Conformal Geometric Algebra in Stochastic Optimization Problems ...

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224 CHAPTER 8. APPLICATIONS IN OMNIDIRECTIONAL VISION is evaluated. This gives � = {x 1...9 } and � = {y 1...9 }, respectively. Every x-y-pair must satisfy equation (8.9) which can be rephrased as vec(x y T ) T vec(E) = 0. Recall that vec(·) reshapes a matrix into a column vector. Hence the best leastsquares approximation of vec(E) is the right-singular vector to the smallest singular value of the matrix consisting of the row vectors vec(x iy T i )T , 1 ≤ i ≤ 9. Let E ⋄ be the so estimated essential matrix. The left- and right-singular vectors to the smallest singular value of E ⋄ are then the sought approximations to the 3D-epipoles, as described above. The epipoles then serve as a first guess for the stochastic epipole estimation explained in the following section. 8.4.4 Stochastic Epipole Estimation Now it is being detailed how the GH-method can be invoked. The role of the previously derived initial estimates in the actual stochastic estimation will thereby become evident, too. Estimating the epipoles is simply done by estimating the motor M, parameterized by p ∈ � 8 : knowing M the directions to the 3D-epipoles can be extracted from the points MF � M and � MFM, respectively, as can be inferred from figure 8.14. The former point, for instance, equals F ′ . As input data all N pairs of corresponding points are used, i.e. � := {x 1...N } and � := {y 1...N }. The sets are computed by means of equation (8.12). An observation is a pair (x i, y i ), 1 ≤ i ≤ N. Note that internally the compound observation vector b i := [x i;y i ] ∈ � 6 is used. The related uncertainties 11 can be computed either by equation (8.12) or more directly by equation (8.11), where independence of x i and y i is assumed, i.e. Σbi,bi = ⎡ 0 ⎣ Σxixi 0 Σy y i i ⎤ ⎦ ∈ � 6×6 , 1 ≤ i ≤ N. The functional model, derived from geometric considerations, is self-evidently given by equation (8.8) for t = t ⋄ . Hence the G-constraint is g t i (p,x i, y i ) = p r p s x k i y l i Ot kc G a rl G c ab R b s, 1 ≤ i ≤ N, (8.13) Differentiating with respect to b i and p yields the required matrices V and U, respectively. Again, the standard H-constraint (7.7) can be used. But additionally it has to be constrained that M does not converge to the identity element M = 1. Otherwise, the condition equation (8.7) would become G = F ∧ X ∧ F ∧ Y being zero at all times. This is achieved by constraining the e-component of F ′ = MF � M, 11 The distribution of the acquired pixel coordinates is assumed to be i.i.d., as usual.
8.4. ESTIMATING EPIPOLES 225 F = eo, to be 0.5. Thus the distance d between F and F ′ is set to one12 . The needed conformal geometric algebra expression is simply d 2 � = −2eo · (M eo � � M) if M was an outright motor. In the iterative estimation process M must effectively be considered a general multivector with structural restrictions imposed by the RBM mask (7.4). Using the matrix representation of CGA, as outlined in chapter 6, a motor p = Φ(M) has the structure ⎡ ⎤ ⎢ M ≡ ⎢ ⎣ p1 + p7 −p4 − p8 p3 + p5 −p2 + p6 −p7 p8 −p5 −p6 p4 + p8 p1 + p7 p2 − p6 p3 + p5 −p8 −p7 p6 −p5 −p3 + p5 −p2 − p6 p1 − p7 p4 − p8 −p5 p6 p7 p8 p2 + p6 −p3 + p5 −p4 + p8 p1 − p7 −p6 −p5 −p8 p7 p7 −p8 p5 p6 p1 − p7 −p4 + p8 p3 − p5 −p2 − p6 p8 p7 −p6 p5 p4 − p8 p1 − p7 p2 + p6 p3 − p5 p5 −p6 −p7 −p8 −p3 − p5 −p2 + p6 p1 + p7 p4 + p8 p6 p5 p8 −p7 p2 − p6 −p3 − p5 −p4 − p8 p1 + p7 This makes it possible to derive the constraint via simple matrix calculations. Since, in the present case, M eo � M can only give vector and 5-vector components, the anti-commutator can be used to expand the inner product as Q := −2eo · (M eo � M) = −(eoM eo � M + M eo � M eo). (8.14) In CGA the reverse operation always amounts to negating the 20 bivector and trivector components of a multivector. Hence � M can be obtained by applying a sign change to the components {p 2...7 } in the above M-matrix. Evaluating equation (8.14) gives a multivector Q with only three non-zero components in q = Φf(Q): q1 : 4(p 2 5 + p2 6 + p2 7 + p2 8 ) q27 : 2(p1p8 − p2p5 − p3p6 − p4p7) q28 : −q27 ! = 1 ! = 0 ! = 0. The rightmost column shows the value the component should take. Note that the last two constraints are already part of the M � M = 1 constraint (7.6), cf. page 196. Adding the constraint p2 5 + p26 + p27 + p28 = 1/4 and differentiating ultimately gives ⎡ ⎤ ⎢ H(p) = 2⎢ ⎣ p1 p2 p3 p4 0 0 0 0 0 0 0 0 p5 p6 p7 p8 p8 −p5 −p6 −p7 −p2 −p3 −p4 p1 It remains to enlighten the usage of the pre-estimated 3D-epipoles. But this issue coincides with the requirement to provide an initial estimate for the GH-method: the initial estimate for p is the unit length translator TE - being a special motor - along the direction of the initial estimate for the 3D-epipole E from section 8.4.3, i.e. p [0] := Φ(TE). 12 This can be done as the true distance between F and F ′ cannot be recovered from the image data. ⎥ ⎦ . ⎥ . ⎥ ⎦
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INSTITUT FÜR INFORMATIK Conformal
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Conformal Geometric Algebra in Stoc
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Dedicated to my beloved wife, Steph
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ACKNOWLEDGMENT This thesis as is ca
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3 The Conformal Geometric Algebra 8
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8 Applications in Omnidirectional V
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2 CHAPTER 1. INTRODUCTION with a si
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4 CHAPTER 1. INTRODUCTION three poi
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6 CHAPTER 1. INTRODUCTION Spherical
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8 CHAPTER 1. INTRODUCTION Fig. 1.3:
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14 CHAPTER 1. INTRODUCTION
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20 CHAPTER 2. GEOMETRIC ALGEBRA 2.1
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146 CHAPTER 5. PARAMETER ESTIMATION
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