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1、 Project 2 questionsAnnouncementsProjective geometryReadings Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992, (read 23.1 - 23.5, 23.10) available online: http:/www.cs.cmu.edu/ph/869/papers/zisser
2、-mundy.pdfAmes RoomProjective geometrywhats it good for?Uses of projective geometry Drawing Measurements Mathematics for projection Undistorting images Focus of expansion Camera pose estimation, match move Object recognitionApplications of projective geometry Vermeers Music LessonReconstructions by
3、Criminisi et al.12341234Measurements on planesApproach: unwarp then measure What kind of warp is this?Image rectificationTo unwarp (rectifycorrect,adjust) an image solve for homography H given p and p solve equations of the form: wp = Hp linear in unknowns: w and coefficients of H H is defined up to
4、 an arbitrary scale factor how many points are necessary to solve for H?ppwork out on boardSolving for homographiesSolving for homographiesAh0Defines a least squares problem:2n 992n Since h is only defined up to scale, solve for unit vector Solution: = eigenvector of ATA with smallest eigenvalue Wor
5、ks with 4 or more points(0,0,0)The projective planeWhy do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection, multi-view relationships What is the geometric intuition直觉? a point in the image is a ray in projective space(sx,sy,s) Each point (x,y) on th
6、e plane is represented by a ray (sx,sy,s) all points on the ray are equivalent: (x, y, 1) (sx, sy, s)image plane(x,y,1)-yx-zProjective linesWhat does a line in the image correspond to in projective space? A line is a planeflat surface of rays through origin all rays (x,y,z) satisfying: ax + by + cz
7、= 0 A line is also represented as a homogeneous 3-vector llplPoint and line duality二元性 A line l is a homogeneous 3-vector It is to every point (ray) p on the line: l p=0p1p2What is the intersection of two lines l1 and l2 ? p is to l1 and l2 p = l1 l2 Points and lines are dual in projective space giv
8、en any formula, can switch the meanings of points and lines to get another formulal1l2pWhat is the line l spanned by rays p1 and p2 ? l is to p1 and p2 l = p1 p2 l is the plane normalIdeal points and linesIdeal point (“point at infinity”) p (x, y, 0) parallelnva.平行 to image plane It has infinite ima
9、ge coordinates(sx,sy,0)-yx-zimage planeIdeal line l (a, b, 0) parallel to image plane(a,b,0) -yx-zimage plane Corresponds to a line in the image (finite coordinates) goes through image origin (principle point)Homographies of points and linesComputed by 3x3 matrix multiplication To transform a point:
10、 p = Hp To transform a line: lp=0 lp=0 0 = lp = lH-1Hp = lH-1p l = lH-1 lines are transformed by postmultiplication of H-13D projective geometryThese concepts generalize naturally to 3D Homogeneous coordinates Projective 3D points have four coords: P = (X,Y,Z,W) Duality A plane N is also represented
11、 by a 4-vector Points and planes are dual in 3D: N P=0 Projective transformations Represented by 4x4 matrices T: P = TP, N = N T-13D to 2D: “perspective” projectionMatrix Projection:What is not preserved under perspective projection?What IS preserved?Vanishing pointsVanishing point projection of a p
12、oint at infinityimage planecamera centerground planevanishing pointVanishing points (2D)image planecamera centerline on ground planevanishing pointVanishing pointsProperties Any two parallel lines have the same vanishing point v The ray from C through v is parallel to the lines An image may have mor
13、e than one vanishing point in fact every pixel is a potential vanishing pointimage planecamera center Cline on ground planevanishing point Vline on ground planeVanishing linesMultiple Vanishing Points Any set of parallel lines on the plane define a vanishing point The union of all of these vanishing
14、 points is the horizon line also called vanishing line Note that different planes define different vanishing linesv1v2Vanishing linesMultiple Vanishing Points Any set of parallel lines on the plane define a vanishing point The union of all of these vanishing points is the horizon line also called va
15、nishing line Note that different planes define different vanishing linesComputing vanishing pointsPropertiesP is a point at infinity, v is its projection They depend only on line direction Parallel lines P0 + tD, P1 + tD intersect at PVP0DComputing vanishing linesPropertiesl is intersection of horizontal plane through C with image plane Compute l from two sets of parallel lines on ground plane All points at same height as C project to l points higher than C project above l Provides way of comparing height of objects in the sceneground planelCFun with vanishing pointsPerspective cue
