Computer Vision Multiple View Geometry Stereo Marc Pollefeys

Computer Vision Multiple View Geometry & Stereo Marc Pollefeys COMP 256

Computer Vision Last class: epipolar geometry p Underlying structure in set of matches for rigid scenes C 1 L 2 p 1 M L 1 l T 1 l 2 e 1 e 2 Fundamental matrix (3 x 3 rank 2 matrix) Canonical representation: p 2 l 2 C 2 1. Computable from corresponding points 2. Simplifies matching 3. Allows to detect wrong matches 4. Related to calibration

Tentative class schedule Computer Vision Aug 26/28 - Introduction Cameras Radiometry Sources & Shadows Color Sep 16/18 Linear filters & edges (Isabel hurricane) Sep 23/25 Pyramids & Texture Multi-View Geometry Stereo Project proposals - Optical flow Oct 14/16 Tracking - Oct 21/23 Silhouettes/carving Structure from motion Oct 28/30 - Camera calibration Project update Segmentation Nov 11/13 Fitting Probabilistic segm. &fit. Nov 18/20 Matching templates Matching relations Nov 25/27 Range data (Thanksgiving) Final project Sep 2/4 Sep 9/11 Sep 30/Oct 2 Oct 7/9 Nov 4/6 Dec 2/4

Computer Vision Multiple Views (Faugeras and Mourrain, 1995)

Computer Vision Two Views Epipolar Constraint

Computer Vision Three Views Trifocal Constraint

Computer Vision Four Views Quadrifocal Constraint (Triggs, 1995)

Computer Vision Geometrically, the four rays must intersect in P. .

Computer Vision Quadrifocal Tensor and Lines

Computer Vision Quadrifocal tensor • determinant is multilinear thus linear in coefficients of lines ! • There must exist a tensor with 81 coefficients containing all possible combination of x, y, w coefficients for all 4 images: the quadrifocal tensor

Computer Vision STEREOPSIS • The Stereopsis Problem: Fusion and Reconstruction • Human Stereopsis and Random Dot Stereograms • Cooperative Algorithms • Correlation-Based Fusion • Multi-Scale Edge Matching • Dynamic Programming • Using Three or More Cameras Reading: Chapter 11.

Computer Vision An Application: Mobile Robot Navigation The INRIA Mobile Robot, 1990. The Stanford Cart, H. Moravec, 1979. Courtesy O. Faugeras and H. Moravec.

Computer Vision Reconstruction / Triangulation

Computer Vision (Binocular) Fusion

Computer Vision Reconstruction • Linear Method: find P such that • Non-Linear Method: find Q minimizing

Computer Vision Rectification All epipolar lines are parallel in the rectified image plane.

Computer Vision Image pair rectification simplify stereo matching by warping the images Apply projective transformation so that epipolar lines correspond to horizontal scanlines e e map epipole e to (1, 0, 0) try to minimize image distortion problem when epipole in (or close to) the image

Computer Vision

Computer Vision

Computer Vision Polar rectification (Pollefeys et al. ICCV’ 99) Polar re-parameterization around epipoles Requires only (oriented) epipolar geometry Preserve length of epipolar lines Choose so that no pixels are compressed original image Works for all relative motions Guarantees minimal image size rectified image

Computer Vision polar rectification: example

Computer Vision polar rectification: example

Computer Vision Example: Béguinage of Leuven Does not work with standard Homography-based approaches

Computer Vision Example: Béguinage of Leuven

Computer Vision Reconstruction from Rectified Images Disparity: d=u’-u. Depth: z = -B/d.

Computer Vision Stereopsis Figure from US Navy Manual of Basic Optics and Optical Instruments, prepared by Bureau of Naval Personnel. Reprinted by Dover Publications, Inc. , 1969.

Computer Vision Human Stereopsis: Reconstruction d=0 Disparity: d = r-l = D-F. In 3 D, the horopter. d<0

Computer Vision Human Stereopsis: experimental horopter…

Computer Vision Iso-disparity curves: planar retinas X 0 X 1 Xi Xj C 1 C 2 X∞ the retina act as if it were flat!

Computer Vision Human Stereopsis: Reconstruction What if F is not known? Helmoltz (1909): • There is evidence showing the vergence angles cannot be measured precisely. • Humans get fooled by bas-relief sculptures. • There is an analytical explanation for this. • Relative depth can be judged accurately.

Computer Vision Human Stereopsis: Binocular Fusion How are the correspondences established? Julesz (1971): Is the mechanism for binocular fusion a monocular process or a binocular one? ? • There is anecdotal evidence for the latter (camouflage). • Random dot stereograms provide an objective answer BP!

Computer Vision A Cooperative Model (Marr and Poggio, 1976) Excitory connections: continuity Inhibitory connections: uniqueness Iterate: C = S Ce - w. S C i + C 0. Reprinted from Vision: A Computational Investigation into the Human Representation and Processing of Visual Information by David Marr. 1982 by David Marr. Reprinted by permission of Henry Holt and Company, LLC.

Computer Vision Correlation Methods (1970 --) Slide the window along the epipolar line until w. w’ is maximized. Normalized Correlation: minimize instead. Minimize |w-w’|. 2

Computer Vision Correlation Methods: Foreshortening Problems Solution: add a second pass using disparity estimates to warp the correlation windows, e. g. Devernay and Faugeras (1994). Reprinted from “Computing Differential Properties of 3 D Shapes from Stereopsis without 3 D Models, ” by F. Devernay and O. Faugeras, Proc. IEEE Conf. on Computer Vision and Pattern Recognition (1994). 1994 IEEE.

