“ Clustering by Composition ” – Unsupervised Discovery of Image Categories Attached Material

Proof: Let R1 and R2 denote the instances of a region R in I1 and I2. In order to detect the entire region R, at least one descriptor d1 ∈ R1 has to randomly sample its correct match d2 ∈ R2 (following which the entire region will be ‘grown’ due to the propagation phase of the Region Growing Algorithm described in Section 4). So, the probability of detecting a region is equal to the probability that at least one of the descriptors d1 ∈ R1 will randomly sample its correct match d2 ∈ R2. The probability of a single descriptor d1 ∈ R1 to randomly fall on its correct match d2 ∈ R2 is 1 N (where N is the size of the image). Therefore, the probability that it will NOT fall on d2 is (1− 1 N ). The probability that NONE of its S samples will fall on d2 is (1− 1 N ) S . Therefore, the probability that NONE of the descriptors in R1 will randomly fall on their correct match is q , ( 1− 1 N )S|R1| = ( 1− 1 N )S|R| . Thus the probability of detecting the shared region R is p , (1− q).