1. © 2010 by Davi GeigerComputer Vision March 2010 L1.1 Binocular Stereo Left Image Right Image

2. © 2010 by Davi GeigerComputer Vision March 2010 L1.2 There are various different methods of extracting relative depth from images, some of the “passive ones” are based on (i) relative size of known objects, (ii)texture variations, (iii) occlusion cues, such as presence of T-Junctions, (iv) motion information, (v) focusing and defocusing, (vi) relative brightness Moreover, there are active methods such as (i)Radar, which requires beams of sound waves or (ii)Laser, uses beam of light Stereo vision is unique because it is both passive and accurate. Binocular Stereo

3. © 2010 by Davi GeigerComputer Vision March 2008 L1.3 Julesz’s Random Dot Stereogram. The left image, a black and white image, is generated by a program that assigns black or white values at each pixel according to a random number generator. The right image is constructed from by copying the left image, but an imaginary square inside the left image is displaced a few pixels to the left and the empty space filled with black and white values chosen at random. When the stereo pair is shown, the observers can identify/match the imaginary square on both images and consequently “see” a square in front of the background. It shows that stereo matching can occur without recognition. Human Stereo: Random Dot Stereogram

4. © 2010 by Davi GeigerComputer Vision March 2008 L1.4 Not even the identification/matching of illusory contour is known a priori of the stereo process. These pairs gives evidence that the human visual system does not process illusory contours/surfaces before processing binocular vision. Accordingly, binocular vision will be thereafter described as a process that does not require any recognition or contour detection a priori. Human Stereo: Illusory Contours Here not only illusory figures on left and right images don’t match, but also stereo matching yields illusory figures not seen on either left or right images alone. Stereo matching occurs in the presence of illusory.

5. © 2010 by Davi GeigerComputer Vision March 2008 L1.5 Human Stereo: Half Occlusions Left Right An important aspect of the stereo geometry are half-occlusions. There are regions of a left image that will have no match in the right image, and vice-versa. Unmatched regions, or half-occlusion, contain important information about the reconstruction of the scene. Even though these regions can be small they affect the overall matching scheme, because the rest of the matching must reconstruct a scene that accounts for the half-occlusion. Leonardo DaVinci had noted that the larger is the discontinuity between two surfaces the larger is the half-occlusion. Nakayama and Shimojo in 1991 have first shown stereo pair images where by adding one dot to one image, like above, therefore inducing occlusions, affected the overall matching of the stereo pair.

6. © 2010 by Davi GeigerComputer Vision March 2008 L1.6 Projective Camera Let be a point in the 3D world represented by a “world” coordinate system. Let be the center of projection of a camera where a camera reference frame is placed. The camera coordinate system has the z component perpendicular to the camera frame (where the image is produced) and the distance between the center and the camera frame is the focal length,. In this coordinate system the point is described by the vector and the projection of this point to the image (the intersection of the line with the camera frame) is given by the point, where z x P o =(X o,Y o,Z o ) p o =(x o,y o,f) f y f

7. © 2010 by Davi GeigerComputer Vision March 2008 L1.7 We have neglected to account for the radial distortion of the lenses, which would give additional intrinsic parameters. Equation above can be described by the linear transformation where the intrinsic parameters of the camera,, represent the size of the pixels (say in millimeters) along x and y directions, the coordinate q x, q y in pixels of the image (also called the principal point) and the focal length of the camera. Projective Camera and Image Coordinate System x y O pixel coordinates

8. © 2010 by Davi GeigerComputer Vision March 2008 L1.8 A 3D point P, view in the cyclopean coordinate system, projected on both cameras. The same point P described by a coordinate system in the left eye is P l and described by a coordinate system in the right eye is P r. The translation vector T brings the origin of one the left coordinate system to the origin of the right coordinate system. Two Projective Cameras OlOl xlxl ylyl P=(X,Y,Z) p l =(x o,y o,f) xrxr yryr zrzr OrOr zlzl f p r =(x o,y o,f) f

9. © 2010 by Davi GeigerComputer Vision March 2008 L1.9 OlOl xlxl ylyl P=(X,Y,Z) p l =(x o,y o,f) xrxr yryr zrzr OrOr zlzl f p r =(x o,y o,f) f The transformation of coordinate system, from left to right is described by a rotation matrix R and a translation vector T. More precisely, a point P described as P l in the left frame will be described in the right frame as Two Projective Cameras: Transformations OlOl xlxl ylyl P=(X,Y,Z) p l =(x o,y o,f) xrxr yryr zrzr OrOr zlzl f p r =(x o,y o,f) f

10. © 2010 by Davi GeigerComputer Vision March 2008 L1.10 OlOl xlxl ylyl P=(X,Y,Z) xrxr yryr zrzr elel epipolar lines OrOr zlzl p r =(x o,y o,f) erer Each 3D point P defines a plane. This plane intersects the two camera frames creating two corresponding epipolar lines. The line will intersect the camera planes at and, known as the epipoles. The line is common to every plane PO l O l and thus the two epipoles belong to all pairs of epipolar lines (the epipoles are the “center/intersection” of all epipolar lines.) p l =(x o,y o,f) Two Projective Cameras Epipolar Lines

