## 1128

Challenges:

1. Mathematical representation of model
2. learning models parameters from data

Overview:

• generative face model
• space of faces: goals
• each face represented as high dimensional vector
• each vector in high dimensional space represents a face
• Each face consists of
• shape vector Si
• Texture vector Ti
• Shape and texture vectors:
• Assumption: known point to point corrsp between faces
• construction: for eaxample 2D parameterization using few manually selected feature points
• shape, texture vecs: sampled at m common locations in parameterization
• Typically m = few thousand
• shape texture vecs 3m dimensional (x, y, z & r, g, b)
• *feature points may not be sampled.
• linear face model
• linear combinations of faces in database
• s = sum(ai * Si)
• t = sum(bi * Ti)
• Basis vectors Si, Ti
• Avg face Savg = 1/n sum(Si)
• Avg face Tavg = 1/n sum(Ti)
• Probabilistic modeling
• PDF over space of faces gives probability to encounter certain face in a population
• Sample the PDF generates random new faces.
• Ovservation
• Shape & tex vecs are not suitable for probabilistic modeling
• Too much redundancy
• many vecs do not resemble real faces
• real faces occupy
• AssumptionL faces occupy linear subspace of high dimensional space.
• Faces lie on hyperplane
• Illustration in 3D
• faces lie on 2D plane in 3D
• How to determine hyperplane?
• PCA:
• find orthogonal sets of vecs that best represent input data points
• first basis vec: largest variance along its direction
• Following basis vecs: orthogonal, ordered according to variance along their directions
• Dimensionality reduction:
• discard basis vectors with low variance
• represent each data point using remaining k basis vecs (k-dim hyperplane)
• can show: hyperplanes obtained via PCA minimize L2 distance to plane
• First basis vec maximized variance along its direction
• w{1} = argmanx{sum((t1)i^2)} – argmax sum(xi * w)^2 = argmax{||Xw||^2} = argmax{w’X’Xw}
• data points as row vecs xi , zero mean
• matrix X consists of row vecs xi
• w{1} is the eigenvec corespp to the max eigenvalue of X’X
• Properties;
• Matrix X’X is proportional to so-called sample covariance matrix of dataset X
• if dataset has multi-variance normal distribution, maximum likelihood estimation of distribution is
• f(x) = (2* pi) ….
• Node
• X is very large X is n x m matrix
• n is # data vec
• m is length of data vec
• m>> n in general
• X is m x m matrix, very large
• SVD of X
• X = U sigma V’
• X’X  = V sigma’ UU’ sigma V’ = V sigma^2 V’
• right singular vecs V of X are eigenvec of X’X
• singular values sigma(k) of X are square roots of eigenvalues lambda(k) of X’X
• change of bassis into orthogonal PCA basis by projection onto eigenvectors
• T = XV = Usigma V’V = uSigma
• left singular vectors U multiplied by singular velues in sigma
• Dimensionality reduction
• only consider l eigenvectors/ singular vectors correp to l largest singular values
• Tl = XVl = U l * sigma l
• Matrix Tl in R (mxl)  contains coord of m samples in reduced number l of dim
• computation of only l components directly via truncated SVD
• multivariate normal distribution in reduced space has covariance matrix
• Diagonal matrix, l largest eigenvalues of X’X
• PCA diagonalizes covariance matrix decorrelates data
• attribute based modeling
• mnually define attributes, label each face i with a weight mui for each attribute mu
• attribute vecs
• add/ subtract multiple of attribute vecs
• model fitting, tracking
• Assume a parametric shape model
• Given parameters,can generate shape
• model fitting, tracking problem
• given some observation, find shape
• parameters that most likely to produce the observation
• bayes theorem
• Matching to images
• Model parameters to generate an image
• Shape vec: alpha
• tex vec: beta
• rendering parameters (projection, lighting) rou
• Given image, what are the most likely rendering parameters that generate that image
• MAP
• BAyes
• compute max p
• Discussion
• linear mdoel may not be accurate
• linear model not suitable for large geometric deformations (rotations)

## deformation

• Reference
• Deformation energy
• Geometric energy to strtch and band thin shell from one shape t another as difference between first and second fundamental form
• First: stretching
• second: bending
• Approach:
• Given constraints (handle position / orientation, find surface that min deformation energy)
• Linear Approximation
• Energy based on fundamental forms in non-linear function of displacements
• Hard to minimize
• linear approximation using partial derivatives of displacement function.
• Assume parameterization of displacement field d(u,v)
• Bending:
• linear energy:
• laplace = 0 minimize surface area
• Variational calculus, Euler0-Lagrange equations:
• laplace of laplace: make it smooth (the derivative of surface change continuously and is minimized)
• So, apply bi-laplacian on mesh
• Skeletal animation

## 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