Lecture 8
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This document discusses edge detection techniques in digital image processing. It begins by defining edges as areas of abrupt intensity change and explains that differentiation is used to detect edges locally. It then discusses the importance of edge detection for image segmentation and defect detection. The document goes on to explain that edge detection involves high-pass spatial domain filtering to eliminate low and medium frequencies while passing or enhancing high frequencies. It covers various first and second order derivative approaches to edge detection, including Roberts, Prewitt, Sobel, and Laplacian methods. Thresholding and boolean filtering techniques are also summarized for precise edge detection.
Lecture 8
- 1. SPATIAL DOMAIN FILTERING EDGE DETECTION CS-467 Digital Image Processing 1
- 2. Edges • Edge pixels are pixels at which the intensity changes abruptly • Edges are sets of connected edge pixels • Often edges are called points of discontinuity of the intensity function • To detect edges, differentiation in a local neighborhood should be used 2
- 3. Importance of Edge Detection • Edge detection is important for localization of image details, especially the smallest details • It can be used, for example, in non- destructive inspection, to localize possible defects; in medical imaging, to localize important details • Edge detection is also used in image segmentation – a process that partitions an image into disjoint regions corresponding to different textures 3
- 4. High-Pass Filtering • In terms of filtering, edge detection is a spatial domain high-pass filtering • Differentiating an image, high-pass filters eliminate low and medium frequencies, passing and possibly enhancing high frequencies. 4
- 5. Three Major Types of Intensity Change 5 Step Ramp Roof © 1992-2008 R.C. Gonzalez & R.E. Woods
- 6. Derivatives of Digital Functions • The derivatives of a digital image function are defined in terms of differences. They have to meet the following requirements: • A 1st derivative 1) Must be 0 in areas of constant intensity 2) Must be nonzero at the onset of an intensity step or ramp; 3) Must be nonzero along ramps 6
- 7. Derivatives of Digital Functions • The derivatives of a digital image function are defined in terms of differences. They have to meet the following requirements: • A 2nd derivative 1) Must be 0 in areas of constant intensity 2) Must be nonzero at the onset and end of an intensity step or ramp; 3) Must be zero along ramps of constant slope 7
- 8. Derivatives of 1-D Digital Functions • A basic definition of the first-order derivative of the digital function • A basic definition of the second-order derivative of the digital function 8 ( ) ( )1 f f x f x x ∂ = + − ∂ ( )f x ( )f x ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 1 1 1 1 2 f f x f x f x f x x f x f x f x ∂ = + − − − − = ∂ = + + − −
- 9. Derivatives of 1-D Digital Functions 9 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 10. Derivatives of 2-D Digital Functions • A basic definition of the first-order derivatives of the function • A basic definition of the second-order derivatives of the function 10 ( ) ( ) ( ) ( )1, , ; , 1 , f f f x y f x y f x y f x y x y ∂ ∂ = + − = + − ∂ ∂ ( ),f x y ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1, 1, 2 , ; , 1 , 1 2 , f f x y f x y f x y x f f x y f x y f x y y ∂ = + + − − ∂ ∂ = + + − − ∂ ( ),f x y
- 11. The Laplacian and Second Order Derivative Edge Detection • The Laplacian (as it is known from calculus and partial differential equations) is • Hence, Laplacian of the image intensity function (“Laplacian 1”) 11 ( ) 2 2 2 2 2 , f f f x y x y ∂ ∂ ∇ = + ∂ ∂ ( )f x ( ) ( ) ( ) ( ) ( ) ( ) 2 , 1, 1, , 1 , 1 4 , f x y f x y f x y f x y f x y f x y ∇ = + + − + + + + − −
