EECS 442 Discussion. Arash Ushani. September 20, Arash Ushani EECS 442 Discussion September 20, / 14
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1 EECS 442 Discussion Arash Ushani September 20, 2016 Arash Ushani EECS 442 Discussion September 20, / 14
2 Projective Geometry Projective Geometry For more detail, see HZ Chapter 2 Arash Ushani EECS 442 Discussion September 20, / 14
3 Projective Geometry Lines Representing Lines How do we represent a line? Arash Ushani EECS 442 Discussion September 20, / 14
4 Projective Geometry Lines Representing Lines How do we represent a line? ax + by + c = 0 l = (a, b, c) Arash Ushani EECS 442 Discussion September 20, / 14
5 Projective Geometry Lines Representing Lines How do we represent a line? ax + by + c = 0 l = (a, b, c) Is this a unique representation? Arash Ushani EECS 442 Discussion September 20, / 14
6 Projective Geometry Lines Representing Lines How do we represent a line? ax + by + c = 0 l = (a, b, c) Is this a unique representation? (ka)x + (kb)y + kc = 0 l = (ka, kb, kc) Equivalance Classes Arash Ushani EECS 442 Discussion September 20, / 14
7 Projective Geometry Lines Points on Lines How can we tell if a point (x, y) is on the line l = (a, b, c) Arash Ushani EECS 442 Discussion September 20, / 14
8 Projective Geometry Lines Points on Lines How can we tell if a point (x, y) is on the line l = (a, b, c) ax + by + c = 0 Arash Ushani EECS 442 Discussion September 20, / 14
9 Projective Geometry Lines Points on Lines How can we tell if a point (x, y) is on the line l = (a, b, c) ax + by + c = 0 x = (x, y, 1) x l = 0 l x = 0 Arash Ushani EECS 442 Discussion September 20, / 14
10 Projective Geometry Lines Intersection of Lines How do we find the intersection x of lines l 1 and l 2 Arash Ushani EECS 442 Discussion September 20, / 14
11 Projective Geometry Lines Intersection of Lines How do we find the intersection x of lines l 1 and l 2 x l 1 = 0 x l 2 = 0 Arash Ushani EECS 442 Discussion September 20, / 14
12 Projective Geometry Lines Intersection of Lines How do we find the intersection x of lines l 1 and l 2 x l 1 = 0 x l 2 = 0 x = l 1 l 2 Arash Ushani EECS 442 Discussion September 20, / 14
13 Projective Geometry Lines Line through two points How do we find the line l that goes through x 1 and x 2 Arash Ushani EECS 442 Discussion September 20, / 14
14 Projective Geometry Lines Line through two points How do we find the line l that goes through x 1 and x 2 x 1 l = 0 x 2 l = 0 Arash Ushani EECS 442 Discussion September 20, / 14
15 Projective Geometry Lines Line through two points How do we find the line l that goes through x 1 and x 2 x 1 l = 0 x 2 l = 0 l = x 1 x 2 Arash Ushani EECS 442 Discussion September 20, / 14
16 Projective Geometry Lines HW Hints: Lines and transformations Helpful identity: (Hx) (Hy) = (det H)H (x y) Arash Ushani EECS 442 Discussion September 20, / 14
17 Projective Geometry Lines HW Hints: Lines and transformations Helpful identity: (Hx) (Hy) = (det H)H (x y) Alernatively, if l x = 0, what can you say about l H 1 Hx if H is a projective transformation? Arash Ushani EECS 442 Discussion September 20, / 14
18 Projective Geometry Lines MATLAB Exercise Go to CTools Resources Discussion matlab.zip Arash Ushani EECS 442 Discussion September 20, / 14
19 DLT For more detail, see chapter 4 section 1 in HZ Arash Ushani EECS 442 Discussion September 20, / 14
20 DLT We have a set of point correspondences x i to x i We want to find the transformation H such that x i = Hx i Arash Ushani EECS 442 Discussion September 20, / 14
