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Recovery of affine and metric properties from images in 2D Projective space Ko Dae-Won

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Affine properties(line at infinity) - Parallelism - Parallel length ratios Metric properties(circular points) - Angles - Length ratios Recover the original shape 2 Recovery of affine and metric properties from images in 2D Projective space

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Homogeneous coordinates 1. Recovery of affine properties 3 Recovery of affine and metric properties from images in 2D Projective space equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 (0,0,0) T forms P 2 Homogeneous representation of points on if and only if The point x lies on the line l if and only if x T l=l T x = 0 Homogeneous coordinates Inhomogeneous coordinates but only 2DOF

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Points from lines and vice-versa 1. Recovery of affine properties 4 Recovery of affine and metric properties from images in 2D Projective space Intersections of lines The intersection of two lines and is Line joining two points The line through two points and is Intersections of parallel lines Ideal points Line at infinity

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1. Recovery of affine properties 5 Recovery of affine and metric properties from images in 2D Projective space Duality Duality principle: To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem

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1. Recovery of affine properties 6 Recovery of affine and metric properties from images in 2D Projective space Conics Curve described by 2 nd -degree equation in the plane or homogenized or in matrix form with

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1. Recovery of affine properties 7 Recovery of affine and metric properties from images in 2D Projective space Tangent lines to conics The line l tangent to C at point x on C is given by l=Cx l x C

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1. Recovery of affine properties 8 Recovery of affine and metric properties from images in 2D Projective space Dual conics A line tangent to the conic C satisfies Dual conics = line conics = conic envelopes In general :

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1. Recovery of affine properties 9 Recovery of affine and metric properties from images in 2D Projective space Projective transformations A projectivity is an invertible mapping h from P 2 to itself such that three points x 1,x 2,x 3 lie on the same line if and only if h(x 1 ),h(x 2 ),h(x 3 ) do. Definition: A mapping h:P 2 P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 represented by a vector x it is true that h(x)=Hx Theorem: Definition: Projective transformation or projectivity=collineation=projective transformation=homography

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The line at infinity 1. Recovery of affine properties 10 Recovery of affine and metric properties from images in 2D Projective space

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Affine properties from images 1. Recovery of affine properties 11 Recovery of affine and metric properties from images in 2D Projective space

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1. Recovery of affine properties 12 Recovery of affine and metric properties from images in 2D Projective space

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1. Recovery of affine properties 13 Recovery of affine and metric properties from images in 2D Projective space

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Distance ratio 1. Recovery of affine properties 14 Recovery of affine and metric properties from images in 2D Projective space

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Distance ratio 1. Recovery of affine properties 15 Recovery of affine and metric properties from images in 2D Projective space

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2. Recovery of metric properties 16 Recovery of affine and metric properties from images in 2D Projective space

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The circular points 2. Recovery of metric properties 17 Recovery of affine and metric properties from images in 2D Projective space

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The circular points 2. Recovery of metric properties 18 Recovery of affine and metric properties from images in 2D Projective space

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Conic dual to the circular points 2. Recovery of metric properties 19 Recovery of affine and metric properties from images in 2D Projective space

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Angles 2. Recovery of metric properties 20 Recovery of affine and metric properties from images in 2D Projective space

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Length ratios 2. Recovery of metric properties 21 Recovery of affine and metric properties from images in 2D Projective space

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2. Recovery of metric properties 22 Recovery of affine and metric properties from images in 2D Projective space

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Length ratios 2. Recovery of metric properties 23 Recovery of affine and metric properties from images in 2D Projective space

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Metric from affine 2. Recovery of metric properties 24 Recovery of affine and metric properties from images in 2D Projective space

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Metric from projective 2. Recovery of metric properties 25 Recovery of affine and metric properties from images in 2D Projective space

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