A Variational Framework for Curve Shortening in Various Geometric Domains

Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path <inline-formula><tex-math notation="LaTeX">\gamma (t)=\mathbf {x...

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Bibliographic Details
Published in:IEEE transactions on visualization and computer graphics Vol. 29; no. 4; pp. 1951 - 1963
Main Authors: Yuan, Na, Wang, Peihui, Meng, Wenlong, Chen, Shuangmin, Xu, Jian, Xin, Shiqing, He, Ying, Wang, Wenping
Format: Journal Article
Language:English
Published: United States IEEE 01.04.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Approximation algorithms
Costs
geodesics
Geodesy
Heating systems
least-cost paths
Parameterization
Point cloud compression
Shortest path problem
shortest paths
Surface waves
Three-dimensional displays
Variational method
ISSN:1077-2626, 1941-0506, 1941-0506
Online Access:Get full text
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Summary:Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path <inline-formula><tex-math notation="LaTeX">\gamma (t)=\mathbf {x}(u(t),v(t))</tex-math> <mml:math><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant="bold">x</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq1-3135021.gif"/> </inline-formula> on a surface <inline-formula><tex-math notation="LaTeX">\mathbf {x}=\mathbf {x}(u,v)</tex-math> <mml:math><mml:mrow><mml:mi mathvariant="bold">x</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="bold">x</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq2-3135021.gif"/> </inline-formula> with <inline-formula><tex-math notation="LaTeX">\gamma (0)=p</tex-math> <mml:math><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq3-3135021.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">\gamma (1)=q</tex-math> <mml:math><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq4-3135021.gif"/> </inline-formula>. We formulate the two-point geodesic problem into a minimization problem <inline-formula><tex-math notation="LaTeX">\int _0^1 H(\Vert \mathbf {x}_uu^{\prime }(t)+\mathbf {x}_vv^{\prime }(t)\Vert)\text{d}t</tex-math> <mml:math><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mo>∥</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>∥</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mtext>d</mml:mtext><mml:mi>t</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq5-3135021.gif"/> </inline-formula>, where <inline-formula><tex-math notation="LaTeX">H(s)</tex-math> <mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq6-3135021.gif"/> </inline-formula> satisfies <inline-formula><tex-math notation="LaTeX">H(0)=0,H^{\prime }(s)>0</tex-math> <mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq7-3135021.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">H^{\prime \prime }(s)\geq 0</tex-math> <mml:math><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>'</mml:mo><mml:mo>'</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq8-3135021.gif"/> </inline-formula> for <inline-formula><tex-math notation="LaTeX">s>0</tex-math> <mml:math><mml:mrow><mml:mi>s</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq9-3135021.gif"/> </inline-formula>. In our implementation, we choose <inline-formula><tex-math notation="LaTeX">H(s)=e^{s^2}-1</tex-math> <mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq10-3135021.gif"/> </inline-formula> and show that it has several unique advantages over other choices such as <inline-formula><tex-math notation="LaTeX">H(s)=s^2</tex-math> <mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq11-3135021.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">H(s)=s</tex-math> <mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq12-3135021.gif"/> </inline-formula>. It is also a minimizer of the traditional geodesic length variational and able to guarantee the uniqueness and regularity in terms of curve parameterization. In the discrete setting, we construct the initial path by a sequence of moveable points <inline-formula><tex-math notation="LaTeX">\lbrace x_i\rbrace _{i=1}^n</tex-math> <mml:math><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup></mml:math><inline-graphic xlink:href="xin-ieq13-3135021.gif"/> </inline-formula> and minimize <inline-formula><tex-math notation="LaTeX">\sum _{i=1}^{n} H(\Vert x_i - x_{i+1}\Vert)</tex-math> <mml:math><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mo>∥</mml:mo></mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>∥</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="xin-ieq14-3135021.gif"/> </inline-formula>. The resulting points are evenly spaced along the path. It's obvious that our algorithm can deal with parametric surfaces. Considering that meshes, point clouds and implicit surfaces can be transformed into a signed distance function (SDF), we also discuss its implementation on a general SDF. Finally, we show that our method can be extended to solve a general least-cost path problem. We validate the proposed algorithm in terms of accuracy, performance and scalability, and demonstrate the advantages by extensive comparisons.
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ISSN:1077-2626
1941-0506
1941-0506
DOI:10.1109/TVCG.2021.3135021