Dense Motion Estimation

1. Dense Motion Estimation Reading: Szeliski, Chapter 8

2. Dense Motion Estimation

3. Dense Motion Estimation • 2D motion in video sequence • Object tracking • Image stabilization

4. Motion Estimation • Error metric • Compare images • Search technique • Full search -- simple but slow • Hierarchical coarse-to-fine • Fourier transforms • Incremental methods • Optical flow • Multiple independent motions

5. Translational Alignment • Alignment between two images or image patches Where they are located in

6. Translational Alignment • Minimum of Sum of Squared Difference (SSD) • Assumption: corresponding pixel values remains the same in the two images • ---- Brightness constancy constraint Residual error (displaced frame difference) u=(u,v): displacement

7. Robust Error Metrics • Robust norm of error (Huber 1981; Hampel, Ronchetti, Rousseeuw et al. 1986; Black and Anandan 1996; Stewart 1999) • Sum of Absolute Difference (L1 norm) Grows less quickly than the quadratic penalty associated with least squares ESAD is NOT differentiable at the origin, not well suited to gradient descent approaches

8. Robust Error Metric • Smoothly varying function (Black and Rangarajan (1996) ) • Quadratic for small values but • grows more slowly away from the origin • Geman–McClure function a: constant that can be thought of as an outlier threshold

9. Spatially Varying Weights • Pixels that may lie outside of the boundaries • Partially or completely downweight the contribution of certain pixels • Erase moving object for background alignment • Multiple moving objects Weighted (or Windowed) SSD function

10. Weighted SSD • Large range of potential motion • Bias towards smaller overlap solutions Overlap area

11. Bias and Gain (Exposure Differences) • For images being aligned were not taken with the same exposure • Simple model of linear intensity variation --- Bias and Gain model Bias Gain

12. Bias and Gain • Least Squares with Bias and Gain • Linear regression • Color image • Estimate bias and gain for each color channel • Weighted prediction in video codecs

13. Correlation • Cross-Correlation • Taking intensity difference • Maximize the produce of two aligned images Is Bias and Gain modeling unnecessary? Bright patch exists in images

14. Normalized Cross-Correlation • NCC in [-1,1] • Works well when matching images taken with different exposure • Degrades for noisy low-contrast regions (Zero variance) Mean images of the corresponding patches

15. Normalized Cross-Correlation • Normalized SSD score(Criminisi, Shotton, Blake et al., 2007). • Produce comparable results to NCC • More efficient when applied to a large number of overlapping patches using a moving average technique

16. Hierarchical Motion Estimation • How can we find its minimum? • Full search over some range of shifts • Often used for block matching in motion compensated video compression • Simple to implement but slow • To accelerate the search process • Hierarchical motion estimation

17. Hierarchical Motion Estimation • Steps • Construct image pyramid • At coarser levels, search over a smaller number of discrete pixels • Motion estimation at coarse level is used to initialize a smaller local search at the next finer level • Not guaranteed to produce the same results as a full search, but works almost as well and much faster

18. Hierarchical Motion Estimation • Image downsampling • Coarsest level: search for the best that minimize the difference between • Full search over the range • Predict a likely displacement • Search over displacement is repeated at the finer level over a much narrower range • Incremental refinement step with warped image

19. Incremental Refinement • Nearest pixel – integer pixel • Higher accuracy is required for stabilization or stitching • Sub-pixel estimates • Evaluate several values (u,v) around the best value • Interpolate the matching score to find the analytic minimum • Gradient descent on SSD energy function

20. Incremental Refinement Lucas and Kanade (1981) • SSD energy and Taylor series expansion Image gradient or Jacobian at (x+u) Current intensity error (residual error)

21. Incremental Refinement Spatial derivative temporal derivative Optical flow constraint or brightness constancy constraint

22. Incremental Refinement Gaussian-Newton approximation of the Hessian Gradient-weighted residual vector

23. Incremental Refinement • For efficiency • Precompute the Hessian and Jacobian image: save significant computation • Precompute the inner product between the gradient field and shifted version of I1 allows the iterative re-computation of eito be performed in constant time (independent of the number of pixels)

24. Incremental Refinement • Iterations • The effectiveness relies on the quality of Taylor series approximation • When far away from the true displacement (say, 1–2 pixels), several iterations may be needed • It is possible to estimate a value for J_1 using a least squares fit to a series of larger displacements in order to increase the range of convergence (Jurie and Dhome 2002) or to “learn” a special-purpose recognizer for a given patch

25. Incremental Refinement • Stopping criterion • monitor the magnitude of the displacement correction |u| and to stop when it drops below a certain threshold (say, 1/10of a pixel) • For larger motions • combine the incremental update rule with a hierarchical coarse-to-fine search strategy

26. Incremental Refinement • Poorly conditioned because of lack of two-dimensional texture in the patch being aligned

27. Uncertainty Modeling • Capture the reliability of a particular patch-based motion estimate • Simplest model: covariance matrix • Captures the expected variance in the motion estimate in all possible directions • Under small amounts of additive Gaussian noise The variance of the additive Gaussian noise

28. Uncertainty modeling • For larger amounts of noise, the linearization performed by the Lucas–Kanade algorithm is only approximate • The minimum and maximum eigenvalues of the Hessian A can now be interpreted as the (scaled) inverse variances in the least-certain and most-certain directions of motion.

