Skip to content
- 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