1. MAN-522: Computer Vision Edge detection Feature and blob detection
Hough transform
Fitting
Image alignment
MAN-522: COMPUTER VISION

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

4. 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

5. Image gradient The gradient of an image:
The gradient points in the direction of most rapid increase in intensity
The gradient direction is given by
how does this relate to the direction of the edge?
The edge strength is given by the gradient magnitude
give definition of partial derivative: lim h->0 [f(x+h,y) – f(x,y)]/h
Source: Steve Seitz

6. Differentiation and convolution
Recall, for 2D function, f(x,y):
This is linear and shift invariant, so must be the result of a convolution.
We could approximate this as
(which is obviously a convolution)
I tend not to prove “Now this is linear and shift invariant, so must be the result of a convolution” but
leave it for people to look up in the chapter. -1 1
Source: D. Forsyth, D. Lowe

7. Finite difference filters
Other approximations of derivative filters exist:
Source: K. Grauman

8. Finite differences: example
Which one is the gradient in the x-direction (resp. y-direction)?

9. 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?
How to fix?
Source: S. Seitz

10. Effects of noise Finite 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 is to be done?
Source: D. Forsyth

11. Effects of noise Finite 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 is to be done?
Smoothing the image should help, by forcing pixels different to their neighbors (=noise pixels?) to look more like neighbors
Source: D. Forsyth

12. Solution: smooth firstg
f * g
To find edges, look for peaks in
Source: S. Seitz

13. Derivative theorem of convolution
Differentiation is convolution, and convolution is associative:
This saves us one operation: f
Source: S. Seitz

14. Derivative of Gaussian filter
* [1 -1] =
Is this filter separable?

15. Derivative of Gaussian filter
x-direction
y-direction
Which one finds horizontal/vertical edges?

16. Summary: Filter mask properties
Filters act as templates
Highest response for regions that “look the most like the filter”
Dot product as correlation
Smoothing masks
Values positive
Sum to 1 → constant regions are unchanged
Amount of smoothing proportional to mask size
Derivative masks
Opposite signs used to get high response in regions of high contrast
Sum to 0 → no response in constant regions
High absolute value at points of high contrast
Source: K. Grauman

17. 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

18. Implementation issues
The gradient magnitude is large along a thick “trail” or “ridge,” so how do we identify the actual edge points?
How do we link the edge points to form curves?
Figures show gradient magnitude of zebra at two different scales
Source: D. Forsyth

19. Edge finding
We wish to mark points along the curve where the magnitude is biggest.
We can do this by looking for a maximum along a slice normal to the curve (non-maximum suppression). These points should form a curve. There are then two algorithmic issues: at which point is the maximum, and where is the next one?
Source: D. Forsyth

20. Non-maximum 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.
Source: D. Forsyth

21. Predicting the next edge point
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).
Source: D. Forsyth

22. Designing an edge detector
Criteria for an “optimal” edge detector:
Good detection: the optimal detector must minimize the probability of false positives (detecting spurious edges caused by noise), as well as that of false negatives (missing real edges)
Good localization: the edges detected must be as close as possible to the true edges
Single response: the detector must return one point only for each true edge point; that is, minimize the number of local maxima around the true edge
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
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: , 1986.
Source: L. Fei-Fei

24. Source: D. Lowe, L. Fei-Fei
Canny edge detector
Filter image with derivative of Gaussian
Find magnitude and orientation of gradient
Non-maximum suppression:
Thin multi-pixel wide “ridges” down to single pixel width
Linking and thresholding (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

25. The Canny edge detector
original image (Lena)

26. The Canny edge detector
norm of the gradient

27. The Canny edge detector
thresholding

28. The Canny edge detector
thinning
(non-maximum suppression)

29. Hysteresis thresholding
original image
high threshold
(strong edges)
low threshold
(weak edges)
hysteresis threshold
Source: L. Fei-Fei

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

31. Edge detection is just the beginning…
image
human segmentation
gradient magnitude
Berkeley segmentation database:

32. Next time: Corner and blob detection

33. Feature extraction: Corners and blobs

34. Review: Linear filtering and edge detection
Name two different kinds of image noise
Name a non-linear smoothing filter
What advantages does median filtering have over Gaussian smoothing?
What is aliasing?
How do we find edges?
Why do we need to smooth before computing image derivatives?
What are some characteristics of an “optimal” edge detector?
What is nonmaximum suppression?
What is hysteresis thresholding?

