Part II: Modeling Aspects

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1 Yosemite test sequence Illumination changes Motion discontinuities Variational Optical Flow Estimation Part II: Modeling Aspects Discontinuity Di ti it preserving i smoothness th tterms Robust data terms Image features for matching Large displacements Occlusion O l i Funded by ONR-MURI and the German Academic Exchange Service (DAAD) Limitations of Horn-Schunck method Cloud region g not well estimated due to illumination changes Large displacements L di l t Part II - 2 Agenda of this part: challenges we would like to resolve Motion discontinuities blurred Motion discontinuities Occlusions Illumination changes Large g displacements p Outliers due to occlusion Underestimation of large displacements Really large displacements Part II - 3 Part II - 4

2 Motion discontinuities Motion discontinuities and the smoothness assumption Fact: smoothness assumption not satisfied everywhere Quadratic penalizer Gaussian error distribution Too much influence given to outliers blurring Different, independently moving objects cause discontinuities in the correct flow field Remedy: y robust error norm (Huber 1981, Cohen 1993, Schnörr 1994, Black-Anandan 1996, Mémin-Pérez 1998) Usually we do not know the object boundaries Part II - 5 Non-quadratic smoothness term Part II - 6 Minimization: Euler-Lagrange equations 16 Applying a robust function: Interpretation: adaptive smoothing with a joint diffusivity For example: Total variation (TV) norm penalty 10 8 TV norm: 6 Advantage: still convex global optimum! 4 Diff Diffusivity i it d depends d on th the unknown k flflow nonlinear system of equations deviation Part II - 7 Part II - 8

3 Resolve nonlinearity: fixed point iterations 1. Keep fixed for an (initial) solution Result: improvements at motion discontinuities (fixed point) 2. Solve resulting linear system to obtain a new fixed point (see Part I+III) 3. Update, iterate Non-quadratic smoothness term (6.36 ) Horn-Schunck (7.17 ) Linear system need not be solved exactly in each iteration Scheme also called lagged diffusivity or lagged nonlinearity Ground truth Part II - 9 Occlusions Part II - 10 Dealing with occlusions and non-gaussian noise Robust function applied to the data term: (Bl k A (Black-Anandan d , Mé Mémin-Pérez i Pé 1998) Euler-Lagrange equations: Some pixels of frame 1 no longer visible in frame 2 Interpretation: less importance given to pixels with high matching cost Pixels matched to most similar pixels in frame 2 erroneous motion vectors Nonlinear system can again be solved with fixed point iterations (lagged nonlinearity) Part II - 11 Part II - 12

4 Result: Better treatment of occlusions Both terms non-quadratic (5.97 ) Illumination changes Non-quadratic smoothness term (6.36 ) Schefflera from Middlebury benchmark Two frames from a Miss Marple movie Typically caused by: Ground truth Shadows light source flickering self-adaptive p cameras different viewing angles and non-lambertian surfaces Horn-Schunck (7.17 ) Part II - 13 Image features for matching Matching the gradient Gray value constancy: matching the gray value of a single pixel Gradient constancyy assumption Linearization leads to the well-known data term Two important positive effects: 1 More information: two more equations Invariant to additive brightness changes We can make use of the pixel s color: Negative effects: 1. Not rotation invariant 2. More noise sensitive Part II - 14 (Uras et al. 1988, Schnörr 1994, Brox et al. 2004) Does D it make k sense tto use more complex l iimage ffeatures? t? Part II - 15 Part II - 16

5 Gradient constancy assumption Combination of features Linearization: Combine with classic optic flow constraint: New data term: Separate robust penalizer for each feature (Bruhn-Weickert 2005) automaticallyy p picks the most reliable feature Euler-Lagrange equations and numerical scheme are straightforward extensions of the versions with classic OFC Other higher order constancy assumptions feasible, but usually ll nott advantageous d t (P (Papenberg b ett al. l 2006) Can be written in compact form using the motion tensor notation Part II - 17 Result: Illumination changes no longer a problem Part II - 18 Brightness changes: alternatives 1. Structure-texture-decomposition (Aujol et al. 2005, Wedel et al. 2008) Gradient constancy (3.5 ) Both terms non-quadratic (5.97 ) Input image Ground truth Texture residual for matching Smooth the image by edge preserving diffusion (TV flow) Consider residual image for matching Horn-Schunck (7.17 ) Part II - 19 Part II - 20

6 Brightness changes: alternatives Why not using more descriptive features (Patches,SIFT,HOG)? 2. Spherical color model (HSV) (Mileva et al. 2007) Possible, but not advantageous: Hue (H) denotes the color Invariant to multiplicative illumination changes, shadows, shading, and specularities Saturation (S) is invariant to multiplicative illumination changes, shadows, and shading Value ((V)) is the actual brightness g (no invariance) with with separate robust functions Lower accuracy at motion discontinuities Descriptive power of no use in the gradient descent like optimization Exploiting descriptive features for large displacement matching needs special treatment in the optimization Liu et al. ECCV 2008 (belief propagation) Brox et al. CVPR 2009 Steinbruecker et al al. ICCV 2009 picks invariant components when needed (Zimmer et al. 2009) Part II - 21 Large displacements Part II - 22 Linearization and large displacements Constancy assumptions have been linearized Linearization only valid for very small displacements (subpixel displacements) Fails in p practice if displacements p are larger g than ~ 5 p pixels (depends on smoothness of image data) What happens if we do not linearize? (Nagel-Enkelmann 1986, Alvarez et al. 2000) Horn&Schunck Brox et al Part II - 24 Part II - 25

