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Conics 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

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Conic representation conic = plane curve described by quadratic polynomial = section of cone –what are the conics? important primitives for vision and graphics conic in Euclidean space: ax^2 + bxy + cy^2 + dx + ey + f = 0 homogenize to translate to projective space 2-minute exercise: how do you homogenize this polynomial and thereby translate to projective space? conic in projective space: ax^2 + bxy + cy^2 + dxw + eyw + fw^2 = 0 –observation: homogenize a polynomial by the replacement x x/w and y y/w (ring a bell?) this conic in projective space is encoded by a symmetric 3x3 matrix: –x^t C x =0 –C = [a, b/2, d/2;b/2, c, e/2;d/2, e/2, f] –ijth entry encodes ijth coefficient (where x=1,y=2,w=3) 2-minute exercise: how many degrees of freedom does a conic have?

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Conic representation 2 matrix C is projective (only defined up to a multiple) conic has 5dof: ratios {a:b:c:d:e:f} or 6 matrix entries of C minus scale –kC represents the same conic as C (equivalent to multiplying equation by a constant) in P^2, all conics are equivalent –that is, can transform from a conic to any other conic using projective transforms –not true in Euclidean space: cannot transform from ellipse to parabola using linear transformation transformation rule: –if point x Hx, then conic C (H -1 )^t C H -1 HZ30-31, 37

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Conic tangents another calculation is made simple in projective space (and reduces to matrix computation) Result: The tangent of the conic C at the point x is Cx. Proof: –this line passes through x since x^t Cx = 0 (x lies on the conic) –Cx does not contain any other point y of C, otherwise the entire line between x and y would lie on the conic : if y^t C y = 0 (y lies on conic) and x^t C y = 0 (Cx contains y), then x+\alpha y (the entire line between x and y) also lies on C –thus, Cx is the tangent through x (a tangent is a line with only one point of contact with conic) HZ31

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Metric rectification with circular points 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

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Material for metric rectification circular points (34) similarity iff fixed circular points (52) conic (55) dual conic degenerate conic C * ∞ : conic dual to circular points (or just dual conic) angle from dual conic image of dual conic under homography then we’re ready for the algorithm of Example 2.26 HZ

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Line conic we have considered point conic points are dual to lines: let’s dualize suppose conic is nondegenerate (not 2 lines or repeated line), so C is nonsingular 3x3 matrix C encodes the points of a conic –x^t C x = 0 if x lies on conic matrix C -1 encodes the tangent lines of that conic: –L is a tangent line of the conic iff L^t C -1 L = 0 –proof: tangent L = Cx for some point x on conic; x^t C x = 0 so substituting x = C -1 L yields (C -1 L)^t C (C -1 L) = L^t C -1 L = 0 (recall that C is symmetric) called a dual conic or line conic; conic envelope transformation rule: if point x Hx, then dual conic C* H C* H^t HZ31,37

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Degenerate conics there are degenerate point conics and degenerate line conics recall the outer product if L and M are lines, LM^t + ML^t is the degenerate point conic consisting of these two lines –notice that the matrix is singular (rank 2) if p and q are points, pq^t + qp^t is the degenerate line conic consisting of all lines through p or q –[2 pencils of lines, drawing] –notice that this is a line conic (or dual conic)

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‘The’ dual conic C * ∞ two circular points I and J make a degenerate line conic out of them C * ∞ = IJ^t + JI^t = (1,0,0; 0,1,0; 0,0,0) –all lines through the circular points C * ∞ will be called ‘the’ dual conic –although it is the line conic dual to the circular points like circular points, C * ∞ is fixed under a projective transform iff it is a similarity interesting fact: null vector of C * ∞ = line at infinity HZ explore relationship to angle

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Computing angle from C * ∞ how can we measure angle in a photograph? angle is typically measured using dot product but dot product is not invariant under homography consider two lines L and M in projective space (viewed as conventional columns, not correct rows) L^t M is replaced by a normalized L^t C * ∞ M –(L^t C * ∞ M) / \sqrt( (L^t C * ∞ L) (M^t C * ∞ M)) this is invariant to homography note that L^t C * ∞ M is equivalent to dot product in the original space –(a1,a2,1) C * ∞ (b1,b2,1) = a1b1 + a2b2 Result: (L^t C * ∞ M) / \sqrt( (L^t C * ∞ L) (M^t C * ∞ M)) is the appropriate measure of angle in projective space. since angle = arccos (A.B), this is of course a measure of cos(angle), not angle corollary: angle can be measured once C * ∞ is known HZ54-55

