The Wave kernel Signature: A Quantum Mechanical Approach to shape Analysis

相对于HKS, WKS是模拟粒子在某一时刻tt,出现在地点为xx的概率,表示为ψ(x,t)\psi(x,t)
HKS的缺点是它强调了低频部分, 而忽略了高频的影响. 为了解决这个问题WKS在方程里面加了一个ii, 变成复数.

{引用[1],66页
A mathematical solution is to multiply the Laplace-Beltrami operator by the imaginary
number ’i’. That way the eigenvalues of the operator are complex and the contribution
of the different frequencies will not be attenuated over time.}

所以这就是方程4的来源

1) 图1, 描述了WKS长啥样, 注意横轴为能量, 竖轴为概率, 然后他说了一句 While the two points of the bottom are quite similar for large scales (small values of the energy), 首先, the bottom的两点是指模型中的下面两点, 即左边两个图表. 然后他说small energy对应large scale, 这里跟HKS不一样small time 对应 small scale. 怎么理解, 看图2, 小能量粒子在曲率大处挣扎不出来, 而大能量粒子可以挣扎出来.

2) 关于公式(4)的解(5)的推导过程可以参见[1],67页, 只是注意公式(3.17)在代换的时候
Δϕk(x)=Ekϕk(x)\Delta \phi_k(x) = E_k\phi_k(x)

3)从公式(6)到公式(7)的推导可以参见[1], 69页,注意
limT→∞1T∫T0eiEkte−iEltdt\displaystyle \lim_{T\rightarrow\infty} \frac{1}{T}\int^T_0e^{iE_kt}e^{-iE_lt}dt
由于正交只有当k=lk = l的时候式子不为0, 此时积分为1.

4) 公式(7)中fE(Ek)f_E(E_k)表示能量为EkE_k的概率, ϕk(x)\phi_k(x)表示能量为EkE_k点在x的概率

5) 2.2 部分主要是解决如何去选择(7)式中能量分布函数去对付perturbations of eigenenergies under non-rigid deformation of the shape. 推了一大堆, 我没细看了. 图3主要说明公式(8)(9)的分布.

6)2.3 是WKS的定义, 2.4讲了它的一些性质, 其中图4是为了说明WKS的稳定性, 但又能抓住变动的细节, 并且还能指出是哪个能量段的变动

7) 图5说明WKS的informative的性质, 使得匹配不局限于feature points.

8) 图6说明WKS的健壮性, 与噪声,简化,有洞的匹配

9) 图7描述3.4算法的结果

10) 图8横轴为每点匹配候选数的百分比, k running from 1(0%) to N/100(100%), 竖轴为feature points 匹配命中成功的百分比

[1] Representing 3D models for alignment and recognition

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