1. EE 4780
Edge Detection

2. Detection of Discontinuities
Matched Filter Example
>> a=[ ];
>> figure; plot(a);
>> h1 = [ ]/10;
>> b1 = conv(a,h1); figure; plot(b1);
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3. 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
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4. Detection of Discontinuities
Point Detection
Detected point
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5. Detection of Discontinuities
Line Detection Example:
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6. Detection of Discontinuities
Line Detection Example:
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7. 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.
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8. 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
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9. Profiles of image intensity edges
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10. 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
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11. 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)
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12. Effects of noise Consider a single row or column of the image
Plotting intensity as a function of position gives a signal
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13. Solution: smooth first
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Look for peaks in

14. Derivative theorem of convolution
This saves us one operation:
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15. Laplacian of Gaussian Consider Laplacian of Gaussian operator
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Zero-crossings of bottom graph

16. 2D edge detection filters
Laplacian of Gaussian
Gaussian
derivative of Gaussian
is the Laplacian operator:
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17. Edge Detection
Possible filters to find gradients along vertical and horizontal directions:
Averaging provides noise suppression
This gives more importance to the center point.
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18. Edge Detection
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19. Edge Detection
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20. Edge Detection
The Laplacian of an image f(x,y) is a second-order derivative defined as
Digital approximations:
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21. Edge Detection
One simple method to find zero-crossings is black/white thresholding:
1. Set all positive values to white
2. Set all negative values to black
3. Determine the black/white transitions.
Compare (b) and (g):
Edges in the zero-crossings image is thinner than the gradient edges.
Edges determined by zero-crossings have formed many closed loops.
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22. Edge Detection The Laplacian of a Gaussian filter
A digital approximation: 1 2
-16
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23. The Canny edge detector
original image (Lena)
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24. The Canny edge detector
norm of the gradient
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25. The Canny edge detector
thresholding
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26. The Canny edge detector
thinning
(non-maximum suppression)
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27. Edge detection by subtraction
original
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28. Edge detection by subtraction
smoothed (5x5 Gaussian)
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29. Edge detection by subtraction
Why does
this work?
smoothed – original
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30. Gaussian - image filter
delta function
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Laplacian of Gaussian