35. Why extract features? Motivation: panorama stitching
We have two images – how do we combine them?

36. Why extract features? Motivation: panorama stitching
We have two images – how do we combine them?
Step 2: match features
Step 1: extract features

37. Why extract features? Motivation: panorama stitching
We have two images – how do we combine them?
Step 1: extract features
Step 2: match features
Step 3: align images

38. Characteristics of good features
Repeatability
The same feature can be found in several images despite geometric and photometric transformations
Saliency
Each feature has a distinctive description
Compactness and efficiency
Many fewer features than image pixels
Locality
A feature occupies a relatively small area of the image; robust to clutter and occlusion

39. Applications Feature points are used for: Motion tracking
Image alignment
3D reconstruction
Object recognition
Indexing and database retrieval
Robot navigation

40. Finding Corners
Key property: in the region around a corner, image gradient has two or more dominant directions
Corners are repeatable and distinctive
C.Harris and M.Stephens. "A Combined Corner and Edge Detector.“ Proceedings of the 4th Alvey Vision Conference: pages  

41. The Basic Idea
We should easily recognize the point by looking through a small window
Shifting a window in any direction should give a large change in intensity
“flat” region: no change in all directions
“edge”: no change along the edge direction
“corner”: significant change in all directions
Source: A. Efros

42. Harris Detector: Mathematics
Change of intensity for the shift [u,v]:
Intensity
Window function
Shifted intensity or
Window function w(x,y) =
Gaussian
1 in window, 0 outside
Source: R. Szeliski

43. Harris Detector: Mathematics
Change of intensity for the shift [u,v]:
Second-order Taylor expansion of E(u,v) about (0,0)
(bilinear approximation for small shifts):

44. Harris Detector: Mathematics
The bilinear approximation simplifies to
where M is a 22 matrix computed from image derivatives: M

45. Interpreting the second moment matrix
First, consider an axis-aligned corner:

46. Interpreting the second moment matrix
First, consider an axis-aligned corner:
This means dominant gradient directions align with x or y axis
If either λ is close to 0, then this is not a corner, so look for locations where both are large.
Slide credit: David Jacobs

47. General Case Since M is symmetric, we have
We can visualize M as an ellipse with axis lengths determined by the eigenvalues and orientation determined by R
direction of the slowest change
direction of the fastest change
(max)-1/2
(min)-1/2
Ellipse equation:

48. Visualization of second moment matrices

49. Visualization of second moment matrices

50. Interpreting the eigenvalues
Classification of image points using eigenvalues of M: 2
“Edge” 2 >> 1
“Corner” 1 and 2 are large, 1 ~ 2; E increases in all directions
1 and 2 are small; E is almost constant in all directions
“Edge” 1 >> 2
“Flat” region 1

51. Corner response function
α: constant (0.04 to 0.06)
“Edge” R < 0
“Corner” R > 0
|R| small
“Edge” R < 0
“Flat” region

52. Harris Detector: Steps

53. Harris Detector: Steps
Compute corner response R

54. Harris Detector: Steps
Find points with large corner response: R>threshold

55. Harris Detector: Steps
Take only the points of local maxima of R

56. Harris Detector: Steps

57. Harris detector: Summary of steps
Compute Gaussian derivatives at each pixel
Compute second moment matrix M in a Gaussian window around each pixel
Compute corner response function R
Threshold R
Find local maxima of response function (nonmaximum suppression)

58. Invariance
We want features to be detected despite geometric or photometric changes in the image: if we have two transformed versions of the same image, features should be detected in corresponding locations

59. Models of Image Change Geometric Photometric Rotation Scale
Affine valid for: orthographic camera, locally planar object
Photometric
Affine intensity change (I  a I + b)

60. Harris Detector: Invariance Properties
Rotation
Ellipse rotates but its shape (i.e. eigenvalues) remains the same
Corner response R is invariant to image rotation

61. Harris Detector: Invariance Properties
Affine intensity change
Only derivatives are used => invariance to intensity shift I  I + b
Intensity scale: I  a I R
x (image coordinate)
threshold
Partially invariant to affine intensity change

62. Harris Detector: Invariance Properties
Scaling
Corner
All points will be classified as edges
Not invariant to scaling

