10.3 Thresholding Foundation Athresholdedimageg(x,y)isdefinedas

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1 Image segmentation( to ) SLIDE 1/ Thresholding Foundation Athresholdedimageg(x,y)isdefinedas g(x,y)= { 1, iff(x,y)>t 0, iff(x,y)<t, where1isobjectand0isbackground Global thresholding: T is constant applicable over whole image Variable(local/regional) thresholding: T changes over an image Dynamic(adaptive) thresholding: T depends on spatial coordinates(x, y) Multiple thresholding: g(x,y)= a, iff(x,y)>t 2 b, ift 1 <f(x,y)<t 2, c, iff(x,y)<t 1 Segmentation requiring more than two thresholds is very difficult and variable thresholding(10.3.7) or region growing(10.4) is often preferred

2 Image segmentation( to ) SLIDE 2/17 Width and depth of valleys(in histogram) affect success of thresholding Properties of valleys are affected by: (1) separation between peaks; (2) noise content; (3) relative sizes of objects and background; (4) uniformity of illumination source; (5) uniformity of reflectance properties of image

3 Image segmentation( to ) SLIDE 3/17 The role of noise in image thresholding

4 Image segmentation( to ) SLIDE 4/17 The role of illumination and reflectance Options for correcting non-uniform illumination: (1) Multiply with inverse of pattern by imaging flat surface with constant intensity;(2) Processing using top-hat transformation(sec 9.6.3);(3) Variable thresholding(sec )

5 Image segmentation( to ) SLIDE 5/ Basic global thresholding Iterative algorithm for automatic estimation of threshold T: (1)SelectaninitialestimateforT (2)SegmentimageusingT GroupG 1 (values>t) GroupG 2 (values<t) (3)ComputeaverageintensityvaluesforG 1,G 2 m 1,m 2 (4)ComputeanewthresholdvalueT = 1 2 (m 1+m 2 ) (5) Repeat(2) through(4) until the difference in T in successive iterations issmallerthan T Average intensity is good initial estimate for T

6 Image segmentation( to ) SLIDE 6/17 Example 10.15: Global thresholding Startwithaveragegrayleveland T =0 Algorithmresultsin T =125.4after3iterations,soletT =125

7 Image segmentation( to ) SLIDE 7/ Optimal global thresholding using Otsu s method Otsu s method(1979) maximizes between-class variance Based entirely on computations performed on histogram(1-d) of image Normalizedhistogram: p i = n L 1 i MN,,...,L 1,with p i =1, p i >0 SelectthresholdT(k)tosegmentimage ClassC 1 (values[0,k]) ClassC 2 (values[k+1,l 1]) ProbofpixelassignedtoC 1 (ieofc 1 occuring): P 1 (k)= k p i ProbofpixelassignedtoC 2 (ieofc 2 occuring): P 2 (k)= L 1 i=k+1 p i =1 P 1 (k)

8 Image segmentation( to ) SLIDE 8/17 MeanvalueofpixelsassignedtoC 1 : m 1 (k) = = k k ip(i/c 1 ) i = 1 P 1 (k) { =1 }} { P(C 1 /i) P(i) / }{{} =p i k ip i =P 1 (k) {}}{ P(C 1 ) (Bayes formula) MeanvalueofpixelsassignedtoC 2 : m 2 (k) = L 1 i=k+1 = 1 P 2 (k) ip(i/c 2 ) L 1 i=k+1 ip i

9 Image segmentation( to ) SLIDE 9/17 Meanintensityuptolevelk: m(k)= Globalmean: m G = L 1 ip i k ip i P 1 m 1 +P 2 m 2 =m G and P 1 +P 2 =1 (kstemporarilyomitted) Goodness of threshold at level k evaluated by dimensionless metric: η= σ2 B σ 2 G σ 2 G= L 1 (i m G ) 2 p i (Globalvariance) σ 2 B=P 1 (m 1 m G ) 2 +P 2 (m 2 m G ) 2 (Between-classvariance) Also: σ 2 B=P 1 P 2 (m 1 m 2 ) 2 = (m GP 1 m) 2 P 1 (1 P 1 ) most efficient

10 Image segmentation( to ) SLIDE 10/17 Reintroduce k ; final results: η(k)= σ2 B (k) σ 2 G σ 2 B(k)= [m GP 1 (k) m(k)] 2 P 1 (k)[1 P 1 (k)] Optimumthresholdisk thatmaximizesσ 2 B (k): Segmentation is as follows: σ 2 B(k )= max k [0,L 1] σ2 B(k) g(x,y)= { 1, iff(x,y)>k 0, iff(x,y)<k, The metricη(k )canbe usedtoobtainaquantativeestimateoftheseparabilityoftheclassesandhasvaluesintherange: η(k ) [0,1]

11 Image segmentation( to ) SLIDE 11/17 Summary of Otsu s algorithm (1)Computenormalizedhistogramoftheimage,p i = n i MN,,...,L 1 (2)Computecumulativesums,P 1 (k)= (3) Compute cumulative means, m(k) = (4)Computeglobalintensitymean,m G = k p i, k=0,...,l 1 k L 1 ip i, k=0,...,l 1 (5)Computebetween-classvariance,σB 2(k)=[m GP 1 (k) m(k)] 2, k=0,.,l 1 P 1 (k)[1 P 1 (k)] (6)ObtaintheOtsuthreshold, k, thatis thevalueofkforwhichσ 2 B (k )is amaximum ifthismaximumisnotunique,obtaink byavaragingthe values of k that correspond to the various maxima detected ip i (7)Obtaintheseparabilitymeasureη(k )= σ2 B (k ) σ 2 G

12 Image segmentation( to ) SLIDE 12/17 Example 10.16: Optimal global thresholding using Otsu s method For the above image... Threshold found by basic algorithm: T = 161; ThresholdfoundbyOtsu salgorithm: T =181(Sepmeasure: η=0.467) Forfingerprintimage... basicandotsu salgorithm: T =125(η=0.944)

13 Image segmentation( to ) SLIDE 13/ Using image smoothing to improve global thresholding

14 Image segmentation( to ) SLIDE 14/17 Small object thresholding fails even after smoothing

15 Image segmentation( to ) SLIDE 15/ Using edges to improve global thresholding Strategy to obtain a histogram of which the peaks are tall, narrow, symmetric, and separated by deep valleys: Consideronlythosepixelsthatlieonorneartheedgesbetweenobjectsand the background Algorithm (1)Computeanedgeimageaseitherthemagnitudeofthegradient,orthe absolutevalueofthelaplacian,off(x,y) (2) Specify a threshold value, T (3)Threshold the image from step (1) using the threshold from step (2) to produce a binary image, g T (x,y), which is used as a mask image in the following step to select pixels from f(x, y) corresponding to strong edge pixels (4)Computeahistogramusingonlythepixelsinf(x,y)thatcorrespondto thelocationsofthe1-valuedpixelsing T (x,y) (5)Use the histogram from step (4) to segment f(x,y) globally using, for example, Otsu s method

16 Image segmentation( to ) SLIDE 16/17 Example 10.17

17 Image segmentation( to ) SLIDE 17/17 Example 10.18

A Fast Algorithm for Multilevel Thresholding

JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 17, 713-727 (2001) A Fast Algorithm for Multilevel Thresholding PING-SUNG LIAO, TSE-SHENG CHEN * AND PAU-CHOO CHUNG + Department of Electrical Engineering