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Unformatted text preview: ons • Displacement Models
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– 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 nonzero 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 3vectors, 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 2dimensional 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.
 Winter '13
 SohaibKhan

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