Transcript EE 7730: Lecture 1 - Louisiana State University

EE 4780
Edge Detection
Detection of Discontinuities
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Matched Filter Example
>> a=[0 0 0 0 1 2 3 0 0 0 0 2 2 2 0 0 0 0 1 2 -2 -1 0 0 0 0];
>> figure; plot(a);
>> h1 = [-1 -2 2 1]/10;
>> b1 = conv(a,h1); figure; plot(b1);
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Detection of Discontinuities
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Point Detection Example:
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Apply a high-pass filter.
A point is detected if the response is larger than a positive
threshold.
| R | T
Threshold
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The idea is that the gray level of an isolated point will be quite
different from the gray level of its neighbors.
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Detection of Discontinuities
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Point Detection
Detected point
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Detection of Discontinuities
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Line Detection Example:
R1
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R2
R3
R4
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Detection of Discontinuities
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Line Detection Example:
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Detection of Discontinuities
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Edge Detection:
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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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Origin of Edges
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity
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Edges are caused by a variety of factors
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Profiles of image intensity edges
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Image gradient
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The gradient of an image:
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The gradient points in the direction of most rapid change in
intensity
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The gradient direction is given by:
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The edge strength is given by the gradient magnitude
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The discrete gradient
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How can we differentiate a digital image f[x,y]?
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Option 1: reconstruct a continuous image, then
take gradient
Option 2: take discrete derivative (finite difference)
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Effects of noise
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Consider a single row or column of the image
 Plotting intensity as a function of position gives a signal
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Solution: smooth first
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Look for peaks in
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Derivative theorem of convolution
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This saves us one operation:
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Laplacian of Gaussian
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Consider
Laplacian of Gaussian
operator
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Zero-crossings of bottom graph
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2D edge detection filters
Laplacian of Gaussian
Gaussian
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derivative of Gaussian
is the Laplacian operator:
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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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Edge Detection
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Edge Detection
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Edge Detection
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The Laplacian of an image f(x,y) is a second-order
derivative defined as
2 f 2 f
 f  2  2
x
y
2
Digital approximations:
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Edge Detection
One simple method to find zerocrossings 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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Edge Detection
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The Laplacian of a Gaussian filter
A digital approximation:
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0
0
1
0
0
0
1
2
1
0
1
2
-16
2
1
0
1
2
1
0
0
0
1
0
0
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The Canny edge detector
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original image (Lena)
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The Canny edge detector
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norm of the gradient
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The Canny edge detector
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thresholding
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The Canny edge detector
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thinning
(non-maximum suppression)
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Edge detection by subtraction
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original
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Edge detection by subtraction
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smoothed (5x5 Gaussian)
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Edge detection by subtraction
Why does
this work?
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smoothed – original
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Gaussian - image filter
Gaussian
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delta function
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Laplacian of Gaussian