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- 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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97ad0938a279c4846d42a4bbd212f6a1f0ca4c0f)
- 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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2aab0c0d32f0438d2ccf5bf779458053ba2bd9)
- 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