• 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: {\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx,}
      • spacial domain–>frequency domain F(epsilon) complex amplitude
      • Inverse transform:{\displaystyle 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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