1

Tal Nir

Alfred M. Bruckstein

Ron Kimmel

Over-Parameterized Variational Optical Flow

Technion, Israel institute of technology

Haifa 32000

ISRAEL

2

What is optic flow?

• Optic flow relates to the perception of motion.• Optic flow – the apparent motion of objects in the scene

as seen on the 2D image plane.

3

An image

4

Warped image

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The corresponding optical flow

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Applications of optic flow

An important pre-processing for many visual tasks• Tracking.• Segmentation.• Compression.• Super-resolution – requires high accuracy.• 3D reconstruction (structure from motion).

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Basic equations

, , 1 ( , , )I x u y v t I x y t

u,v are the optic flow components between frame t and t+1

0I u I v Ix y t

Linearized brightness constancy equation

Brightness constancy equation

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The aperture problem

0x y t

x y t

t

I u I v I

uI I I

v

uI I

v

��������������

Linearized brightness constancy equation

From an algebraic point of view this is an ill-posed problem

An image with N pixels: N equations with 2N unknowns.

I�������������� ,s u v

Only the flow component in the gradient direction can be determined (normal flow).

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Going around the aperture problem

Looking for locations where the image has • “Multiple” gradient directions,• Discontinuous first image derivatives,• “Corners”.

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I��������������

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The Lucas-Kanade method

0 0

2

( , ) ( , )

Assume constant motion in a region

( , , 1) ( , , )x y N x y

J I x u y v t I x y t

2),(),( 00

),,()1,,(

yxNyx

kk tyxItdvvyduuxIJ

0 0

2

( , ) ( , )x y z

x y N x y

J I du I dv I

dvvv

duuu

kk

kk

1

1

B. D. Lucas and T. Kanade, “An iterative image registration technique with an application to stereo vision,” Proc. DARPA Image Understanding Workshop, April, 1981.

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Lucas-Kanade continued

2

2

x zx x y

y zx y y

I II I I du

I II I I dv

0 0

2

( , ) ( , )

( , , 1)

( , , 1)

( , , 1) ( , , )

x y zx y N x y

x k k

y k k

z k k

J I du I dv I

I I x u y v tx

I I x u y v ty

I I x u y v t I x y t

Solve the linear 2x2 system of equations

1. The “aperture problem” can occur in certain regions (zero eigenvalue).

2. Typically, the aperture problem does not appear in an exact sense.

3. Method may yield a sparse flow field estimate.

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Neighborhood based methods

• The flow in the patch can be described by a constant, affine, or other model.

M. Irani, B. Rousso, S. Peleg, “Recovery of Ego-Motion Using Region Alignment” . IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI), Vol. 19, No. 3, pp. 268--272, March 1997

• The smoothness within the patch is inherently enforced.• Discontinuities of the model within the patch may cause

inaccuracies.• The resulting problem is over-constrained.

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Optical flow estimation – an ill posed

problem

Motion in a patch –Over constrained

solution (Lucas-Kanade)

Over-parameterized Variational

Our work

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The variational approachB. K. P. Horn and B. G. Schunck, "Determining optical flow,"

Artificial Intelligence, vol. 17, pp. 185--203, 1981.

( , )min ( , )

u vE u v

2

2 2 2 2

( , ) ( , ) ( , )

( , )

( , )

D S

D x y t

S x y x y

E u v E u v E u v

E u v I u I v I dxdy

E u v u u v v dxdy

Find the flow which minimizes the functional

Composed of a data and smoothness (regularization) term

The resulting Euler-Lagrange equations

0)(

0)(

yyxxytyx

yyxxxtyx

vvIIvIuI

uuIIvIuI

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Variational approach. Cont.’

• Dense optical flow field (i.e. a vector at each pixel).• The smoothness (regularization) term enables the

completion of the flow in locations with insufficient information.

• Global solution – incorporates all the available information.

• The best results are achieved by modern variational approaches.

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T. Brox, A. Bruhn, N. Papenberg, J. Weickert“High Accuracy Optical Flow Estimation Based on a Theory

for Warping”, ECCV 2004.

22( , ) (x+w) (x) (x+w) (x) x

x ( , , )

( , ,1)

DE u v I I I I d

x y t

w u v

2 2( , ) xSE u v u v d

2 2 2s s

• L1 non-linear data term with a gradient constancy term

• L1 smoothness term in x,y,t space (3D)

2 22' div ' 0z x zI I I u u u

( , ) ( , ) ( , )D SE u v E u v E u v

Euler-Lagrange equation for u (Γ=0)

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Brox et. al. “High Accuracy Optical Flow Estimation”. Cont.’

