There are many cues in monocular vision which suggests that vision in stereo starts very early from two similar 2D images. Lets see a few...

Transcription

1 STEREO VISION The slides are from several sources through James Hays (Brown); Srinivasa Narasimhan (CMU); Silvio Savarese (U. of Michigan); Bill Freeman and Antonio Torralba (MIT), including their own slides.

2 There are many cues in monocular vision which suggests that vision in stereo starts very early from two similar 2D images. Lets see a few...

3 Depth cues: Linear perspective

4 Depth cues: Aerial perspective

5 Depth ordering cues: Occlusion Source: J. Koenderink

6 Shape cues: Texture gradient

7

8 the two cameras are calibrated

9 Triangulation: the geometry behind Find shortest segment connecting the two viewing rays and let X be the midpoint of that segment X K known x x 2 1 K' known O 1 cameras calibrated O 2 R and t also known

10 Triangulation: is a linear approach λ 1 x1 = 1 2 x 2 = 2 P X λ P X P X 0 1 P X = 0 [x1 ]P1 X = 0 x [x 2 ]P2 X = 0 x 1 x 2 2 = Vector product written as matrix multiplication. The P_1 and P_2 (3x4) are known. Cross product as matrix multiplication: The x_1 and x_2 (3x1) are also known. Two independent 0 equations a aper point, b in terms of three unknown entries of X (the 4th coord. = 1) x

11 Stereo photography and stereo viewers Take two pictures of the same subject from two slightly different viewpoints and display so that each eye sees only one of the images. Invented by Sir Charles Wheatstone, 1838 a few stereo images blue/red stereo viewer

12 two images of the Eiffel tower superposed

13 Random dot stereograms Julesz Can we identify without an object a 3D structure from two images? Experiment. Pair of synthetic identical images and moving a small rectangular part in the middle one pixel to the left. When viewed monocularly they appear random, when viewed stereoscopically, is a 3D structure. Human binocular fusion appears early in the central nervous system.

14 Depth without objects Julesz, 1971 Bela Julesz

15 Stereo vision P p p O 1 O 2 Goal: estimate the position of P given the observation of P from two view points. Assumptions: known camera parameters and position (K, R, T).

16 Stereo vision calibrated cameras P p p O 1 O 2 Subgoals: - Solve the correspondence problem between the images; p, p'. - Use corresponding observations to triangulate in 3D.

17 If two images become parallel... P p p O O when views are parallel these two steps becomes much easier -- correspondence and triangulate.

18 E matrix with parallel images P v v e 1 p p e 2 y u u K O z x O K` Parallel epipolar lines. Epipoles at infinity. v = v Rectification: making two images parallel.

19 Making the two images parallel. P H p p H' O H, H' homographies O GOAL: Estimate the perspective transformations that makes images parallel. Impose v =v Leaves degrees of freedom for determining H and H'!!! If not appropriate H (H') is chosen, severe projective distortions are present.

20 First frame will call the "camera", the second frame the "world". X 2is the inhomogeneous coordinates in the world coordinates X 1is the inhomogeneous coordinates in the camera coordinates R and t take from world coordinate system to the camera c.s. describing the camera with respect the world coordinates. By choice, the zero in the second coor. system is the first origin T of the camera O = [ ] and e' is the epipole in second c.s. The origin of the camera center O in second c.s. is related by T X 1= R (X 2 - t) (inverse R, t) to O' measured in the first c.s. Thus, the coordinates of the world origin in the first coor. system T T is the point O' = [ -R t 1] and e is the epipole of O' in first c.s. X 2= [0 0 0] T X -> O' first c.s. 1

21 P K I 0 P P P K R T P p p K e e K O R,T O Compute epipoles. e K R T T [e 1 sign does not matter T T e = K[I 0] [-R t 1] e' = K' [R t] [0 1] T e 2 1] T e K T

22 P. p p e e O O Map e to the x-axis at location [1,0,1] T (normalization). T two 3x3 matrix e [e e ] 0 1 T H R T 1 H H

23 P p p e O O Send epipole to infinity. H e T T

24 P H = H 2 H 1 H' e p p e O O p <==> p' epipolar lines are l = e x p l' = e' x p' can be computed. Matched pair of transformation. Parallel in each image To have v = v' need to find the final separately. transformations.

25 P H = H 2 H 1 H' p p O O' Align the epipolar lines across the two images based of the matched transformation. Descriped also in [Hartley and Zisserman] Chapter 11 (Sec )... more advanced.

26 IMAGE RECTIFICATION Denote E from image 2 to 1 and is given. If the images are in parallel, and the t is known, the matrix E becomes E = [1 0 0] x = [i] x. The two 3 3 homographies are H and H. Let p and p be the rectified homogeous points p [i] x p = 0 related to the original points by p = Hp p = H p. The fundamental matrix F is F = H [i] x H p Fp = 0. The two homographies have more degrees of freedom then we need to solve the alignment.