29. Bias and gain, weighting, and robust error metrics • 4*4 system of equations to estimate • Weighed SSD using Lucus-Kanade algorithm • Robust Error metrics • solved using the iteratively reweighted least squares technique

30. 8.2 Parametric Motion • More sophisticated motion models • Affine, has 4 unknowns • Full search over possible range is impractical • Lucas-Kanade algorithm  parametric motion models (Lucas and Kanade 1981; Rehg and Witkin 1991; Fuh and Maragos 1991; Bergen, Anandan, Hanna et al. 1992; Shashua and Toelg 1997; Shashua and Wexler 2001; Baker and Matthews 2004).

31. Parametric Motion • Instead of using a single constant translation u • Use a spatially varying motion field or correspondence map

32. Parametric Motion Jacobian of corresponding field Image Gradient Hessian and Gradient-weighted residual vector are

33. Incremental Refinement Translational motion Parametric motion • Jacobian • (Gauss-Newton) Hessian • Gradient weighted residual vector

34. Patch-based Approximation • Expensive computation of A, b • N pixels and n parameters: O(n^2N) • Image to sub-blocks Pj, only accumulate the simpler 2x2 quantities

35. Compositional Approach • Complex parametric motion such as homography • Warp target image I_1 to the current estimate

36. Compositional Approach • and are assumed to be fairly similar, then only an incremental parametric motion is required, i.e. the incremental motion can be evaluated around Szeliski and Shum (1997)

37. Compositional Approach • Homography

38. Compositional Approach • If the appearance of the warped and template images is similar enough, we can replace the gradient of with the gradient of • Pre-computate the Hessian matrix • The residual vector b can also be partially precomputed, i.e., the steepest descent images can can be precomputed and stored for later multiplication with the ea error images

39. Inverse Compositional Algorithm Baker and Matthews (2004) • Rather than (conceptually) re-warping the warped target image I_1(x), they instead warp the template image I_0(x) and minimize • Identical to the forward warped algorithm with • Gradients are replaced by • Difference sign of e_i

40. Inverse Compositional Algorithm

41. Non-Linear Least Sequares • Solve using • Update • The parameter is an additional damping parameter used to ensure that the system takes a “downhill” step in energy (squared error) and is an essential component of the Levenberg–Marquardt algorithm

42. 8.4 Optical Flow • Optical flow or optic flow is the pattern of apparent motion of objects, surfaces, and edges in a visual scene caused by the relative motion between an observer (an eye or a camera) and the scene. • The concept of optical flow was first studied in the 1940s and ultimately published by American psychologist James J. Gibson[4] as part of his theory of affordance. • Optical flow techniques utilize this motion of the objects surfaces, and edges • motion detection, object segmentation, time-to-collision and focus of expansion calculations, motion compensated encoding, and stereo disparity measurement

43. 8.4 Optical Flow • Independent estimate of motion at each pixel • Number of variables is twice the number of measurements -- underconstrained problem • two typical approaches • Patch-based or window-based approach • Add smoothness the terms on {ui} using regularization or Markov random fields and to search for a global minimum

44. Optical Flow http://en.wikipedia.org/wiki/Optical_flow • Phase correlation – inverse of normalized cross-power spectrum • Block-based methods – minimizing sum of squared differences or sum of absolute differences, or maximizing normalized cross-correlation • Differential methods of estimating optical flow, based on partial derivatives of the image signal and/or the sought flow field and higher-order partial derivatives, such as: • Lucas–Kanade Optical Flow Method – regarding image patches and an affine model for the flow field • Horn–Schunck method – optimizing a functional based on residuals from the brightness constancy constraint, and a particular regularization term expressing the expected smoothness of the flow field • Buxton–Buxton method – based on a model of the motion of edges in image sequences[9] • Black–Jepson method – coarse optical flow via correlation[6] • General variational methods – a range of modifications/extensions of Horn–Schunck, using other data terms and other smoothness terms. • Discrete optimization methods – the search space is quantized, and then image matching is addressed through label assignment at every pixel, such that the corresponding deformation minimizes the distance between the source and the target image.[10] The optimal solution is often recovered through min-cut max-flow algorithms, linear programming or belief propagation methods.

45. Optical Flow • Regularization-based framework Horn and Schunck (1981) • Instead of solving for each motion (or motion update) independently • Simultaneously minimized over all flow vectors {u_i} • Smoothness constraints • Brightness constancy constraint

46. Optical Flow • Combine local and global flow estimation • Using a locally aggregated Hessian as the brightness constancy term • Replace per-pixel Hessian and with aggregated version

47. Optical Flow • Combine global (parametric) and local motion models • Estimate either per-image or per-segment affine motion models combined with per-pixel residual corrections • Image brightness varying • Gradient descent and coarse-to-fine continuation methods to minimize the global energy function • Combinatorial optimization methods based on Markov random fields

48. Multi-frame Motion Estimation • Filter the spatio-temporal volume using oriented or steerable filters (Heeger 1988) • Spatio-temporal filtering uses a 3D volume around each pixel to determine the best orientation in space–time, which corresponds to a pixel’s velocity

49. Multi-frame Motion Estimation • Spatio-temporal filters have moderately large extents, which severely degrades the quality of their estimates near motion discontinuities • An alternative to full spatio-temporal filtering is to estimate more local spatio-temporal derivatives and use them inside a global optimization framework to fill in texturelessregions(Bruhn,Weickert, and Schnorr 2005; Govindu 2006).

50. 8.5 Layered Motion • Global smoothness? Local neighborhood constraints? • Visual motion is caused by the movement of a number of objects at different depths • Pixels are grouped into appropriate objects or layers • The pixel motions can be described more succintly and estimated more reliably