Computer Vision Multi-Scale Edge Matching (Marr, Poggio and Grimson, 1979 -81) • Edges are found by repeatedly smoothing the image and detecting the zero crossings of the second derivative (Laplacian). • Matches at coarse scales are used to offset the search for matches at fine scales (equivalent to eye movements).

Computer Vision Multi-Scale Edge Matching (Marr, Poggio and Grimson, 1979 -81) One of the two input images Image Laplacian Zeros of the Laplacian Reprinted from Vision: A Computational Investigation into the Human Representation and Processing of Visual Information by David Marr. 1982 by David Marr. Reprinted by permission of Henry Holt and Company, LLC.

Computer Vision Multi-Scale Edge Matching (Marr, Poggio and Grimson, 1979 -81) Reprinted from Vision: A Computational Investigation into the Human Representation and Processing of Visual Information by David Marr. 1982 by David Marr. Reprinted by permission of Henry Holt and Company, LLC.

Computer Vision The Ordering Constraint In general the points are in the same order on both epipolar lines. But it is not always the case. .

Computer Vision Dynamic Programming (Baker and Binford, 1981) Find the minimum-cost path going monotonically down and right from the top-left corner of the graph to its bottom-right corner. • Nodes = matched feature points (e. g. , edge points). • Arcs = matched intervals along the epipolar lines. • Arc cost = discrepancy between intervals.

Computer Vision Dynamic Programming (Ohta and Kanade, 1985) Reprinted from “Stereo by Intra- and Intet-Scanline Search, ” by Y. Ohta and T. Kanade, IEEE Trans. on Pattern Analysis and Machine Intelligence, 7(2): 139 -154 (1985). 1985 IEEE.

Computer Vision Three Views The third eye can be used for verification. .

Computer Vision More Views (Okutami and Kanade, 1993) Pick a reference image, and slide the corresponding window along the corresponding epipolar lines of all other images, using inverse depth relative to the first image as the search parameter. Reprinted from “A Multiple-Baseline Stereo System, ” by M. Okutami and T. Kanade, IEEE Trans. on Pattern Analysis and Machine Intelligence, 15(4): 353 -363 (1993). copyright 1993 IEEE. Use the sum of correlation scores to rank matches.

Computer Vision Stereo matching Similarity measure (SSD or NCC) Optimal path (dynamic programming ) Constraints • epipolar • ordering • uniqueness • disparity limit • disparity gradient limit Trade-off • Matching cost (data) • Discontinuities (prior) (Cox et al. CVGIP’ 96; Koch’ 96; Falkenhagen´ 97; Van Meerbergen, Vergauwen, Pollefeys, Van. Gool IJCV‘ 02)

Computer Vision Hierarchical stereo matching Allows faster computation Disparity propagation (Gaussian pyramid) Downsampling Deals with large disparity ranges (Falkenhagen´ 97; Van Meerbergen, Vergauwen, Pollefeys, Van. Gool IJCV‘ 02)

Computer Vision image I(x, y) Disparity map D(x, y) (x´, y´)=(x+D(x, y) image I´(x´, y´)

Computer Vision Example: reconstruct image from neighboring images

Computer Vision I 1 I 2 I 10 Reprinted from “A Multiple-Baseline Stereo System, ” by M. Okutami and T. Kanade, IEEE Trans. on Pattern Analysis and Machine Intelligence, 15(4): 353 -363 (1993). copyright 1993 IEEE.

Computer Vision Multi-view depth fusion Pollefeys and Van Gool. ECCV‘ 98) • Compute depth for (Koch, every pixel of reference image – Triangulation – Use multiple views – Up- and down sequence – Use Kalman filter Allows to compute robust texture

Computer Vision Real-time stereo on graphics hardware Ruigang Yang and Marc Pollefeys • Computes Sum-of-Square-Differences • Hardware mip-map generation used to aggregate results over support region • Trade-off between small and large support window Shape of a kernel for summing up 6 levels 140 M disparity hypothesis/sec on Radeon 9700 pro e. g. 512 x 20 disparities at 30 Hz

Computer Vision near Sample Re-Projections far

Computer Vision Combine multiple aggregation windows using hardware mipmap and multiple texture units in single pass (1 x 1) (1 x 1+2 x 2 +4 x 4+8 x 8) (1 x 1+2 x 2 +4 x 4+8 x 8 +16 x 16)

Computer Vision Live stereo demo PC with ATI Radeon 9700 pro Bumblebee (Point Grey) Open. GL 1. 4 (Fragment programs) Code available at: http: //www. cs. unc. edu/~ryang/research/View. Syn/realtime. htm

Computer Vision Cool ideas • Space-time stereo (varying illumination, not shape)

Computer Vision More on stereo … The Middleburry Stereo Vision Research Page http: //cat. middlebury. edu/stereo/ Recommended reading D. Scharstein and R. Szeliski. A Taxonomy and Evaluation of Dense Two. Frame Stereo Correspondence Algorithms. IJCV 47(1/2/3): 7 -42, April-June 2002. PDF file (1. 15 MB) - includes current evaluation. Microsoft Research Technical Report MSR-TR 2001 -81, November 2001. PDF file (1. 27 MB).

Computer Vision Next class: project proposals!