1. EDGE DETECTION Presentation by Sumit Tandon
Department of Electrical Engineering
University of Texas at Arlington
Course # EE6358 Computer Vision

2. CONTENTS Introduction Types of Edges Steps in Edge Detection
Methods of Edge Detection
First Order Derivative Methods
First Order Derivative Methods - Summary
Second Order Derivative Methods
Second Order Derivative Methods - Summary
Optimal Edge Detectors
Canny Edge Detection
Edge Detector Performance
Line Detection
Convolution based technique
Hough transform
Application areas
September 13, 2005
EE Computer Vision

3. INTRODUCTION
Edge - Area of significant change in the image intensity / contrast
Edge Detection – Locating areas with strong intensity contrasts
Use of Edge Detection – Extracting information about the image. E.g. location of objects present in the image, their shape, size, image sharpening and enhancement
September 13, 2005
EE Computer Vision

4. TYPES OF EDGES Variation of Intensity / Gray Level Step Edge Ramp Edge
Line Edge
Roof Edge
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EE Computer Vision

5. Steps in Edge Detection
Filtering – Filter image to improve performance of the Edge Detector wrt noise
Enhancement – Emphasize pixels having significant change in local intensity
Detection – Identify edges - thresholding
Localization – Locate the edge accurately, estimate edge orientation
September 13, 2005
EE Computer Vision

6. Noisy Image Example of Noisy Image September 13, 2005
EE Computer Vision

7. METHODS OF EDGE DETECTION
First Order Derivative / Gradient Methods
Roberts Operator
Sobel Operator
Prewitt Operator
Second Order Derivative
Laplacian
Laplacian of Gaussian
Difference of Gaussian
Optimal Edge Detection
Canny Edge Detection
September 13, 2005
EE Computer Vision

8. First Derivative
At the point of greatest slope, the first derivative has maximum value
E.g. For a Continuous 1-dimensional function f(t)
September 13, 2005
EE Computer Vision

9. Gradient
For a continuous two dimensional function Gradient is defined as
September 13, 2005
EE Computer Vision

10. Gradient
Approximation of Gradient for a discrete two dimensional function
Convolution Mask
Gx=
Gy =
Differences are computed at the interpolated points [i, j+1/2] and [i+1/2, j]
Demo -1 1 -1 1 1 -1 1 -1
September 13, 2005
EE Computer Vision

11. Gradient Methods – Roberts Operator
Provides an approximation to the gradient
Convolution Mask
Gx=
Gy =
Differences are computed at the interpolated points [i+1/2, j+1/2] and not [i, j]
Demo 1 -1 -1 1
September 13, 2005
EE Computer Vision

12. Roberts Operator - Example
The output image has been scaled by a factor of 5
Spurious dots indicate that the operator is susceptible to noise
September 13, 2005
EE Computer Vision

13. Gradient Methods – Sobel Operator
The 3X3 convolution mask smoothes the image by some amount , hence it is less susceptible to noise. But it produces thicker edges. So edge localization is poor
Convolution Mask
Gx = Gy=
The differences are calculated at the center pixel of the mask.
Demo -1 1 -2 2 1 2 -1 -2
September 13, 2005
EE Computer Vision

14. Sobel Operator - Example
Compare the output of the Sobel Operator with that of the Roberts Operator:
The spurious edges are still present but they are relatively less intense compared to genuine lines
Roberts operator has missed a few edges
Sobel operator detects thicker edges
Will become more clear with the final demo
Outputs of Sobel (top) and Roberts operator
September 13, 2005
EE Computer Vision

15. Gradient Methods – Prewitt Operator
It is similar to the Sobel operator but uses slightly different masks
Convolution Mask
Px =
Py =
Demo -1 1 1 -1
September 13, 2005
EE Computer Vision

16. First Order Derivative Methods - Summary
Noise – simple edge detectors are affected by noise – filters can be used to reduce noise
Edge Thickness – Edge is several pixels wide for Sobel operator– edge is not localized properly
Roberts operator is very sensitive to noise
Sobel operator goes for averaging and emphasizes on the pixel closer to the center of the mask. It is less affected by noise and is one of the most popular Edge Detectors.
September 13, 2005
EE Computer Vision

17. Second Order Derivative Methods
Zero crossing of the second derivative of a function indicates the presence of a maxima
September 13, 2005
EE Computer Vision

18. Second Order Derivative Methods - Laplacian
Defined as
Mask
Demo
Very susceptible to noise, filtering required, use Laplacian of Gaussian 1 -4
September 13, 2005
EE Computer Vision

19. Second Order Derivative Methods - Laplacian of Gaussian
Also called Marr-Hildreth Edge Detector
Steps
Smooth the image using Gaussian filter
Enhance the edges using Laplacian operator
Zero crossings denote the edge location
Use linear interpolation to determine the sub-pixel location of the edge
September 13, 2005
EE Computer Vision

