EECS 556 Image Processing W 09
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1 EECS 556 Image Processing W 09 Motion estimation Global vs. Local Motion Block Motion Estimation Optical Flow Estimation (normal equation) Man slides of this lecture are courtes of prof Milanfar (UCSC)
2 Video Images A pair of images from a video scene appears to be ver similar: f(t-1) f(t) 2
3 The Motion Model Illustration f(t-1) 3
4 The Motion Model Illustration f(t) 4
5 The Motion Model Illustration The difference image f(t) f(t-1) f(t)-f(t-1) 5
6 The Motion Model Description For a pair of such images (almost) ever pixel in one image can be found in the other in a slightl different location. f(t-1) f(t) 6
7 8 Motion estimation Problem Motion Estimation Module = ) ( ) ( ) ( t x v t x v t x v x ( ) ( ) 1 ) ( ) ( = t t x v t x v x f t x f x
8 Displacement field Vector field function of the spatio temporal image brightness variations
9 Wh Estimate Motion? 10
10 De Noising Additive noise in video sequences can be removed b spatial & temporal filtering. A tpical such filter applies a weighted average of the 3D neighborhood in order to compute the corrected pixel value. In appling temporal smoothing the correspondence information is crucial for good performance. 11
11 De Noising f(xt) f(xt-1) f(xt-2) t 12
12 Interpolation Signal in intermediate frames can be reconstructed (interpolated) Again the correspondence information is crucial 13
13 Interpolation f(xt) f(xt-1)
14 Compression In compressing video images if we have alread compressed f(t 1) we know much about f(t). Tpical approach building a predicted image for f(t) based on f(t 1). Much better compression is achieved if we appl motion compensation. 15
15 Compression Current Image Predicted Image Error Image Previous image Prediction MC previous image Prediction
16 Tracking Courtes of Jean Yves Bouguet Vision Lab California Institute of Technolog
17 Tracking Courtes of Jean Yves Bouguet Vision Lab California Institute of Technolog
18 Super resolution Given a set of low qualit images of the same object taken from slightl different locations a better (super resolution) image can be recovered. The recover process requires the knowledge of the motion with sub pixel accurac. 19
19 Super resolution Example: A set of low qualit images 20
20 Super resolution Each of these images looks like this: 21
21 Super resolution The recover result: 22
22 Other Applications There are numerous other applications for motion in images. Ever application comes with a different set of requirements regarding: Motion representation Accurac Complexit. 23
23 24 The Motion Estimation Problem Motion Estimation Module ) ( ) ( t x v t x v x ( ) ( ) 1 ) ( ) ( = t t x v t x v x f t x f x
24 Motion Representation Methods: Global: Assume global motion behavior and represent the motion b the parameters of the global mapping. Local: Define a motion vector for ever block of M b M pixels. Dense: Define a motion vector for ever pixel v(xt). 25
25 Example For an image of 1000 b 1000 pixels: Dense representation requires unknowns to be computed. Using blocks of 10 b 10 pixels we need numbers. If we assume that the motion consists of rotation and translation we need 3 numbers onl! Which is better? Depends on the application. 26
26 Global Motion Assumption In the general case we assume the existence of the following relation: x t = x p t 1 Parameters The function p is a simple relation that relates previous position to the current one. A set of parameters governs the specific tpe of motion. 27
27 Global Motion Translation x = x t t 1 + a b a=2.1 b=4.7 28
28 x Global Motion Similarit = c 0 0 cos d sin ( θ ) sin ( θ ) x a + ( θ ) cos( θ ) b t t 1 a=5 b=5 c=0.7 d=1.5 θ=10 29
29 30 Global Motion Affine + = b a x d f e c x t t 1 a=-45 b=15 c=1 d=1.3 e=0.7 f=-0.1
30 Global Motion Estimation Parametric Model Motion Estimation Module Motion Parameters 31
31 Motion Representation Methods: Global: Assume global motion behavior and represent the motion b the parameters of the global mapping. Local: Define a motion vector for ever block of M b M pixels. Dense: Define a motion vector for ever pixel v(xt). 32
32 Wh Blocks? More flexible than global but help reduce: # of unknowns to be estimated storage space Data to be transmitted (coding) # of unknowns. Solution: Divide the image to blocks and assign a motion vector per each block. 33
33 The Basics Stage 1 Divide the image f(xt) into nonoverlapping blocks of M b M pixels. B { M ( m 1) + x M ( n 1) + t } x [1 ] [ m n ][ x ] = f M { 3(2 1) + x3(2 1) + t } = f { 3 + x3 t } B[ 22][ x ] = f +
34 The Basics Stage 2 For ever block we perform a search process in the previous image f(xt 1). 35
35 The Basics Stage 3 The search in f(t 1) is done b computing a matching factor to ever possible displacement. E[ m n] = 3 3 x= 1 = 1 [ { }] 2 B [ x ] f 3 + x + m3 + + n t [ 22] 1 36
36 The Basics Stage 4 The indices which result with the smallest error E returns the (estimated) motion vector [v x v ] [ v v ] = x ArgMin [ m n] E[ m n] 37
37 Basic Assumption In the Block Matching algorithm we assume that all the pixels in ever block have the same motion vector. Hence each block is assumed to have global translational motion. We can assume more general parameteric model for the block motion such as affine. Computational complexit is an issue! 38
