Over-Parameterized Variational Optical Flow Tal Nir Alfred M. Bruckstein Ron Kimmel Technion, Israel institute of technology Haifa 32000 ISRAEL

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.

An image

Warped image

The corresponding optical flow

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).

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

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.

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

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.

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.

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.

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

The resulting Euler-Lagrange equations The variational approach B. K. P. Horn and B. G. Schunck, "Determining optical flow," Artificial Intelligence, vol. 17, pp. 185--203, 1981. Find the flow which minimizes the functional Composed of a data and smoothness (regularization) term The resulting Euler-Lagrange equations

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.

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)

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.

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.

Results – Brox et al.

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

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.

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

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

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

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

Affine over-parameterization model Six basis functions

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

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.

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.

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

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

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

Experimental results The parameters were set experimentally to the following values

Synthetic piecewise affine flow example

Synthetic piecewise affine flow – ground truth

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

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

Our method - affine model Ground truth

Yosemite without clouds sequence

The End

+39% +35% +16% +15%

Yosemite without clouds – ground truth

Images of the angular error

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

Yosemite - Solution of the affine parameters

Noise sensitivity results

“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

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.

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.

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.

Office sequence – Frame 7

Office sequence – Frame 8

Office sequence – Frame 9

Office sequence – Frame 10

Office sequence – Optic flow at frame 9

Experimental results - Office sequence

Office sequence results - Cont’.

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

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.

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

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…