1CSE152, Spr 04 Intro Computer VisionImage Formation and CamerasIntroduction to Computer VisionCSE 152Lecture 4CSE152, Spr 04 Intro Computer VisionAnnouncements• Assignment 1: posted to web page, due on Thursday• Read Trucco & Verri: pp. 15-40 CSE152, Spr 04 Intro Computer VisionPinhole Camera: Perspective projection• Abstract camera model - box with a small hole in itForsyth&PonceCSE152, Spr 04 Intro Computer VisionGeometric Aspects of Perspective Projection• Points project to points• Lines project to lines• Angles & distances (or ratios) are NOT preserved under perspective• Vanishing pointImage planeCSE152, Spr 04 Intro Computer VisionThe equation of projectionCartesian coordinates:• We have, by similar triangles, that (x, y, z) -> (f x/z, f y/z, -f)• Ignore the third coordinate, and get(x, y, z) → ( fxz, fyz)CSE152, Spr 04 Intro Computer VisionEuclidean -> Homogenous-> Euclidean In 2-D• Euclidean -> Homogenous: (x, y) -> k (x,y,1)(can just take k=1)• Homogenous -> Euclidean: (x, y, z) -> (x/z, y/z)In 3-D• Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1)(can just take k=1)• Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w)2CSE152, Spr 04 Intro Computer VisionThe camera matrixTurn into homogenous coordinates– HC’s for 3D point are (X,Y,Z,1)– HC’s for point in image are (U,V,W)=1010000100001ZYXfWVU(x, y, z) → ( fxz, fyz)CSE152, Spr 04 Intro Computer VisionAffine Camera Model• Take Perspective projection equation, and perform Taylor Series Expansion about (some point (x0,y0,z0).• Drop terms of higher order than linear.• Resulting expression is called affine camera model.• Properties– Pts. map to pts, lines map to lines– Parallel lines map to parallel lines (no vanishing point –at infinity)– Ratios of distance/angles preservedCSE152, Spr 04 Intro Computer Vision==yyxx''Orthographic projectionStart with affine camera model, and take Taylor series about (x0, y0, zo)= (0, 0, z0) – a point on optical axis Depth (z) is lostCSE152, Spr 04 Intro Computer VisionThe projection matrix for orthographic projection=1100000100001ZYXWVUCSE152, Spr 04 Intro Computer VisionCoordinate Changes: Pure TranslationsOBP = OBOA + OAP ,BP = AP + BOACSE152, Spr 04 Intro Computer VisionCoordinate Changes: Pure Rotations[][]PRPzyxzyxOPABABBBBBBBAAAAAA=⇒== kjikjiA rotation matrix R has the following properties:• Its inverse is equal to its transpose R-1= RT• Its determinant is equal to 1: det(R)=1.Or equivalently:• Rows (or columns) of R form a right-handed orthonormal coordinate system.3CSE152, Spr 04 Intro Computer VisionA rotation matrix R has the following properties:• Its inverse is equal to its transpose R-1= RT• its determinant is equal to 1: det(R)=1.Or equivalently:• Rows (or columns) of R form a right-handedorthonormal coordinate system.CSE152, Spr 04 Intro Computer VisionRotation: Homogenous Coordinates• About z axisx'y'z'1=xyz1cos θsin θ00-sin θcos θ0000100001rot(z,θ)xyzpp'θCSE152, Spr 04 Intro Computer VisionRotation• About x axis:• About y axis:x'y'z'1=xyz10cos θsin θ00-sin θcos θ010000001x'y'z'1=xyz1cos θ0-sin θ0sin θ0cos θ001100001CSE152, Spr 04 Intro Computer VisionRoll-Pitch-Yaw),ˆ(),ˆ(),ˆ(ϕβαkrotjrotirotR =),ˆ(),'ˆ(),''ˆ(ϕβαkrotjrotkrotR =Euler AnglesCSE152, Spr 04 Intro Computer VisionRotation• About (kx, ky, kz), a unit vector on an arbitrary axis(Rodrigues Formula)x'y'z'1=xyz1kxkx(1-c)+ckykx(1-c)+kzskzkx(1-c)-kys00001kzkx(1-c)-kzskzkx(1-c)+ckzkx(1-c)-kxs0kxkz(1-c)+kyskykz(1-c)-kxskzkz(1-c)+c0where c = cos θ & s = sin θRotate(k, θ)xyzθkCSE152, Spr 04 Intro Computer VisionCoordinate Changes: Rigid TransformationsABABABOPRP +=4CSE152, Spr 04 Intro Computer VisionBlock Matrix Multiplication==2221121122211211BBBBBAAAAAWhat is AB ?++++=22221221212211212212121121121111BABABABABABABABAABHomogeneous Representation of Rigid Transformations==+=11111PTPOROPRPABAATABBAABABAB0Transformation represented by 4 by 4 MatrixCSE152, Spr 04 Intro Computer VisionCamera parameters• Issue– camera may not be at the origin, looking down the z-axis• extrinsic parameters (Rigid Transformation)– one unit in camera coordinates may not be the same as one unit in world coordinates• intrinsic parameters - focal length, principal point, aspect ratio, angle between axes, etc.UVW        =Transformationrepresenting intrinsic parameters        100001000010        Transformationrepresentingextrinsic parameters        XYZT          3 x 34 x 4CSE152, Spr 04 Intro Computer Vision, estimate intrinsic and extrinsic camera parameters• See Text book for how to do it.Camera CalibrationCSE152, Spr 04 Intro Computer VisionGetting more light – Bigger ApertureCSE152, Spr 04 Intro Computer VisionLimits for pinhole camerasCSE152, Spr 04 Intro Computer VisionPinhole Camera Images with Variable Aperture 1mm.35 mm.07 mm.6 mm2 mm.15 mm5CSE152, Spr 04 Intro Computer VisionThe reason for lensesCSE152, Spr 04 Intro Computer VisionThin LensO• Rotationally symmetric about optical axis.• Spherical interfaces.Optical axisCSE152, Spr 04 Intro Computer VisionThin Lens: CenterO• All rays that enter lens along line pointing at O emerge in same direction.FCSE152, Spr 04 Intro Computer VisionThin Lens: Focus OParallel lines pass through the focus, FFCSE152, Spr 04 Intro Computer VisionThin Lens: Image of Point OAll rays passing through lens and starting at Pconverge upon P’FPP’CSE152, Spr 04 Intro Computer VisionThin Lens: Image of Point OFPP’Z’fZfzz11'1=−6CSE152, Spr 04 Intro Computer VisionThin Lens: Image Plane OFPP’Image PlaneQ’QA price: Whereas the image of P is in focus,the image of Q isn’t.CSE152, Spr 04 Intro Computer VisionThin Lens: Aperture OPP’Image Plane• Smaller Aperture-> Less Blur• Pinhole -> No BlurCSE152, Spr 04 Intro Computer VisionField of View OField of ViewImage PlanefCSE152, Spr 04 Intro Computer VisionDeviations from