38 Complexit Search Zone Instead of computing E for ever possible position in f(xt 1) assume: D v v + D x This wa E is computed onl (2D+1) 2 times. 39
39 Complexit Error Comp. Instead of defining the error E as a Mean Square Error define it as Mean Absolute Differences: E[ m n] = 3 3 x= 1 = 1 { 3+ x + m3 + + n t } B[ 22] [ x ] f 1 40
40 Complexit Thresholding Start b computing E[00] and if it is smaller than some threshold assume no motion and do not continue: E[00] 3 3 = x= 1 = 1 { 3+ x3 + t } T B[ 22] [ x ] f 1 41
41 Complexit Spiral Search Start from the [00] and compare E[uv] (while computed) to the minimum so far. If bigger no need to continue. E[ m n] = 3 3 x= 1 = 1 { 3+ x + m3 + + n t } B[ 22] [ x ] f 1 42
42 Example v8o1bv6a0
43 Applications Block Motion Estimation (BME) is extensivel used in video coding (MPEG1 MPEG2 MPEG4 H261 & H263). Tpicall blocks of 8 b 8 or 16 b 16 are used. In regular video coding motion estimation consumes 50 80% of the CPU time. 46
44 Motion Representation Methods: Global: Assume global motion behavior and represent the motion b the parameters of the global mapping. Local: Define a motion vector for ever block of M b M pixels. Dense: Define a motion vector for ever pixel (optical flow) 47
45 Estimating optical flow I(xt 1) I(xt) Given two subsequent frames estimate the apparent motion field u(x) v(x) between them Ke assumptions Brightness constanc: projection of the same point looks the same in ever frame Small motion: points do not move ver far Spatial coherence: points move like their neighbors
46 t x I x v I x u I t x I t u u x I ) ( ) ( 1) ( ) ( Brightness Constanc Equation: ) ( 1) ( ) ( ) ( t x x v u x I t x I + + = Linearizing the right side using Talor expansion: The brightness constanc constraint I(xt 1) I(xt) t x I v I u I Hence Image derivative along x [ ] 0 I v u I t T = + t x I x v I x u I t x I t u u x I + + = + + ) ( ) ( 1) ( ) (
47 The brightness constanc constraint Can we use this equation to recover image motion (uv) at each pixel? I [ ] T u v + I = 0 How man equations and unknowns per pixel? One equation (this is a scalar equation!) two unknowns (uv) The component of the flow perpendicular to the gradient (i.e. parallel to the edge) cannot be measured t If (u v ) satisfies the equation so does (u+u v+v ) if I [ u' v' ] T = 0 gradient (uv) (u+u v+v ) (u v ) edge
48 The aperture problem Actual motion
49 The aperture problem Perceived motion
50 The barber pole illusion
51 The barber pole illusion
52 Aperture problem cont d 55 * From Marc Pollefes COMP
53 Solving the ambiguit B. Lucas and T. Kanade. An iterative image registration technique with an application to stereo vision. In Proceedings of the International Joint Conference on Artificial Intelligence pp How to get more equations for a pixel? Spatial coherence constraint: Assume the pixel s neighbors have the same (uv) If we use a 5x5 window that gives us 25 equations per pixel
54 Solving the ambiguit Least squares problem: When is this sstem solvable? What if the window contains just a single straight edge?
55 Conditions for solvabilit Bad case: single straight edge
56 Lucas Kanade flow Overconstrained linear sstem Least squares solution for d given b The summations are over all pixels in the K x K window
57 Conditions for solvabilit Optimal (u v) satisfies Lucas Kanade equation When is This Solvable? A T A should be invertible A T A should not be too small due to noise eigenvalues λ 1 and λ 2 of A T A should not be too small A T A should be well conditioned λ 1 / λ 2 should not be too large (λ 1 = larger eigenvalue) Does this remind anthing to ou?
58 M = A T A is the second moment matrix! (Harris corner detector ) Eigenvectors and eigenvalues of A T A relate to edge direction and magnitude The eigenvector associated with the larger eigenvalue points in the direction of fastest intensit change The other eigenvector is orthogonal to it
59 Interpreting the eigenvalues Classification of image points using eigenvalues of the second moment matrix: λ 2 Edge λ 2 >> λ 1 Corner λ 1 and λ 2 are large λ 1 ~ λ 2 λ 1 and λ 2 are small Flat region Edge λ 1 >> λ 2 λ 1
60 Edge gradients ver large or ver small large λ 1 small λ 2
61 Low texture region gradients have small magnitude small λ 1 small λ 2
62 High texture region gradients are different large magnitudes large λ 1 large λ 2
63 What are good features to track? Can measure qualit of features from just a single image Hence: tracking Harris corners (or equivalent) guarantees small error sensitivit! Implemented in Open CV
64 Recap Ke assumptions (Errors in Lucas Kanade) Small motion: points do not move ver far Brightness constanc: projection of the same point looks the same in ever frame Spatial coherence: points move like their neighbors
65 Is small motion assumption correct? Is this motion small enough? Probabl not it s much larger than one pixel (2 nd order terms dominate) How might we solve this problem? * From Khurram Hassan-Shafique CAP5415 Computer Vision 2003
66 Aliasing Temporal aliasing causes ambiguities in optical flow because images can have man pixels with the same intensit. I.e. how do we know which correspondence is correct? actual shift estimated shift nearest match is correct (no aliasing) nearest match is incorrect (aliasing) 69
67 Coarse to fine estimation * From Khurram Hassan-Shafique CAP5415 Computer Vision 2003
68 Next lecture Interpolation methods Splines
69
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