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Lec04.2DTransformations2 - Groups A group is a set G...

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Groups A group is a set G together with an operation that combines two elements of a and b to form another element a b. To form a group, ( G, ) must satisfy the following axioms Closure For all a , b in G, a b is also in G Associativity For all a, b, c in G , ( a b ) c = a ( b c ) Identity There exists an element e in G s.t. a e = e a = a Inverse For each a in G , there exists an element b in G s.t. a b = e
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Hierarchy of Transformation Groups Translation Rigid Body Transformation Similarity Each higher group completely contains the lower group
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Affine Group General 2 x 3 linear transform Contains rotation, scaling, shear, translation and any combination thereof Preserves Parallel lines [Proof?] Ref: Steve Mann & Rosalind W. Picard, “Video Orbits of the Projective Group: A simple approach to featureless estimation of parameters”, IEEE Trans. on Image Processing, Vol. 6, No. 9, September 1997
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Order of Transformations Rotation/Scaling/Shear, followed by Translation Translation, followed by Rotation/Scaling/Shear Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ = Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 2 4 3 1 2 1 4 3 2 1 2 1 b a a b a a a a a a b b Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ + + = Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Φ Τ ΢ ΢ ΢ Σ Ρ 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 2 4 1 3 4 3 2 2 1
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