1. Edges and Lines

2. Edge detection
• Goal: Identify sudden
changes (discontinuities) in an
image
– Intuitively, most semantic and
shape information from the
image can be encoded in the
edges
– More compact than pixels
• Ideal: artist’s line drawing (but
artist is also using object-level
knowledge)
Source: D. Lowe

3. Why do we care about edges?
• Extract information,
recognize objects
Vertical vanishing
point
(at infinity)
Vanishing
• Recover geometry and line
viewpoint Vanishing Vanishing
point point

4. Origin of Edges
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity
• Edges are caused by a variety of factors
Source: Steve Seitz

5. Closeup of edges
Source: D. Hoiem

6. Closeup of edges
Source: D. Hoiem

7. Closeup of edges
Source: D. Hoiem

8. Closeup of edges
Source: D. Hoiem

9. Characterizing edges
• An edge is a place of rapid change in the image
intensity function
intensity function
image (along horizontal scanline) first derivative
edges correspond to
extrema of derivative

10. With a little Gaussian noise
Gradient
Source: D. Hoiem

11. 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?
Source: S. Seitz

12. Effects of noise
• Difference filters respond strongly to noise
– Image noise results in pixels that look very different
from their neighbors
– Generally, the larger the noise the stronger the
response
• What can we do about it?
Source: D. Forsyth

13. Solution: smooth first
f
g
f*g
d
( f ∗ g)
dx
d
• To find edges, look for peaks in ( f ∗ g)
dx Source: S. Seitz

14. Derivative theorem of convolution
• Differentiation is convolution, and convolution is
d d
associative: ( f ∗ g) = f ∗ g
dx dx
• This saves us one operation:
f
d
g
dx
d
f∗ g
dx
Source: S. Seitz

15. Derivative of Gaussian filter
* [1 -1] =

16. 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

17. 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

18. Canny edge detector
• This is probably the most widely used edge
detector in computer vision
• Theoretical model: step-edges corrupted by
additive Gaussian noise
• Canny has shown that the first derivative of
the Gaussian closely approximates the
operator that optimizes the product of signal-
to-noise ratio and localization
J. Canny, A Computational Approach To Edge Detection, IEEE
Trans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986.
Source: L. Fei-Fei

19. Example
original image (Lena)

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

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

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

23. 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

24. Before Non-max Suppression

25. After non-max suppression

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

27. Hysteresis thresholding
• Check that maximum value of gradient value
is sufficiently large
– drop-outs? use hysteresis
• use a high threshold to start edge curves and a low
threshold to continue them.
Source: S. Seitz

28. Final Canny Edges

29. Canny edge detector
1. Filter image with x, y derivatives of Gaussian
2. Find magnitude and orientation of gradient
3. Non-maximum suppression:
– Thin multi-pixel wide “ridges” down to single pixel width
4. 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
• MATLAB: edge(image, ‘canny’)
Source: D. Lowe, L. Fei-Fei

30. Effect of σ (Gaussian kernel spread/size)
original Canny with Canny with
The choice of σ depends on desired behavior
• large σ detects large scale edges
• small σ detects fine features
Source: S. Seitz

31. Learning to detect boundaries
image human segmentation gradient magnitude
• Berkeley segmentation database:
http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/

32. Finding straight lines
• One solution: try many possible lines and see
how many points each line passes through
• Hough transform provides a fast way to do this

33. Hough transform
• An early type of voting scheme
• General outline:
– Discretize parameter space into bins
– For each feature point in the image, put a vote in
every bin in the parameter space that could have
generated this point
– Find bins that have the most votes
Image space Hough parameter space
P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc.
Int. Conf. High Energy Accelerators and Instrumentation, 1959

34. Parameter space representation
• A line in the image corresponds to a point in
Hough space
Image space Hough parameter space
Source: S. Seitz

35. Parameter space representation
• What does a point (x0, y0) in the image space
map to in the Hough space?
Image space Hough parameter space

36. Parameter space representation
• What does a point (x0, y0) in the image space
map to in the Hough space?
– Answer: the solutions of b = –x0m + y0
– This is a line in Hough space
Image space Hough parameter space

