Image Segmentation and Edge Detection Digital Image Processing Instructor: Dr. Cheng-Chien LiuCheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 21 October 2003 Chapter 7

Introduction  Image Segmentation and Edge Detection Purpose  extract information (outlines)  division (color, brightness)  automatic vision system The simplest method of division  Histogramming and thresholding  One threshold  label (classified) image  e.g. Fig 7.1 Hysteresis thresholding  Two thresholds  e.g. Fig 7.2  Principle  minimize the number of misclassified pixels  p-tile method

The minimum error threshold method  Total error (Fig 7.3) E(t) =   -  t p 0 (x)dx + (1 –  t  p b (x)dx   : the fraction of the pixels that make up the object  1-  : the fraction of the pixels that make up the background  E/  t =  p 0 (t) – (1 –  p b (t)  Example 7.1: E(t)   E/  t  B7.1: the Leibnitz rule  Example 7.2: draw p 0 (x) and p b (x)  Example 7.3: given p 0 (x), p b (x) and   t  Example 7.4: given p 0 (x), p b (x) and   t  E(t)

The minimum error threshold method (cont.)  Drawbacks Need the prior knowledge of p 0 (x), p b (x) and   Approximate p 0 (x) and p b (x) by normal distributions  still need to estimate the parameters  and   Two solutions of t  t 1 < x < t 2 (Example 7.6)  Example 7.5: the result of optimal thresholding is worse than that obtained by hysteresis thresholding with two heuristically chosen thresholds (Fig 7.4d)

Otsu’s threshold method  Derivation Fraction  Background pixels:  (t)  Object pixels: 1 –  (t) Mean gray value  The whole image:   Background:  b  Object:  o Variance  The whole image:  T 2  Background:  b 2  Object:  o 2

Otsu’s threshold method (cont.)  Derivation (cont.)  T 2 =  W 2 +  B 2  The within-class variance:  W 2 =  (t)  b 2 + (1 –  (t))  o 2   he between-class variance:  B 2 = (  b –  ) 2  (t) + (  o –  ) 2 (1 –  (t)) Otsu’s thresholding:  Optimizing t to maximize  B and minimize  W  If work with  B (Example 7.7)   B  (t) –  (t)    (t)  (t) 

Otsu’s threshold method (cont.)  Drawbacks Assume  and  are sufficient in representing p 0 (x) and p b (x) Break down when p 0 (x) and p b (x) are very unequal Assume the histogram of the image is bimodal Dividing the image into two classes is not valid under variable illumination

Variable illumination  p z (u) =    p r (u – i)p i (i)di f(x, y) = r(x, y) i(x, y)  An image f(x, y) is a product of a reflectance function r(x, y) and an illumination function i(x, y) ln f(x, y) = ln r(x, y) + ln i(x, y)  Multiplicative  additive f(x, y) = r(x, y) + i(x, y) z = P z (u) = probability of z  u  P(z  u) =     r  u-i p ri (r, i)drdi p z (u) = dP z (u)/du =    p ri (u-i, i)di =    p r (u – i)p i (i)di If i = const  i = const  p i (i) =  (i – i o )  p z (u) = p r (u) If i  const  the thresholding methods break down

Variable illumination (cont.)  Solution for non-uniform illumination Divide the image into (more or less) uniformly illuminated patches (Fig 7.8) Correcting the effect of illumination  Pure illumination field  i(x, y)  Image of an uniform reflectance surface  f(x, y)  f(x, y) / i(x, y)  Subtract i(x, y) from z(x, y)  Multiply f(x, y) / i(x, y) with a reference value, say i(0, 0) to bring the whole image under the same illumination

Shortcomings of the thresholding methods  The spatial proximity of the pixels in the image is not considered at all Fig 7.8 Fig 7.9  Solutions Region growing method  Seed pixels  attach neighboring pixels based on the predefined range  scan and assign all pixels to a region Split and merge method  Test the original image  split into four quadrants if LV < attribute < HV  test for each quadrant  split  …  merge the region with the same attribute (Fig 7.10)  Favored when the image is square with N = 2 n

Pattern recognition  Texture region Regions are not uniform in terms of their grey values but are perceived as uniform  For segmentation purposes Characterize a pixel  Its GL and the variation of GL in a small patch around it  Not just a scalar (GL), but a vector (feature)  Pattern recognition Multidimensional histograms  clustering Beyond the scope of this book

Edge detection  Measurement Convolve the image with a window  Slide a window  calculate the statistical properties  compare the difference  specify the boundary  e.g. 8  8 image in Fig 7.11 The smallest window  two pixels  the first derivative   f x = f(i+1, j) - f(i, j)   f y = f(i, j+1) - f(i, j)  The dual grid Non-maxima suppression  The process of identifying the local maxima as candidate edge pixels (edgels)  If there is no noise in the image  pick up the discontinuities in intensity

Edge detection (cont.)  Noise Smooth the image with a lowpass filter before detecting the edges (Fig 7.12, 7.13)  1D case A i  (I i-1 + I i + I i+1 ) / 3 F i  (A i+1 – A i ) + (A i – A i-1 ) / 2 F i  (I i+2 + I i+1 – I i-1 – I i-2 ) / 6 The larger the mask used, the better is the smoothing, the more blurred and more inaccurate its position will be (Fig 7.14)

Edge detection (cont.)  2D case (3  3 mask) Consider  f y only (rotating 90 0 to calculate  f x ) Symmetry: left  right Local difference = front – behind Zero response for a smooth image   a ij = 0 Differentiate in the direction of columns for a smooth image  0 for each column  a 21 = 0 a 11 a 12 a 13  a 11 a 12 a 11  a 12 a 11  a 12 a 11  a 12 a 11 a 21 a 22 a 23 a 21 a 22 a 21 a 22 a a 21 a a 31 a 32 a 33 a 31 a 32 a 31 -a 11 -a 12 -a 11 -a 12 -a 11 -a 12 -a 11

Edge detection (cont.)  2D case (cont.) Divide by a 11  one parameter mask 1K K

Sobel mask  Sobel mask Differentiating an image along two directions  Choose K = 2  B7.2  Strength: E(i, j) = [  f x 2 +  f y 2 ] 1/2  Orientation: a(i, j) = tan -1 [  f y /  f x ]  Specify K  keep E and a to response the true values of the non-discretized image Example 7.9:  Expression of Sobel mask at (i, j) Example 7.10:  Constructing a 9  9 matrix to calculate the i-gradient of a 3  3 matrix Example 7.11:  Implementation of Example 7.10