutsvangwa TEM. Review of 2-D/3-D Reconstruction Using Statistical Shape and Intensity Models and X-Ray Image Synthesis: Toward a Unified Framework. IEEE Rev Biomed Eng 2018; 12:269-286. [PMID: 30334808 DOI: 10.1109/rbme.2018.2876450] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
Patient-specific three-dimensional (3-D) bone models are useful for a number of clinical applications such as surgery planning, postoperative evaluation, as well as implant and prosthesis design. Two-dimensional-to-3-D (2-D/3-D) reconstruction, also known as model-to-modality or atlas-based 2-D/3-D registration, provides a means of obtaining a 3-D model of a patient's bones from their 2-D radiographs when 3-D imaging modalities are not available. The preferred approach for estimating both shape and density information (that would be present in a patient's computed tomography data) for 2-D/3-D reconstruction makes use of digitally reconstructed radiographs and deformable models in an iterative, non-rigid, intensity-based approach. Based on a large number of state-of-the-art 2-D/3-D bone reconstruction methods, a unified mathematical formulation of the problem is proposed in a common conceptual framework, using unambiguous terminology. In addition, shortcomings, recent adaptations, and persisting challenges are discussed along with insights for future research.
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Hanaoka S, Masutani Y, Nemoto M, Nomura Y, Miki S, Yoshikawa T, Hayashi N, Ohtomo K, Shimizu A. Landmark-guided diffeomorphic demons algorithm and its application to automatic segmentation of the whole spine and pelvis in CT images. Int J Comput Assist Radiol Surg 2016; 12:413-430. [DOI: 10.1007/s11548-016-1507-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2016] [Accepted: 11/16/2016] [Indexed: 10/20/2022]
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