Edge detection. Gradient-based edge operators

Transcription

1 Edge detection Gradient-based edge operators Prewitt Sobel Roberts Laplacian zero-crossings Canny edge detector Hough transform for detection of straight lines Circle Hough Transform Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 1

2 Gradient-based edge detection Idea (continous-space): local gradient magnitude indicates edge strength Digital image: use finite differences to approximate derivatives Edge templates Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection

3 Practical edge detectors Edges can have any orientation Typical edge detection scheme uses K= edge templates Some use K> e 1 x, y s[ x, y] e x, y Combination, k e k e.g., x, y or edge magnitude M x, y Max k e k x, y e K x, y α x, y (edge orientation) Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 3

4 Gradient filters (K=) Central Difference [ 0] [ 0] Roberts [ 0] [ 1] Prewitt [ 0] [ ] Sobel [ 0] [ ] Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 4

5 Kirsch operator (K=8) Kirsch [ 0] [ 0] [ 0] [ 0] [ 0] [ 0] [ 0] [ 0] Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 5

6 Prewitt operator example Magnitude of image filtered with [ 0] (log display) Original 104x710 Magnitude of image filtered with [ ] (log display) Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 6

7 Prewitt operator example (cont.) Sum of squared horizontal and vertical gradients (log display) threshold = 900 threshold = 4500 threshold = 700 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 7

8 Sobel operator example Sum of squared horizontal and vertical gradients (log display) threshold = 1600 threshold = 8000 threshold = 1800 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 8

9 Roberts operator example Magnitude of image filtered with Original 104x710 Magnitude of image filtered with Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 9

10 Roberts operator example (cont.) Sum of squared diagonal gradients (log display) threshold = 100 threshold = 500 threshold = 800 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 10

11 Edge orientation Central Difference [ 0] [ 0] y-component Gradient scatter plot x-component Roberts [ 0] [ 1] y-component x-component Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 11

12 Edge orientation Prewitt [ 0] [ ] y-component Gradient scatter plot x-component Sobel [ 0] [ ] y-component x-component Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 1

13 Edge orientation Gradient scatter plot 5x5 consistent gradient operator [Ando, 000] y-component x-component Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 13

14 Gradient consistency problem Calculate this value using gradient field Same result, regardless of path? Known gray value Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 14

15 Laplacian operator Detect edges by considering second derivative Isotropic (rotationally invariant) operator Zero-crossings mark edge location Detect zero crossing Edge profile f(x) f (x) f (x) Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 16

16 Approximations of Laplacian operator by 3x3 filter H 6 4 H ω y ω x ω y ω x Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 17

17 Zero crossings of Laplacian Sensitive to very fine detail and noise blur image first Responds equally to strong and weak edges suppress zero-crossings with low gradient magnitude Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 18

18 Laplacian of Gaussian Filtering of image with Gaussian and Laplacian operators can be combined into convolution with Laplacian of Gaussian (LoG) operator σ = LoG( x, y)= 1 1 x + y πσ 4 σ e x + y σ Y X 0 4 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 19

19 Discrete approximation of Laplacian of Gaussian σ = Y X 0 4 H ω y - - ω x 0 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 0

20 Zero crossings of LoG σ = σ = σ = 4 σ = 8 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 1

21 Zero crossings of LoG gradient-based threshold σ = σ = σ = 4 σ = 8 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection

22 Canny edge detector 1. Smooth image with a Gaussian filter. Approximate gradient magnitude and angle (use Sobel, Prewitt...) M x, y f x + f y f α x, y tan 1 y f x 3. Apply nonmaxima suppression to gradient magnitude 4. Double thresholding to detect strong and weak edge pixels 5. Reject weak edge pixels not connected with strong edge pixels [Canny, IEEE Trans. PAMI, 1986] Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 3

23 Canny nonmaxima suppression Quantize edge normal to one of four directions: horizontal, -45 o, vertical, +45 o If M[x,y] is smaller than either of its neighbors in edge normal direction suppress; else keep. [Canny, IEEE Trans. PAMI, 1986] Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 4

24 Canny thresholding and suppression of weak edges Double-thresholding of gradient magnitude Typical setting: Region labeling of edge pixels Reject regions without strong edge pixels [Canny, IEEE Trans. PAMI, 1986] Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 5

25 Canny edge detector σ = σ = σ = 4 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 6

26 Hough transform Problem: fit a straight line (or curve) to a set of edge pixels Hough transform (196): generalized template matching technique Consider detection of straight lines y = mx + c x m edge pixel y c Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 8

27 Hough transform (cont.) Subdivide (m,c) plane into discrete bins, initialize all bin counts by 0 Draw a line in the parameter space [m,c] for each edge pixel [x,y] and increment bin counts along line. Detect peak(s) in [m,c] plane x m detect peak y c Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 9

28 Hough transform (cont.) Alternative parameterization avoids infinite-slope problem θ ρ edge pixel x ρ y xcosθ + ysinθ = ρ π π π θ Similar to Radon transform Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 30

29 Hough transform example Original image Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 31

30 Hough transform example Original image Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 3

31 Hough transform example Original image Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 33

32 Hough transform example ρ 1000 Paper De-skewed (364x448) Paper deg θ deg Global thresholding Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 34

33 Circle Hough Transform Find circles of fixed radius r x Circle center x 0 y edge pixel Equivalent to convolution (template matching) with a circle y 0 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 35

34 Circle Hough Transform for unknown radius 3-d Hough transform for parameters ( x 0, y 0,r) -d Hough transform aided by edge orientation spokes in parameter space x x 0 Circle center y edge pixel y 0 Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 36

35 Original coins image Example: circle detection by Hough transform Prewitt edge detection Detected circles Digital Image Processing: Bernd Girod, 013 Stanford University -- Edge Detection 37