Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition

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1 Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition VO KU 1

2 Optical Flow (I) Content Introduction Local approach (Lucas Kanade) Global approaches (Horn-Schunck, TV-L1) Coarse-to-fine warping 2

3 What is Optical Flow? Optical Flow is a major task of every biological and artificial visual system Is the aparent motion in images sequenzes. Can be seen as a velocity field that transforms one image to the next image in a sequence 3

4 The apparent motion Optical flow is not the true 3D motion of the objects It is the 3D motion projected to the camera plane The true 3D motion is called the scene flow and additionally requires 3D information of the scene 4

5 5

6 Applications of Optical Flow Tracking Video Compression (Recent MPEG Standard) 3D Reconstruction (Stereo) Segmentation Object Detection Video Interpolation in time (The Matrix) 6

7 History Computing Optical Flow started in the early 80 s and is still a hot research topic 1980: Horn and Schunck (global approach) 1981: Lucas and Kanade (local approach) 1989: Shulman and Herve (discontinuity preserving) 1993: Black and Anandan (robust optical flow) 1999: Alvarez et al. (PDE model) 2003: Bruhn et al. (realtime optical flow using mulitgrid) 2004: Brox et al. (high accuracy using warping) 2007: Zach, Pock, Bischof: (duality based minimization) 7

8 Evaluation For a long time, the so-called Yosemite sequence (Barron et al. 1994) was used to evaluate the algorithms Yosemite Yosemite with clouds 8

9 Evaluation Middlebury optical flow benchmark (Baker et al. 2007) 9

10 Evaluation SINTEL open source synthetic movie, Butler et al

11 Basic assumptions Brightness constancy assumption The intensities remain constant, although the location might change. Problems by changing illumination Can be generalized to a feature constancy assumption Spatial coherence assumption Neighbouring pixels are likely to have the same motion Difficult to find a good model Temporal persistence Motion changes gradually over time Only useful in case of high frame rates (small motion) 11

12 Brightness constancy assumption We assume that intensity patterns only change their positions frame t frame t+1 12

13 Difficulties of the brightnes constancy assumption Aperture Problem: Only the normal flow can be estimated Untextured areas: No information in untextured areas 13

14 Changing illumination A changing illumination induces an optical flow that does not correspond to the motion of the object A changing illumination causes an optical flow although the object does not move 14

15 What to do if the brightness constancy assumption is violated? If the illumination changes from frame to frame, the brightness constancy assumption may be violated A simple idea is to perform a structure-texture decomposition Although the absolute intensity values might change, the texture part stays the same 1. Low-pass filter the images (e.g. total variation smoothing) 2. Subtract the low-pass filtered images from the original images 3. Work with the resulting images 15

16 Structure - Texture Decomposition 16

17 Structure - Texture Decomposition 17

18 Beyond the brightness constancy assumption What can we do if we want to compute the optical flow between such images? We can work on feature transforms such as SIFT Each pixel in the images is replaced by its SIFT feature. SIFT-Flow [Liu, Yuen, Torralba, 2011] 18

19 Spatial coherence assumption Describes the a-priori assumption about flow fields Can be learned from statistics of natural flow fields without spatial coherence with spatial coherence 19

20 Temporal persistency Idea: The motion of objects does not suddenly change for example, one can assume linear motion model between three frames Can be generalized to higher-order models Sometimes, it does not improve too much in practice 20

21 The optical flow constraint (OFC) Brightness constancy assumption Taylor development leads to the linearized Brightness Constancy Assumption Error function over the whole image Underdetermined problem 21

22 Warping In practice, we are given two images of an image sequence and an often an initial flow field The differential quantities are computed by Warping: Geometrically transform the second image by Compute on the warped second image 22

23 Linearization of the Image 23

24 The Linearized Data Term (p=2) 24

25 The Linearized Data Term (p=1) 25

26 General outlook We will now discuss three methods Lukas Kanade method Horn Schunck method TV-L1 method All methods can only be used to estimate small motion We will first assume that we only have small motion Large motion can be computed using a coarse-to-fine warping framework Finally we will discuss Matlab implementations of all three methods 26

27 The Lucas Kanade (LK) method 27

28 A local approach Impossible to compute the flow by only using the optical flow constraint, hence we need additional constraints Assume that all pixels in a patch have the same motion This gives the following equations for each pixel in the patch Example: 5x5 patch would give us 25 equations instead of one! A: 25x2 d: 2x1 b: 25x1 28

29 Lucas-Kanade optical flow We have more equations than unknowns Find least squares solution Solution is given by the pseudo inverse Note that the system to be solved is only 2x2 29

30 Conditions for solvability Optimal (u, v) satisfies Lucas-Kanade equation When is this system solvable? is recognized to be the structure tensor Invertible, if both eigenvalues are sufficiently larger than zero 30

31 The Lucas Kanade (LK) method Advantages Only one parameter (window size) Very fast to compute (easily realtime) Can be done dense or sparse Disadvantages Each patch is independent, no global consensus The local window assumes a constant motion Sometimes bad results 31