7 Non-linearized constancy assumption Non-linearized constancy assumption Energy functional: Euler-Lagrange equations: Correct description of the matching criterion (no approximation) Two problems: 1. Not convex in multiple local minima 2. Further source of nonlinearity in Euler-Lagrange equations Highly nonlinear system of equations Another fixed point iteration loop necessary Local minima combination with a multi-scale strategy Part II - 26 Part II - 27 Image warping (dates back to Lucas-Kanade 1981) Compute expressions of type for given Gauss-Newton method Goal: solve nonlinear system: Second image (at ), warped by means of Compute Points may fall between grid points bilinear interpolation Start with an initial fixed point Linearize with a first order Taylor expansion Part II - 28 Part II - 29

8 Gauss-Newton method Yields nonlinear system in increment Solve for by an inner fixed point iteration loop (as in case of the linearized i OFC) Continuation method Energy functional has multiple local minima How to find the global minimum? Yields a new fixed point Difference to linearization in the energy functional: linearization must hold only for the increment, not the flow vector itself Heuristic: continuation method Start with a smoothed version of the energy (by smoothing the input images) Find minimum of this energy Use solution as initialization for refinement Part II - 30 Part II - 31 Optical flow with an efficient continuation method Create a pyramid of downsampled images Using downsampling factors around 0.95 instead of 0.5 increases accuracy (Brox et al. 2004) Start outer fixed point iteration loop at coarsest level In each such iteration, ti refine the level l Two birds killed with one stone: Continuation method avoids local minima Multi-scale implementation provides speedup Summary of numerical scheme (Brox et al. 2004) Two nested fixed point loops + linear solver Outer loop: Removes nonlinearity due to non-linearized constancy assumptions y y p Leads to solving for an increment in each iteration Requires warping of the second input image Coupled with coarse-to-fine strategy: First iteration starts at the coarsest level Every new iteration is executed at the next finer level Inner loop: Removes the remaining nonlinearity in the increment due to nonquadratic penalizers Lagged nonlinearity: factors kept fixed in each iteration Iterative solver for the remaining linear system Part II - 32 Part II - 33

9 Result: higher accuracy, large displacements possible Non-linearized constancy (2.44 ) Gradient constancy (3.5 ) Ground truth Horn-Schunck (7.17 ) Other results: zoom into a scene Part II - 34 Car in a movie Part II - 35 Persons in another movie Part II - 36 Part II - 37

10 Taxi sequence anno 2009 Middlebury benchmark: quantitative comparison This tutorial Variational approaches ICCV 2009 papers Part II - 38 Part II - 39 How to reach first rank on Middlebury (Zimmer et al. EMMCVPR 2009) Build upon previous Brox et al model Use HSV color model with separate robust functions: Normalization Standard data term comprises implicit weighting by image gradient Large gradients might correspond to noise Multiply with normalization factors: Normalize the linearized constraints (Simoncelli et al. 1991, Lai-Vemuri 1998, Zimmer et al. 2009) Use an anisotropic regularizer (Nagel-Enkelmann 1986, Sun et al. ECCV 2008, Zimmer et al. 2009, Werlberger et al. BMVC 2009) Part II - 41 Part II - 42

11 Anisotropic smoothness terms Jointly image- and flow driven regularizer (Sun et al. 2008) Basic idea: reduced/no smoothing across edges, strong smoothing along edges (Nagel-Enkelmann 1986) Smoothing direction defined by structure tensor (as before) Image-driven: I di edge d iinformation f ti ffrom iimage cues Amount of smoothing defined by flow structure tensor eigen decomposition Penalize motion discontinuities onlyy along g image g edges: g Rotationally invariant version (Zimmer et al. al 2009) Drawback: no smoothing across internal edges Part II - 43 Complementary regularization (Zimmer et al. 2009) Part II - 44 Urban3 sequence from Middlebury dataset Directional information from motion tensor (used in data term) Motion tensor Amount of smoothing across edge from flow, full smoothing along edge: RGB instead of HSV Part II - 45 Ground truth Zimmer et al. without normalization li ti structure tensor iinstead t d off motion ti tensor Part II - 46

12 Still problematic: Fast motion of small structures Incorporating correspondences from descriptor matching BruhnWeickert 2005 Warping method Brox et al. CVPR 2009 point correspondences by descriptor matching Multiple hypotheses Matching score Robust function Flow = correspondence vector Brox et al. CVPR 2009 Warping method Brox et al. CVPR 2009 Part II - 47 Coarse-to-fine optimization Part II - 48 Ball example from Middlebury dataset Evolution over scales With point correspondences Bruhn Bruhn-Weickert BroxWeickert et al al. CVPR ICCV Without Part II - 49 Part II - 50

13 Analyzing challenging sport sequences Comparison of ways to compute large displacement flow Interpolated region correspondences Part II - 51 Shot from a movie Brox et al. ECCV 2004 Brox et al. CVPR 2009 SIFT flow Liu et al. ECCV 2008 Part II - 52 Other modeling aspects not considered here Spatio-temporal smoothness (Nagel ECCV 1990, Weickert-Schnörr JMIV 2001) Process a whole image sequence at once Enforce smoothness in temporal p direction Over-parametrization (Nir et al. IJCV 2008) Represent R t fl flow vector t by b an affine ffi model d l Enforce smoothness of affine parameters Affine transformations not penalized Enforcing rigid body motion (Valgaerts et al. DAGM 2008, Wedel et al. ICCV 2009) Miss Marple Miss Marple A pocket A pocket full of rye full of (slow rye motion) Estimate fundamental matrix from optical flow Soft constraint: correspondences close to epipolar lines Part II - 53 Part II - 54