4. [15 points] One method to test whether two adjacent regions should be merged together is by computing the likelihood ratio L defined as L=p(g1,g2,H0)p(g1,g2,H1) where H0 and H1 are the hypotheses - H0 : Both regions belong to the same object. Intensities are all drawn from a single Gaussian distributions with parameters (0,0) - H1 : Regions belong to different objects. Intensities of each region are drawn from separate Gaussian distributions with parameters (1,1) and (2,2). Starting with the equation of a normal distribution p(gi)=21e22(gi)2 where gi is the gray level value of pixel i, show that the joint probability density under hypothesis H1 is p(g1,g2,,gm1,gm1+1,.,gm1+m2H1)=(21)m11e2m1(22)m21e2m2 where m1 and m2 are the number of pixels in the two adjacent regions under hypothesis H1