10/5/2001 CS 461, Copyright G.D. HagerComputer Vision, Lecture 9Professor Hagerhttp://www.cs.jhu.edu/~hager10/5/2001 CS 461, Copyright G.D. HagerOutline for Today• Geometric Image Formation• Camera Models10/5/2001 CS 461, Copyright G.D. HagerPinhole cameras• Abstract camera model - box with a small hole in it• Pinhole cameras work in practice10/5/2001 CS 461, Copyright G.D. HagerReal Pinhole CamerasPinhole too big -many directions areaveraged, blurring theimagePinhole too small-diffraction effects blurthe imageGenerally, pinhole cameras are dark, becausea very small set of raysfrom a particular pointhits the screen.10/5/2001 CS 461, Copyright G.D. HagerThe reason for lensesLenses gather andfocus light, allowingfor brighter images.10/5/2001 CS 461, Copyright G.D. HagerThe thin lens1z'−1z=1fThin Lens Properties:1. A ray entering parallel to optical axisgoes through the focal point.2. A ray emerging from focal point is parallel to optical axis3. A ray through the optical center is unaltered10/5/2001 CS 461, Copyright G.D. HagerThe thin lens1z'−1z=1fNote that, if the image plane is verysmall and/or z >> z’, then z’ ≈ f10/5/2001 CS 461, Copyright G.D. HagerField of View• The effective diameter of a lens (d) is the portion of a lens actually reachable by light rays.• The effective diameter and the focal length determine the field of view:• w is the half the total angular “view” of a lens system.• Another fact is that in practice points at different distances are imaged, leading to so-called “circles of confusion” of size d/z | z’-z| where z is the nominal image plane and z’ is the focusing distance given by the thin lens equation.• The “depth of field” is the range of distances that produce acceptably focused images. Depth of field varies inversely with focal length and lens diameter.)2/(tan fdw =10/5/2001 CS 461, Copyright G.D. HagerLens RealitiesReal lenses have a finite depth of field, and usuallysuffer from a variety of defectsvignettingSpherical Aberration10/5/2001 CS 461, Copyright G.D. HagerStandard Camera Coordinates• Optical axis is z axis pointing outward• X axis is parallel to the scanlines (rows) pointing to the right!• By the right hand rule, the Y axis must point downward• Note this corresponds with indexing an image from the upper leftto the lower right, where the X coordinate is the column index and the Y coordinate is the row index.10/5/2001 CS 461, Copyright G.D. HagerThe equation of projection• Equating z’ and f– We have, by similar triangles, that (x, y, z) -> (-f x/z, -f y/z, -f)– Ignore the third coordinate, and flip the image around to get:(x, y, z) → ( fxz, fyz)10/5/2001 CS 461, Copyright G.D. HagerDistant objects are smaller10/5/2001 CS 461, Copyright G.D. HagerParallel lines meetcommon to draw film planein front of the focal pointA Good Exercise: Show this is the case!10/5/2001 CS 461, Copyright G.D. HagerParallel lines meet• First, show how lines project to images.• Second, consider lines that have the same direction (are parallel)• Third, consider the degenerate case of lines parallel in the image– (by convention, the vanishing point is at infinity!)A Good Exercise: Show this is the case!10/5/2001 CS 461, Copyright G.D. HagerVanishing points• Another good exercise (really follows from the previous one): show the form of projection of *lines* into images.• Each set of parallel lines (=direction) meets at a different point– The vanishing point for this direction• Sets of parallel lines on the same plane lead to collinear vanishing points. – The line is called the horizon for that plane10/5/2001 CS 461, Copyright G.D. HagerGeometric properties of projection• Points go to points• Lines go to lines• Planes go to whole image• Polygons go to polygons• Degenerate cases– line through focal point to point– plane through focal point to line10/5/2001 CS 461, Copyright G.D. HagerPolyhedra project to polygons• (because lines project to lines)10/5/2001 CS 461, Copyright G.D. HagerJunctions are constrained• This leads to a process called “line labelling”– one looks for consistent sets of labels, bounding polyhedra– disadv - can’t get the lines and junctions to label from real images10/5/2001 CS 461, Copyright G.D. HagerThe Camera Matrix• Homogenous coordinates for 3D– four coordinates for 3D point– equivalence relation (X,Y,Z,T) is the same as (k X, k Y, k Z,k T) • Turn previous expression into HC’s– 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          ),(),(),,( vuWVWUWVU =→10/5/2001 CS 461, Copyright G.D. HagerOrthographic projectionyvxu==Suppose I let f go to infinity; then10/5/2001 CS 461, Copyright G.D. HagerThe model for orthographic projectionUVW        =100001000001        XYZT         10/5/2001 CS 461, Copyright G.D. HagerWeak perspective• Issue– perspective effects, but not over the scale of individual objects– collect points into a group at about the same depth, then divide each point by the depth of its group– Adv: easy– Disadv: wrong*/ Zfssyvsxu===10/5/2001 CS 461, Copyright G.D. HagerThe model for weak perspective projection=TZYXfZWVU/*0000010000110/5/2001 CS 461, Copyright G.D. HagerGeometric TransformsIn general, a point in n-D space transforms byP’ = rotate(point) + translate(point)In 2-D space, this can be written as a matrix equation:+−=tytxyxCosSinSinCosyx)()()()(''θθθθIn 3-D space (or n-D), this can generalized as a matrix equation:p’ = R p + T or p = Rt(p’ – T)10/5/2001 CS 461, Copyright G.D. HagerGeometric TransformsNow, using the idea of homogeneous transforms,we can write:pTRp=1000'R and T both require 3 parameters. These correspondto the 6 extrinsic parameters needed for camera calibration10/5/2001 CS 461, Copyright G.D. HagerIntrinsic ParametersIntrinsic Parameters describe the conversion fromunit focal length metric to pixel coordinates (and the reverse)xmm= - (xpix–ox) sx Æ -/sxxmm–ox= -xpixymm= - (ypix–oy) sy Æ -/syymm–oy=