Image Segmentation and Image Segmentation and Edge DetectionEdge Detection

Digital Image ProcessingInstructor: Dr. Cheng-Chien Liu

Department of Earth Sciences

National Cheng Kung University

Last updated: 21 October 2003

Chapter 7Chapter 7

IntroductionIntroduction

Image Segmentation and Edge DetectionImage Segmentation and Edge Detection• Purpose extract information (outlines) division

(color, brightness) automatic vision system• The simplest method of division

Histogramming and thresholding One thresholdlabel (classified) imagee.g. Fig 7.1

• Hysteresis thresholding Two thresholdse.g. Fig 7.2Principle minimize the number of misclassified pixelsp-tile method

The minimum error threshold The minimum error threshold methodmethod

Total error (Fig 7.3)Total error (Fig 7.3)• E(t) = -

t p0(x)dx + (1 – t pb(x)dx

: the fraction of the pixels that make up the object1-: the fraction of the pixels that make up the background

• E/t = p0(t) – (1 – pb(t)Example 7.1: E(t) E/tB7.1: the Leibnitz ruleExample 7.2: draw p0(x) and pb(x)

Example 7.3: given p0(x), pb(x) and t

Example 7.4: given p0(x), pb(x) and t E(t)

The minimum error threshold The minimum error threshold method (cont.)method (cont.)

DrawbacksDrawbacks• Need the prior knowledge of p0(x), pb(x) and

Approximate p0(x) and pb(x) by normal distributions still need to estimate the parameters and

Two solutions of t t1 < x < t2 (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 methodOtsu’s threshold method

DerivationDerivation• Fraction

Background pixels: (t)Object pixels: 1 – (t)

• Mean gray valueThe whole image: Background: b

Object: o

• VarianceThe whole image: T

2

Background: b2

Object: o2

Otsu’s threshold method (cont.)Otsu’s threshold method (cont.)

Derivation (cont.)Derivation (cont.)• T

2 = W2 + B

2 The within-class variance:

W2 = (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.)Otsu’s threshold method (cont.)

DrawbacksDrawbacks• Assume and are sufficient in representing

p0(x) and pb(x)

• Break down when p0(x) and pb(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 illuminationVariable illumination

ppzz((uu) = ) = pprr((uu – – ii))ppii((ii))ddii

• 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 = Pz(u) = probability of z u P(z u)

= r

u-i pri(r, i)drdi• pz(u) = dPz(u)/du =

pri(u-i, i)di = pr(u – i)pi(i)di

• If i = const i = const pi(i) = (i – io) pz(u) = pr(u)• If i const the thresholding methods break down

Variable illumination (cont.)Variable illumination (cont.)

Solution for non-uniform illuminationSolution for non-uniform illumination• Divide the image into (more or less) uniformly

illuminated patches (Fig 7.8)

• Correcting the effect of illuminationPure 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 Shortcomings of the thresholding methodsmethods

The spatial proximity of the pixels in the The spatial proximity of the pixels in the image is not considered at allimage is not considered at all• Fig 7.8• Fig 7.9

SolutionsSolutions• Region growing method

Seed pixels attach neighboring pixels based on the predefined range scan and assign all pixels to a region

• Split and merge methodTest 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 = 2n

Pattern recognitionPattern recognition

Texture regionTexture region• Regions are not uniform in terms of their grey values

but are perceived as uniform

For segmentation purposesFor segmentation purposes• Characterize a pixel

Its GL and the variation of GL in a small patch around itNot just a scalar (GL), but a vector (feature)

Pattern recognitionPattern recognition• Multidimensional histograms clustering• Beyond the scope of this book

Edge detectionEdge detection

MeasurementMeasurement• 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 derivativefx = f(i+1, j) - f(i, j)fy = f(i, j+1) - f(i, j)The dual grid

• Non-maxima suppressionThe 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.)Edge detection (cont.)

NoiseNoise• Smooth the image with a lowpass filter before

detecting the edges (Fig 7.12, 7.13)

1D case1D case• Ai (Ii-1 + Ii + Ii+1) / 3• Fi (Ai+1 – Ai) + (Ai – Ai-1) / 2• Fi (Ii+2 + Ii+1 – Ii-1 – Ii-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.)Edge detection (cont.)

2D case (3 2D case (3 3 mask) 3 mask)• Consider fy only (rotating 900 to calculate fx)

• Symmetry: left right

• Local difference = front – behind

• Zero response for a smooth image aij = 0

• Differentiate in the direction of columns for a smooth image 0 for each column a21 = 0

aa1111 aa1212 aa1313

aa1111 aa1212 aa1111

aa1111 aa1212 aa1111

aa1111 aa1212 aa1111

aa1111 aa1212 aa1111

aa2121 aa2222 aa2323 aa2121 aa2222 aa2121 aa2121 aa2222 aa2121 aa2121 --22aa2121 aa2121 00 00 00

aa3131 aa3232 aa3333 aa3131 aa3232 aa3131 -a-a1111 -a-a1212 -a-a1111 -a-a1111 -a-a1212 -a-a1111 -a-a1111 -a-a1212 -a-a1111

Edge detection (cont.)Edge detection (cont.)

2D case (cont.2D case (cont.))• Divide by a11 one parameter mask

11 KK 11

00 00 00

--11 -K-K --11

Sobel maskSobel mask

Sobel maskSobel mask• Differentiating an image along two directions

Choose K = 2B7.2

Strength: E(i, j) = [fx2 + fy2]1/2

Orientation: a(i, j) = tan-1[fy/fx] 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 99 matrix to calculate the i-gradient of a 33 matrix

• Example 7.11:Implementation of Example 7.10