20. Laplacian of Gaussian – contd.
Defined as
Greater the value of s, broader is the Gaussian filter, more is the smoothing
Too much smoothing may make the detection of edges difficult
September 13, 2005
EE Computer Vision

21. Laplacian of Gaussian - contd.
Also called the Mexican Hat operator
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EE Computer Vision

22. Laplacian of Gaussian – contd.
Mask
Demo
Discrete approximation to LoG function with Gaussian   = 1.4
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EE Computer Vision

23. Second Order Derivative Methods - Difference of Gaussian - DoG
LoG requires large computation time for a large edge detector mask
To reduce computational requirements, approximate the LoG by the difference of two LoG – the DoG
September 13, 2005
EE Computer Vision

24. Difference of Gaussian – contd.
Advantage of DoG
Close approximation of LoG
Less computation effort
Width of edge can be adjusted by changing s1 and s2
September 13, 2005
EE Computer Vision

25. Second Order Derivative Methods - Summary
Second Order Derivative methods especially Laplacian, are very sensitive to noise
Probability of false and missing edges remain
Localization is better than Gradient Operators
September 13, 2005
EE Computer Vision

26. Optimal Edge Detector Optimal edge detector depending on
Low error rate – edges should not be missed and there must not be spurious responses
Localization – distance between points marked by the detector and the actual center of the edge should be minimum
Response – Only one response to a single edge
One dimensional formulation
Assume that 2D images have constant cross section in some direction
September 13, 2005
EE Computer Vision

27. Mathematical Formulation of Performance Criteria
Notation
Edge : G(x)
Noise : n(x)
Impulse response of Filter : f(x)
Assumptions
The edge is located at x=0
Filter has finite response bounded by [-W, W]
Noise is of Gaussian nature
n0 : Root mean squared noise amplitude per unit length
September 13, 2005
EE Computer Vision

28. First Criterion: Edge Detection
Response of filter to the edge:
RMS response of filter to noise :
First criterion: output Signal to Noise Ratio
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EE Computer Vision

29. Second Criterion: Edge Localization
A measure that increases as the localization increases is needed
Reciprocal of RMS distance of the marked edge from center of true edge is taken as the measure of localization
Localization is defined as:
September 13, 2005
EE Computer Vision

30. Maximizing the product
The design problem is reduced to maximization of the product:
September 13, 2005
EE Computer Vision

31. Third Criterion – Elimination of multiple responses
In presence of noise several maxima are detected – it is difficult to separate noise from edge
We try to obtain an expression for the distance between adjacent noise peaks
The mean distance between the adjacent maxima in the output is twice the distance between the adjacent zero crossings in the derivative of output operator
September 13, 2005
EE Computer Vision

32. Elimination of Multiple Responses – contd.
Expected number of noise maxima in the region of width 2W is given by
Fixing k will restrict the number of noise maxima that could lead to false response – this is a constraint imposed on the design
September 13, 2005
EE Computer Vision

33. Finding Optimal Detectors by Numerical Optimization
It is not possible to find a function f which will maximize the SNR, Localization product in the presence of the multiple response constraint
If the function f is discrete then the computation is simplified to four inner products
Additional constraints can be included by the use of Penalty Function – it has non-zero value when the constraint is violated
f is then found out for maximizing the following expression
September 13, 2005
EE Computer Vision

34. Detector for Step Edges
G(x) = Au-1(x)
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EE Computer Vision

35. Spatial scaling We can scale the filter width
Detection improves with width, localization degrades with increase in width
The uncertainty principle is
September 13, 2005
EE Computer Vision

36. Efficient approximation
Depending on the above principles, several optimal edge detectors are calculated
Best approximation to the above detectors is the First Derivative of Gaussian
It is chosen because of the ease of computation in 2 dimensions
September 13, 2005
EE Computer Vision

37. Noise estimation
Important to estimate the amount of noise in the image to set thresholds
Noise component can be efficiently isolated using Weiner Filtering – requires the knowledge of the autocorrelation of individual components and their cross-correlation
Noise strength is estimated by Global Histogram Estimation
September 13, 2005
EE Computer Vision

38. Thresholding
Broken edges due to fluctuation of operator output above and below the threshold – results in Streaking
Use double thresholding to eliminate streaking
September 13, 2005
EE Computer Vision

39. Two Dimensional Edge Detection
In two dimensions edge has both position and direction
A 2-D mask is created by convolving a linear edge detection function aligned normal to the edge direction with a projection function parallel the edge direction
Projection function is Gaussian with same deviation as the detection function
The image is convolved with a symmetric 2-D Gaussian and then differentiated normal to the edge direction
September 13, 2005
EE Computer Vision

