1. Lecture 4 Edge Detection
Slides by:
David A. Forsyth
Clark F. Olson
Steven M. Seitz
Linda G. Shapiro

2. Image edges Points of sharp change in an image are interesting:
changes in reflectance
changes in object
changes in illumination
noise
These are sometimes called edge points or edge pixels.
We want to find the edges generated by scene elements and not by noise.

3. Edge detection Convert a 2D image into a set of curves:
Extracts salient features of the scene
More compact than pixels 3

4. Origins of edges Edges are caused by a variety of factors.
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity 4

5. Edge detection
How can you tell whether a pixel is on an edge? 5

6. Edge detection
Basic idea: look for a neighborhood with lots of change.
Questions:
What is the best neighborhood size?
How should change be detected?

7. Finding edges General strategy:
Determine image gradients after smoothing
(Gradients are directional derivatives computed using finite differences.)
Mark points where the gradient magnitude is large with respect to neighboring points
Ideally this yields curves of edge points.

8. Image gradients
We use the image gradient to determine whether a pixel is an edge.
Two components: [gx, gy]
Both components use finite differencing to approximate derivatives
Gradients have magnitude and orientation
Vertical edges respond strongly to the x component
Horizontal edges respond strongly to the y component
Diagonal edges will respond less strongly, but to both components
Overall magnitude should be the same (on edge of same contrast)

9. Sobel operator The Sobel operator is a simple example that is common.
Sx = Sy =
On a pixel of the image I:
let gx be the response to Sx
let gy be the response to Sy
Then the gradient is
I = [gx gy] T
g = (gx + gy ) is the gradient magnitude.
 = atan2(gy, gx) is the gradient direction. 2 2
1/2

10. Smoothing and differentiation
Issue: noise
Need to smooth image before determining image gradients
Should we perform two convolutions (smooth, then differentiate)?
Not necessarily: we can use a derivative of Gaussian filter
Differentiation is convolution and convolution is associative
D * (G * I) = (D * G) * I What are D, G, and I?
Important to point out that the derivative of gaussian kernels I’ve illustrated “look like” the effects
that they’re trying to identify.
Gaussian
Gaussian derivative in x
Plot of Gaussian derivative

11. Smoothing and differentiation
Shape of Gaussian derivative:
Light on one side (positive values)
Dark on other side (negative values)
Values fall off from horizontal center line
After initial peaks, values fall off from vertical center line
Gaussian derivative in x
Plot of Gaussian derivative

12. Smoothing and differentiation
Important implementation trick – we don’t need to convolve by a 2D kernel!
A 2D Gaussian function is “separable.”
Gσ(x, y) = Gσ(x) * Gσ(y)
This means we can convolve the image with two 1D functions (rather than one 2D function).
This results in considerable savings for an n x n image and k x k kernel:
1 2D kernel: approximately n2k2 multiplications and additions
2 1D kernels: approximately 2n2k
The gradient operator is convolved with the appropriate 1D kernel or applied in succession.

13. Gradient magnitudes after smoothing
Original
Sigma = 1
Sigma = 5
As the scale (sigma) increases, finer features are lost, but diffuse edges are gained.
Note that the gradient magnitude encompasses horizontal, vertical, and diagonal edges.

14. Gradient magnitudes after smoothing
Original
Sigma = 1
Sigma = 5
There are three major issues:
1) The gradient magnitude at different scales is different; which should we choose?
2) The gradient magnitude is large along thick trail; how do we identify the
significant points?
3) How do we link the points up into curves?

15. Non-maxima suppression
We wish to mark points along the curve where the gradient magnitude is largest.
We can do this by looking for a maximum along a slice along the gradient direction. These points should form a curve. There are two algorithmic issues: at which point is the maximum, and where is the next one along the curve?

16. Non-maxima suppression
At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.

17. Non-maxima suppression
At q, the gradient Gq is a vector perpendicular to the edge direction.
The locations p and r are one pixel in the direction of the gradient and the opposite direction.
One pixel in the gradient direction is: g = [Gx/Gmag, Gy/Gmag].
Recall that Gmag is the length of the gradient vector [Gx, Gy].
r = q + g and p = q - g

18. Non-maxima suppression
At p and r, the gradient magnitude should be interpolated from the surrounding four pixels.
If the gradient magnitude at q is larger than the interpolated value at p and r, then q is marked as an edge.

19. Predicting the next edge pixel
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).
Only necessary if following edges.

20. Remaining issues
Must check that the gradient magnitude is sufficiently large.
A common problem is that at some points along the curve the gradient magnitude will drop below the threshold, but not at others.
Use hysteresis: a high threshold to start edge curves and a lower threshold to continue them.
Accuracy at corners is poor.

21. Canny edge detector
The Canny edge detector (1986) is still used most often in practice. It is essentially what we have discussed:
Smooth and differentiate the image using derivative of Gaussian filters in x and y
Detect initial candidates by thresholding the gradient magnitude
Apply non-maxima suppression at the candidates
Aggregate edge pixels into contours by following edges perpendicular to the gradient
When aggregating, allow contour gradient magnitude to fall below initial threshold, but must remain above lower threshold
Note that this detector (and others) is sensitive to the parameters used (sigma, thresholds)

22. Zero-crossing detectors
Edge detection using the zero-crossing of the 2nd derivative is historically important.
Performance at corners is poor, but zero-crossings always form closed contours.
step edge
smoothed
1st derivative
zero crossing
2nd derivative

23. Example
Original
image

24. Example Fine scale (sigma=1), Medium threshold, No hysteresis
Much detail (and noise) that disappears at coarser scales.

25. Example Coarse scale (sigma=4), High threshold, No hysteresis
Curves are often broken, not closed contours.

26. Example Coarse scale (sigma=4), Low threshold, No hysteresis
Additional edges found are questionable.