37. Parameter space representation
• Where is the line that contains both (x0, y0)
and (x1, y1)?
Image space Hough parameter space
(x1, y1)
(x0, y0)
b = –x1m + y1

38. Parameter space representation
• Where is the line that contains both (x0, y0)
and (x1, y1)?
– It is the intersection of the lines b = –x0m + y0 and
Image –x m +
b = space y1 Hough parameter space
1
(x1, y1)
(x0, y0)
b = –x1m + y1

39. Parameter space representation
• Problems with the (m,b) space:
– Unbounded parameter domain
– Vertical lines require infinite m

40. Parameter space representation
• Problems with the (m,b) space:
– Unbounded parameter domain
– Vertical lines require infinite m
• Alternative: polar representation
x cosθ + y sinθ = ρ
Each point will add a sinusoid in the (θ,ρ) parameter space

41. Algorithm outline
• Initialize accumulator H
to all zeros
• For each edge point (x,y) ρ
in the image
For θ = 0 to 180
ρ = x cos θ + y sin θ
H(θ, ρ) = H(θ, ρ) + 1 θ
end
end
• Find the value(s) of (θ, ρ) where H(θ, ρ) is a
local maximum
– The detected line in the image is given by
ρ = x cos θ + y sin θ

42. Basic illustration
features votes

43. Other shapes
Square Circle

44. Several lines

45. A more complicated image
http://ostatic.com/files/images/ss_hough.jpg

46. Effect of noise
features votes

47. Effect of noise
features votes
• Peak gets fuzzy and hard to locate

48. Effect of noise
• Number of votes for a line of 20 points with
increasing noise:

49. Random points
features votes
• Uniform noise can lead to spurious peaks in the array

50. Random points
• As the level of uniform noise increases, the
maximum number of votes increases too:

51. Dealing with noise
• Choose a good grid / discretization
– Too coarse: large votes obtained when too many
different lines correspond to a single bucket
– Too fine: miss lines because some points that are not
exactly collinear cast votes for different buckets
• Increment neighboring bins (smoothing in
accumulator array)
• Try to get rid of irrelevant features
– Take only edge points with significant gradient
magnitude

52. Incorporating image gradients
• Recall: when we detect an
edge point, we also know its
gradient direction
• But this means that the line
is uniquely determined!
• Modified Hough transform:
• For each edge point (x,y)
θ = gradient orientation at (x,y)
ρ = x cos θ + y sin θ
H(θ, ρ) = H(θ, ρ) + 1
end

53. Hough transform for circles
• How many dimensions will the parameter space
have?
• Given an oriented edge point, what are all
possible bins that it can vote for?

54. Hough transform for circles
image space Hough parameter space
y r
( x , y ) + r∇ I ( x , y )
(x,y)
x
( x , y ) − r∇ I ( x , y )
x
y

55. Hough transform for circles
• Conceptually equivalent procedure: for each
(x,y,r), draw the corresponding circle in the
image and compute its “support”
r
x
y
Is this more or less efficient than voting with features?

56. Finding straight lines
• Another solution: get connected components of
pixels and check for straightness

57. Finding line segments using connected
components
1. Compute canny edges
– Compute: gx, gy (DoG in x,y directions)
– Compute: theta = atan(gy / gx)
2. Assign each edge to one of 8 directions
3. For each direction d, get edgelets:
– find connected components for edge pixels with directions in {d-1, d,
d+1}
• Compute straightness and theta of edgelets using eig of x,y 2nd
moment matrix of their points
∑ ( x − µ x ) ( y − µ y )
Larger eigenvector
 ∑( x − µx )2
M=  [ v, λ] = eig(Μ )
∑ ( x − µ x ) ( y − µ y ) ∑( y − µ ) θ = atan 2( v(2,2), v(1,2))
2
 y 

conf = λ2 / λ1
• Threshold on straightness, store segment
Slides from Derek Hoiem

58. 1. Image  Canny

59. 2. Canny lines  … straight edges

60. Comparison
Hough Transform Method Connected Components Method