Nov14. Mesh Smoothing – Xiaoxu Meng

Mesh smoothing:
- local averaging
- minimize local gradient energy in 3 dimensions
- Fourier transform (low pass filter) similar to local averaging idea
  - image convolution
  - F(A*B) = F(A) * F(B)
Spectral Analysis
- In general: extending eigenvalues, eigenvectors to linear operators on (continuous) functions.
- Fourior transform:
  - approximate signal as weighted sum (linear combination) of sines and cosines of different frequencies.
  - change of basis using eigenfunctions of Laplace operator (complex exponentials including sines and cosines)
  - Fourier transform function: ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx,$
  - spacial domain–>frequency domain F(epsilon) complex amplitude
  - Inverse transform: $f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi ix\xi }\,d\xi ,$
  - denoising: fourier transform–>filter out high frequency–>fourior inverse transform
- For mesh:
  - Intuition: Fourior transform by projecting onto eigenfunctions of Laplacian
  - mesh laplacian L is n x n matrix, n is number of vertices
    - Use PSD L (not normalized by vertex valence of voronoi area)
    - eigenvectors orthogonal
  - Project geometry onto eigenvectors.
  - reconstruction from eigenvectors associated with low frequencies
  - Chanllenge:
    - Too complex!
    - Too much computation!
Diffusion
- Laplace smoothing
  - Laplace is second derivative.
  - Smooth with Gaussian kernel.
  - backward Euler
    - solve p’ = p + mu * dt *L * p’
    - (I – mu * dt * L) p’ = p, identity matrix I
    - solve linear system for p’ in each step
    - Advantages: Allow larger time steps, no numerical stability problems.
Energy minimization
Alternatives

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