Level Set Methods Based on Distance Function
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摘要: 讨论了有关水平集方法的基本问题,如保持为距离函数的方法,水平集方程解的存在性和唯一性。主要贡献是证明了,在距离函数约束下,水平集方程在初始零水平集附近有唯一解,它是关于演化界面的有向距离函数。并且用到了一些处理技巧:如注意到原始方程的任意解都是距离函数,将原始方程变化为另一简单形式。由于新的方程组不是一个经典方程组,则它被变换为一个普通形式,其中隐函数方法被采用。Abstract: Some basic problems on the levelset methods were discussed,such as the method used to preserve the distance function,the existence and uniqueness of solution for the level set equations.The main contribution is to prove that in a neighborhood of the initial zero level set,the level set equations with the restriction of the distance function have a unique solution,which must be the signed distance function with respect to the evolving surface.Some skillful approaches were used: Noticing that any solution for the original equation was a distance function,the original level set equations were transformedinto a simpler alternative form.Moreover,since the new system was not a classical one,the system was transformed into an ordinary one,for which the implicit function method was adopted.
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