1. Edges and Contours– Chapter 7
continued

2. Edge detection
All of the previously studied edge detection filters worked on finding local gradients (1st derivative)
As such, all had to make decisions regarding the actual existence of an edge based on local information

3. Sobel edges – threshold on magnitude

4. Sobel edges – threshold on magnitude
The preceding images were edge-detected with a Sobel operator then binarized with various thresholds
The overall trend is predictable but the individual results are not
This threshold dependency is the curse of image processing
Difficult to isolate “real” edges
Difficult to find accurate edge location
We need smarter algorithms

5. Scale space processing
One method of dealing with the threshold dependency is through scale-space processing
An image is processed at various scales (sizes/resolution) revealing different sets of features at each scale
Features common to all [or many] sets are deemed “important”

6. The Gaussian Pyramid
Select a set of Gaussian filter parameters (sigma and width) and blur the image
Save the image
Reduce it’s size by subsampling (eliminating every other row/column)
Do it again

7. The Gaussian Pyramid
You end up with a series of images (a pyramid) each exposing different features
Coarse features at the top of the pyramid
Fine features at the bottom
You can now use the coarse features to select where to concentrate your processing of the fine features

8. Better (different) algorithms
Another method of dealing with the threshold dependency is through more complex algorithms

9. A better kernel
The Sobel mask is a derivative mask (computes the weighted difference of neighborhood pixels)
So is the Laplacian (negative in the middle, positive around the rest)
From calculus: the 2nd derivative of a signal is zero when the derivative magnitude is extremal

10. A better kernel
Combining these observations we can derive the Laplacian-Gaussian filter due to Marr and Hildreth
Also known as a DoG (Difference of Gaussians) because it can be approximated by subtracting two Gaussian kernels of same width and differing sigmas

11. The Gaussian kernel
Sigma 1.6, Filter Width 15

12. The Laplacian-Gaussian kernel
Bypassing the math and jumping to the results of applying the rules of calculus…
Similar in form to the Gaussian kernel

13. The Laplacian-Gaussian kernel
In 1 dimension…
Can be approximated by difference of two Gaussians

14. The Laplacian-Gaussian kernel
Convolution with this kernel results in edge magnitudes of both positive and negative values
We’re interested in where the sign changes occur (the zero crossing)
These are where the edges are (recall the calculus result)

15. Sobel vs. Laplacian-Gaussian
L-G does not detect as much “noise” as Sobel – why not?

16. Still something missing…
What are we ignoring in the following edge detection processes?
Gaussian filter
Sobel edge detection
Threshold based binarization of Sobel magnitude
And
Laplacian-Gaussian filter
Zero crossing detection

17. We need a smarter algorithm
We merely selected each edge based on its magnitude and zero-crossing
We completely ignored its immediate surroundings
Why is this bad?
Detects edges we don’t want
Prone to noise
Prone to clutter
Doesn’t detect edges we do want
Weak magnitude edges are thresholded (thresheld?) away
We need a smarter algorithm!

18. The Canny edge detector
John Canny – MIT 1986
Developed a process (series of steps) for identifying edges in an image
Gaussian smoothing of input image
Local derivative operator to detect edge candidates
Removal of edges that are clustered around a single location
Adaptive thresholding algorithm for final edge selection

19. Concepts Gaussian smooth original image Local derivative operator
Reduce the effects of noise
Local derivative operator
Simple neighbor pixel subtraction to detect gradients (1st derivative)
Non-maximal suppression
Thin out the edges around a given point
Threshold
Use two thresholds combined with spatial continuity

20. Spatial continuity aka contour following
Apply the high threshold to get “obvious” edges
Search for a neighboring edge above the low threshold in the direction of the strong edge
We’re using a strong edge as an anchor point for a contour of weak and strong edges – known as Hysteresis

21. Canny results
With very conservative high/low thresholds

22. Results
All edges
Weak edges
Contours
Strong edges

23. Summary
Through more complex processing we arrive at a set of edges in which we have confidence
Minimal reliance on thresholds
The cost is complex (slow, memory intensive) processing
We must trade resources for accuracy!

24. ImageJ Download/install plugins for Plugins->FeatureJ->Edges
Sobel, Prewitt, Roberts, Kirsch, Nevatia-Babu
Plugins->FeatureJ->Edges
Canny implementation

25. Edge Sharpening An process for making an image look “better”
In this case “better” means “sharper”
The general approach is to amplify the high-frequency components (the edges) to create a “perceived” sharpness

26. Method 1 – subtract 2nd derivative
This causes a step edge to undershoot at the bottom and overshoot at the top
The Laplacian works well for the 2nd derivative

27. ImageJ
Note that this is difficult to do in ImageJ since the convolution operation clamps negative pixel values to 0

28. Unsharp mask
Steps
Subtract a Gaussian-smoothed version of the image from the original creating a mask
Add a weighted version of the mask to the original image

29. ImageJ
Plugins->Chapter07->Filter UnsharpMask