Polygon Mesh Processing 笔记

the input surface is obtained by discrete sampling(evaluation / probing)
|
|
establish continuity ( to generate a mathematical surface representation). This requires building a consistent neighborhood relation between the samples.

Since our surface representations are supposed to support efficient processing, a natural choice is to restrict functions to the class of polynomials because those can be evaluated by elementary arithmetic operations.

An implicit surface does not have any holes as long as the defining function F is continuous


>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
由于parametric representation和 implicit representation各自的缺点，产生了相互转换的方法。
它们各自的优点说明，parametric representation反映的是metric性质，implicit representation反映的是topology的性质。

我还是以前的观点，计算机科学里令人称赞的空间划分结构(e.g. kd-tree, bsp, ...)， 本质上是因为我们从数学上没有找到更优秀的解决办法。计算机科学里的这些算法是为当前那些愚蠢的数学方法而擦屁股的；算法的改进，无非是比以前擦得更好一些而已。

什么叫好的方法？能用尽可能少的输入(原始)数据去表示更多的数据，让输入(原始)数据和其他数据产生联系。比如halfedge结构算是个例子：通过遍历vertex 的halfedge，可以获得(调用相应的函数)vertex相邻的点、边、面。
相反的方法，可以把vertex和相邻的点、边、面直接存储。
这两种方法相比，前者象是一种压缩和解压缩的过程----让压缩后的数据尽可能的少；在解压缩的过程里，压缩后的数据被当作输入数据，解压缩的过程就是把原始数据影射为实际需要的数据。
可以把调用相应的函数获得vertex相邻的点、边、面类比为解压缩。这个过程会有时间开销。但从历史发展来看，这种开销是值得的(cpu越来越快就是在帮我们这个忙)。

类比人类的认知过程：人们倾向与使用尽可能少的假设、尽可能多的规律去解释尽可能多的现象。为了解释同样多的物理现象，如果我能用尽可能多的规律换取尽可能少的假设，这被认为是进步。规律是我们理解了的事情，我们不理解的那些，暂时用假设来表示。用规律作为函数，把假设影射到现象。
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
Radial basis functions [Light 92] are a prominent example,  
as are partition of unity implicits [Ohtake et al. 03] 
point-based representations [Pauly 03, Kobbelt and Botsch 04]

The tangent vector x' thus encodes the metric of the curve.

While any regularcurve can be parameterized with respect to arc length, we will see in Chapter 5 that such a canonical parameterization cannot in general be defined for surfaces.

for a 2D square, 
the eigenfunctions of the Laplace-Beltrami operator ~ the basis functions of the discrete cosine transform (used by the JPEG format) and, 
for a sphere, 
the eigenfunctions of the Laplace-Beltrami operator ~ spherical harmonics. 
Therefore, the eigenfunctions of the Laplace-Beltrami are called 【manifold harmonics】.
。
uniform Laplacian tries to move each vertex to the barycenter of its one-ring neighbors --->isotropic remeshing
.
higher-order Laplacian flows provide better low-pass filtering properties.
.
∆L=0   ----> minimize area surfaces (membrane surfaces)
∆2L=0 ----> minimizing curvature surface (thin-plate surfaces)
∆3L=0 ----> minimum variation surfaces
.
surface fairing and diffusion flow:
Laplacian flows converge to fair surfaces
.
Diffusion flow, as discussed so far, is an isotropic smoothing scheme since it diffuses high frequency noise equally in all directions.
anisotropic diffusion tries to preserve features
by adjusting the direction of diffusion, such that smoothing happens along,
but not across features.
.
The Laplacian measures the regularity (or irregularity) of a function.for a linear function the Laplacian is equal to zero. Therefore,
minimizing the Laplacian of u and v results in smooth parametric coordinates; in other words, this also minimizes the distortion of the parameterization.
.
5.4.3 Conformal Maps and Harmonic Maps
the real part and the imaginary part of a conformal map
are two harmonic functions

area of the surface = Dirichlet’s energy + conformal energy
.
conformal: angle-preserving
equiareal: area-preserving
isometric: length-preserving(equiareal & conformal)
only developable surfaces admit an isometric parameterization

.
The Jacobian matrix of the parameterization function x corresponds to the linear map that transforms a vector w' in parameter space into a tangent vector w on the surface.

w1T w2 = (J w1')T (J w2') = w1'T (JT J) w2' := w1'T (I) w2'
I 衡量的是(切空间)夹角的变化
II衡量的是 normal curvature

two unit direction vectors in the parameter space
.
E(S') =∫∫Ω  ks*FrobeniusNorm( I'(u,v) - I(u,v) )^2  +  kb*FrobeniusNorm( II'(u,v) - II(u,v) )^2  dudv
ks: stretching stiffness parameters
kb: bending stiffness parameters
可见:
I(u,v)可以用来衡量stretching stiffness, (angle ~ inner product ~stretch)
II(u,v)可以用来衡量bending stiffness, (normal direction variation ~normal curvature~ blend)

. RBF  φ(r) = r^3 <==> ∆3 d=0 . 当矩阵很大的时候，而且是分块矩阵，会用图论来研究它！----每一块看作一个node