Edge Detection. Image Processing - Computer Vision

Transcription

1 Image Processing - Lesson 10 Edge Detection Image Processing - Computer Vision Low Level Edge detection masks Gradient Detectors Compass Detectors Second Derivative - Laplace detectors Edge Linking Image Processing Computer Vision representation, compression,transmission image enhancement edge/feature finding image "understanding" High Level UFO - Unidentified Flying Object Point Detection Convolution with: Large Positive values = light point on dark surround Large Negative values = dark point on light surround Eample: * =

2 Edge Definition Edge Detection Line Edge Line Edge Detectors gray value Step Edge edge edge Step Edge Detectors gray value edge edge Eample Edge Detection by Differentiation Step Edge detection by differentiation: 1D image f() gray value 1st derivative f'() f'() - threshold threshold Piels that passed the threshold are Edge Piels

3 Gradient Edge Detection Gradient Edge - Eamples Gradient f((,y) = f f y f((,y) = f, 0 Gradient Magnitude f f y ( ) + ( ) Gradient Direction tg-1 ( f f y / ) f((,y) = f 0, y Differentiation in Digital Images Eample Edge horizontal - differentiation approimation: F A = f(,y) = f(,y) - f(-1,y) convolution with [ 1-1 ] vertical - differentiation approimation: Gradient-X Gradient-Y F B = f(,y) y = f(,y) - f(,y-1) convolution with 1-1 Gradient (F A, F B ) Gradient-Magnitude Gradient-Direction Magnitude Appro. Magnitude ((F A ) + (F B ) ) 1/ F A + F B

4 Roberts Edge Detector Sobel Edge Detector F A = f(,y) - f(-1,y-1) F B = f(-1,y) - f(,y-1) A = - 0 B = A = B = Roberts and other operators are sensitive to noise. Isotropic Edge Detector A = Prewitt Edge Detector B = A = - 0 B = Smoothed operators Edge Detector - Comparison Compass Operators N NW W SW Roberts S SE E NE Sobel Given k operators, g k (,y) is the image obtained by convolving f(,y) with the k th operator. The gradient is defined as: Prewitt g(,y) = ma g k (,y) k k defines the edge direction Magnitude Negative Magnitude

5 Various Compass Operators: Edge Detector - Comparison Kirsch Edge Detector Sobel Kirsch (Compass) Direction Map Magnitude Negative Magnitude Derivatives Laplacian Operators Approimation of second derivative (horizontal): f() f (,y) = f''(,y) = f'(+1,y) - f'(,y) = = [f(+1,y) - f(,y)] - [f(,y) - f(-1,y)] = f(+1,y) - f(,y) + f(-1,y) convolution with: [ 1-1] f () Approimation of second derivative (vertical): convolution with: 1-1 f () Laplacian Operator = ( + ) y zero crossing convolution with:

6 Variations on Laplace Operators: Eample of Laplacian Edge Detection All are approimations of: Laplacian ~ Difference of gaussians Laplacian Operator (Image Domain) Laplacian Filter (Frequency Domain) - = FFT DOG = Difference of Gaussians - = FFT - =

7 Enhancement Using the Laplacian Operator Eamples of enhanced images Laplacian Image Edge Image f() 1 st derivative f nd derivative (Laplacian) f f - Scaled Laplacian Image Enhanced Image f() - f Mach Bands Edge Detection - Noise Issues Edge Detection - Noise Issues f f ( ) g d f ( ) d g f ( g f ) Peaks in ( g f ) mark the edge

8 Edge Detection - Noise Issues Edge Detection - Noise Issues From the convolution theorem we have: ( h f ) = h f f f g h g f h f Zero Crossings in ( g f ) mark the edge Canny Edge Detector 1) Convolve image with derivative of a Gaussian ) Perform NonMaimum suppression. 3) Perform Hysteresis Thresholding. Canny Edge Detector - Step 1 Convolve image with derivative of a Gaussian f(,y) * G'(,y) f(,y) * G' (,y) + f(,y) * G' y (,y) F a = f(,y) * G (,y) * D (,y) = f(,y) * * 1-1 F b = f(,y) * G y (,y) * D y (,y) = f(,y) * * 1-1 Magnitude ((F A ) + (F B ) ) 1/

9 Canny Edge Detector - Step 1 Canny Edge Detector - Step NonMaimum suppression. Remove edges that are NOT local Maima in the gradient direction. f * G f * G y Local Maima Local Maima F a = f*g *D F b = f*g y *D y Magnitude ((F A ) + (F B ) ) 1/ Canny Edge Detector - Step NonMaimum Suppression. Canny Edge Detector - Step 3 Hysteresis Thresholding. Magnitude NonMaimum Supression Edge Piel High Threshold Possible Edge Piel Low Threshold Non-Edge Piel Include possible edge piels that are adjacent to edge piels.

10 Canny Edge Detector - Step 3 Magnitude (NonMa Suppressed) Effect of Scale Effect of σ (Gaussian kernel size) original Low Threshold High Threshold Canny with σ = 1 Canny with σ = Hysteresis Threshold Scale Space Witkin 83 Edge Completion first derivative peaks Gaussian filtered signal Properties of Scale Space: Position of edges may change with scale. Edges may merge with increase in scale. Edges do NOT split with increase in scale.

11 Edge Linking (,y) is an edge piel. Search for neighboring edge piels that are "similar". Eamples of edge linking Sobel Vertical Similarity: Similarity in Edge Orientation Similarity in Edge strength (Gradient Amplitude) Perform Edge Following along similar edge piels. (as in Contour Following in binary images). Sobel Horizontal Linked Edges Edge Points linked: Gradient Value > 5 Gradient direction within 15% Problem: Edges are not lines even when linked. Image Domain Hough Domain Edge piels are not ellipses even when linked. y s straight line s (, s) y = a+b s = cos()+ysin()

12 Image Domain Hough Domain y = a+b y s straight line s (, s) y many points on a line = many lines in the Hough transform space which intersect at 1 point. s y s single point = many possible lines Eample Eample Edges square image s (s,) space Results1 Results Results3 Reconstructed line segments

13 for Circles Eample Image Domain Hough Domain y ( 0,y 0 ) r y 0 r ( 0,y 0,r) circle 0 Edges r = (- 0 ) +(y-y 0 ) Result Nice demo: 3D Perception - Depth Perception Impossible Figures (Escher)