To extend Otsu’s thresholding method to multi-level thresholding the between class variance equation becomes:
Please check out Deng-Yuan Huang, Ta-Wei Lin, Wu-Chih Hu, Automatic
Multilevel Thresholding Based on Two-Stage Otsu’s Method with Cluster
Determination by Valley Estimation, Int. Journal of Innovative
Computing, 2011, 7:5631-5644 for more information.
Here is my C# implementation of Otsu Multi for 2 thresholds:
/* Otsu (1979) - multi */
Tuple < int, int > otsuMulti(object sender, EventArgs e) {
//image histogram
int[] histogram = new int[256];
//total number of pixels
int N = 0;
//accumulate image histogram and total number of pixels
foreach(int intensity in image.Data) {
if (intensity != 0) {
histogram[intensity] += 1;
N++;
}
}
double W0K, W1K, W2K, M0, M1, M2, currVarB, optimalThresh1, optimalThresh2, maxBetweenVar, M0K, M1K, M2K, MT;
optimalThresh1 = 0;
optimalThresh2 = 0;
W0K = 0;
W1K = 0;
M0K = 0;
M1K = 0;
MT = 0;
maxBetweenVar = 0;
for (int k = 0; k <= 255; k++) {
MT += k * (histogram[k] / (double) N);
}
for (int t1 = 0; t1 <= 255; t1++) {
W0K += histogram[t1] / (double) N; //Pi
M0K += t1 * (histogram[t1] / (double) N); //i * Pi
M0 = M0K / W0K; //(i * Pi)/Pi
W1K = 0;
M1K = 0;
for (int t2 = t1 + 1; t2 <= 255; t2++) {
W1K += histogram[t2] / (double) N; //Pi
M1K += t2 * (histogram[t2] / (double) N); //i * Pi
M1 = M1K / W1K; //(i * Pi)/Pi
W2K = 1 - (W0K + W1K);
M2K = MT - (M0K + M1K);
if (W2K <= 0) break;
M2 = M2K / W2K;
currVarB = W0K * (M0 - MT) * (M0 - MT) + W1K * (M1 - MT) * (M1 - MT) + W2K * (M2 - MT) * (M2 - MT);
if (maxBetweenVar < currVarB) {
maxBetweenVar = currVarB;
optimalThresh1 = t1;
optimalThresh2 = t2;
}
}
}
return new Tuple(optimalThresh1, optimalThresh2);
}
And this is the result I got by thresholding an image scan of soil with the above code:
(T1 = 110, T2 = 147).
Otsu’s original paper: “Nobuyuki Otsu, A Threshold Selection Method
from Gray-Level Histogram, IEEE Transactions on Systems, Man, and
Cybernetics, 1979, 9:62-66” also briefly mentions the extension to
Multithresholding.https://engineering.purdue.edu/kak/computervision/ECE661.08/OTSU_paper.pdf
Hope this helps.