11. © 2010 by Davi GeigerComputer Vision March 2008 L1.11 Estimating Epipolar Lines and Epipoles The two vectors,, span a 2 dimensional space. Their cross product,, is perpendicular to this 2 dimensional space. Therefore where F is known as the fundamental matrix and needs to be estimated

12. © 2010 by Davi GeigerComputer Vision March 2008 L1.12 “Eight point algorithm”: (i)Given two images, we need to identify eight points or more on both images, i.e., we provide n  8 points with their correspondence. The points have to be non-degenerate. (ii)Then we have n linear and homogeneous equations with 9 unknowns, the components of F. We need to estimate F only up to some scale factors, so there are only 8 unknowns to be computed from the n  8 linear and homogeneous equations. (i) If n=8 there is a unique solution (with non-degenerate points), and if n > 8 the solution is overdetermined and we can use the SVD decomposition to find the best fit solution. Computing F (fundamental matrix)

13. © 2010 by Davi GeigerComputer Vision March 2008 L1.13 Each potential match is represented by a square. The black ones represent the most likely scene to “explain” the images, but other combinations could have given rise to the same images (e.g., red) Stereo Correspondence: Ambiguities What makes the set of black squares preferred/unique is that they have similar disparity values, the ordering constraint is satisfied and there is a unique match for each point. Any other set that could have given rise to the two images would have disparity values varying more, and either the ordering constraint violated or the uniqueness violated. The disparity values are inversely proportional to the depth values

14. © 2010 by Davi GeigerComputer Vision March 2010 L1.14 AB C D E F A B A C D D C F F E Stereo Correspondence: Matching Space Right boundary no match Boundary no match Left depth discontinuity Surface orientation discontinuity A B C D F FED CAFED CA Note 2: Due to pixel discretization, points A and C in the right frame are neighbors. Note 1: Depth discontinuities and very tilted surfaces can/will yield the same images ( with half occluded pixels) In the matching space, a point (or a node) represents a match of a pixel in the left image with a pixel in the right image Left Right

15. © 2010 by Davi GeigerComputer Vision March 2008 L1.15 Cyclopean Eye The cyclopean eye “sees” the world in 3D where x represents the coordinate system of this eye and w is the disparity axis For manipulating with integer coordinate values, one can also use the following representation Restricted to integer values. Thus, for l,r=0,…,N-1 we have x=0,…2N-2 and w=-N+1,.., 0, …, N-1 Note: Not every pair (x,w) have a correspondence to (l,r), when only integer coordinates are considered. For “x+w even” we have integer values for pixels r and l and for “x+w odd” we have supixel locations. Thus, the cyclopean coordinate system for integer values of (x,w) includes a subpixel image resolution w w=2 Right Epipolar Line l-1 l=3 l+1 r+1 r=5 r-1 x

16. © 2010 by Davi GeigerComputer Vision March 2008 L1.16 The Uniqueness-Opaque Constraint There should be only one disparity value, one depth value, associated to each cyclopean coordinate x (see figure). The assumption is that objects are opaque and so a 3D point P, seen by the cyclopean coordinate x and disparity value w will cause all other disparity values not to be allowed. Closer points than P, along the same x coordinate, must be transparent air and further away points will not be seen since P is already been seen (opaqueness). However, multiple matches for left eye points or right eye points are allowed. This is indeed required to model tilt surfaces and occlusion surfaces as we will later discuss. This constraint is a physical motivated one and is easily understood in the cyclopean coordinate system. Uniqueness-Opaque Left Epipolar Line w w=2 Right Epipolar Line x x=8 r+1 r=5 r-1 l-1 l=3 l+1 NO: Uniqueness x=8 YES, multiple match for the right eye YES, multiple match for the left eye NO: Uniqueness Given that the l=3 and r=5 are matched (blue square), then the red squares represent violations of the uniqeness-opaqueness constraint while the yellow squares represent unique matches, in the cyclopean coordinate system but multiple matches in the left or right eyes coordinate system.

17. © 2010 by Davi GeigerComputer Vision March 2010 L1.17 Left Epipolar Line w w=2 Right Epipolar Line x Smoothness Surface Constraints I Smoothness : In nature most surfaces are smooth in depth compared to their distance to the observer, but depth discontinuities also occur. r+1 r=5 r-1 l-1 l=3 l+1 x=8 YES Given that the l=3 and r=5 are matched (blue square), then the red squares represent violations of the ordering constraint while the yellow squares represent smooth matches.