- 12. The Laplacian • A global edge detector must be spatially isotropic, it should not depend on any rotation of an image and particular gradient direction • To detect diagonal edges, we should involve the diagonal second-order derivatives (Laplacian 2) 12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , 1, 1, , 1 , 1 1, 1 1, 1 1, 1 1, 1 8 , f x y f x y f x y f x y f x y f x y f x y f x y f x y f x y ∇ = + + − + + + − + + − − + + − + − + + + + − − , f f x y y x ∂ ∂ ∂ ∂ ∂ ∂
- 13. Laplacian Edge Detector • Laplacian 1 Laplacian 2 13 Downward Upward © 1992-2008 R.C. Gonzalez & R.E. Woods
- 14. Laplacian Detector of Lines Oriented in Various Directions 14 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 15. First- and Second- Order Derivatives and Edge Detection • Laplace edge detectors are based on second- order derivatives • There are also edge detectors based on first- order derivatives • Second-order derivatives better detect edges corresponding to the smallest details, they also detect and enhance isolated pixels • First-order derivatives detect and possibly enhance the brightness jumps “as they are” 15
- 16. First- and Second- Order Derivatives and Edge Detection 16 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 17. Distinction between First- and Second- Order Derivatives in Edge Detection 17 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 18. First-Order Derivatives in Edge Detection • First-order derivatives in image processing are implemented using the magnitude of the gradient • The magnitude (length) of vector is 18 ( ) x y f g x f grad f fg y ∂ ∂ ∇= = = ∂ ∂ f∇ ( ) ( ) 2 2 , x y x yM x y mag f g g g g= ∇ = + ≈ +
- 19. Gradient and Edge 19 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 20. First-Order Derivatives by Roberts (1965) 20 1 2 3 4 5 6 7 8 9 z z z z z z z z z ( ) ( ) ( ) 9 5 8 6 9 5 8 6 ; , x yg z z g z z M x y z z z z =− =− ≈ − + − ( ) ( ) ( ) 8 5 6 5 8 5 8 5 ; , x yg z z g z z M x y z z z z =− =− ≈ − + − Robert’s derivatives and gradients Disadvantage: no center of symmetry
- 21. First-Order Derivatives by Prewitt (1970) 21 1 2 3 4 5 6 7 8 9 z z z z z z z z z ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 7 8 9 1 2 3 3 6 9 1 4 7 7 8 9 1 2 3 3 6 9 1 4 7 , x y f g z z z z z z x f g z z z z z z y M x y z z z z z z z z z z z z ∂ = = + + − + + ∂ ∂ = = + + − + + ∂ ≈ + + − + + + + + + − + +
- 22. First-Order Derivatives by Sobel (1970) 22 1 2 3 4 5 6 7 8 9 z z z z z z z z z ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 7 8 9 1 2 3 3 6 9 1 4 7 7 8 9 1 2 3 3 6 9 1 4 7 2 2 2 2 2 2 2 , 2 x y f g z z z z z z x f g z z z z z z y M x y z z z z z z z z z z z z ∂ = = + + − + + ∂ ∂ = = + + − + + ∂ ≈ + + − + + + + + + − + +
- 23. Roberts, Prewitt, and Sobel masks 23 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 24. Some Disadvantages of Classical Edge Detectors • They do not differ (except Laplace) between upward (from “dark” to “bright”) and downward (from “bright” to “dark”) brightness jumps • This is resulted in “bold” (“thick”) edges where the same brightness jump can be detected twice • All of them (except Prewitt) give a preference to the pixel of interest (the symmetry center). This may help to suppress noise, to avoid its detection, but simultaneously some edges can be emphasized, while some other can be suppressed 24
- 25. Nonlinear Edge Detectors • In fact, even Sobel and Prewitt edge detectors are nonlinear, because they operate with absolute values of gradients. |…| function is only piecewise-linear; thus, it is nonlinear. • Nonlinear edge detectors are based either on nonlinear operation applied to digital derivatives or nonlinear transformation of an image followed by some kind of differentiation 25
