Lec06.Factorization-Duality-3DTransforms

Projective space points in 2d point will lie on line

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Unformatted text preview: ons •  Displacement Models –  –  –  –  –  Rigid / Euclidean Similarity Affine Projective Billinear, biquadratic etc •  Recovering the best affine transformation –  Least Squared Error solution –  Pseudo inverse •  Image Warping Lines and Points in 2D Lines in 2D •  Equation of line in 2D •  Thus, a line can be represented by vector •  and mean the same line for •  Thus lines can be represented by equivalence classes of vectors in ie. Projective space Points in 2D •  Point will lie on line if •  Can be written as inner product •  Any non-zero k can be added multiplied to the point, without loss of generality Point on Line •  Point x lies on line l iff •  Even though x and l are 3-vectors, they have 2 degrees of freedom each Intersection of Two Lines •  Two lines will intersect at a point •  Let l and l¶ intersect at point x •  Then •  Proof: Line Joining Two Points •  Two points lie on a line •  Let x and x¶ lie on line l •  Then •  Proof: Duality Duality Theorem: 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 Buy ONE Get ONE in the original theorem FREE! 3D Transformations 3D Translation •  Point in 3D given by (X1 Y1 Z1) •  Translated by (dx dy dz) X2 = X1 + dx Y2 = Y1 + dy Z2 = Z1 + dz Translation •  In matrix form ⎡ྎ X 2 ⎤ྏ ⎡ྎ1 ⎢ྎ Y ⎥ྏ ⎢ྎ0 ⎢ྎ 2 ⎥ྏ = ⎢ྎ ⎢ྎ Z 2 ⎥ྏ ⎢ྎ0 ⎢ྎ ⎥ྏ ⎢ྎ ⎣ྏ 1 ⎦ྏ ⎣ྏ0 0 0 dx ⎤ྏ ⎡ྎ X 1 ⎤ྏ ⎥ྏ ⎢ྎ Y ⎥ྏ 1 0 dy ⎥ྏ ⎢ྎ...
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This note was uploaded on 03/02/2014 for the course CS 436 taught by Professor Sohaibkhan during the Winter '13 term at Lahore University of Management Sciences.

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