Edge Detection Lecture 03 Computer Vision

Transcription

1 Edge Detection Lecture 3 Computer Vision

2 Suggested readings Chapter 5 Linda G. Shapiro and George Stockman, Computer Vision, Upper Saddle River, NJ, Prentice Hall,. Chapter David A. Forsyth and Jean Ponce, Computer Vision A Modern Approach, nd edition, Prentice Hall, Inc., 3.

3 Material Citations Dr George Stockman Professor Emeritus, Michigan State University Dr Mubarak Shah Professor, University of Central Florida The Robotics Institute Carnegie Mellon University

4 Online personal info update Those who have not updated their data online will not be able to get their deliverables marked

5 Recap

6 Example f ( x) 5 5 f ( x) f ( x) Derivative Masks Backward difference Forward difference Central difference [- ] [ -] [- ]

7 (, ) f x y y x f f y y x f x y x f y x f ), ( ), ( ), ( ), ( y f x f y x f y x f f tan Given function Gradient vector Gradient magnitude Gradient direction Derivatives in -D Image is D so we will have partial derivatives

8 Derivatives of images Derivative masks f x 3 f y 3 Averaging is one way to get rid of noise in any pixel I I x (-++) + (-++) + (-++) = 3

9 Derivatives of images I I y f y 3

10 Gaussian Filter g x ( x) e gx ( ) x=-3 x=- x=- x= x= x= x=3 ( x y ) g( x, y) e

11 Properties of Gaussian Most common natural model Smooth function, it has infinite number of derivatives Fourier Transform of Gaussian is Gaussian Convolution of a Gaussian with itself is a Gaussian There are cells in eye that perform Gaussian filtering

12 Averages Mean I I I I n n i n n I i Weighted mean I w I wni n i wi n n w I n i i

13 Image filtering Modify pixels in an image based on some function of a local neighborhood of the pixels Local image data Some function f p 7 Modified image data Replace each pixel by a linear combination of its neighbors We don t want to only do this at a single pixel, of course, but want instead to run the kernel over the whole image.

14 Correlation f h f ( k, l) h( i k, j l) k l Go through each row Given a row, go through each column f = image h = kernel/filter f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 h h h 3 h 4 h 5 h 6 h 7 h 8 h 9 f h f h f h f h 3 3 f h f h f h f h f h f h

15 Convolution f * h f ( k, l) h( i k, j l) k l f = image h = kernel/filter h 7 h 8 h 9 h 4 h 5 h 6 X-flip h h h 3 h 4 h 5 h 6 h h h 3 h 7 h 8 h 9 Y-flip If h is symmetric here eg. Gaussian, correlation and convolution will have same results f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 h 9 h 8 h 7 h 6 h 5 h 4 h 3 h h f * h f h f h f h f h f h f h f h f h f h

16 Lets Start!!!

17 Example

18 What is an object? How can we find it? An Application

19 Edge Detection in Images Can occur due to different sources Shadows Texture Edges show discontinuity and change in shape and color etc

20 What is an Edge? Discontinuity of intensities in the image Edge models Intensity plots Step Roof Ramp Spike Step Ramp Roof Spike

21 Detecting Discontinuities Image derivatives f x lim f x f x f x f x f x n x Convolve image with derivative filters Backward difference Forward difference Central difference [- ] [ -] [- ]

22 Definition Approximation Convolution kernels y x f y x f x y x f,, lim, y x f y x f y y x f,, lim, x y x f y x f x y x f m n m n,,,,,, n m n m f x y f x y f x y y y f x f y Derivative in Two-Dimensions

23 Image Derivatives Image I I x I * I y I *

24 Derivatives and Noise Strongly affected by noise obvious reason: image noise results in pixels that look very different from their neighbors The larger the noise is the stronger the response What is to be done? Neighboring pixels look alike Pixel along an edge look alike Image smoothing should help Force pixels different to their neighbors (possibly noise) to look like neighbors

25 Derivatives and Noise Increasing noise Adding here zero mean additive Gaussian noise

26 Image Smoothing Expect pixels to be like their neighbors Relatively few reflectance changes Generally expect noise to be independent from pixel to pixel Smoothing suppresses noise

27 Gaussian Smoothing ( x y ) g( x, y) e Scale of Gaussian As increases, more pixels are involved in average As increases, image is more blurred As increases, noise is more effectively suppressed

28 Edge Detectors Gradient operators Prewit Sobel Laplacian of Gaussian (Marr-Hildreth) Gradient of Gaussian (Canny)

29 Prewitt and Sobel Edge Detector Compute derivatives In x and y directions Find gradient magnitude Threshold gradient magnitude

30 image blurred edges in x average smoothing in x derivative filtering in x and image blurred edges in x average smoothing in y derivative filtering in y and Prewitt Edge Detector

31 Image I * * I dx d I dy d I dy d I dx d Threshold Edges Sobel Edge Detector Center row has more weight

32 Sobel Edge Detector d I dx d dy I

33 Sobel Edge Detector Magnitude d dx I d dy I Threshold

34 Marr Hildreth Edge Detector Smooth image by Gaussian filter S Apply Laplacian to S Used in mechanics, electromagnetics, wave theory, quantum mechanics and Laplace equation Find zero crossings and declare edge point Calculus first derivative is maxima and second derivative is a zero Scan along each row, record an edge point at the location of zerocrossing Repeat above step along each column

