1. Lecture 8: Edges and Feature Detection
CS4670/5670: Computer Vision
Kavita Bala
Lecture 8: Edges and Feature Detection

2. Announcements PA 1 (A) is PA 1 Winter break
A quiz some time after break

3. Characterizing edges
An edge is a place of rapid change in the image intensity function
intensity function (along horizontal scanline)
image
first derivative
edges correspond to extrema of derivative
Source: L. Lazebnik

4. Image derivatives How can we differentiate a digital image F[x,y]?
Take discrete derivative (finite difference)
give definition of partial derivative: lim h->0 [f(x+h,y) – f(x,y)]/h
How would you implement this as a linear filter? 1 -1 -1 1 : :
Source: S. Seitz

5. Effects of noise Where is the edge? Noisy input image Source: S. Seitz
How to fix?
Where is the edge?
Source: S. Seitz

6. Solution: smooth first
f * h
To find edges, look for peaks in
Source: S. Seitz

7. Associative property of convolution
Differentiation is a convolution
Convolution is associative:
This saves us one operation: f
Source: S. Seitz

9. The 1D Gaussian and its derivatives

10. Derivative of Gaussian filter
x-direction
y-direction

11. Tradeoff between smoothing and localization
1 pixel
3 pixels
7 pixels
Smoothed derivative removes noise, but blurs edge. Also finds edges at different “scales”.
Source: D. Forsyth

12. The Sobel operator Common approximation of derivative of Gaussian
A mask (not a convolution kernel) -1 1 -2 2 1 2 -1 -2
Q: Why might these work better?
A: more stable when there is noise
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 magnitude 12

13. Sobel operator: example
Image, x derivative, y derivative, magnitude
Source: Wikipedia

15. The 1D Gaussian and its derivatives

16. 2D edge detection filters
How many 2nd derivative filters are there? There are four 2nd partial derivative filters.
In practice, it’s handy to define a single 2nd derivative filter—the Laplacian
is the Laplacian operator:

17. 2D edge detection filters
Laplacian of Gaussian
Gaussian
derivative of Gaussian
How many 2nd derivative filters are there? There are four 2nd partial derivative filters.
In practice, it’s handy to define a single 2nd derivative filter—the Laplacian
is the Laplacian operator:

18. Sharpening revisited = = What does blurring take away? – + α
original
smoothed (5x5)
detail = –
Let’s add it back:
original
detail
+ α
sharpened =
Source: S. Lazebnik

19. unit impulse (identity)
Sharpen filter
blurred image
image
unit impulse (identity)
Gaussian
scaled impulse
Laplacian of Gaussian
f + a(f - f * g) = (1+a)f-af*g = f*((1+a)e-g)

20. Sharpen filter
unfiltered
filtered

21. Super-resolution

22. Designing an edge detector
Criteria for a good edge detector:
Good detection: the optimal detector should find all real edges, ignoring noise or other artifacts
Good localization
the edges detected must be as close as possible to the true edges
the detector must return one point only for each true edge point
Cues of edge detection
Differences in color, intensity, or texture across the boundary
Continuity and closure
High-level knowledge
Source: L. Fei-Fei

23. Canny edge detector
This is probably the most widely used edge detector in computer vision
Theoretical model: step-edges corrupted by additive Gaussian noise
J. Canny, A Computational Approach To Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8: , 1986.
22,000 citations!
Source: L. Fei-Fei

24. Example
original image (Lena)

25. Compute Gradients (DoG)
X-Derivative of Gaussian
Y-Derivative of Gaussian
Gradient Magnitude

26. Get Orientation at Each Pixel
Threshold at minimum level
Get orientation
theta = atan2(gy, gx)

27. Non-maximum suppression for each orientation
At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.
Source: D. Forsyth

28. Before Non-max Suppression

29. After Non-max Suppression

30. Hysteresis thresholding
Threshold at low/high levels to get weak/strong edge pixels
Do connected components, starting from strong edge pixels

31. Final Canny Edges

32. Source: D. Lowe, L. Fei-Fei
Canny edge detector
Filter image with x, y derivatives of Gaussian
Find magnitude and orientation of gradient
Non-maximum suppression:
Thin multi-pixel wide “ridges” down to single pixel width
Thresholding and linking (hysteresis):
Define two thresholds: low and high
Use the high threshold to start edge curves and the low threshold to continue them
Source: D. Lowe, L. Fei-Fei

33. Feature detection and matching

34. Reading
Szeliski: 4.1

35. Motivation: Automatic panoramas
Credit: Matt Brown