03 - Image Transforms

03 - Image Transforms - Image Transforms copyright 2002H....

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Image Transforms copyright 2002©H. R. Myler Fourier Transform Pair: ( ) 2 () ( ) j ux vy Ff d d π ∞∞ + 2 (, ) (,) ju x v y f x y F u v e dudv + −∞ −∞ = ∫∫ ( , )( , Fuv f x y e dxdy −∞ −∞ = copyright 2002©H. R. Myler Same properties as in 1D apply
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Fourier Transform Pair: We can show that for a “box” image of finite size X by Y and of a constant value A: () ( ) 22 2 sin sin (,) XY ju x v y uX vY Fuv Ae dxdy AXY uX vY π ππ −+ −− ⎤⎡ == ⎥⎢ ⎦⎣ ∫∫ The magnitude (spectrum) is: copyright 2002©H. R. Myler ( ) ( ) sin sin uX vY F u v AXY uX vY = maxima of: cos 2 π (ux+vy) copyright 2002©H. R. Myler
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Discrete Fourier Transform Pair: kernels 11 2 00 (,) (, ) ux vy MN j M N xy Fuv f xye π ⎛⎞ −− −+ ⎜⎟ ⎝⎠ == = ∑∑ 2 ux vy j M N fxy Fuve + = copyright 2002©H. R. Myler For an M x N image The Fourier Transform kernel is separable : exp -2 π jux N exp -2 π jvy N Separability gives rise to processing advantages in the computer. The transform maybepe r fo rmedasas ing led imens iona l computation on the rows and columns. • The Fourier matrix can be factored into the produc o 2 lo N spars and diagona copyright 2002©H. R. Myler product of 2 log 2 N sparse and diagonal matrices (observed by Good in 1958). This is the basis for the Fast Fourier Transform.
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03 - Image Transforms - Image Transforms copyright 2002H....

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