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Edge detection
- Goal: map image from 2d array of pixels to a set of curves
- r line segments or contours.
- Why?
- Main idea: look for strong gradients, post-process
Figure from J. Shotton et al., PAMI 2007
- ? Edge detection Goal : map - - PowerPoint PPT Presentation
- ? Edge detection Goal : map - - PowerPoint PPT Presentation
- ? Edge detection Goal : map image from 2d array of pixels to a set of curves or line segments or contours. Why? Figure from J. Shotton et al., PAMI 2007 Main idea :
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What can cause an edge?
Depth discontinuity:
- bject boundary
Change in surface
- rientation: shape
Cast shadows Reflectance change: appearance information, texture
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Recall : Images as functions
- Edges look like steep cliffs
Source: S. Seitz
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Derivatives and edges
image intensity function (along horizontal scanline) first derivative edges correspond to extrema of derivative
Source: L. Lazebnik
An edge is a place of rapid change in the image intensity function.
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Differentiation and convolution
For 2D function, f(x,y), the partial derivative is: For discrete data, we can approximate using finite differences: To implement above as convolution, what would be the associated filter?
) , ( ) , ( lim ) , ( y x f y x f x y x f
1 ) , ( ) , 1 ( ) , ( y x f y x f x y x f
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- First, consider a signal in 1D…
- Let’s replace each pixel with an average of all
the values in its neighborhood
- Moving average in 1D:
Side note: Filters and Convolutions
Source: S. Marschner
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Weighted Moving Average
- Can add weights to our moving average
- Weights [1, 1, 1, 1, 1] / 5
Source: S. Marschner
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Weighted Moving Average
- Non-uniform weights [1, 4, 6, 4, 1] / 16
Source: S. Marschner
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Moving Average In 2D
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
Source: S. Seitz
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Moving Average In 2D
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 10 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
Source: S. Seitz
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Moving Average In 2D
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 10 20 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
Source: S. Seitz
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Moving Average In 2D
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 10 20 30 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
Source: S. Seitz
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Moving Average In 2D
10 20 30 30 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
Source: S. Seitz
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Moving Average In 2D
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 10 20 30 30 30 20 10 20 40 60 60 60 40 20 30 60 90 90 90 60 30 30 50 80 80 90 60 30 30 50 80 80 90 60 30 20 30 50 50 60 40 20 10 20 30 30 30 30 20 10 10 10 10
Source: S. Seitz
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Correlation filtering
Say the averaging window size is 2k+1 x 2k+1:
Loop over all pixels in neighborhood around image pixel F[i,j] Attribute uniform weight to each pixel
Now generalize to allow different weights depending on neighboring pixel’s relative position:
Non-uniform weights
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Correlation filtering
Filtering an image: replace each pixel with a linear combination of its neighbors. The filter “kernel” or “mask” H[u,v] is the prescription for the weights in the linear combination. This is called cross-correlation, denoted
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Convolution
- Convolution:
– Flip the filter in both dimensions (bottom to top, right to left) – Then apply cross-correlation
Notation for convolution
- perator
F H
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Convolution vs. correlation
Convolution Cross-correlation
Back to our question: To implement the derivates, what would be the associated filter?
1 ) , ( ) , 1 ( ) , ( y x f y x f x y x f
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Partial derivatives of an image
Which shows changes with respect to x?
- 1
1
- 1 1
x y x f ) , ( y y x f ) , (
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Assorted finite difference filters
>> My = fspecial(‘sobel’); >> outim = imfilter(double(im), My); >> imagesc(outim); >> colormap gray;
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Image gradient
The gradient of an image: The gradient points in the direction of most rapid change in intensity The gradient direction (orientation of edge normal) is given by: The edge strength is given by the gradient magnitude
Slide credit S. Seitz
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Another issue: The effects of noise
Consider 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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Smoothing with a Gaussian
Parameter σ is the “scale” / “width” / “spread” of the Gaussian kernel, and controls the amount of smoothing.
…
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Effect of σ on derivatives
The apparent structures differ depending on Gaussian’s scale parameter. Larger values: larger scale edges detected Smaller values: finer features detected
σ = 1 pixel σ = 3 pixels
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So, what scale to choose?
It depends what we’re looking for. Too fine of a scale…can’t see the forest for the trees. Too coarse of a scale…can’t tell the maple grain from the cherry.
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Edge detection is just the beginning…
Berkeley segmentation database:
http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/
image human segmentation gradient magnitude
Source: L. Lazebnik