1. Edge detection

2. Edge Detection in Images Finding the contour of objects in a scene

3. Edge Detection in Images What is an object? It is one of the goals of computer vision to identify objects in scenes.

4. Edge Detection in Images Edges have different sources.

5. What is an Edge Lets define an edge to be a discontinuity in image intensity function. Edge Models –Step Edge –Ramp Edge –Roof Edge –Spike Edge

6. Detecting Discontinuities Discontinuities in signal can be detected by computing the derivative of the signal.

7. Differentiation and convolution Recall Now this is linear and shift invariant, so must be the result of a convolution. We could approximate this as (which is obviously a convolution with Kernel ; it’s not a very good way to do things, as we shall see)

8. Finite Difference in 2D Discrete Approximation Definition Convolution Kernels

9. Finite differences

10. Frequency Response of Differential Kernel Fourier Transform Fourier Transform

11. Noise Simplest noise model –independent stationary additive Gaussian noise –the noise value at each pixel is given by an independent draw from the same normal probability distribution Issues –this model allows noise values that could be greater than maximum camera output or less than zero –for small standard deviations, this isn’t too much of a problem - it’s a fairly good model –independence may not be justified (e.g. damage to lens) –may not be stationary (e.g. thermal gradients in the ccd)

12. Finite differences and noise Finite difference filters respond strongly to noise –obvious reason: image noise results in pixels that look very different from their neighbours Generally, the larger the noise the stronger the response What is to be done? –intuitively, most pixels in images look quite a lot like their neighbours –this is true even at an edge; along the edge they’re similar, across the edge they’re not –suggests that smoothing the image should help, by forcing pixels different to their neighbours (=noise pixels?) to look more like neighbours

13. Finite differences responding to noise Increasing noise -> (this is zero mean additive gaussian noise)

14. Smoothing reduces noise Generally expect pixels to “be like” their neighbours –surfaces turn slowly –relatively few reflectance changes Generally expect noise processes to be independent from pixel to pixel Implies that smoothing suppresses noise, for appropriate noise models Scale –the parameter in the symmetric Gaussian –as this parameter goes up, more pixels are involved in the average –and the image gets more blurred –and noise is more effectively suppressed

15. The effects of smoothing Each row shows smoothing with gaussians of different width; each column shows different realisations of an image of gaussian noise.

16. Classical Operators Prewitt’s Operator Smooth Differentiate

17. Classical Operators Sobel’s Operator Smooth Differentiate

18. Gaussian Filter

19. Sobel Edge Detector Detecting Edges in Image Image I Threshold Edges any alternative ?

20. Quality of an Edge Detector Robustness to Noise Localization Too Many/Too less Responses Poor robustness to noise Poor localization Too many responses True Edge

21. Canny Edge Detector Criterion 1: Good Detection: The optimal detector must minimize the probability of false positives as well as false negatives. Criterion 2: Good Localization: The edges detected must be as close as possible to the true edges. Single Response Constraint: The detector must return one point only for each edge point.

22. Canny Edge Detector Difficult to find closed-form solutions.

23. Canny Edge Detector Convolution with derivative of Gaussian Non-maximum Suppression Hysteresis Thresholding

24. Canny Edge Detector Smooth by Gaussian Compute x and y derivatives Compute gradient magnitude and orientation

25. Canny Edge Operator

26. Canny Edge Detector

28. 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? Non-Maximum Suppression

29. Suppress the pixels in ‘Gradient Magnitude Image’ which are not local maximum

30. Non-Maximum Suppression

32. Hysteresis Thresholding

33. If the gradient at a pixel is above ‘High’, declare it an ‘edge pixel’ If the gradient at a pixel is below ‘Low’, declare it a ‘non-edge-pixel’ If the gradient at a pixel is between ‘Low’ and ‘High’ then declare it an ‘edge pixel’ if and only if it is connected to an ‘edge pixel’ directly or via pixels between ‘Low’ and ‘ High’

34. Hysteresis Thresholding

35. an image its detected edges