# Nov14. Mesh Smoothing

• 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: • spacial domain–>frequency domain F(epsilon) complex amplitude
• Inverse transform: • 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