Three loops of iteration• Outer loop k.• Inner loop fixed point

iteration in order to deal with the nonlinearity in Ψ.

• Gauss-Seidel iterations are used in order to solve the resulting sparse linear system of equations.

1

1 1

(x ) (x)

;

(x )

(x )

(x ) ( , , )

k k kx y t

k k k k k k

kx

ky

kt

I w I I du I dv I

u u du v v dv

I I wx

I I wy

I I w I x y t

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Brox et. al. “High Accuracy Optical Flow Estimation”. Advantages

• Solution in Multi-scale helps to avoid being trapped in local minima – large motion (reduction factor of 0.95).

• The 3D smoothness term solves the problem in the volume in contrast to the 2D (two frames) solution.

• The gradient constancy term reduces the sensitivity to brightness changes.

• Choosing Ψ as an approximately L1 function:• In the smoothness term it allows discontinuities in the

flow field.• In the data term it reduces the sensitivity to outliers.• The addition of ε is for numerical reasons.

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Results – Brox et al.

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Our motivation

Our motivation stems from the smoothness term

Penalty for changes in the optical flow

Penalty for changes from an optical flow model

Weighted spatio-temporal gradient

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The proposed over-parameterization model

The different roles of the coefficients and basis functions• The basis functions are selected a-priori, the

coefficients are estimated.• The regularization is applied only to the coefficients.

1

( , , ) ( , , )n

i ii

u A x y t x y t

• Basis functions of the flow model

• Space and time varying coefficients

1

( , , ) ( , , )n

i ii

v A x y t x y t

The optical flow is now estimated via the coefficients

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Over-parameterization - one frame

1

u

v*

*

u

*

*+v

Coefficients

Basis functions

Basis functions

Conventional representation

Over-parameterized representation

n

1

1A

nA1

n

+

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Over-parameterized functional2

1 1

, , 1 ( , , ) xn n

D i i i ii i

E I x A y A t I x y t d

The new regularization term penalizes for changes in the model parameters.

2

1

xn

S ii

E A d

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Euler-Lagrange equations

22

1

ˆ' div ' 0n

z z x q y q i qi

I I I I A A

The Euler-Lagange equation for the coefficient Aq

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Over-Parameterization models Constant motion model

• This case reduces to the regular variational approach of solving directly for u and v.

1 2

1 2

1 ; 0

0 ; 1

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( , , ) ( , , )n

i ii

u A x y t x y t A

21

( , , ) ( , , )n

i ii

v A x y t x y t A

The number of coefficients is n=2

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Affine over-parameterization model• Six basis functions

is a relative weight constant.

1

( , , ) ( , , )n

i ii

u A x y t x y t

1

( , , ) ( , , )n

i ii

v A x y t x y t

1 2 3

4 5 6

ˆ ˆ1; ;

0; 0; 0

x y

1 2 3

4 5 6

0; 0; 0

ˆ ˆ1; ;x y

0

0

0

0

ˆ

ˆ

x xx

x

y yy

y

2

1

xn

S ii

E A d

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Rigid motion over-parameterization model

• The optic flow of a rigid body

1 1 2 2 3 3 4 1 5 2 6 3; ; ; ; ;A A A A A A

21 2 3 4 5 6ˆ ˆˆ ˆ ˆ1; 0; ; ; 1 ;x xy x y

21 2 3 4 5 6ˆ ˆ ˆˆ ˆ0; 1; ; 1 ; ;y y xy x

21 3 1 2 3

22 3 1 2 3

ˆ ˆˆ ˆ ˆ1

ˆ ˆ ˆˆ ˆ1

u x xy x y

v y y xy x

1 2 3, ,T is the translation vector divided by the depth (z)

1 2 3, ,T is the rotation vector

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Rigid motion, cont…’

• In a seminal paper

• The optical flow calculation is a pre-processing followed by motion and structure estimation.

• In our formulation, the rigid motion model is used directly in the optical flow estimation process.

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Pure translation over-parameterization model

• Rigid motion with pure translation

1 3

2 3

ˆ

ˆ

u x

v y

Use only the first three coefficients and basis functions of the general rigid motion model.

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Numerical scheme

• Multi-resolution necessary to deal with large displacements.

• At each resolution, three loops of iterations are applied.

