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

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

3. Linear Systems The system H is called shift invariant if
for all fi(x) {f(x)} and for all x0.
This means that offsetting the independent variable of the input by x0 causes the same offset in the independent variable of the output. Hence, the input-output relationship remains the same.
Bahadir K. Gunturk

4. 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.
Bahadir K. Gunturk

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

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

7. Linear Systems The term is called the impulse response of H.
From the previous slide
It states that, if the response of H to a unit impulse [i.e., h(x, )], is known, then response to any input f can be computed using the preceding integral. In other words, the response of a linear system is characterized completely by its impulse response.
Bahadir K. Gunturk

8. Linear Systems If H is a shift-invariant system, then
and the integral becomes
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.
Bahadir K. Gunturk

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

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

11. Convolution Example 1 -1 2 2 3 1 1 -1 2 h f Rotate
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

12. Convolution Example Step 1 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1h 1 3 2 2 1 1 1 -1 3 2 1 4 2 5 -1 -2 1 f
f*h
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

13. Convolution Example Step 2 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1h 1 3 2 2 1 1 1 3 2 1 -2 4 2 5 4 -2 -1 3 f
f*h
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

14. Convolution Example Step 3 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1h 1 3 2 2 1 1 1 3 2 1 -2 4 3 4 5 -1 -3 3 f
f*h
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

15. Convolution Example Step 4 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1h 1 3 2 2 1 1 1 3 2 1 -2 6 1 4 -2 5 -3 -3 1 f
f*h
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

16. Convolution Example Step 5 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1h 1 3 2 2 1 3 2 1 2 2 4 9 -2 5 -1 4 1 -1 -2 2 f
f*h
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

17. Convolution Example Step 6 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1h 1 3 2 2 2 3 2 1 2 2 6 4 9 -2 5 -2 2 3 -2 -2 1 f
f*h
From C. Rasmussen, U. of Delaware
Bahadir K. Gunturk

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

19. Example = *
Bahadir K. Gunturk

20. Example = *
Bahadir K. Gunturk

21. MATLAB Review your matrix-vector knowledge
Matlab help files are helpful to learn it
Exercise:
f = [1 2; 3 4]
g = [1; 1]
g = [1 1] g’
z = f * g’
n=0:10
plot(sin(n));
plot(n,sin(n)); title(‘Sinusoid’); xlabel(‘n’); ylabel(‘Sin(n)’);
n=0:0.1:10
plot(n,sin(n));
grid;
figure; subplot(2,1,1); plot(n,sin(n)); subplot(2,1,2); plot(n,cos(n));
Bahadir K. Gunturk

22. MATLAB Some more built-ins a = zeros(3,2) b = ones(2,4)
c = rand(3,3) %Uniform distribution
help rand
help randn %Normal distribution
d1 = inv(c)
d2 = inv(rand(3,3))
d3 = d1+d2
d4 = d1-d2
d5 = d1*d2
d6 = d1.*d3
e = d6(:)
Bahadir K. Gunturk

23. MATLAB Image processing in Matlab x=imread(‘cameraman.tif’); figure;
imshow(x);
[h,w]=size(x);
y=x(0:h/2,0:w/2);
imwrite(y,’man.tif’);
% To look for a keyword
lookfor resize
Bahadir K. Gunturk

24. MATLAB M-file Save the following as myresize1.m
function [y]=myresize1(x)
% This function downsamples an image by two
[h,w]=size(x);
for i=1:h/2,
for j=1:w/2,
y(i,j) = x(2*i,2*j);
end
Compare with myresize2.m
function [y]=myresize2(x)
for i=0:h/2-1,
for j=0:w/2-1,
y(i+1,j+1) = x(2*i+1,2*j+1);
Compare with myresize3.m
function [y]=myresize3(x)
% This function downsamples an image by two
y = x(1:2:end,1:2:end);
We can add inputs/outputs
function [y,height,width]=myresize4(x,factor)
% Inputs:
% x is the input image
% factor is the downsampling factor
% Outputs:
% y is the output image
% height and width are the size of the output image
y = x(1:factor:end,1:factor:end);
[height,width] = size(y);
Bahadir K. Gunturk

25. 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=[ ; ; ; ]/16;
g=conv2(f3,h);
figure;imshow(uint8(g));
Bahadir K. Gunturk

26. EE 7730
Edge Detection

27. Detection of Discontinuities
Matched Filter Example
>> a=[ ];
>> figure; plot(a);
>> h1 = [ ]/10;
>> b1 = conv(a,h1); figure; plot(b1);
Bahadir K. Gunturk

