Day x: Infinity DLT alg HZ 4.1 Rectification HZ 2.7 Hierarchy of maps Invariants HZ 2.4 Projective transform HZ 2.3 Behaviour at infinity Primitives pt/line/conic HZ 2.2

Ideal points points at infinity now have a natural representation: set w = 0 –(x,y,0) represents the point at infinity in the direction (x,y) points at infinity (x,y,0) are called ideal points –no different in treatment than finite points all one happy family (finite and infinite points) –in a projective transform (linear map of projective space), points at infinity can be mapped to finite points, which is exactly what happens in photography intersection of parallel lines is consistent (not a special case) –e.g., x=1 and x=2

Line at infinity all of the ideal points in 2-space combine to form the line at infinity it is the line (0,0,1) –after all, it contains all the points (x,y,0) (0,0,1). (x,y,0) = 0 HZ28 the line at infinity is often mapped to a finite line in the act of photography finding the line at infinity in the image is crucial to removal of distortion (rectification) Exercise: where does the line (a,b,c) meet the line at infinity? take 2 minutes to discuss in pairs

Answer to exercise line (a,b,c) meets line (0,0,1) at the point (b,-a,0) [apply the intersection rule] –this encodes its tangent vector or direction –recall that (a,b) is the normal of the line ax+by+c = 0

Circular points two special points at infinity are the circular points (1,i,0) and (1,-i,0) they are also crucial in rectification (getting true measurements from an image) every circle contains the circular points –proof: a circle has the implicit representation (x-aw)^2 + (y-bw)^2 – r^2w^2 = 0: the circular points satisfy this equation –also note that setting w=0 in this equation yields x^2+y^2=0; w=0 and the circular points are the solutions of this equation explains why circle is defined by 3 points, rather than 5 explains why 2 circles have 2 intersections, not 4 HZ5, HZ52

Finding the line at infinity the line at infinity can be mapped to a finite line in an image, which is one of the sources of distortion in the image vanishing line = imaged line at infinity let’s find it; good example is Houckgeest image (FL206) we shall intersect the images of parallel lines to find vanishing points (imaged points at infinity) prerequisite: 2 pairs of identified parallel lines in the image, the 1 st pair not parallel to the 2 nd pair 2 minute exercise: develop an algorithm to find the vanishing line and answer ‘how many cross products are required’? HZ50

Translate into projective geometry: how many cross products?

Algorithm for vanishing line 1.mark 2 points on 1 st parallel line, say P1 and P2 2.mark 2 points on 2 nd parallel line, say Q1 and Q2 3.parallel lines are L=P1xP2 and M=Q1xQ2 4.a point A on the vanishing line = LxM (intersection) 5.find another point B on the vanishing line using the same approach with a 2 nd pair of parallel lines (not parallel to the 1 st pair) 6.vanishing line = AxB (join) 7.2 to get lines, 1 to get intersection, 3 more for B, 1 for join = 7

Removal of projective distortion now map vanishing line (L1,L2,L3) to (0,0,1), using the projective transformation represented by the matrix M = [1,0,0; 0,1,0; L1,L2,L3] to map points (see below for relationship of line transformation and point transformation) –actually any affine transformation can be added to this projective transform (H_a M) –HZ49 motivates an understanding of projective transforms

HW1 See handout. This homework has a dual purpose: it is a warmup for building metric reconstructions in 3D (we will do it in 2D) and it is an exploration of some of the key ideas of projective geometry through implementation. suggested warmup: input lines and intersect (the first stage of affine rectification) –does not require an image –allows you to develop your functions for projective geometry (either C functions or C++ class) –allows you to explore OpenGL mouse reading 1.affine rectification: implement algorithm described above (and Example 2.18 HZ50) to find a vanishing line and map it to the line at infinity (0,0,1) in order to affinely rectify an image should work with any image use image of Figure 2.13 for preliminary testing (available at my website) 2.metric rectification: move circular points to their proper position using Algorithm of Example 2.26 (described in detail next lecture); advise delaying start until Monday 3.Prove that if a projective transform fixes the circular points, then it must be a similarity. [we proved the other direction]