1. © 2006 by Davi GeigerComputer Vision April 2006 L1.1 Binocular Stereo Left Image Right Image

2. © 2006 by Davi GeigerComputer Vision April 2006 L1.2 Each potential match is represented by a square. The black ones represent the most likely scene to “explain” the image, but other combinations could have given rise to the same image (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

3. © 2006 by Davi GeigerComputer Vision April 2006 L1.3 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 F D C B A AC DEFAC DEF 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

4. © 2006 by Davi GeigerComputer Vision April 2006 L1.4 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. Indeed, the integer coordinate system (x,w) exhibit subpixel accuracy. For x+w even we have integer values for pixels r and l and for x+w odd we have supixel locations. x

5. © 2006 by Davi GeigerComputer Vision April 2006 L1.5 Surface Constraints Smoothness : In nature most surfaces are smooth in depth compared to their distance to the observer, but depth discontinuities also occur. Usually smoothness implies an ordering constraint, where points to the right of match point to the right of Uniqueness: There should be only one disparity value associated to each cyclopean coordinate x. Note, multiple matches for left eye points or right eye points are allowed. Left Epipolar Line w

6. © 2006 by Davi GeigerComputer Vision April 2006 L1.6 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 Let us develop formulas for both probability terms on the numerator. The denominator is a normalization to make the probability sum to 1.

7. © 2006 by Davi GeigerComputer Vision April 2006 L1.7 C(e,x,w) Є [0,1], x+w even, represents how good is a match between a point (e,l) in the left image and a point (e,r) in the right image (where x=l+r is the cyclopean eye coordinate and w=r-l is the disparity.) The epipolar lines are indexed by e (for the homework, they are just the horizontal lines). C(e,x,w) Є [0,1], x+w odd, represents how good is a match between an edge (e,l - > l+1) in the left image and an edge (e,r ->r+1) in the right image The parameter  reduces the effect of the gradient values.

8. © 2006 by Davi GeigerComputer Vision April 2006 L1.8 w w=2 Right Epipolar Line l-1 l=3 l+1 r+1 r=5 r-1 x x=8 Epipolar interaction: the higher the intensity edges the less the cost (the higher the probability) to have disparity changes across epipolar lines

9. © 2006 by Davi GeigerComputer Vision April 2006 L1.9 w w=2 Right Epipolar Line l-1 l=3 l+1 r+1 r=5 r-1 x x=8

10. © 2006 by Davi GeigerComputer Vision April 2006 L1.10 Limit Disparity The matrix is updated only 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.

11. © 2006 by Davi GeigerComputer Vision April 2006 L1.11 Stereo Correspondence: Belief Propagation (BP) We want to obtain/compute the marginal We have finally the posterior distribution for disparity values (surface {w(x,e)}) These are exponential computations on the size of the grid N

12. © 2006 by Davi GeigerComputer Vision April 2006 L1.12 “Horizontal” Belief Tree “Vertical” Belief Tree Kai Ju’s approximation to BP We use Kai Ju’s Ph.D. thesis work to approximate the (x,e) graph/lattice by horizontal and vertical graphs, which are singly connected. Thus, exact computation of the marginal in these graphs can be obtained in linear time. We combine the probabilities obtained for the horizontal and vertical graphs, for each lattice site, by “picking” the “best” one (the ones with lower entropy, where.)

13. © 2006 by Davi GeigerComputer Vision April 2006 L1.13 Result

14. © 2006 by Davi GeigerComputer Vision April 2006 L1.14 Regi on A Regi on B Region A Left Region A Right Region B Left Region B Right Junctions and its properties (false matches that reveal information from vertical disparities (see Malik 94, ECCV) Some Issues in Stereo: