Announcements. Stereo Vision II. Midterm. Example: Helmholtz Stereo Depth + Normals + BRDF. Stereo
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1 Announcements Stereo Vision II Introduction to Computer Vision CSE 15 Lecture 13 Assignment 3: Due today. Extended to 5:00PM, sharp. Turn in hardcopy to my office 3101 AP&M No Discussion section this week. Guest lecturer, Dr. Jeff Ho, next week. Today Midterm summary Shape from X Stereo Vision I High: 87 Low: 31 Mean: 57 Median: 55 Midterm Problem regrades: Must be requested today in class and in writing. Shape-from-X (i.e., Reconstruction Methods for estimating 3-D shape from image data. X can be one (or more of many cues. Stereo (two or more iews, known iewpoints Motion (moing camera or object Shading Changing lighting (Photometric Stereo Texture ariation Focus/blur Prior knowledge/context structured light/lasers Example: Helmholtz Stereo Depth + Normals + BRDF Stereo
2 Binocular Stereopsis: Mars Gien two images of a scene where relatie locations of cameras are known, estimate depth of all common scene points. An Application: Mobile Robot Naigation The INRIA Mobile Robot, Two images of Mars The Stanford Cart, H. Moraec, Courtesy O. Faugeras and H. Moraec. Commercial Stereo Heads Stereo can work well Trinocular stereo Binocular stereo Need for correspondence Triangulation Nalwa Fig. 7. Truco Fig. 7.5
3 Stereo Vision Outline Offline: Calibrate cameras & determine B epipolar geometry Online 1. Acquire stereo images C. Rectify images to conenient epipolar geometry D 3. Establish correspondence A 4. Estimate depth DISPARITY (X L -X R Z = (f/x L X Z= (f/x R (X-d (f/x L X = (f/x R (X-d X = (X L d / (X L -X R X = Z = d X L (X L -X R d f (X L -X R BINOCULAR STEREO SYSTEM Estimating Depth Z=f Z X L X R (0,0 (d,0 X (Adapted from Hager Reconstruction: General 3-D case Linear Method: find P such that Two Approaches A From each image, process monocular image to obtain cues. B Establish correspondence between cues. Directly compare image regions between the two images. Non-Linear Method: find Q minimizing Human Stereopsis: Binocular Fusion How are the correspondences established? Random Dot Stereograms Julesz (1971: Is the mechanism for binocular fusion a monocular process or a binocular one?? There is anecdotal eidence for the latter (camouflage. Random dot stereograms proide an objectie answer BP!
4 Random Dot Stereograms A Cooperatie Model (Marr and Poggio, 1976 Epipolar Constraint Epipolar Geometry Potential matches for p hae to lie on the corresponding epipolar line l. Potential matches for p hae to lie on the corresponding epipolar line l. Epipolar Plane Epipoles Epipolar Lines Baseline Family of epipolar Planes (standard approach JEFF, HERE s WHERE I ENDED Jeff, I coered up to this point. I suggest that you reiew the pictorial part of the epipolar geometry, discuss the essentnial matrix, showing how it can be computed from R&T, and that this can come from calibration. Discuss rectification in qualitatie way, showing that result is epipolar lines become parallel lines. Then discuss matching, mostly SSD & SAD metric. The mention issue with half occluded regions. Perhaps mention some challenges, dynamic programming, etc.
5 Family of epipolar Planes Epipolar Constraint: Calibrated Case (standard approach Essential Matrix (Longuet-Higgins, 1981 Properties of the Essential Matrix Calibration T E p is the epipolar line associated with p. T E T p is the epipolar line associated with p. E e =0 and E T e=0. E is singular. E has two equal non-zero singular alues (Huang and Faugeras, Determine intrinsic parameters and extrinsic relation of two cameras The Eight-Point Algorithm (Longuet-Higgins, 1981 Epipolar geometry example Set F 33 to 1 Minimize: under the constraint F =1.
6 Example: conerging cameras Example: motion parallel with image plane (simple for stereo rectification courtesy of Andrew Intro Zisserman Computer Vision courtesy of Andrew Intro Zisserman Computer Vision Example: forward motion Rectification Gien a pair of images, transform both images so that epipolar lines are scan lines. e e courtesy of Andrew Intro Zisserman Computer Vision Rectification Image pair rectification simplify stereo matching by warping the images Apply projectie transformation so that epipolar lines correspond to horizontal scanlines e e All epipolar lines are parallel in the rectified image plane. 1 0 = He 0 map epipole e to (1,0,0 try to minimize image distortion Note that rectified images usually not rectangular
7 Rectification Gien a pair of images, transform both images so that epipolar lines are scan lines. Rectification Gien a pair of images, transform both images so that epipolar lines are scan lines. Input Images Rectified Images See Section for specific method Features on same epipolar line Mobi: Stereo-based naigation Truco Fig. 7.5 Epipolar correspondence A challenge: Multiple Interpretations This ersion is feature-based: detect edges in 1-D signal, and use dynanic progrmaming toe find correspondences that minimize an error function. Each feature on left epipolar line match one and only one feature on right epipolar line.
8 Multiple Interpretations Multiple Interpretations Each feature on left epipolar line match one and only one feature on right epipolar line. Each feature on left epipolar line match one and only one feature on right epipolar line. Multiple Interpretations Dense Correspondence: A Photometric constraint Same world point has same intensity in both images (Constant Brightness Constraint Lambertian fronto-parallel Issues: Noise Specularity Foreshortening Each feature on left epipolar line match one and only one feature on right epipolar line. Using epipolar & constant Brightness constraints for stereo matching Comparing Windows: =? f g For each epipolar line For each pixel in the left image compare with eery pixel on same epipolar line in right image pick pixel with minimum match cost This will neer work, so: Most popular For each window, match to closest window on epipolar line in other image. Improement: match windows (Camps (Seitz
9 Match Metric Summary MATCH METRIC DEFINITION Correspondence Search Algorithm (simple ersion for Cross Correlation Normalized Cross-Correlation (NCC Sum of Squared Differences (SSD Normalized SSD Sum of Absolute Differences (SAD Zero Mean SAD Rank Census ( I1( I1 ( I( u + d, I ( I1( I1 ( I( u + d, I ( I1 ( I ( u + d, ( I1( I1 ( I ( u + d, I ( I ( u I ( I ( u + d I 1, 1, I1 ( I ( u + d, ( I1 ( I1 ( I ( u + d, I I k ( = I k ( m, n < I k ( m, n ( I1 ( I ( u + d, Ik ( = BITSTRINGm, n( Ik ( m, n < Ik ( HAMMING( I1 (, I ( u + d, These two are actually the same u d For i = 1:nrows for j=1:ncols best(i,j = -1 for k = mindisparity:maxdisparity c = CC(I 1 (i,j,i (i,j+k,winsize if (c > best(i,j best(i,j = c disparities(i,j = k end end end end I 1 I O(nrows * ncols * disparities * winx * winy Stereo results Data from Uniersity of Tsukuba Results with window correlation Scene (Seitz Ground truth Window-based matching (best window size (Seitz Ground truth Results with better method Window size State of the art method Boyko et al., Fast Approximate Energy Minimization ia Graph Cuts, International Conference on Computer Vision, September (Seitz Ground truth Effect of window size (Seitz W = 3 W = 0 Better results with adaptie window T. Kanade and M. Okutomi, A Stereo Matching Algorithm with an Adaptie Window: Theory and Experiment,, Proc. International Conference on Robotics and Automation, D. Scharstein and R. Szeliski. Stereo matching with nonlinear diffusion. International Journal of Computer Vision, 8(: , July 1998
10 CONSTRAINT Stereo Constraints BRIEF DESCRIPTION Problem of Occlusion 1-D Epipolar Search Arbitrary images of the same scene may be rectified based on epipolar geometry such that stereo matches lie along onedimensional scanlines. This reduces the computational complexity and also reduces the likelihood of false matches. Monotonic Ordering Points along an epipolar scanline appear in the same order in both stereo images, assuming that all objects in the scene are approximately the same distance from the cameras. Image Brightness Constancy Match Uniqueness Disparity Continuity Disparity Limit Fronto-Parallel Surfaces Feature Similarity Structural Grouping Assuming Lambertian surfaces, the brightness of corresponding points in stereo images are the same. For eery point in one stereo image, there is at most one corresponding point in the other image. Disparities ary smoothly (i.e. disparity gradient is small oer most of the image. This assumption is iolated at object boundaries. The search space may be reduced significantly by limiting the disparity range, reducing both computational complexity and the likelihood of false matches. The implicit assumption made by area-based matching is that objects hae fronto-parallel surfaces (i.e. depth is constant within the region of local support. This assumption is iolated by sloping and creased surfaces. Corresponding features must be similar (e.g. edges must hae roughly the same length and orientation. Corresponding feature groupings and their connectiity must be consistent. (From G. Hager Stereo Matching using Dynamic Programming 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 (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(: ( IEEE. (From Pollefeys (Cox et al. CVGIP 96; Koch 96; Falkenhagen 97; Van Meerbergen,Vergauwen,Pollefeys,VanGool IJCV 0 Some Challenges & Problems Photometric issues: specularities strongly non-lambertian BRDF s Variations on Binocular Stereo 1. Trinocular Stereopsis. Helmholtz Reciprocity Stereopsis Surface structure lack of texture repeating texture within horopter bracket Geometric ambiguities as surfaces turn away, difficult to get accurate reconstruction (affine approximate can help at the occluding contour, likelihood of good match but incorrect reconstruction
11 Trinocular Epipolar Constraints More on stereo The Middleburry Stereo Vision Research Page Recommended reading D. Scharstein and R. Szeliski. A Taxonomy and Ealuation of Dense Two-Frame Stereo Correspondence Algorithms. IJCV 47(1//3:7-4, April-June 00. PDF file (1.15 MB - includes current ealuation. Microsoft Research Technical Report MSR-TR , Noember 001. These constraints are not independent! Myron Z. Brown, Darius Burschka, and Gregory D. Hager. Adances in Computational Stereo. IEEE Transactions on Pattern Analysis and Machine Intelligence, 5(8: , 003.
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