1. EE 4780: Introduction to Computer Vision Linear Systems

2. Bahadir K. Gunturk2 Review: Linear Systems We define a system as a unit that converts an input function into an output function. System operator Independent variable

3. Bahadir K. Gunturk3 Linear Systems Then the system H is called a linear system. where f i (x) is an arbitrary input in the class of all inputs {f(x)}, and g i (x) is the corresponding output. Let If A linear system has the properties of additivity and homogeneity.

4. Bahadir K. Gunturk4 Linear Systems for all f i (x)  {f(x)} and for all x 0. The system H is called shift invariant if This means that offsetting the independent variable of the input by x 0 causes the same offset in the independent variable of the output. Hence, the input-output relationship remains the same.

5. Bahadir K. Gunturk5 Linear Systems The operator H is said to be causal, and hence the system described by H is a causal system, if there is no output before there is an input. In other words, A linear system H is said to be stable if its response to any bounded input is bounded. That is, if where K and c are constants.

6. Bahadir K. Gunturk6 Linear Systems (a)(a) a x  (x-a) A unit impulse function, denoted  (a), is defined by the expression

7. Bahadir K. Gunturk7 Linear Systems The response of a system to a unit impulse function is called the impulse response of the system. h(x) = H[  (x)]

8. Bahadir K. Gunturk8 Linear Systems If H is a linear shift-invariant system, then we can find its reponse to any input signal f(x) as follows: This expression is called the convolution integral. It states that the response of a linear, fixed-parameter system is completely characterized by the convolution of the input with the system impulse response.

9. Bahadir K. Gunturk9 Linear Systems Convolution of two functions is defined as In the discrete case

10. Bahadir K. Gunturk10 Linear Systems is a linear filter. In the 2D discrete case

11. Bahadir K. Gunturk11 Convolution Example From C. Rasmussen, U. of Delaware 1 12 111 2223 2133 2212 1322 Rotate 1 12 111 h f

12. Bahadir K. Gunturk12 Convolution Example From C. Rasmussen, U. of Delaware Step 1 3 2 1 2 2 1 3 2 32 21 22 325 3 2 1 2 2 1 3 2 32 21 22 32 1-2 24 111 f f*h h 1 12 111

13. Bahadir K. Gunturk13 Convolution Example From C. Rasmussen, U. of Delaware Step 2 3 2 1 2 2 1 3 2 32 21 22 3245 3 2 1 2 2 1 3 2 32 21 22 32 3-2 24 111 f f*h h 1 12 111

14. Bahadir K. Gunturk14 Convolution Example From C. Rasmussen, U. of Delaware Step 3 3 2 1 2 2 1 3 2 32 21 22 32445 3 2 1 2 2 1 3 2 32 21 22 32 3-3 34-2 111 f f*h h 1 12 111

15. Bahadir K. Gunturk15 From C. Rasmussen, U. of Delaware Convolution Example Step 4 3 2 1 2 2 1 3 2 32 21 22 3244-25 3 2 1 2 2 1 3 2 32 21 22 32 1-3 16-2 111 f f*h h 1 12 111

16. Bahadir K. Gunturk16 From C. Rasmussen, U. of Delaware Convolution Example Step 5 3 2 1 2 2 1 3 2 32 21 22 3244 9 -25 3 2 1 2 2 1 3 2 32 21 22 32 2 14 221 f f*h h 1 12 111

17. Bahadir K. Gunturk17 From C. Rasmussen, U. of Delaware Convolution Example Step 6 3 2 1 2 2 1 3 2 32 21 22 32 6 44 9 -25 3 2 1 2 2 1 3 2 32 21 22 32 1 32 222 f f*h h 1 12 111

18. Bahadir K. Gunturk18 From C. Rasmussen, U. of Delaware Convolution Example and so on…

19. Bahadir K. Gunturk19 Example * =

20. Bahadir K. Gunturk20 Example * =

21. Bahadir K. Gunturk21 Try MATLAB f=imread(‘saturn.tif’); figure; imshow(f); [height,width]=size(f); f2=f(1:height/2,1:width/2); figure; imshow(f2); [height2,width2=size(f2); f3=double(f2)+30*rand(height2,width2); figure;imshow(uint8(f3)); h=[1 1 1 1; 1 1 1 1; 1 1 1 1; 1 1 1 1]/16; g=conv2(f3,h); figure;imshow(uint8(g));

22. Bahadir K. Gunturk22 Gaussian Lowpass Filter

23. Bahadir K. Gunturk23 Gaussian Lowpass Filter  = 2  = 4 From Forsyth & Ponce Original