中国科学技术大学：《数字几何处理 Digital Geometry Processing》课程教学资源（课件讲义）03 Mesh Smoothing

Mesh Smoothing Xiao-Ming Fu

Mesh Smoothing Xiao-Ming Fu

Denoising Removing the noise (the high frequencies)and keeping the overall shape (the low frequencies) Physical scanning process 。Feature VS Noise

Denoising • Removing the noise (the high frequencies) and keeping the overall shape (the low frequencies) • Physical scanning process • Feature VS Noise

Smoothing From wiki In statistics and image processing,to smooth a data set is to create an approximating function that attempts to capture important patterns in the data,while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing,the data points of a signal are modified so individual points (presumably because of noise)are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal

Smoothing – From wiki • In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. • In smoothing, the data points of a signal are modified so individual points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal

Outline Filter-based methods Optimization-based methods 。Data-driven methods

Outline • Filter-based methods • Optimization-based methods • Data-driven methods

Outline Filter-based methods Optimization-based methods 。Data-driven methods

Outline • Filter-based methods • Optimization-based methods • Data-driven methods

Laplacian smoothing Diffusion flow:a mathematically well-understood model for the time- dependent process of smoothing a given signal f(x,t). Heat diffusion,Brownian motion 。Diffusion equation: afx,边=a△fx,t Ot 1.A second-order linear partial differential equation; 2.Smooth an arbitrary function f on a manifold surface by using Laplace-Beltrami Operator. 3. Discretize the equation both in space and time

Laplacian smoothing • Diffusion flow: a mathematically well-understood model for the time￾dependent process of smoothing a given signal 𝑓(𝒙,𝑡). • Heat diffusion, Brownian motion • Diffusion equation: 𝜕𝑓 𝒙,𝑡 𝜕𝑡 = 𝜆∆𝑓(𝒙,𝑡) 1. A second-order linear partial differential equation; 2. Smooth an arbitrary function 𝑓 on a manifold surface by using Laplace-Beltrami Operator. 3. Discretize the equation both in space and time

Spatial discretization Sample values at the mesh vertices f(t)=(f(v1,t),...,f(vn,t))T Discrete Laplace-Beltrami using either the uniform or cotangent formula. The evolution of the function value of each vertex: of(vi, Ot 2=△f(x,t) Matrix form: of(t) at =入·Lf(t)

Spatial discretization • Sample values at the mesh vertices 𝒇(𝑡) = 𝑓 𝑣1,𝑡 , … , 𝑓 𝑣𝑛,𝑡 𝑇 • Discrete Laplace-Beltrami using either the uniform or cotangent formula. • The evolution of the function value of each vertex: 𝜕𝑓 𝑣𝑖 ,𝑡 𝜕𝑡 = 𝜆∆𝑓(𝒙𝑖 ,𝑡) Matrix form: 𝜕𝒇 𝑡 𝜕𝑡 = 𝜆 ∙ 𝐿𝒇(𝑡)

Temporal discretization Uniform sampling:(t,t h,t 2h,... Explicit Euler integration: f化+=fO+hf@=f0+haf阳 at 1.Numerically stability:a sufficiently small time step h. Implicit Euler integration: f(t+h)=f(t)+hλ·Lf(t+h) →(I-hn·L)f(t+h)=f(t)

Temporal discretization • Uniform sampling: (𝑡,𝑡 + ℎ,𝑡 + 2ℎ, … ) • Explicit Euler integration: 𝒇 𝑡 + ℎ = 𝒇 𝑡 + ℎ 𝜕𝒇 𝑡 𝜕𝑡 = 𝒇 𝑡 + ℎ𝜆 ∙ 𝐿𝒇(𝑡) 1. Numerically stability: a sufficiently small time step ℎ. • Implicit Euler integration: 𝒇 𝑡 + ℎ = 𝒇 𝑡 + ℎ𝜆 ∙ 𝐿𝒇(𝑡 + ℎ) ⟺ 𝑰 − ℎ𝜆 ∙ 𝐿 𝒇 𝑡 + ℎ = 𝒇 𝑡

Laplacian smoothing ·Arbitrary function→vertex positions ·f→(x1,,xn)T Laplacian smoothing: xi←-xi+hM·△xi 1.Ax =-2Hn vertices move along the normal direction by an amount determined by the mean curvature H. 2. mean curvature flow

Laplacian smoothing • Arbitrary function ⟹ vertex positions • 𝒇 ⟹ 𝒙𝟏, … , 𝒙𝒏 𝑻 • Laplacian smoothing: 𝒙𝑖 ⟵ 𝒙𝑖 + ℎ𝜆 ∙ ∆𝒙𝑖 1. ∆𝒙 = −2𝐻𝒏 ⟶ vertices move along the normal direction by an amount determined by the mean curvature 𝐻. 2. mean curvature flow

Figure 4.5.Curvature flow smoothing of the bunny mesh (left),showing the result after ten iterations (center)and 100 iterations (right).The color coding shows the mean curvature.(Model courtesy of the Stanford Computer Graphics Laboratory.)