Volume 28, Issue 1
Construction of Geometric Partial Differential Equations for Level Sets

Chong Chen & Guoliang Xu

J. Comp. Math., 28 (2010), pp. 105-121.

Published online: 2010-02

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  • Abstract

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations (second-order, fourth-order and sixth- order) into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.

  • Keywords

Geometric partial differential equations Level set Differential geometry operators

  • AMS Subject Headings

65D17.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-105, author = {}, title = {Construction of Geometric Partial Differential Equations for Level Sets}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {1}, pages = {105--121}, abstract = {

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations (second-order, fourth-order and sixth- order) into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m2971}, url = {http://global-sci.org/intro/article_detail/jcm/8510.html} }
TY - JOUR T1 - Construction of Geometric Partial Differential Equations for Level Sets JO - Journal of Computational Mathematics VL - 1 SP - 105 EP - 121 PY - 2010 DA - 2010/02 SN - 28 DO - http://doi.org/10.4208/jcm.2009.09-m2971 UR - https://global-sci.org/intro/article_detail/jcm/8510.html KW - Geometric partial differential equations KW - Level set KW - Differential geometry operators AB -

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations (second-order, fourth-order and sixth- order) into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.

Chong Chen & Guoliang Xu. (2019). Construction of Geometric Partial Differential Equations for Level Sets. Journal of Computational Mathematics. 28 (1). 105-121. doi:10.4208/jcm.2009.09-m2971
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