28. Detection of Discontinuities
Point Detection Example:
Apply a high-pass filter.
A point is detected if the response is larger than a positive threshold.
The idea is that the gray level of an isolated point will be quite different from the gray level of its neighbors.
Threshold
Bahadir K. Gunturk

29. Detection of Discontinuities
Point Detection
Detected point
Bahadir K. Gunturk

30. Detection of Discontinuities
Line Detection Example:
Bahadir K. Gunturk

31. Detection of Discontinuities
Line Detection Example:
Bahadir K. Gunturk

32. Detection of Discontinuities
Edge Detection:
An edge is the boundary between two regions with relatively distinct gray levels.
Edge detection is by far the most common approach for detecting meaningful discontinuities in gray level. The reason is that isolated points and thin lines are not frequent occurrences in most practical applications.
The idea underlying most edge detection techniques is the computation of a local derivative operator.
Bahadir K. Gunturk

33. Origin of Edges Edges are caused by a variety of factors
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity
Edges are caused by a variety of factors
Bahadir K. Gunturk

34. Profiles of image intensity edges
Bahadir K. Gunturk

35. Image gradient The gradient of an image:
The gradient points in the direction of most rapid change in intensity
The gradient direction is given by:
The edge strength is given by the gradient magnitude
Bahadir K. Gunturk

36. Edge Detection
The gradient vector of an image f(x,y) at location (x,y) is the vector
The magnitude and direction of the gradient vector are
is also used in edge detection in addition to the magnitude of the gradient vector.
Bahadir K. Gunturk

37. The discrete gradient How can we differentiate a digital image f[x,y]?
Option 1: reconstruct a continuous image, then take gradient
Option 2: take discrete derivative (finite difference)
Bahadir K. Gunturk

38. Effects of noise Consider a single row or column of the image
Plotting intensity as a function of position gives a signal
Bahadir K. Gunturk

39. Solution: smooth first
Bahadir K. Gunturk
Look for peaks in

40. Derivative theorem of convolution
This saves us one operation:
Bahadir K. Gunturk

41. Laplacian of Gaussian Consider Laplacian of Gaussian operator
Bahadir K. Gunturk
Zero-crossings of bottom graph

42. 2D edge detection filters
Laplacian of Gaussian
Gaussian
derivative of Gaussian
is the Laplacian operator:
Bahadir K. Gunturk

43. The Canny edge detector
original image (Lena)
Bahadir K. Gunturk

44. The Canny edge detector
norm of the gradient
Bahadir K. Gunturk

45. The Canny edge detector
thresholding
Bahadir K. Gunturk

46. The Canny edge detector
thinning
(non-maximum suppression)
Bahadir K. Gunturk

47. Non-maximum suppression
Check if pixel is local maximum along gradient direction
requires checking interpolated pixels p and r
Bahadir K. Gunturk

48. Predicting the next edge point
Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).
Bahadir K. Gunturk

49. Non-maximum suppression
Bahadir K. Gunturk

50. Hysteresis
The threshold used to find starting point may be large in following the edge.
This leads to broken edge curves.
The trick is to use two thresholds: A large one when starting an edge chain, a small one while following it.
Bahadir K. Gunturk

51. Edge detection by subtraction
original
Bahadir K. Gunturk

52. Edge detection by subtraction
smoothed (5x5 Gaussian)
Bahadir K. Gunturk

53. Edge detection by subtraction
Why does
this work?
smoothed – original
(scaled by 4, offset +128)
Bahadir K. Gunturk
filter demo

54. Gaussian - image filter
delta function
Bahadir K. Gunturk
Laplacian of Gaussian

55. Edge Detection
Bahadir K. Gunturk

56. Edge Detection
Bahadir K. Gunturk

57. Edge Detection
Bahadir K. Gunturk

58. Edge Detection
Bahadir K. Gunturk

59. Edge Detection
Bahadir K. Gunturk

60. Edge Detection
The Laplacian of an image f(x,y) is a second-order derivative defined as
Bahadir K. Gunturk

61. Edge Detection
Bahadir K. Gunturk

62. Corners contain more edges than lines.
A point on a line is hard to match.
Bahadir K. Gunturk

63. Corners contain more edges than lines.
A corner is easier
Bahadir K. Gunturk

64. Corner Detector
Locate points where intensity is varying in two directions.
Bahadir K. Gunturk

65. Reading
Chris Harris and Mike Stephens. A combined corner and edge detector. In M. M. Matthews, editor, Proceedings of the 4th ALVEY vision conference, pages , University of Manchester, England, September 1988.
Bahadir K. Gunturk