Digital Image Processing “Edge Detection”

1. Digital Image Processing“Edge Detection”

2. Edge Detection • What are edges in an image? • Edge Detection • Edge Detection Methods • Edge Operators • MatlabProgram

3. What are edges in an image? • Edges are those places in an image that correspond to object boundaries. • Edges are pixels where image brightness changes abruptly. • Edges are used frequently for segmenting images based on local changes in intencity Brightness Spatial Coordinates

4. Edge Detection • Convert 2D image into a set of curves • Edge may represents an image and contain shape information

5. Edge is Where Change Occurs • Edge information in an image is found by looking at the relationship a pixel has with its neighborhoods. • If a pixel’s gray-level value is similar to those around it, there is probably not an edge at that point. • If a pixel’s has neighbors with widely varying gray levels, it may present an edge point. • Change is measured by derivative in 1D • Biggest change, derivative has maximum magnitude • Or 2nd derivative is zero.

6. Edge detectors • locate sharp changes in the intensity function • edges are pixels where brightness changes abruptly. • Calculus describes changes of continuous functions using derivatives; an image function depends on two variables - partial derivatives. • A change of the image function can be described by a gradient that points in the direction of the largest growth of the image function. • An edge is a property attached to an individual pixel and is calculated from the image function behavior in a neighborhood of the pixel.

7. The gradient direction gives the direction of maximal growth of the function, e.g., from black (f(i,j)=0) to white (f(i,j)=255). • This is illustrated below; closed lines are lines of the same brightness. • Boundary and its parts (edges) are perpendicular to the direction of the gradient.

9. 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 Image gradient • The gradient of an image: • The gradient points in the direction of most rapid change in intensity

10. The discrete gradient • How can we differentiate a digital image f[x,y]? • Option 1: reconstruct a continuous image, then take gradient • Option 2: take discrete derivative (finite difference) How would you implement this as a cross-correlation?

11. The Sobel operator • Better approximations of the derivatives exist • The Sobel operators below are very commonly used • 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 value, however

12. Gradient operators (a): Roberts’ cross operator (b): 3x3 Prewitt operator (c): Sobel operator (d) 4x4 Prewitt operator

13. Roberts Operator • Mark edge point only • No information about edge orientation • Work best with binary images • Primary disadvantage: • High sensitivity to noise • Few pixels are used to approximate the gradient

14. Roberts Operator (Cont.) • First form of Roberts Operator • Second form of Roberts Operator

15. Prewitt Operator • Looks for edges in both horizontal and vertical directions, then combine the information into a single metric. Edge Magnitude = Edge Direction =

16. Sobel Operator • Similar to the Prewitt, with different mask coefficients: Edge Magnitude = Edge Direction =

17. 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?

18. Look for peaks in Solution: smooth first Where is the edge?

19. Derivative theorem of convolution • This saves us one operation:

20. Laplacian of Gaussian • Consider Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph

21. 2D edge detection filters • is the Laplacian operator: Laplacian of Gaussian Gaussian derivative of Gaussian

22. Laplacian Operators • Edge magnitude is approximated in digital images by a convolution sum. • The sign of the result (+ or -) from two adjacent pixels provide edge orientation and tells us which side of edge brighter

23. Laplacian Operators (Cont.) • Masks for 4 and 8 neighborhoods • Mask with stressed significance of the central pixel or its neighborhood

24. A Matlab Quick Note • Matlab’s image processing toolbox provides edge function to find edges in an image: I = imread('rice.tif'); BW1 = edge(I,'prewitt'); BW2 = edge(I,'canny'); imshow(BW1) figure, imshow(BW2) • Edge function supports six different edge-finding methods: Sobel, Prewitt, Roberts, Laplacian of Gaussian, Zero-cross, and Canny.