eee508_lect2 - EEE 508 Digital Image Processing and...

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EEE 508 - Digital Image Processing and Compression 2D DSP Basics: 2D Systems EEE 508 - Lecture 2
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2D Systems x ( n 1 , n 2 ) T[ ] y ( n 1 , n 2 ) = T [ x ( n 1 , n 2 )] Linearity ¾ Additivity: x 2 ( n 1 , n 2 ) T y 2 ( n 1 , n 2 ) = T [ x 2 ( n 1 , n 2 )] If Then x 1 ( n 1 , n 2 ) + x 2 ( n 1 , n 2 ) T y ( n 1 , n 2 ) = y 1 ( n 1 , n 2 ) + y 2 ( n 1 , n 2 ) ¾ Homogeneity: x 1 ( n 1 , n 2 ) T y 1 ( n 1 , n 2 ) = T [ x 1 ( n 1 , n 2 )] If ax 1 ( n 1 , n 2 ) y ( n 1 , n 2 ) = ay 1 ( n 1 , n 2 ) = aT [ x 1 ( n 1 , n 2 )] T Then Linearity ĺ superposition principle holds a i x i ( n 1 , n 2 ) a i y i ( n 1 , n 2 ), where y i ( n 1 , n 2 ) = T [ x i ( n 1 , n 2 )] T ¦ i ¦ i EEE 508 - Lecture 2
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2D Systems Shift invariance Shift in the input results in a corresponding shift in the output x ( n 1 , n 2 ) T y ( n 1 , n 2 ) If Exp. x ( n 1 - s 1 , n 2 -s 2 ) T y ( n 1 - s 1 , n 2 - s 2 ) Then EEE 508 - Lecture 2
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2D Systems Why LSI Systems? ¾ Most frequently used ¾ Easy to design and analyze ¾ Need to simplify ¾ Mathematical tractable and rich theory BUT ¾ Superposition not always useful for images, e.g. occlusion EEE 508 - Lecture 2
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2D Systems Impulse response ¾ What is it? G ( n 1 , n 2 ) T h ( n 1 , n 2 ) General system (Not SI) ( n 1 - s 1 , n 2 -s 2 ) h s 1 ,s 2 ( n 1 , n 2 ) T Shift Invariant (SI) system ( n 1 - s 1 , n 2 -s 2 ) h ( n 1 - s 1 , n 2 - s 2 ) T EEE 508 - Lecture 2
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Linear and Shift Invariant (LSI) Systems Impulse response h ( n 1 , n 2 ) uniquely characterizes system ff for input x ( n 1 , n 2 ) can compute output using h ( n 1 , n 2 ) since: ¦ ¦ ±f ±f 12 ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 1 kk k n k n k k x n n x G f f > @ > @ ¦ ¦ ±f ±f ± ± ) , ( ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 1 2 1 k n k n T k k x n n x T n n y ¦ ¦ ± ± ) , ( ) , ( 2 2 1 1 2 1 k n k n h k k x ±f ±f ) , ( 2 1 n n x ** ) , ( 2 1 n n h 2D Convolution EEE 508 - Lecture 2
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2D Systems x ( n 1 , n 2 ) h ( n 1 , n 2 ) y ( n 1 , n 2 ) = x ( n 1 , n 2 ) ±± h ( n 1 , n 2 ) Exp. 2D filter LSI Let x ( n 1 , n 2 ) be a noisy image ² choose impulse response h ( n 1 , n 2 ) of a low-pass filter to remove high frequency noise; ² y : enhanced or restored image LSI System is characterized by: ¾ impulse response h ( n 1 , n 2 )in t ime /space doma in ¾ frequency response H ( Z 1 , 2 ) in frequency domain, where ^` ¦¦ f f ³ ³ 2 2 1 1 ) , ( ) , ( ) , ( 2 1 2 1 2 1 n j n j e e n n h n n h DTFT H EEE 508 - Lecture 2 ³f ³f 12 n n
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2D Systems
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