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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)**

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

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**1. Recovery of affine properties**

Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Homogeneous coordinates equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T forms P2 Homogeneous representation of points on if and only if The point x lies on the line l if and only if xTl=lTx=0 Homogeneous coordinates Inhomogeneous coordinates but only 2DOF

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**1. Recovery of affine properties**

Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Points from lines and vice-versa 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**

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

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

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**1. Recovery of affine properties**

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

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

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**1. Recovery of affine properties**

Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Projective transformations Definition: A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3 lie on the same line if and only if h(x1),h(x2),h(x3) do. A mapping h:P2P2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 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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**1. Recovery of affine properties**

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

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**1. Recovery of affine properties**

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

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**1. Recovery of affine properties**

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

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**1. Recovery of affine properties**

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

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**1. Recovery of affine properties**

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

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**1. Recovery of affine properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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**2. Recovery of metric properties**

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

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Recovering metric and affine properties from images

Recovering metric and affine properties from images

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