1. Over-Parameterized Variational Optical Flow
Tal Nir
Alfred M. Bruckstein
Ron Kimmel
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

5. The corresponding optical flow

6. 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).

7. Basic equations Brightness constancy equation
u,v are the optic flow components between frame t and t+1
Linearized brightness constancy equation

8. The aperture problem
Only the flow component in the gradient direction can be determined (normal flow).
From an algebraic point of view this is an ill-posed problem
An image with N pixels: N equations with 2N unknowns.

9. Going around the aperture problem
Looking for locations where the image has
“Multiple” gradient directions,
Discontinuous first image derivatives,
“Corners”.

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

11. Lucas-Kanade continued
Solve the linear 2x2 system of equations
The “aperture problem” can occur in certain regions (zero eigenvalue).
Typically, the aperture problem does not appear in an exact sense.
Method may yield a sparse flow field estimate.

12. 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 , 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.

13. Optical flow estimation – an ill posed problem Our work
Motion in a patch –
Over constrained
solution
(Lucas-Kanade)
Optical flow estimation – an ill posed problem
Our work
Over-parameterized Variational

14. The resulting Euler-Lagrange equations
The variational approach B. K. P. Horn and B. G. Schunck, "Determining optical flow," Artificial Intelligence, vol. 17, pp , 1981.
Find the flow which minimizes the functional
Composed of a data and smoothness (regularization) term
The resulting Euler-Lagrange equations

15. 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.

16. L1 smoothness term in x,y,t space (3D)
T. Brox, A. Bruhn, N. Papenberg, J. Weickert “High Accuracy Optical Flow Estimation Based on a Theory for Warping”, ECCV 2004.
L1 non-linear data term with a gradient constancy term
L1 smoothness term in x,y,t space (3D)
Euler-Lagrange equation for u (Γ=0)

17. 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.

18. 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.

19. Results – Brox et al.

20. Our motivation Our motivation stems from the smoothness term
Weighted spatio-temporal gradient
Penalty for changes in the optical flow
Penalty for changes from an optical flow model

21. The proposed over-parameterization model
Basis functions of the flow model
Space and time varying coefficients
The optical flow is now estimated via the coefficients
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.

22. Over-parameterization - one frame
Conventional representation u u + v *
Basis functions * *
Coefficients
Basis functions *
Over-parameterized representation + v

23. Over-parameterized functional
The new regularization term penalizes for changes in the model parameters.

24. Euler-Lagrange equations
The Euler-Lagange equation for the coefficient Aq

25. Over-Parameterization models Constant motion model
This case reduces to the regular variational approach of solving directly for u and v.
The number of coefficients is n=2

26. Affine over-parameterization model
Six basis functions

27. Rigid motion over-parameterization model
The optic flow of a rigid body
is the translation vector divided by the depth (z)
is the rotation vector

28. 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.

29. Pure translation over-parameterization model
Rigid motion with pure translation
Use only the first three coefficients and basis functions of the general rigid motion model.

30. 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 to our over-parameterization model

31. Outer loop k Euler-Lagrange equations, q=1...n
Insert first order Taylor approximation to the brightness constancy equation

32. 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

33. Experimental results
The parameters were set experimentally to the following values

34. Synthetic piecewise affine flow example

35. Synthetic piecewise affine flow – ground truth

36. Our method is better in the AAE by 68%
Results
Our method is better in the AAE by 68%

37. Piecewise affine test case
The estimated affine parameters are approximately piecewise constant

38. Our method - affine model
Ground truth

39. Yosemite without clouds sequence

46. The End

47. +39%
+35%
+16%
+15%

48. Yosemite without clouds – ground truth

49. Images of the angular error

50. Histogram of the angular error
Our method – pure translation model
Brox et. al.

51. Yosemite - Solution of the affine parameters

52. Noise sensitivity results

53. “Variational Joint optic-flow Computation and Video Restoration” T
“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

54. 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.

55. Solution strategy Iterations between optic flow and denoising.
Initialization: zero optic flow and initial sequence.
Solve for the optic flow.
Perform denoising.
Iterate steps 2,3 until convergence.

56. 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.

57. Office sequence – Frame 7

58. Office sequence – Frame 8

59. Office sequence – Frame 9

60. Office sequence – Frame 10

61. Office sequence – Optic flow at frame 9

62. Experimental results - Office sequence

63. Office sequence results - Cont’.

64. A. Borzi, K. Ito, K. Kunisch: “Optimal control formulation for determining optical flow”, SIAM J. Sci. Comp. 24(3), , (2002)
Minimize with respect to u,v,I
Subject to the constraints

65. Comparison with Borzi Borzi Our method
Constrain the first image to equal the measurement.
Symmetric - all the sequence is denoised.
Linearized brightness constancy equation as a constraint
Non-linear brightness constancy penalty.
Comparative results reported on simple synthetic examples.
Comparison on sequences run by the best results available from the literature.
First to suggest the idea of changing the images together with the flow.

66. What is the actual gap between L1 and L2?
Cloudy
STD
AAE
No clouds
30.28
32.43
~19.16
~26.14 HS
16.41
11.26
HS modified. σ= 1.5
9.14
6.25
2.38
Multiscale + re-linearization
σ=0.8
9.01
5.90
1.96
1.86
+Smoothness 3D
6.02
1.94
1.17
0.98
L1 – Brox

67. 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…