32 The Horn and Schunck (HS) Method 32

33 The Horn and Schunck (HS) Method Global energy to be minimized is the regularization parameter The image is of size are column vectors is a finite differences approximation of the gradient operator 33

34 Optimality conditions: The linear system The optimality condition can be re-arranged as the following linear system System is large but very sparse Suitable solvers are Gauss-Seidel, CG, or Matlab \ 34

35 The Horn and Schunck (HS) Method Advantages Easy and fast to solve due to quadratic functions Easy to implement Disadvantages Quadratic smoothnes term does not allow for sharp discontinuities in the motion field Quadratic data term does not allow for outliers in the optical flow constraint 35

36 The TV-L1 approach 36

37 The TV-L1 approach Due to the quadratic functions, the HS method does not allow for discontinuities in the flow field A good idea is to replacing the quadratic functions by norms Leads to the so-called TV-L1 approach The first term is the so-called total variation of the flow field, the second term is the norm of the OFC 37

38 Minimizing the TV-L1 energy The TV-L1 energy is convex but non-differentiable Standard gradient descent cannot be applied We can rely on recent advances in convex optimization We can apply two strategies: Smooth the total variation term and apply the FISTA algorithm Compute a saddle-point formulation of the energy and apply a primal-dual algorithm 38

39 The FISTA Algorithm Fast Iterative Shrinkage Thresholding Algorithm Proposed in 2008 by Beck and Teboulle Can be applied to the following class of convex optimization problems The function has a Lipschitz continuous gradient The function can be non-smooth but has a simple to compute proximal operator 39

40 FISTA Convergence rate: 40

41 Application to the TV-L1 energy (1) Smoothing of the TV term to make its gradient Lipschitz where denotes the Huber function, has a Lipschitz continuous gradient with 41

42 Application to the TV-L1 energy (2) The non-smooth function is given by the norm of the data term The solution of the proximal operator is given by where and 42

43 Primal-dual optimization A first-order primal-dual algorithm proposed in Chambolle, Pock, 2011 Can be used to find a saddle point of the following class of convex-concave saddle-point problems where is a linear operator, and are convex (non-smooth) functions and have simple proximal operators Corresponds to a saddle-point formulation of the primal and dual problems 43

44 The algorithm Step sizes:, Proximal operators 44

45 Saddle-point formulation of TV-L1 The total variation can be written as (convex conjugate) The TV-L1 optical flow model is written as with Exactly falls into the class of the primal-dual algorithm 45

46 The proximal operators The proximal operator with respect to is given by the same soft-shrinkage formula as before The proximal operator with respect to is a projection onto the Euclidean ball 46

47 The TV-L1 approach Advantages TV regularization allows for discontinuities in the flow field The L1 data term allows for outliers (occlusions) Reasonable fast to compute (GPU leads to realtime) Disadvantages The method requires an iterative solver to compute the minimizer (FISTA, or Primal-Dual) FISTA requires to smooth the TV Primal-Dual can also deal with the pure TV Hard to find a good stopping criterion 47

48 Coarse-to-fine warping framework Due to the restrictions of the OFC, the discussed methods are only able to recover small motion How to extend the method for large motion? 1. Solving, warping, re-linearization, solving, Implement the method on an image pyramide 48

49 Coarse-to-fine warping framework Compute image pyramids init Initialize the flow field solve Transform the moving image by the given flow field (warping) Perform linearization Apply one of the three methods (LK, HS, TV-L1) prolongate Initialize the flow field on the next finer level using interpolation and rescaling of the motion vectors 49

50 Coarse-to-fine warping framework Advantages Allows to compute large motion Due to the logarithmic nature of pyramids, not much more to compute Disadvantages Large motion that is not captured at a coarse scale cannot be found on finer levels Dilemma: Large motion of small objects Different interpolation, rescaling, filtering schemes lead to different results Can lead to unstable results in practice BUT: What is the alternative? 50

51 Comparison of the three methods In the following, we will show an comparison of the three methods we covered so far We use the Army sequence of the Middlebury benchmark Input frames Ground truth flow 51

52 Structure-texture decomposition The data set contains changing illumination and shadows 52

53 Lucas-Kanade 53

54 Horn-Schunck 54

55 TV-L1 55

56 Real-time implementation Implementation of the TV-L1 method on a GPU allows for real-time computation 56

57 Flow Games Jakob Santner et al. 57

58 Feature Flow How can we compute the motion between challenging sequences Large displacements Different modalities Image taken at different time points Images from objects of the same category 58

59 Feature Flow The idea is simple Replace each pixel in both images by a feature vector SIFT descriptor LBP Feature constancy assumption Linearize each feature channel individually 59

60 Some results using SIFT 60

61 Some results using SIFT 61

62 Some results using SIFT 62

63 Some results using SIFT 63

64 Some results using SIFT 64

65 Some results using SIFT 65

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