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 timedependent 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.)