- 26. Threshold Boolean Filtering • Let X be (M+1)-valued gray scale image • Let be a binary plane • Then filtering operation is determined by where F is the processing Boolean function, is a local window around a pixel of interest 26 ( ) 1 , if ( , ) ( , ) ; 1,..., 0 , otherwise k x i j k x i j k M ≥ = = ( ) ( )( ) 1 , M k ij k XFy i j = = ∑ ( )k ijX
- 27. Edge Detecting Boolean Functions • The Boolean function for detection of upward brightness jumps • The Boolean function for detection of downward brightness jumps 27 ( ) 1 2 3 4 5 6 5 1 2 3 4 6 7 8 9 7 8 9 &u x x x F f x x x x x x x x x x x x x x x = = ∨ ∨ ∨ ∨ ∨ ∨ ∨ ( ) 1 2 3 4 5 6 5 1 2 3 4 6 7 8 9 7 8 9 &d x x x F f x x x x x x x x x x x x x x x = = ∨ ∨ ∨ ∨ ∨ ∨ ∨
- 28. Edge Detecting Boolean Functions 28 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 Upward Edge passes through the center of the window, there is no downward edge Downward Edge passes through the center of the window, there is no upward edge
- 29. TBF (Stack) Edge Detection Filter • Detection of upward brightness jumps is equivalent to • Detection of downward brightness jumps is equivalent to 29 ( ) ( ) ( )( )max , 1, 1 ,if 0 , 0, otherwise m f x y f x y m g x y = − ± ± > = ( ) ( ) ( )( )max 1, 1 , ,if 0 , 0, otherwise m f x y f x y m g x y = ± ± − > =
- 30. TBF Edge Detection 30 35 210 100 112 120 80 100 221 220 35 210 100 112 120 80 100 221 220 Upward Edge pass through the center of the window, its intensity is 120-35=85 Downward Edge pass through the center of the window, its intensity is 221-120=101
- 31. Edge Detection using Stack Filter • Simultaneous detection of downward and upward brightness jumps • This is resulted in “bold” (“thick”) edges 31 ( ) ( ) ( ), max , 1, 1g x y f x y f x y= − ± ±
- 32. Image Break-up Into Binary Planes • Each intensity value in the range 0…255 is presented by one byte, which consists of 8 bits • Breaking up these bytes into bits, we break up the entire image into 8 binary planes 32 © 1992-2008 R.C. Gonzalez & R.E. Woods
- 33. Precise Edge Detection • Algorithm: 1) Breaking up the image into binary planes; 2) Detection of edges using edge detecting Boolean function; 3) Assembling the resulting image from the binary planes 33
- 34. Precise Edge Detection • Upward Edge Detecting Boolean function • Downward Edge Detecting Boolean function • “Global” Edge Detecting Boolean function 34 1 2 3 4 5 6 7 8 9 x x x f x x x x x x = 5 1 2 3 4 6 7 8 9( )x x x x x x x x x∧ ∨ ∨ ∨ ∨ ∨ ∨ ∨ 5 1 2 3 4 6 7 8 9( )x x x x x x x x x∧ ∨ ∨ ∨ ∨ ∨ ∨ ∨ 1 2 3 4 5 6 7 8 9 x x x f x x x x x x = 1 2 3 4 5 6 7 8 9 x x x f x x x x x x = ( )5 1 2 3 4 6 7 8 9& ( )x x x x x x x x x∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ 5 1 2 3 4 6 7 8 9&( )x x x x x x x x x∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨
- 35. Edged Segmentation using Precise Edge Detection • Edged Segmentation is edge detection with simultaneous emphasizing of areas with different textures bounded by edges • To make segmentation using precise edge detection, it is necessary to detect edges in some binary planes preserving some other binary planes unchanged. For example, a good segmentational effect can be achieved by detecting edges in binary planes 0,1,2,3,4,7 and preserving binary planes 5,6 unchanged 35
- 36. Noise and Edge Detection • Noise prevents detecting clean edges because edge detectors don’t care of what is behind particular intensity jump – noise or useful detail 36 © 1992-2008 R.C. Gonzalez & R.E. Woods