Consider a pattern classification problem where we would like to discriminate be

Question

Consider a pattern classification problem where we would like to discriminate between two materials M_1 and M_2 using a measured color vector (R, G, B) of the unknown material. Assume that the a priori probability of M_1 is 3/7 and that the a priori probability of M_2 is 4/7. Suppose that the probability density for M_1 is uniform over the cube 50 lessthanorequalto R lessthanorequalto 90, 30 lessthanorequalto G lessthanorequalto 70, 40 lessthanorequalto B lessthanorequalto 80. Suppose that the probability density for M_2 is uniform over the cube 70 lessthanorequalto R lessthanorequalto 120, 30 lessthanorequalto G lessthanorequalto 80, 50 lessthanorequalto B lessthanorequalto 100. What is the conditional pdf p(R, G, B|M_1) as a function of (R, G, B)? What is the conditional pdf p(R, G, B|M_2) as a function of (R, G, B)? What is the a posteriori probability P(M_1|R, G, B) as a function of (R, G, B)? What is the a posteriori probability P(M_2|R, G, B) as a function of (R, G, B)? What is the best guess for what material we are looking at as a function of (R, G, B)? What is the probability of error as a function of (R, G, B) if we take the guess in part e)?

Explanation / Answer

a) P (RGB / M1) = i.e probabiltiy of observing the red green blue indices given M1 material is chosen

= P (RGB intersectn M1)/ P (M1)

= (R-50)/(90-50) * (G-30)/(70-30) * (B-40)/(80-40) / 3 /7 = 1/403 (R-50) (G-30)(B-40) * 7/3

b)

P (RGB / M2) = i.e probabiltiy of observing the red green blue indices given M2 material is chosen

= P (RGB intersectn M2)/ P (M2)

= (R-70)/(120-70) * (G-30)/(80-30) * (B-50)/(100-50) / 4 /7 = 1/503 (R-70) (G-30)(B-50) * 7/4

c) P (M1/ RGB) = P (RGB/M1) * P (M1) / { P (RGB/M1) * P (M1) + P (RGB/M2) * P (M2) }

= 1/403 (R-50) (G-30)(B-40) / { 1/403 (R-50) (G-30)(B-40) + 1/503 (R-70) (G-30)(B-50) }

d) similarly P (M2/RGB) = 1/503 (R-70) (G-30)(B-50) / { 1/403 (R-50) (G-30)(B-40) + 1/503 (R-70) (G-30)(B-50) }

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