18. © 2010 by Davi GeigerComputer Vision March 2010 L1.18 Left Epipolar Line w w=2 Right Epipolar Line x r+1 r=5 r-1 l-1 l=3 l+1 x=8 Ordering Violation Discontinuities: Note that in these cases, some pixels will not be matched to any pixel, e.g., “l+1”, and other pixels will have multiple matches, e.g., “r-1”. In fact, the number of pixels unmatched in the left image is the same as the number of multiple matches in the right image. Surface Constraints: Discontinuities and Occlusions x w l r

19. © 2010 by Davi Geiger ? w’=w-2 w’=w+2 x x’=x-2-|w’-w| x Neighborhood structure for a node (e,xw) consisting of flat, tilt, or occluded surfaces. Note that when an occlusion/discontinuity occurs, the contrast matches on the front surface. Jumps “at the right eye” are from back to front, while jumps “at the left eye” are from front to the back. X ? w’=w-2 w’=w+2 x’=x-2-|w’-w| x x x l l l l r r r r x w w Neighborhood (with no subpixel accuracy)

20. © 2010 by Davi GeigerComputer Vision March 2008 L1.20 Limit Disparity The search is within a range of disparity : 2D+1, i.e., The rational is: (i)Less computations (ii)Larger disparity matches imply larger errors in 3D estimation. (iii)Humans only fuse stereo images within a limit. It is called the Panum’s limit. We may start the computations at x=D to avoid limiting the range of w values. In this case, we also limit the computations to up to x=2N-2-D

21. © 2010 by Davi GeigerComputer Vision March 2008 L1.21 Bayesian Formulation The probability of a surface w(x,e) to account for the left and right image can be described by the Bayes formula as where e index the epipolar lines. Let us develop formulas for both probability terms on the numerator. The denominator can be computed as the normalization constant to make the probability sum to 1.

22. © 2010 by Davi GeigerComputer Vision March 2008 L1.22 P(e,x,w) Є [0,1], for x+w even, represents how similar the images are between pixels (e,l) in the left image and (e,r) in the right image, given that they match. We use “left” and “right” windows to account for occlusions. The Image Formation I (special case) left right

23. © 2010 by Davi GeigerComputer Vision March 2008 L1.23 We expand the previous formula to include matching of windows that have other orientations than just q =0 and we expand the formula to any integer value of x+w. Note that when x+w is odd, the coordinates l and r are half integers and so an interpolation/average value for the intensity values need to be computed. For half integer values of l and r we use the formulas The Image Formation I ( intensity ) where is the floor value of x.

24. © 2010 by Davi GeigerComputer Vision March 2008 L1.24 Occlusions are regions where no feature match occur. In order to consider them we introduce an “occlusion field” O(e,x) which is 1 if an occlusion occurs at column (e,x) and zero otherwise. Thus, the likelihood of left and right images given the feature match must take into account if an occlusion occurs. We modify the data probability to include occlusions. The Image Formation II (Occlusions) where O is an occlusion binary variable The cost is introduced as a prior to encourage matches, otherwise it is better to occlude everything.

25. © 2010 by Davi GeigerComputer Vision March 2010 L1.25 w=2 x l=1 l=2 l=3 r=6 r=5 r=4 r=3 x=9 X X The Image Formation III (Occluded Surfaces) X X X Example of a jump/occlusion, w=2 and w’=-1. Three x coordinates will have O(e,x)=1 O(e,x)=1 where and

26. © 2010 by Davi GeigerComputer Vision March 2010 L1.26 Computer Vision March 2010 L1.26 The Image Formation IV (Tilted Surfaces) w=2 x l=1 l=2 l=3 r=6 r=5 r=4 r=3 x=9 X X X X X Example of a jump/occlusion, w=2 and w’=-1. Three x coordinates will have O(e,x)=1 O(e,x)=1 where and

27. © 2010 by Davi GeigerComputer Vision March 2008 L1.27 x Posterior Model If there is a match at (e,xw) then either it is a flat/tilt surface or it is an occluded surface Neighborhood structure for a node (e,xw) consisting of flat, tilt, or occluded surfaces

28. © 2010 by Davi GeigerComputer Vision March 2008 L1.28 Computer Vision March 2008 L1.28 x Posterior Model (with no subpixel accuracy) Neighborhood structure for a node (e,xw) consisting of flat, tilt, or occluded surfaces ? X ? w’=w-2 w’=w+2 x w=D w=-D x’=x-2-|w’-w| x

29. © 2010 by Davi GeigerComputer Vision March 2008 L1.29 Need for prior model: Flat or (Double) Tilted ? w left right left right The data and the probabilities for (a) and (b) are the same. Since we are not considering curvature preferences, the preference for flat surfaces must be built as a prior model of surfaces. w=2 x l=1 l=2 l=3 r=5 r=4 r=3 x=8 X X a.Flat Plane b.Doubled Tilted left right Tilted left right Flat w=2 x l=1 l=2 l=3 The images and the probabilities for (a) and (b) are the same. a.Flat Plane b.Tilted r=5 r=4 r=3 r=2 x=8 X X X X

30. © 2010 by Davi GeigerComputer Vision March 2008 L1.30 Computer Vision March 2008 L1.30 w=D w w=-D ? States (2D+1) ? 1 2 3 x’ … x-1 x... 2N-2 ? Dynamic Programming (pixel-pixel matching only) F[x, w+D, i+1 ]  x[x’,w(x’)+D]  x,w+D]= C [ x, w+D] + min x’=-D-1,..,D+1 {  [x’,w(x’)+D]+ F [x,x’,w,w’]}