Image formationGeometry of perspective imaging■ Coordinate transformations■ Image formation■ Vanishing points■ Stereo imagingImage formationImages of the 3-D world■ What is the geometry of the image of a three dimensional object?– Given a point in space, where will we see it in an image?– Given a line segment in space, what does its image look like?– Why do the images of lines that are parallel in space appear to converge to asingle point in an image?■ How can we recover information about the 3-D world from a 2-D image?– Given a point in an image, what can we say about the location of the 3-D pointin space?– Are there advantages to having more than one image in recovering 3-Dinformation?– If we know the geometry of a 3-D object, can we locate it in space (say for arobot to pick it up) from a 2-D image?Image formationEuclidean versus projective geometry■ Euclidean geometry describes shapes “as they are”– properties of objects that are unchanged by rigidmotions» lengths» angles» parallelism■ Projective geometry describes objects “as they appear”– lengths, angles, parallelism become “distorted” whenwe look at objects– mathematical model for how images of the 3D worldare formedImage formationExample 1■ Consider a set of railroad tracks– Their actual shape:» tracks are parallel» ties are perpendicular to the tracks» ties are evenly spaced along the tracks– Their appearance» tracks converge to a point on thehorizon» tracks don’t meet ties at right angles» ties become closer and closer towardsthe horizonImage formationExample 2■ Corner of a room– Actual shape» three walls meeting at right angles. Total of 270o of angle.– Appearance» a point on which three lines segments are concurrent. Totalangle is 360oImage formationExample 3■ B appears between A and C from pointp■ But from point 1, A appears between Band C■ Apparent displacement of objects due tochange in viewing position is calledparallax shift– Greeks knew that is the earth revolvedaround the sun there would be parallaxshift of the stars– Tycho Brahe looked for this shift, buthis instruments were not accurateenough.BACABCpqImage formationVanishing points and lines■ To begin with, let’s assume we’re lookingat a scene on the plane, σ, which isperpendicular to the image plane ρ.– To each point P on σ we associate thepoint p on ρ corresponding to theintersection of the line CP with theplane ρ.– C is called the center of projection■ The point of intersection of the linethrough the optical center, C, that isperpendicular to the image plane (and soparallel to the object plane) is called theprincipal vanishing point, V.■ The line, v, which is the intersection ofthe picture plane and the plane through Pparallel to the object plane is called thevanishing line, or the horizon line.Image plane, ρObject plane, σCPpVvImage formationVanishing points and lines■ Our mapping between σ and ρ does notdefine a 1-1 correspondence.– Points on v are not the image of anypoint on σ– Points on m (intersection of planethrough C parallel to ρ) have noimage on σ■ We’ll fix this later.■ Add railroad tracks to σ.Image plane, ρObject plane, σCPpVvmImage formationVanishing points and lines■ The images of all the points on L1 lie on theplane π1 defined by L1 and C.– this is because the lines of sight for each point onL1 lie on this plane.■ The image of L1 is then the intersection of π1with the picture plane - l1.■ L1 is parallel to CV since they are bothperpendicular to the picture plane.– But, then CV must also lie on π1, since parallellines are co-planar, and V lies on π1.■ So, the principal vanishing point must lie on l1,since it lies on both π1 and the picture plane.■ Similarly, V must lie on l2, the image of L2.■ In fact, ALL lines in the object plane that areperpendicular to the image plane, image to linespassing through V.– This is the image of the “point at infinity” forthat set of parallel linesImage plane, ρObject plane, σCVvL1L2l2l1Image formationVanishing points and lines■ M1 and M2 are parallel, but not perpendicular tothe image plane.■ The image of Mi is the line mi where the plane,πi, containing C and Mi intersects the imageplane.■ There is a unique line, Λ, through C which isparallel to Mi and so this line is also on πi.■ Since this line is horizontal, it also lies in theplane determined by C and v, the vanishing line.■ Since v and Λ lie in the same plane, and are notparallel, they must intersect at some point Vm.Image plane, ρObject plane, σCVvM1M2m2m1VMΛImage formationVanishing points and lines■ This places Vm on both πi and the imageplane, so it must lie on their intersection,mi.– So, any family of parallel lines on theobject plane will image to a set of linespassing through a point (vanishing point)on the vanishing line.■ As we rotate the set of lines, the vanishingpoint moves along the vanishing line.■ So even though parallel lines don’t “meet” wecan see where they meet in images!Image plane, ρObject plane, σCVvM1M2m1m2VMΛImage formationGeneral case■ Generally, the vanishing line for any object plane is the line of intersection ofthe– image plane– plane parallel to the object plane through C (called the horizon plane).■ The vanishing point for a line on the object plane is the intersection of the– vanishing line– plane containing the original line and the center of projection.vVCρπImage formationVanishing points and linesImage formationHomogeneous coordinates■ Classical Euclidean geometry: through any point not on a given line, thereexists a unique line which is parallel to the given line.– For 2,000 years, mathematician tried to “prove” this from Euclid’spostulates.– In the early 20’th century, geometry was revolutionized whenmathematicians asked: What if this were false?– That is, what if we assumed that EVERY pair of lines intersected?– To do this, we’ll have to add points and lines to the standard Euclideanplane.■ If (x,y) are the rectangular coordinates of a point, P, and if (x1, x2, x3) are anythree real numbers such that:– x1/x3 = x– x2/x3 = ythen (x1, x2, x3) are a set of homogeneous coordinates for (x,y).■ So, in particular, (x,y,1) are a set of homogeneous coordinates for (x,y)Image formationHomogeneous coordinates■ Given the homogeneous coordinates, (x1, x2, x3), the rectangular coordinatescan be