1CSE152, Spr 04 Intro Computer VisionImage Formation and CamerasIntroduction to Computer VisionCSE 152Lecture 3CSE152, 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 VisionImage Formation: Outline• Factors in producing images• Projection• Perspective• Vanishing points• Orthographic•Lenses•Sensors• Quantization/Resolution• Illumination• ReflectanceCSE152, Spr 04 Intro Computer VisionEarliest Surviving Photograph• First photograph on record, “la table service” byNicephore Niepce in 1822.• Note: First photograph by Niepce was in 1816.CSE152, Spr 04 Intro Computer VisionHow Cameras Produce Images• Basic process:– photons hit a detector– the detector becomes charged– the charge is read out as brightness• Sensor types:– CCD (charge-coupled device)• most common• high sensitivity• high power• cannot be individually addressed• blooming–CMOS• simple to fabricate (cheap)• lower sensitivity, lower power• can be individually addressedCSE152, Spr 04 Intro Computer VisionImages are two-dimensional patterns of brightness values.They are formed by the projection of 3D objects.Figure from US Navy Manual of Basic Optics and Optical Instruments, prepared by Bureau of Naval Personnel. Reprinted by Dover Publications, Inc., 1969.2CSE152, Spr 04 Intro Computer VisionEffect of Lighting: MonetCSE152, Spr 04 Intro Computer VisionChange of Viewpoint: MonetHaystack at Chailly at sunrise(1865)CSE152, Spr 04 Intro Computer VisionPinhole Camera: Perspective projection• Abstract camera model - box with a small hole in itForsyth&PonceCSE152, Spr 04 Intro Computer Visionhttp://www.acmi.net.au/AIC/CAMERA_OBSCURA.html (Russell Naughton)Camera Obscura"When images of illuminated objects ... penetrate through a small hole into a very dark room ... you will see [on the opposite wall] these objects in their proper form and color, reduced in size ... in a reversed position, owing to the intersection of the rays".Da VinciCSE152, Spr 04 Intro Computer Vision• Used to observe eclipses (eg., Bacon, 1214-1294)• By artists (eg., Vermeer).CSE152, Spr 04 Intro Computer Visionhttp://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus)Jetty at Margate England, 1898.3CSE152, Spr 04 Intro Computer VisionDistant objects are smaller(Forsyth & Ponce)CSE152, Spr 04 Intro Computer VisionGeometric properties of projection• Points go to points• Lines go to lines• Planes go to whole image or half-plane• Polygons go to polygons• Angles & distances not preserved• Degenerate cases:– line through focal point yields point– plane through focal point yields lineCSE152, Spr 04 Intro Computer VisionParallel lines meet in the image• vanishing pointImage planeCSE152, Spr 04 Intro Computer VisionTake out paper and pencilCSE152, Spr 04 Intro Computer Vision CSE152, Spr 04 Intro Computer VisionVanishing pointsVPLVPRHVP1VP2VP3To different directions correspond different vanishing points4CSE152, Spr 04 Intro Computer VisionVanishing PointsCSE152, 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 VisionHomogenous coordinates• Add an extra coordinate and use an equivalence relation• for 2D– equivalence relationk*(X,Y,Z) is the same as (X,Y,Z) • for 3D– equivalence relationk*(X,Y,Z,T) is the same as (X,Y,Z,T)• Why do this?– Possible to represent points “at infinity”• Where parallel lines intersect• Where parallel planes intersect– Possible to write the action of a perspective camera as a matrixCSE152, Spr 04 Intro Computer VisionEuclidean -> Homogenous-> Euclidean In 2-D• Euclidean -> Homogenous: (x, y) -> k (x,y,1)• Homogenous -> Euclidean: (x, y, z) -> (x/z, y/z)In 3-D• Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1)• Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w)CSE152, Spr 04 Intro Computer VisionThe camera matrixTurn previous expression into Homogenous Coordinates– HC’s for 3D point are (X,Y,Z,T)– HC’s for point in image are (U,V,W)UVW          =10 0 001 0 0001f0            XYZT       CSE152, Spr 04 Intro Computer VisionProjective Geometry• Axioms of Projective Plane1. Every two distinct points define a line2. Every two distinct lines define a point (intersect at a point)3. There exists three points, A,B,C such that C does lie on the line defined by A and B.• Different than Euclidean (affine) geometry• Projective plane is “bigger” than affine plane –includes “line at infinity”• Homogenous coordinates are a way to represent points on a projective space.5CSE152, Spr 04 Intro Computer Vision2-D Affine & 2-D Projective SpaceCSE152, 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 affine camera modelBlack board derivationCSE152, Spr 04 Intro Computer VisionOrthographic projectionTake Taylor series about (0, 0, z0) – a point on optical axis CSE152, Spr 04 Intro Computer Vision==yyxx''Orthographic projectionTake Taylor series about (0, 0, z0) – a point on optical axis CSE152, Spr 04 Intro Computer VisionThe projection matrix for orthographic projectionUVW          =100001000001          XYZT              CSE152, Spr 04 Intro Computer VisionEuclidean Coordinate Systems=⇔++=⇔===zyxzyxOPOPzOPyOPxPkjikji...6CSE152, Spr 04 Intro Computer VisionCoordinate Changes: Pure TranslationsOBP = OBOA + OAP ,BP = AP + BOACSE152, Spr 04 Intro Computer VisionRotation Matrix=BABABABABABABABABABARkkkjkijkjjjiikijii.........[]ABABABkji==TBATBATBAkjiCSE152, 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