63. Scale-invariant feature detection
Goal: independently detect corresponding regions in scaled versions of the same image
Need scale selection mechanism for finding characteristic region size that is covariant with the image transformation

64. Scale-invariant features: Blobs

65. Recall: Edge detectionf
Derivative of Gaussian
Edge = maximum of derivative
Source: S. Seitz

66. Edge detection, Take 2 f Edge
Second derivative of Gaussian (Laplacian)
Edge = zero crossing of second derivative
Source: S. Seitz

67. From edges to blobs Edge = ripple Blob = superposition of two ripples
maximum
Spatial selection: the magnitude of the Laplacian response will achieve a maximum at the center of the blob, provided the scale of the Laplacian is “matched” to the scale of the blob

68. original signal (radius=8)
Scale selection
We want to find the characteristic scale of the blob by convolving it with Laplacians at several scales and looking for the maximum response
However, Laplacian response decays as scale increases:
increasing σ
original signal (radius=8)
Why does this happen?

69. Scale normalization
The response of a derivative of Gaussian filter to a perfect step edge decreases as σ increases

70. Scale normalization
The response of a derivative of Gaussian filter to a perfect step edge decreases as σ increases
To keep response the same (scale-invariant), must multiply Gaussian derivative by σ
Laplacian is the second Gaussian derivative, so it must be multiplied by σ2

71. Effect of scale normalization
Original signal
Unnormalized Laplacian response
Scale-normalized Laplacian response
maximum

72. Blob detection in 2D
Laplacian of Gaussian: Circularly symmetric operator for blob detection in 2D

73. Blob detection in 2D
Laplacian of Gaussian: Circularly symmetric operator for blob detection in 2D
Scale-normalized:

74. Scale selection The 2D Laplacian is given by
Therefore, for a binary circle of radius r, the Laplacian achieves a maximum at
(up to scale)
Laplacian response r
scale (σ)
image

75. Characteristic scale
We define the characteristic scale as the scale that produces peak of Laplacian response
characteristic scale
T. Lindeberg (1998). "Feature detection with automatic scale selection." International Journal of Computer Vision 30 (2): pp

76. Scale-space blob detector
Convolve image with scale-normalized Laplacian at several scales
Find maxima of squared Laplacian response in scale-space

77. Scale-space blob detector: Example

78. Scale-space blob detector: Example

79. Scale-space blob detector: Example

80. Efficient implementation
Approximating the Laplacian with a difference of Gaussians:
(Laplacian)
(Difference of Gaussians)

81. Efficient implementation
David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp , 2004.

82. From scale invariance to affine invariance

83. Affine adaptation Recall:
We can visualize M as an ellipse with axis lengths determined by the eigenvalues and orientation determined by R
direction of the slowest change
direction of the fastest change
(max)-1/2
(min)-1/2
Ellipse equation:

84. Affine adaptation
The second moment ellipse can be viewed as the “characteristic shape” of a region
We can normalize the region by transforming the ellipse into a unit circle

85. Orientation ambiguity
There is no unique transformation from an ellipse to a unit circle
We can rotate or flip a unit circle, and it still stays a unit circle

86. Orientation ambiguity
There is no unique transformation from an ellipse to a unit circle
We can rotate or flip a unit circle, and it still stays a unit circle
So, to assign a unique orientation to keypoints:
Create histogram of local gradient directions in the patch
Assign canonical orientation at peak of smoothed histogram 2 p

87. Affine adaptation
Problem: the second moment “window” determined by weights w(x,y) must match the characteristic shape of the region
Solution: iterative approach
Use a circular window to compute second moment matrix
Perform affine adaptation to find an ellipse-shaped window
Recompute second moment matrix using new window and iterate

88. Iterative affine adaptation
K. Mikolajczyk and C. Schmid, Scale and Affine invariant interest point detectors, IJCV 60(1):63-86, 2004.

89. Affine adaptation example
Scale-invariant regions (blobs)

90. Affine adaptation example
Affine-adapted blobs

91. Summary: Feature extraction
Eliminate rotational
ambiguity
Compute appearance descriptors
Extract affine regions
Normalize regions
SIFT (Lowe ’04)

92. Invariance vs. covariance
features(transform(image)) = features(image)
Covariance:
features(transform(image)) = transform(features(image))
Covariant detection => invariant description