论文标题
显式计算张力多核内核的单数积分
Explicit calculation of singular integrals of tensorial polyadic kernels
论文作者
论文摘要
$ u $：$ \ mathcal {s}（\ mathbb {r}^n）\ rightArrow \ Mathcal {s'}（\ Mathbb {r}^n）$的riesz变换定义为通过单核内核的卷积定义，并且可以很方便地表达出使用傅立叶和简单的多级别。我们将此分析扩展到高阶Riesz的变换，即某种类型的单数积分，包含张力的多核内核，并定义功能的积分转换$ \ MATHCAL {s}（\ MATHBB {r}^n）\ rightArrow \ rightArrow \ rightArrow \ Mathcal \ Mathcal \ Mathcal {s'}（s'}（\ Mathbbbb {\ Mathbb {r} n} $ n {我们表明，转化后的内核也是多核张量，并提出了一种普通方法来明确计算傅立叶Mutlipliper。给出了分析结果以及递归算法，以计算转化核的系数。我们将结果比较到指导数值评估，并讨论$ n = 2 $的情况，并应用于图像分析。
The Riesz transform of $u$ : $\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S'}(\mathbb{R}^n)$ is defined as a convolution by a singular kernel, and can be conveniently expressed using the Fourier Transform and a simple multiplier. We extend this analysis to higher order Riesz transforms, i.e. some type of singular integrals that contain tensorial polyadic kernels and define an integral transform for functions $\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S'}(\mathbb{R}^{ n \times n \times \dots n})$. We show that the transformed kernel is also a polyadic tensor, and propose a general method to compute explicitely the Fourier mutliplier. Analytical results are given, as well as a recursive algorithm, to compute the coefficients of the transformed kernel. We compare the result to direct numerical evaluation, and discuss the case $n=2$, with application to image analysis.
