IEG4160_Part4

# IEG4160_Part4 - IEG 4160 Image and Video Processing...

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Linear & Shift Invariant Systems IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu Linearity and shift invariance Delta function and its properties Point spread function Convolution Discrete LSI systems

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Page 2 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Linearity
Page 3 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Shift invariance 1D 2D Linear, shift invariant systems (LSI systems) I f () , then ( ) ( ) f t g t f tT g →− t t f t gt t t f gt T T T If ( , ) ( , ), then ( , ) ( , ) f x yg x y fx y gx y α βα β −→

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Page 4 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Delta function
Page 5 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Delta function Can be defined as the limit of a sequence of functions ± each member has some area, but getting narrower & higher A “ sharp spike ”atorigin

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Page 6 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Delta function Examples in 1D ± ± 0 1 ( ) lim rect a t t aa δ ⎛⎞ = ⎜⎟ ⎝⎠ t () t 0 1/2 t rect( ) t 1 /2 a /2 a t 1 rect( ) a t a 1/ a
Page 7 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Delta function Examples in 2D ± ± ( , ) lim ( , ) N N xy δ →∞ = 2 rect( , ), | | 1/2 , |y| 1/2 (, ) 0, elsewhere N NN x N y x N N ≤≤ = x y x y (0,0)

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Page 8 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Delta function Properties ± It’s 0 everywhere except at the origin ± Its volume = 1 x y (, ) x y δ (0,0)
Page 9 IEG 4160: Image and Video Processing. Lecturer: Jianzhuang Liu 4. Linear & Shift Invariant Systems Delta function

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IEG4160_Part4 - IEG 4160 Image and Video Processing...

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