Support function representation of convex bodies, its application in geometric computing, and some related representations
Support function representation of convex bodies, its application in geometric computing, and some related representations
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DOI：
10.1006/cviu.1998.0674
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发表时间：
1998-12-01
影响因子：
4.5
通讯作者：
Kumar, KV
中科院分区：
文献类型：
--
作者：
Ghosh, PK;Kumar, KV
支撑函数在凸体的表示、操作和分析中的重要性确实可以与信号处理中的傅里叶变换相比较。直观地说，支撑函数是凸体的支撑平面到原点的有符号距离。在本文中，我们表明，正如简单的乘法在傅立叶变换域原来是两个信号的卷积，同样简单的代数运算的支持函数的结果在各种几何运算相应的几何对象。事实上，由于支撑函数是一个实值函数，这些简单的代数运算只不过是算术运算，如加法、减法、倒数和最大-最小，这就产生了几何运算，如闵可夫斯基加法（膨胀）、闵可夫斯基分解（腐蚀）、极对偶和并交。此外，它已被证明在本文中，一些表示方案（如勒让德变换，扩展高斯图像，斜率图表示，正常的变换，斜率变换），这似乎是非常不同的第一眼，属于同一类的支持函数表示。最后，我们指出一些代数操作的支持函数，导致新的和未知的几何运算。支持功能，如表示非凸对象也表示。(C)北京：科学出版社.
The importance of the support function in representation, manipulation, and analysis of convex bodies can indeed be compared with that of the Fourier transform in signal processing. The support function, in intuitive terms, is the signed distance of a supporting plane of a convex body from the origin point. In this paper we show that, just as simple multiplication in the Fourier transform domain turns out to be the convolution of two signals, similarly simple algebraic operations on support functions result in a variety of geometric operations on the corresponding geometric objects. In fact, since the support function is a real-valued function, these simple algebraic operations are nothing but arithmetic operations such as addition, subtraction, reciprocal, and max-min, which give rise to geometric operations such as Minkowski addition (dilation), Minkowski decomposition (erosion), polar duality, and union-intersection. Furthermore, it has been shown in this paper that a number of representation schemes (such as the Legendre transformation, the extended Gaussian image, slope diagram representation, the normal transform, and slope transforms), which appear to be very disparate at first sight, belong to the same class of the support function representation. Finally, we indicate some algebraic manipulations of support functions that lead to new and unsuspected geometric operations. Support function like representations for nonconvex objects are also indicated. (C) 1998 Academic Press.