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How C * ∞ transforms we want to understand how C * ∞ transforms under a homography (since we don’t want it to transform!) a homography matrix can be decomposed into a projective, affine, and similarity component: –H = [I 0][K 0][sR t] [v^t 1][0 1][01] = H p H a H s –note that K is the affine component recall that line conics transform by C* H C* H^t, so C * ∞ (H p H a H s ) C * ∞ (H p H a H s ) t but the dual conic is fixed under a similarity, so: C * ∞ (H p H a ) C * ∞ (H p H a ) t which reduces as follows using C * ∞ = diag(1,1,0): image of C * ∞ = [KK^t KK^t v ] [v^t KK^tv^t KK^t v ] when line at infinity is not moving, v = 0: image of C * ∞ in an affinely rectified image = [KK^t 0] [0 0] [HZ43 for this decomposition chain of a homography, HZ55]

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Metric rectification algorithm “We have the technology. We can rebuild him!” input: 2 orthogonal line pairs assume that the image has already been affinely rectified (i.e., line at infinity is in correct position) we are within an affinity of a similarity! what is this affinity K? use the known orthogonality (known angle) to solve for K let S = KK^t: we actually solve for S, then retrieve K using Cholesky decomposition –note: S has 2 dof (symmetric 2x2) –use the two line angle constraints to solve for these dof

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Metric rectification continued (L,M) = image of orthogonal line pair –Figure 2.17a –choose two points P1,P2 on L: L = P1xP2 –L = (L1,L2,L3) and M = (M1,M2,M3) orthogonal pair (L,M) satisfies L t C * ∞ M = 0 –L t [S 0; 0 0] M = (L1 L2) S (M1 M2) –this imposes a linear constraint on S (L1 M1, L1M2 + L2M1, L2M2). (s11, s12, s22) = 0 first row of matrix. S = 0 a second orthogonal pair defines the 2 nd row of the system As = 0 [A for angle] –A is 2x3 matrix, s is the vector (s11,s12,s22) S is the null vector of this matrix M Cholesky decompose S to retrieve K the ‘true’ image (up to metric structure) must have been mapped by the affinity [K 0; 0,1] –see ‘how C * ∞ transforms’ above so rectify by mapping image by K -1 Moral: use orthogonal constraint to solve for affinity K, using the dual conic to help you measure angle –note: never actually compute dual conic, just rely on its properties –this is a perfect example of the power of a theoretical construct HZ56

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Synopsis want affinity K want S = KK^t S is available from the image of C * ∞ get at image of C * ∞ using angle relationship use orthogonal pairs to solve for image of C * ∞ back out from image of C * ∞ to S to K

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Numerical aside: solving for null space to solve Ax = 0: 1.compute SVD of A = U D V^t 2.if null space is 1d, x = last column of V (called null vector) 3.in general, last d columns of V span the d- dimensional null space

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Numerical aside: SVD singular value decomposition of A is A = U D V^t, where –mxn A –mxm orthogonal U –mxn diagonal D (containing singular values sorted) –nxn orthogonal V do not compute yourself! compute using CLAPACK or OpenCV

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Other methods for metric rectification the 2-ortho-pair method is called stratified rectification stratification = 2 step approach to removal of distortion, first projective, then affine Note: 2 orthogonal lines are conjugate with respect to the dual conic C * ∞ can also rectify (in a stratified fashion) using an imaged circle image of circle is ellipse: intersect it with vanishing line to find imaged circular points can also rectify using 5 orthogonal line pairs –why would you want to? because it doesn’t assume image has already been affinely rectified method (your responsibility to read) –find the dual conic as the null vector of a 5x6 matrix built up from 5 linear equations –see DLT algorithm below for a similar approach (this is a very popular approach!) –note: solving for entire 5dof conic, not just the 2dof affine component HZ56-57

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On OpenCV review openCVInstallationNotes OpenCV demo programs at my website –cvdemo, cvmousedemo, cvresizedemo, cvpixeldemo opencvlibrary.sourceforge.net –invaluable resource for full documentation, FAQ, examples, Wiki IPL = Image Processing Library depth = # of bits in each data value of image –8 = uchar, 32 = float channels = # of data values for each image pixel –1 = grayscale; 3 = RGB

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Capturing images as test data I would like you to personalize your test data I would also like to gather an image database assignment: find images replete with parallel lines (and optimally, vanishing lines inside the image) and orthogonal lines –also want different directions of parallel and orthogonal lines: e.g., brick wall is not as useful since all parallel lines yield only two distinct ideal points challenge: rectification without gaps challenge: rectification to within a translation using external cues challenge: rectification from other constraints (e.g., in a natural scene with no visible lines or circles) challenge: image contexts with circles and other known nonlinear curves

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