LEFT-INVARIANT PARABOLIC EVOLUTIONS ON SE(2) AND CONTOUR ENHANCEMENT VIA INVERTIBLE ORIENTATION SCORES PART I: LINEAR LEFT-INVARIANT DIFFUSION EQUATIONS ON SE(2)
LEFT-INVARIANT PARABOLIC EVOLUTIONS ON SE(2) AND CONTOUR ENHANCEMENT VIA INVERTIBLE ORIENTATION SCORES PART I: LINEAR LEFT-INVARIANT DIFFUSION EQUATIONS ON SE(2)
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DOI：
10.1090/s0033-569x-10-01172-0
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发表时间：
2010-06-01
影响因子：
0.8
通讯作者：
Franken, Erik
中科院分区：
文献类型：
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作者：
Duits, Remco;Franken, Erik
给出了二维欧氏运动群SE（2）= R-2 × T上线性左不变扩散方程的显式解和相应的预解方程.这些抛物型方程是正向Kolmogorov方程，用于轮廓增强和轮廓完成的众所周知的随机过程。解是用相应的绿色函数的群卷积给出的。在早期的工作中，我们已经解决了前向Kolmogorov方程（或福克-普朗克方程）的随机过程的轮廓完成。在这里，我们主要集中在前向Kolmogorov方程的轮廓增强过程，其中不包括对流。我们推导出绿色函数的显式公式（即，在SE（2）上的热核）。通过应用一个压缩，我们用Heisenberg群的左不变生成元来逼近SE（2）上的左不变向量场，并导出了绿色函数的适当逼近.精确的绿色函数被用于SE（2）上的所谓碰撞分布，这是给定SE（2）上的初始分布的两个左不变预解扩散的乘积。我们使用左不变的进化过程中的自动轮廓增强在嘈杂的医学图像数据中使用所谓的方向得分，这是从一个灰度值图像通过一种特殊类型的酉小波变换。这里，（可逆的）取向分数的真实的部分用作碰撞分布中的初始条件。
We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R-2 x T. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Green's functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Green's functions (i.e., the heat kernels on SE(2)) of the left-invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the left-invariant vector fields on SE(2) by left-invariant generators of a Heisenberg group, and we derive suitable approximations of the Green's functions. The exact Green's functions are used in so-called collision distributions on SE(2), which are the product of two left-invariant resolvent diffusions given an initial distribution on SE(2). We use the left-invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.