# 3fil - 3 Image Enhancement Using Filtering In the Spatial...

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1 3. Image Enhancement Using Filtering In the Spatial Domain Chapter 3 (Image Enhancement in the Spatial Domain) Filtering for: - Removing noise - Sharpening image - Detecting edges

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2 1-D Linear Time Invariant (LTI) Systems 1-D signal x (n) passes through a system T: y(n) = T [x(n)] A system is linear : A system is time invariant : An LTI system can be completely characterized by its impulse response:  ) ( ) ( ) ( ) ( 2 2 1 1 2 2 1 1 n y a n y a n x a n x a T ) ( ) ( m n y m n x T    kk k n x k h k n h k x n y ) ( ) ( ) ( ) ( ) ( 1-D convolution   () hn T n
3 2-D Linear Shift Invariant (LSI) Systems 2-D signal passes through a system T : A system is linear : A system is shift invariant : The LSI system can be completely characterized by its unit impulse response (point spread function): ) , ( 2 1 n n x ) , ( ) , ( 2 1 2 1 n n x T n n y  ) , ( ) , ( ) , ( ) , ( 2 1 2 2 2 1 1 1 2 1 2 2 2 1 1 1 n n y a n n y a n n x a n n x a T ) , ( ) , ( 2 2 1 1 2 2 1 1 m n m n y m n m n x T   ) , ( ) , ( 2 1 2 1 n n T n n h ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 1 12 k n k n h kk k k x n n y    2-D convolution

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4 A Typical 2-D Convolution h(n1,n2) y(n1,n2)=x*h (N+M-2) (N+M-2) (M-1) (M-1) x(n1,n2) (N-1) (N-1)
5 x(n1,n2) h(n1,n2) x(k1,k2) h(k1,k2) h(-k1,-k2) h(n1-k1,n2-k2) y(n1,n2) 2 3 3 1 6 10 10 4 6 10 10 4 4 7 7 3 1 1 1 1 1 1 1 1 1 A 2-D Convolution Example 2 1 4 3 3 4 1 2

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6 A Separable 2-D System Given an M x M system , it is separable if The convolution can be done in two steps:   n n h 2 1 , ) ( ) ( ) , ( 2 2 1 1 2 1 n h n h n n h   1 0 , 0 1 1 1 M for n n h   1 0 , 0 2 2 2 M for n n h ) ( ) ( ) , ( ) , ( 2 2 2 1 1 1 2 1 2 1 12 k n h k n h k k n n kk x y      x k n h k k k n h ) ( ) , ( ) ( 2 2 2 2 1 1 1 1 ) , ( ) ( 2 1 1 1 1 1 n k k n h f k  (1) (2) where and
7 x(n1,n2) f(k1,n2) f(0,n2) f(n1,1) Column-wise convolution y(n1,n2) y(n1,1) Convolution with 2-D Separable Filter

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8 Example of 2-D Separable Convolution x(n1,n2) 3 2 1 6 5 4 9 8 7 h(n1,n2) 1 1 1 1  1 1 h1(n1) h2(n2) [1 -1] f(n1,n2) 3 3 3 9 2 3 3 8 1 3 3 7 3 5 3 1 3 6 6 3 3 6 6 3 9 17 15 7
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## This note was uploaded on 07/08/2011 for the course EE 440 taught by Professor Jenq-nenghwang during the Spring '11 term at University of Washington.

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3fil - 3 Image Enhancement Using Filtering In the Spatial...

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