40. Need for multiple widths and Feature Synthesis
Width is chosen to give best tradeoff between detection and localization
For different edges, width has to be different since SNR is different
Feature Synthesis
Smaller operator responses are used to predict the larger operator response
New edge points are marked if the actual large operator output varies significantly from the predicted output
September 13, 2005
EE Computer Vision

41. Implementation of Canny Edge Detector
Step 1
Noise is filtered out – usually a Gaussian filter is used
Width is chosen carefully
Step 2
Edge strength is found out by taking the gradient of the image
A Roberts mask or a Sobel mask can be used
September 13, 2005
EE Computer Vision

42. Implementation of Canny Edge Detector – contd.
Step 3
Find the edge direction
Step 4
Resolve edge direction
September 13, 2005
EE Computer Vision

43. Canny Edge Detector – contd.
Step 5
Non-maxima suppression – trace along the edge direction and suppress any pixel value not considered to be an edge. Gives a thin line for edge
Step 6
Use double / hysterisis thresholding to eliminate streaking
September 13, 2005
EE Computer Vision

44. Canny Edge Detector – contd.
Compare the results of Sobel and Canny
September 13, 2005
EE Computer Vision

45. Edge Localization – Subpixel Location Estimation
Take Gaussian edge detector output without non-maxima suppression
Error displacement is computed by taking the weighted mean of distance, along the edge gradient on either side of the edge
September 13, 2005
EE Computer Vision

46. Edge Detector Performance
Criteria
Probability of false edges
Probability of missing edges
Error in estimation of edge angle
Mean square distance of edge estimate from true edge
Tolerance to distorted edges and other features such as corners and junctions
September 13, 2005
EE Computer Vision

47. Figure of Merit Basic errors in a Edge Detector Missing Edges
Error in localizing
Classification of noise as Edge
IA : detected edges
II : ideal edges
d : distance between actual and ideal edges
a : penalty factor for displaced edges
September 13, 2005
EE Computer Vision

48. Line Detection Lines in an image present another form of discontinuity
It is worthwhile to consider them separately since they are very frequent
Methods
Use of Convolution Mask
Hough Transform
September 13, 2005
EE Computer Vision

49. Line Detection using Convolution Mask
We need to decide the orientation and width of line to be detected
Following masks are for single pixel wide lines in four directions
September 13, 2005
EE Computer Vision

50. Line Detection using Convolution Mask – contd.
The mask for horizontal orientation is applied
Lines are 5 pixels wide, mask is tuned for detecting 1 pixel wide line
Line detector acts as edge detector
Demo
September 13, 2005
EE Computer Vision

51. Line detection – Hough Transform
Method to isolate the shapes from an image
Performed after edge detection
Not affected by noise or gaps in the edges
Technique
Thresholding is used to isolate pixels with strong edge gradient
Parametric equation of straight line is used to map the edge points to the Hough parameter space
Points of intersection in the Hough parameter space gives the equation of line on actual image
September 13, 2005
EE Computer Vision

52. Hough Transform - contd
Parametric Equation of Straight Line
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EE Computer Vision

53. Hough Transform – contd.
Mapping of Edge Points in Hough Parameter Space
Edge Detected Image
September 13, 2005
EE Computer Vision

54. Applications
Enhancement of noisy images – satellite images, x-rays, medical images like cat scans
Text detection
Mapping of roads
Video surveillance, etc.
September 13, 2005
EE Computer Vision

55. Applications Canny Edge Detector for Remote Sensing Images
Reasons to go for Canny Edge Detector – Remote sensed images are inherently noisy
Other edge detectors are very sensitive to noise
September 13, 2005
EE Computer Vision

56. Edge map of remote sensed image using Canny
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EE Computer Vision

57. Image enhancement and sharpening using canny
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EE Computer Vision

58. MATLAB Demo Get an image from the internet
Apply following operators on the image and display the results
Roberts
Sobel
Prewitt
Laplacian of Gaussian
Canny
September 13, 2005
EE Computer Vision

59. References
Machine Vision – Ramesh Jain, Rangachar Kasturi, Brian G Schunck, McGraw-Hill, 1995
INTRODUCTION TO COMPUTER VISION AND IMAGE PROCESSING - by Luong Chi Mai Department of Pattern Recognition and Knowledge Engineering Institute of Information Technology, Hanoi, Vietnam
The Hypermedia Image Processing Reference -
A Survey and Evaluation of Edge Detection Operators Application to Medical Images – Hanene Trichili, Mohamed-Salim Bouhlel, Nabil Derbel, Lotfi Kamoun, IEEE, 2002
Using The Canny Edge Detector for Feature Extraction and Enhancement of Remote Sensing Images - Mohamed Ali David Clausi, Systems Design Engineering, University of Waterloo, IEEE 2001
A Computational Approach to Edge Detection – John Canny, IEEE, 1986
September 13, 2005
EE Computer Vision