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Announcements• Mailing list: [email protected]
– you should have received messages
• Project 1 out today (due in two weeks)– help session today
• Carpools
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Edge Detection
Today’s reading• Forsyth, chapters 8, 15.1
From Sandlot Science
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Edge detection
Convert a 2D image into a set of curves• Extracts salient features of the scene• More compact than pixels
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Origin of Edges
Edges are caused by a variety of factors
depth discontinuity
surface color discontinuity
illumination discontinuity
surface normal discontinuity
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Edge detection
How can you tell that a pixel is on an edge?
snoop demo
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Image gradientThe gradient of an image:
The gradient points in the direction of most rapid change in intensity
The gradient direction is given by:
• how does this relate to the direction of the edge?
The edge strength is given by the gradient magnitude
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The discrete gradientHow can we differentiate a digital image F[x,y]?
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The discrete gradientHow 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)
How would you implement this as a cross-correlation?
filter demo
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The Sobel operatorBetter approximations of the derivatives exist
• The Sobel operators below are very commonly used
-1 0 1
-2 0 2
-1 0 1
1 2 1
0 0 0
-1 -2 -1
• The standard defn. of the Sobel operator omits the 1/8 term– doesn’t make a difference for edge detection
– the 1/8 term is needed to get the right gradient value, however
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Effects of noiseConsider a single row or column of the image
• Plotting intensity as a function of position gives a signal
Where is the edge?
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Where is the edge?
Solution: smooth first
Look for peaks in
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Derivative theorem of convolution
This saves us one operation:
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Laplacian of Gaussian
Consider
Laplacian of Gaussianoperator
Where is the edge? Zero-crossings of bottom graph
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2D edge detection filters
is the Laplacian operator:
Laplacian of Gaussian
Gaussian derivative of Gaussian
filter demo
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The Canny edge detector
original image (Lena)
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The Canny edge detector
norm of the gradient
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The Canny edge detector
thresholding
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Non-maximum suppression
Check if pixel is local maximum along gradient direction• requires checking interpolated pixels p and r
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The Canny edge detector
thinning(non-maximum suppression)
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Effect of (Gaussian kernel spread/size)
Canny with Canny with original
The choice of depends on desired behavior• large detects large scale edges
• small detects fine features
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Edge detection by subtraction
original
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Edge detection by subtraction
smoothed (5x5 Gaussian)
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Edge detection by subtraction
smoothed – original(scaled by 4, offset +128)
Why doesthis work?
filter demo
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Gaussian - image filter
Laplacian of Gaussian
Gaussian delta function
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An edge is not a line...
How can we detect lines ?
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Finding lines in an imageOption 1:
• Search for the line at every possible position/orientation• What is the cost of this operation?
Option 2:• Use a voting scheme: Hough transform
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Finding lines in an image
Connection between image (x,y) and Hough (m,b) spaces• A line in the image corresponds to a point in Hough space• To go from image space to Hough space:
– given a set of points (x,y), find all (m,b) such that y = mx + b
x
y
m
b
m0
b0
image space Hough space
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Finding lines in an image
Connection between image (x,y) and Hough (m,b) spaces• A line in the image corresponds to a point in Hough space• To go from image space to Hough space:
– given a set of points (x,y), find all (m,b) such that y = mx + b
• What does a point (x0, y0) in the image space map to?
x
y
m
b
image space Hough space
– A: the solutions of b = -x0m + y0
– this is a line in Hough space
x0
y0
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Hough transform algorithmTypically use a different parameterization
• d is the perpendicular distance from the line to the origin is the angle this perpendicular makes with the x axis• Why?
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Hough transform algorithmTypically use a different parameterization
• d is the perpendicular distance from the line to the origin is the angle this perpendicular makes with the x axis• Why?
Basic Hough transform algorithm1. Initialize H[d, ]=0
2. for each edge point I[x,y] in the image
for = 0 to 180
H[d, ] += 1
3. Find the value(s) of (d, ) where H[d, ] is maximum
4. The detected line in the image is given by
What’s the running time (measured in # votes)? Hough line demo
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ExtensionsExtension 1: Use the image gradient
1. same
2. for each edge point I[x,y] in the image
compute unique (d, ) based on image gradient at (x,y)
H[d, ] += 1
3. same
4. same
What’s the running time measured in votes?
Extension 2• give more votes for stronger edges
Extension 3• change the sampling of (d, ) to give more/less resolution
Extension 4• The same procedure can be used with circles, squares, or any
other shape
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ExtensionsExtension 1: Use the image gradient
1. same
2. for each edge point I[x,y] in the image
compute unique (d, ) based on image gradient at (x,y)
H[d, ] += 1
3. same
4. same
What’s the running time measured in votes?
Extension 2• give more votes for stronger edges
Extension 3• change the sampling of (d, ) to give more/less resolution
Extension 4• The same procedure can be used with circles, squares, or any
other shapeHough circle demo