# Face Transfer with Multilinear Models - PowerPoint PPT Presentation

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Face Transfer with Multilinear Models. Daniel Vlasic & Jovan Popovic CSAIL MIT Matthew Brand & Hanspeter Pfister MERL. Outline. Introduction to Multilinear Model Multilinear Face Model Face Transfer. A. I 2. X 1. X 2. X 1 -X 2. X 1. X 2. I 1. B. Y 1. Y 2. Y 1 -Y 2. Y 1. Y 2.

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Face Transfer with Multilinear Models

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## Face Transfer with Multilinear Models

Daniel Vlasic & Jovan Popovic

CSAIL MIT

Matthew Brand & Hanspeter Pfister

MERL

### Outline

• Introduction to Multilinear Model

• Multilinear Face Model

• Face Transfer

A

I2

X1

X2

X1-X2

X1

X2

I1

B

Y1

Y2

Y1-Y2

Y1

Y2

X1-Y1

X2-Y2

(X1-Y1) –

(X2-Y2)

X1

X2

1

0

X1

X2

Y1

Y2

=

1

0

x

Y1

Y2

1

-1

X1-Y1

X2-Y2

U(1)

A = (U(2)x(U(1)xB)T)T

1

0

X1

Y1

X1-Y1

AT

1

0

=

x

A =B x1U(1) x2U(2)

X2

Y2

X2-Y2

1

-1

U(2)

J2

U(2)

I1

J2

A

I1

U(1)

I2

J1

=

B

I1

J1

### Multilinear Model

• Generalization of linear model

• A = B x1U(1)x2U(2)x3U(3)…xnU(n)…xNU(N)

Orthogonal

Transformation

Data Tensor

Core Tensor

### How to Multiply?

• A = B x1 U(1)x2 U(2)x3 U(3)…xn U(n)…

• B xn U(n) :U(n) * B(n)

I2

Tensor Flattening

X1

X2

I1

B

A =B x1U(1) x2U(2)

Y1

Y2

0

0

A(1) =

A=

2

4

1

1

0

2

2

0

2

4

1

-1

2

2

-2

4

1

2

2

0

2

4

0

4

-1

-2

0

0

1

2

1

2

2

4

Data Tensor

### Mathematically…

?

Data

Tensor

Left Singular of SVD

In data reduction, we use PCA as Y = eTX

• SVD => A = USVT

• AAT = USVT(USVT)T = USVT * ((VT)TSUT) = US2UT

• PCA => Find Cov(A) = (AAT)/(n-1)

• AAT = eDeT => U = e

• ### SVD for Multilinear Model

• To find Un, perform SVD on mode n space of the data tensor, i.e., J(n)

• This is not optimal, however, and they use ALS, or Alternating Least Square

• A lot of SIAM papers address this topic, and out of our scope

### Multilinear Face Model

• Bilinear Model (3-mode)

• 30K vertices x 10 expression x 15 identities

• Trilinear Model (4-mode)

• + 5 visemes

• Multilinear model of face geometry

### Arbitrary Interpolation

n

Synthesized Data, f

1

=

n

Original Data, M

m

Weighting, w

m rows data

1

x

f= Mx2w(2)

### Interpolation in Multilinear Model

F = M x2w(2)

Multilinear model of face geometry

f = M x2 w(2) x3 w(3) x4 w(4) …. xN w(N)

### Missing Data

• So far, we dealt with perfect data set

• In practice… NOT the case

• Maximum A Posteriori (MAP) estimation failed

• Probability Principle Component Analysis (PPCA)

### Short Review on PPCA

• t = Wx +μ +ε

• x is N(0, I) , εis isotropic error N(0, σ2I)

• So t is N(μ, WWT + σ2I)

• Given t, we want to estimate W, σ

• Maximize the likelihood function L = p(t) = Πip(ti |x,W)

### Short Review on PPCA

• Maximum Likelihood Estimators (M.L.E) tells us that, by taking log-likelihood

• WML = Uq(Λq – σ2I)1/2R

• σML = 1/(d-q) Σj = q+1 to dλj

• Uq is eigen-vector and Λq eigen-value

• -------------------------------------------------------

• End of review

### Probabilistic Face Model

t = Wx + μ +ε

Likelihood Function

p(t |x,W)

### Missing Data

Tj = mode-j of J

=>

Jj = mode-j of

Mx2U(2)…xj-1U(j-1) xj+1U(j+1) ..xn U(n)