27 Image rectification. Loop and Zhang (2001). The essential matrix is known. Denote E from image 2 to 1. T T p' H' [i] H p = DOF each homography H, H'. X A homography is decomposed in (shearing.similarity).projective -- order starts at right! projective: epipolar lines become parallel in each image; similarity: rotate the points into alignment with line maps in the two images simultaneously. Its enough... shearing: minimizing the horizontal distortion in each image. after projective and similarity a b two intervals with known original aspect ratio

28 Rectification is useful. Slides of Loop, Zhang Projective transformation.

29 Rotate, translate, uniformly scale only. Shearing with u(u') parameters only, reduces the projective distortion separately in both images. E.g., to preserve perpendicularity and aspect ratio of two line segments. The two cameras have some distortion.

30 Geometry for a simple stereo system Assuming by now parallel optical axes, known camera parameters (i.e., calibrated cameras).

31 3D: X_l; X_r X_l = X_r - T The two cameras at T distance along the x axis. T = X_r - X_l = (Z x_r)/f - (Z x_l)/f Sometime disparity is x_l - x_r.

32 Depth from disparity image I(x,y) Disparity map D(x,y) image I (x,y ) (x,y )=(x+d(x,y), y) So if we could find the corresponding points in two images, we could estimate relative depth

33 Example: depth from disparity Venus data pair B = T translation x x B f z x x Stereo pair Disparity map / depth map Disparity map with occlusions

34 Basic stereo matching algorithm Rectify the two stereo images to transform epipolar lines into scanlines For each pixel x in the first image Find corresponding epipolar scanline in the right image. Examine all pixels on the scanline and pick the best match x. Compute disparity x-x and set depth(x) = fb/(x-x ).

35 The epipolar constraint helps, but much ambiguity remains.

36 Correlation Methods (from 1970's on) p...for normalized correlation Two parallel image. Pick up a window around p(u,v).

37 Correlation Methods Pick up a window around p(u,v). Build vector w. Slide the window along v line in image 2 and compute w. Keep sliding until w w (inner product) is maximized.

38 Correlation Methods Normalized correlation has to maximize: (w w)(w (w w)(w w ) w )

39 Correspondence search with... Left Right scanline...maximization of the normalized correlation in the second image. Norm. corr

40 Correspondence search with... Left Right scanline Matching cost disparity...the minimum of the sum squared differences (SSD).

41 Example: Left Right scanline Minimization of the sum squared differences (SSD) in the second image. SSD

42 Image block matching How do we determine correspondences? block matching d is the disparity (horizontal motion) How big should the neighborhood be? CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching

43 Neighborhood size Smaller neighborhood: more details. Larger neighborhood: fewer isolated mistakes. w = 3 w = 20 CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching

44 Issues on stereo matching: T = B! Foreshortening effect large B/z ratio O O Occlusions O O

45 small B/z ratio depth(x) = fb/(x-x ) O O Small error in measurements implies large error in estimating depth.

46 Homogeneous regions Hard to match pixels in these regions.

47 Repetitive patterns Correspondence problem is difficult. Apply nonlocal constraints to help enforce the correspondences.

48 Nonlocal constraints can have other problems Uniqueness For any point in one image, there should be at most one matching gpoint in the other image Ordering Corresponding points should be in the same order in both views Ordering constraint doesn t hold

49 Stereo testing and comparisons D. Scharstein and R. Szeliski. "A Taxonomy and Evaluation of Dense Two- Frame Stereo Correspondence Algorithms," International Journal of Computer Vision, 47 (2002), pp Scene Tsukuba laboratory left image Ground truth submissions, July 2013

50 Stereo best algorithms 94

51 Example: window correlation Window-based matching (best window size) Ground truth not the best method

52 A few of the other methods: - based on edges, corners, gradients; - rank statistics; - dynamic programming; - matching as energy minimization (graph cuts); - multiple-baseline stereo; - space carving stereo etc... None of the methods is the ultimate solution. Stereo (like everything else) works only if the matching is robust.

53 Active stereo (point) P p projector O Replace one of the two cameras by a projector - Single camera. - Projector geometry calibrated too. - What s the advantage of having the projector? Correspondence problem solved!

54 Active stereo (stripe) projector O - If the projector and camera are parallel... correspondence problem solved. Otherwise, transform it into parallel first.

55 High precision: Laser scanning Object Laser sheet Direction of travel CCD image plane Laser Cylindrical lens CCD Digital Michelangelo Project Levoy et al. Optical triangulation Project a single stripe of laser light. across Scan the it across surfacethe of surface the object of the object. This is a very precise version of structured light scanning. Source: S. Seitz

56 Example: laser scanning The Digital Michelangelo Project, Levoy et al. Source: S. Seitz

57 Active stereo (color-coded stripes) L. Zhang, B. Curless, and S. M. Seitz 2002 S. Rusinkiewicz & Levoy 2002 projector - Dense reconstruction. Projector, image are parallel. - Correspondence found by a technique based on the color code. O

58 KINECT Structured infrared light