论文标题
关于几何功能变化的注释
A note on the variation of geometric functionals
论文作者
论文摘要
变异的计算与差异几何形状结合在一起，作为建模和解决图像处理和计算机视觉问题的工具，在20世纪后期和90年代后期引入了计算机视觉。在这些方向上进行的广泛作品的开始是用大地测量轮廓（GAC），Beltrami框架，Osher和Sethian的水平设定方法等作品标记的。陈和兽医的作品仅举几例。在许多情况下，这些功能的优化是通过梯度下降方法通过计算Euler-Lagrange方程来完成的。在梯度下降方案中直接使用所产生的EL方程会导致非几何，在某些情况下是非感觉方程式的。为了获得几何和/或感性方程，修改这些EL方程甚至功能本身是成本的。本注释的目的是指出得出EL和梯度下降方程的正确方法，以使所得的梯度下降方程是几何的，并且是有道理的。
Calculus of Variation combined with Differential Geometry as tools of modelling and solving problems in image processing and computer vision were introduced in the late 80's and the 90s of the 20th century. The beginning of an extensive work in these directions was marked by works such as Geodesic Active Contours (GAC), the Beltrami framework, level set method of Osher and Sethian the works of Charpiat et al. and the works by Chan and Vese to name just a few. In many cases the optimization of these functional are done by the gradient descent method via the calculation of the Euler-Lagrange equations. Straightforward use of the resulted EL equations in the gradient descent scheme leads to non-geometric and in some cases non sensical equations. It is costumary to modify these EL equations or even the functional itself in order to obtain geometric and/or sensical equations. The aim of this note is to point to the correct way to derive the EL and the gradient descent equations such that the resulted gradient descent equation is geometric and makes sense.
