Question

(10 pts) Consider optimal global thresholding using Otsu's method to segment a digital image into two classes Cị and C2. Otsu's method is optimum in the sense that it maximizes the between-class variance oß = P1(m1 – mg)2 + P2(m2 – mg)?, where Pi (i = 1, 2) is the probability that a pixel is assigned to class Cị, mi is the mean intensity value of the pixels assigned to class Cị, and mg is the average intensity of the entire image. Show that the between-class variance can be also written as oß = P1 P2 (mų – m2)2. You may use the fact that P1 + P2 = 1 and mg = Pımı + P2m2.

Answer 1

The following is the hand written proof. If you have any queries,let me know. And please upvote the solution if you find it helpful. Thank you.  

Given - J = P₂ (m, - MG + P₂ (m ₂ - MG) (2 P + P2 mg = P mat fy my T 13 eq ☺ 2 2 on B Take Pq cm, - m² + P2 [me-mo P ( m ² + m2 - 2mg met P2 (m ² + MG-2m₂ mg) = p m ₂ ² - 2 (P, m, tP, ma mo + (P + P2 m + P2 m² = Pm, ² - 2mg + mg + Puma using = P, m? m? It Pm (2) & 13