21 DLT We have a set of point correspondences x i to x i We want to find the transformation H such that x i = Hx i How many degrees of freedom are there in H? Arash Ushani EECS 442 Discussion September 20, / 14
22 DLT We have a set of point correspondences x i to x i We want to find the transformation H such that x i = Hx i How many degrees of freedom are there in H? 8 Arash Ushani EECS 442 Discussion September 20, / 14
23 DLT We have a set of point correspondences x i to x i We want to find the transformation H such that x i = Hx i How many degrees of freedom are there in H? 8 How many correspondences do we need? Arash Ushani EECS 442 Discussion September 20, / 14
24 DLT We have a set of point correspondences x i to x i We want to find the transformation H such that x i = Hx i How many degrees of freedom are there in H? 8 How many correspondences do we need? 4 Arash Ushani EECS 442 Discussion September 20, / 14
25 DLT If x i = Hx i, equivalently x i (Hx i) = 0 Arash Ushani EECS 442 Discussion September 20, / 14
26 DLT If x i = Hx i, equivalently x i (Hx i) = 0 h 1 x i Hx i = h 2 x i where each h j is a row in H h 3 x i Arash Ushani EECS 442 Discussion September 20, / 14
27 DLT If x i = Hx i, equivalently x i (Hx i) = 0 h 1 x i Hx i = h 2 x i where each h j is a row in H h 3 x i If x i = (x i, y i, w i ), then: y x i h3 x i w i h2 x i i (Hx i ) = w i h1 x i x i h3 x i = 0 x i h2 x i y i h1 x i Arash Ushani EECS 442 Discussion September 20, / 14
28 DLT If x i = Hx i, equivalently x i (Hx i) = 0 h 1 x i Hx i = h 2 x i where each h j is a row in H h 3 x i If x i = (x i, y i, w i ), then: y x i h3 x i w i h2 x i i (Hx i ) = w i h1 x i x i h3 x i = 0 x i h2 x i y i h1 x i 0 w i x i y i x i h 1 w i x i 0 x i x i h 2 = 0 y i x i x i x i 0 h 3 Arash Ushani EECS 442 Discussion September 20, / 14
29 DLT [ 0 w i x i y i x i w i x i 0 x i x i ] h 1 h 2 = 0 h 3 Arash Ushani EECS 442 Discussion September 20, / 14
30 DLT [ 0 w i x i y i x i w i x i 0 x i x i ] h 1 h 2 = 0 h 3 A i h 1 h 2 = 0 h 3 Arash Ushani EECS 442 Discussion September 20, / 14
31 DLT [ 0 w i x i y i x i w i x i 0 x i x i ] h 1 h 2 = 0 h 3 A i h 1 h 2 = 0 h 3 With 4 correspondences, we have Ah = 0 where A is rank 8 Arash Ushani EECS 442 Discussion September 20, / 14
32 DLT What if we have more than 4 correspondences? (Why would we have more than 4?) Arash Ushani EECS 442 Discussion September 20, / 14
33 DLT What if we have more than 4 correspondences? (Why would we have more than 4?) Overdetermined solution, find h that minimizes error (with h = 1) It can be shown that the singular vector corresponding to the smallest singular value of A is the solution to h that minimizes Ah Arash Ushani EECS 442 Discussion September 20, / 14
34 DLT What if we have more than 4 correspondences? (Why would we have more than 4?) Overdetermined solution, find h that minimizes error (with h = 1) It can be shown that the singular vector corresponding to the smallest singular value of A is the solution to h that minimizes Ah SVD Decomposition: A = UDV D is a diagonal matrix of the singular values of A V contains the singular vectors of A as column vectors Arash Ushani EECS 442 Discussion September 20, / 14
35 Next Week Next Week Guest Speaker: Steven Parkison, Calibration Expert No Office Hours Arash Ushani EECS 442 Discussion September 20, / 14
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