35 Marr Hildreth Edge Detector Gaussian smoothing smoothed image Gaussian filter image S g * I g x y e Find Laplacian second order second order derivative in x derivative in y S S S x y is used for gradient (derivative) is used for Laplacian

36 Marr Hildreth Edge Detector Deriving the Laplacian of Gaussian (LoG) Laplacian of Gaussian g * I g I S * Smooth I Inverted Mexican hat Smooth I Calculate Laplacian g x y x y e 3

37 LoG Filter G x y x y e 3 Fix sigma and get values of x and y Y x=, y= x=, y= X Can multiply with a constant to get integer values

38 Finding Zero Crossings Four cases of zero-crossings : {+,-} {+,,-} {-,+} {-,,+} Slope of zero-crossing {a, -b} is a+b To check how strong is the change To mark an edge compute slope of zero-crossing Apply a threshold to slope

39 On the Separability of LoG Similar to separability of Gaussian filter Two-dimensional Gaussian can be separated into one-dimensional Gaussians D h( x, y) I( x, y)* g( x, y) n multiplications I( x, y)* g ( x) * g ( ) h( x, y) y n multiplications g..3.6 D D.6.3. g

40 On the Separability of LoG S g * I g * I I * g Requires n multiplications xx yy S I g ( x) g( y) I g ( y) g( x) 4 x D convolutions Requires 4n multiplications

41 Separability Gaussian Filtering Image g(x) g(y) + I g Laplacian of Gaussian Filtering g xx (x) g(y) Image + S g yy (y) g(x)

42 Example I I * g Zero crossingsof S

43 Example 3 6

44 Algorithm Compute LoG Use D filter g( x, y) Use 4 x D filters g( x), g ( x), g( y), g ( y) xx yy Find zero-crossings from each row Find slope of zero-crossings Apply threshold to slope and mark edges

45 Quality of an Edge Robust to noise Localization Too many or too less responses

46 Quality of an Edge Edge points True edge Poor robustness to noise Poor localization Too many responses

47 Recap (Prewitt and Sobel) Prewitt and Sobel edge detectors Compute derivatives In x and y directions Find gradient magnitude Threshold gradient magnitude Difference between Prewitt and Sobel is the derivative filters

48 Prewitt s edges in x direction Prewitt s edges in y direction I x I y Prewitt Edge Detector

49 Sobel s edges in x direction Sobel s edges in y direction I x I y Sobel Edge Detector

50 Canny Edge Detector Criterion : Good Detection: The optimal detector must minimize the probability of false positives as well as false negatives Criterion : 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

51 Canny Edge Detector Steps Smooth image with Gaussian filter Compute derivative of filtered image Find magnitude and orientation of gradient Apply Non-maximum Suppression Apply Hysteresis Threshold

52 Smoothing Derivative I y x g y x g I S ), ( ), ( ), ( y x e y x g I g I g S y x g g y g x g g I g g S y x I g I g y x Canny Edge Detector - First Two Steps

53 Canny Edge Detector - Derivative of Gaussian g x ( x, y) g( x, y) g y ( x, y)

54 Canny Edge Detector - First Two Steps I S x S y

55 Canny Edge Detector - Third Step Gradient magnitude and gradient direction ( S x, S y ) Gradient Vector magnitude direction ( S x tan S S S y y x ) image gradient magnitude

56 Canny Edge Detector - Fourth Step Non maximum suppression 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?

57 Canny Edge Detector - Non-Maximum Suppression Suppress the pixels in S which are not local maximum Direction of gradient is always perpendicular to edge x, y x, y x, y, if,, S x y S x y S x y M x, y & S x, y S x, y otherwise x and x are the neighbors of x along normal direction to an edge Using gradient direction here Object

58 Canny Edge Detector - Non-Maximum Suppression Magnitude after Non-Maximum Suppression S S x S y Magnitude M Further Thresholding For visualization M Threshold 5

59 Canny Edge Detector - Hysteresis Thresholding If the gradient at a pixel is above High, declare it an edge pixel below Low, declare it a non-edge-pixel between low and high Consider its neighbors iteratively then declare it an edge pixel if it is connected to an edge pixel directly or via pixels between low and high.

60 Canny Edge Detector - Hysteresis Thresholding Connectedness x x x 4 connected 8 connected 6 connected

61 Canny Edge Detector - Hysteresis Thresholding Gradient magnitude Pixels above high threshold Edge pixels selected High Connected with pixels above high threshold Edge pixel selected low Above low threshold but not connected with edge pixels above high threshold Edge pixel rejected

62 Canny Edge Detector - Hysteresis Thresholding Scan the image from left to right, top-bottom The gradient magnitude at a pixel is above a high threshold declare that as an edge point Then recursively consider the neighbors of this pixel If the gradient magnitude is above the low threshold declare that as an edge pixel

63 Canny Edge Detector - Hysteresis Thresholding regular M M 5 Hysteresis High 35 Low 5

64 Final project ideas!!!??? I will upload the list this week

65 Assignment Will be uploaded

66 Quiz Next class