We adopt parts of the numerical scheme from T. Brox, A. Bruhn, N. Papenberg, and J. Weickert,“High Accuracy Optical Flow Estimation Based on a Theory for Warping,” ECCV 2004.to our over-parameterization model

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Outer loop k

Euler-Lagrange equations, q=1...n

Insert first order Taylor approximation to the brightness constancy equation

21 1

21 1

1

'

ˆdiv ' 0

k kk kz z x q y q

nk ki q

i

I I I I

A A

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Inner loop – fixed point iteration l

Solves the nonlinearity of the convex function Ψ

At each pixel we have n linear equations with n unknowns: the increments of the coefficients - dAi

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Experimental results

The parameters were set experimentally to the following values

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Synthetic piecewise affine flow example

2 1Brightness constancy equation: ( , , 1) ( , ) ( , ) ( , , )I x u y v t I x u y v I x y I x y t

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Synthetic piecewise affine flow – ground truth

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Results

1

( )

( )

, ,1 , ,1cos

, ,1 , ,1

( , )

( , )

AAE Average Angular Error mean

STD std

u v u v

u v u v

u v Estimated flow

u v Ground truth flow

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Our method is better in the AAE by 68%

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Piecewise affine test case

The estimated affine parameters are approximately piecewise constant

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Ground truthOur method - affine

model

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Yosemite without clouds sequence

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41

42

43

44

45

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The End

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+35%

+15%+16%

+39%

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Yosemite without clouds – ground truth

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Images of the angular error

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Histogram of the angular error

Our method – pure translation model

Brox et. al.

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Yosemite - Solution of the affine parameters

1A 2A3A

4A 5A 6A

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Noise sensitivity results

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“Variational Joint optic-flow Computation and Video Restoration”

T. Nir, A.M. Bruckstein, R. Kimmel

Errors in the data term appear for two reasons:• Errors in the computed flow.• Errors in the image data – noise, blur, interlacing,

lossy compression, …

The proposed functional

2( , , ) (x , , 1) ( ) xDE u v I I u y v t I x d

0( , , ) ( , , ) ( , ) ( , )D S FE u v I E u v I E u v E I I

2 2( , ) xSE u v u v d

2

0Fidelity term : ( ) xFE I I I d

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Variational Joint optic-flow Computation and Video Restoration. Cont’.

• Minimization is performed with respect to the optical flow u,v and the image sequence I.

• The fidelity term requires that the minimization would not deviate too far from the measured sequence, thus avoiding trivial solutions.

• If the expected noise is large, a lower choice of λ is appropriate, allowing larger deviations from the measured sequence.

• For , the sequence is constrained to be equal to the measurement, resulting in a regular optic flow scheme.

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Solution strategy

Optic flowcalculation

Denoising

Iterations between optic flow and denoising.

1. Initialization: zero optic flow and initial sequence.

2. Solve for the optic flow.

3. Perform denoising.

4. Iterate steps 2,3 until convergence.

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The Denoising step

• For the denoising step we use the discrete approximation with bilinear interpolation:

• Minimize with respect to I1,I2,I3,I4 and I is performed by gradient descent (A,B,C,D are constant – frozen flow).

• The denoising step performs smoothing along the optical flow trajectories.

• Remark: Smoothing by total variation is not good for optic flow calculation.

2 2

0

2 2

1 2 3 4 0

( , , 1) ( , , )I x u y v t I x y t I I dxdydt

AI BI CI DI I I I

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Office sequence – Frame 7

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Office sequence – Frame 8

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Office sequence – Frame 9

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Office sequence – Frame 10

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Office sequence – Optic flow at frame 9

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Experimental results - Office sequence

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Office sequence results - Cont’.

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A. Borzi, K. Ito, K. Kunisch: “Optimal control formulation for determining optical flow”, SIAM J.

Sci. Comp. 24(3), 818-847, (2002)

• Minimize with respect to u,v,I

• Subject to the constraints

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Comparison with Borzi

Our methodBorzi

Symmetric - all the sequence is denoised.

Constrain the first image to equal the measurement.

Non-linear brightness constancy penalty.

Linearized brightness constancy equation as a constraint

Comparison on sequences run by the best results available from the literature.

Comparative results reported on simple synthetic examples.

First to suggest the idea of changing the images together with the flow.

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What is the actual gap between L1 and L2?

No clouds

AAE

No clouds

STD

Cloudy

AAE

Cloudy

STD

HS~26.14~19.1632.4330.28

HS modified. σ= 1.5

11.2616.41

Multiscale + re-linearization

σ=0.82.382.386.259.14

+Smoothness 3D

1.861.965.909.01

L1 – Brox0.981.171.946.02

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Summary

• Over-parameterized representation of the optic flow introduces better regularization.

• The per pixel model allows the functional minimization to decide on the locations of discontinuities in the higher dimensional space.

• Significant improvement for both the 2D and 3D cases.• Coupling with our joint optic flow and denoising scheme

gives excellent results under heavy noise.• Future: The improved accuracy of the method has the

potential to improve motion segmentation, video compression, super-resolution…