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filter2 - Fourier Transform Analytic geometry gives a...

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Fourier Transform Analytic geometry gives a coordinate system for describing geometric objects. Fourier transform gives a coordinate system for functions.
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Decomposition of the image function The image can be decomposed into a weighted sum of sinusoids and cosinuoids of different frequency. Fourier transform gives us the weights
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Basis P=(x,y) means P = x(1,0)+y(0,1) Similarly: + + + + = ) 2 sin( ) 2 cos( ) sin( ) cos( ) ( 2 2 1 2 2 1 1 1 θ θ θ θ θ a a a a f
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cosc a sin sin cos ) sin( : such that , , 2 1 2 1 2 1 = = + = + 5 2200 c a a a c a a c θ θ θ
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Orthonormal Basis ||(1,0)||=||(0,1)||=1 (1,0).(0,1)=0 Similarly we use normal basis elements eg: While, eg: = π θ θ θ θ θ 2 0 2 cos ) cos( ) cos( ) cos( d = π θ θ θ 2 0 0 sin cos d
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2D Example
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t i )e A( ϖ ϖ ϖ t i e t ω i t e t i sin cos + = ϖ ϖ Sinusoids and cosinuoids are eigenfunctions of convolution Why are we interested in a decomposition of the signal into harmonic components? Thus we can understand what the system (e.g filter) does to the different components (frequencies) of the signal (image)
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