Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Computer VisionNovember 2006 L1.1© 2006 by Davi GeigerBinocular StereoBinocular Stereo Left Image Right ImageComputer VisionNovember 2006 L1.2© 2006 by Davi GeigerThere 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) occlusion cues, such as presence of T-Junctions,(iii) motion information, (iv) focusing and defocusing, (v) 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 StereoComputer VisionNovember 2006 L1.3© 2006 by Davi GeigerJulesz’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 StereogramComputer VisionNovember 2006 L1.4© 2006 by Davi GeigerNot 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 ContoursHere 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.Computer VisionNovember 2006 L1.5© 2006 by Davi GeigerHuman Stereo: Half Occlusions Left Right 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.Computer VisionNovember 2006 L1.6© 2006 by Davi GeigerProjective CameraLet 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 zxPo=(Xo,Yo,Zo)0PZfpopo=(xo,yo,f)fyOOf),,( ZYXP  ),,(OOOOZYXPPO ),,( fyxpooo),,( ZYXP O0PZfpoComputer VisionNovember 2006 L1.7© 2006 by Davi Geigeryx )1,,(0 yoxoqqqxoyoO,)(;)(0000 yyyxxxsoqysoqx We have neglected to account for the radial distortion of the lenses, which would give an additional intrinsic parameter. Equation above can be described by the linear transformation foossyxyx);,();,(where the intrinsic parameters of the camera, , represent the size of the pixels (say in millimeters) along x and y directions, the coordinate in pixels of the image (also called the principal point) and the focal length of the camera. fossossQyyyxxx0000101qQpo0pQqoffosfosQyyxx1001001Projective Camera Coordinate Systempixel coordinatesComputer VisionNovember 2006 L1.8© 2006 by Davi GeigerOlxlylP=(X,Y,Z)pl=(xo,yo,f)xryrzrOrzlfpr=(xo,yo,f)fA 3D point P projected on both cameras. 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 Pl in the left frame will be described in the right frame as rlOOTlPrP)(1TPRPlrTwo Projective CamerasComputer VisionNovember 2006 L1.9© 2006 by Davi GeigerOlxlylP=(X,Y,Z)xryrzrelepipolar linesOrzlpr=(xo,yo,f)erEach 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 POlOl and thus the two epipoles belong to all pairs of epipolar lines (the epipoles are the “center/intersection” of all epipolar lines.)rlOOlererlOPOrlOOTlP)(1TPRPlrrlOOpl=(xo,yo,f)Two Projective CamerasEpipolar LinesComputer VisionNovember 2006 L1.10© 2006 by Davi