h4 - 11 ECE 253a Digital Image Processing Pamela Cosman...

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# 11 ECE 253a Digital Image Processing Pamela Cosman 10/28/11 HOMEWORK 4 Due Friday November 4 in class 1. Scalar Quantization – optimality conditions for squared error distortion Suppose that a random variable X has the two-sided exponential pdf f X ( x ) = λ 2 e - λ | x | A three level quantizer q for X has the form q ( x ) = + b x > a 0 - a x + a - b x < - a (a) Find a simple expression for b as a function of a so that the centroid condition is met. (b) For what value of a will the quantizer (using b chosen as above) satisfy the nearest neighbor condition for optimality? 2. Lloyd Algorithm for Quantizer Design A 3-level quantizer is to be designed to minimize mean squared error using the Lloyd algorithm with the training set T = { 1 . 0 , 2 . 0 , 3 . 001 , 4 . 0 , 8 . 0 , 9 . 0 , 12 . 0 } In class, starting with the initial codebook { 1 . 0 , 5 . 0 , 9 . 0 } I found the algorithm con- verges at the codebook { 1 . 5 , 3 . 5 , 9 . 7 } . If we start instead with the initial codebook C 1 = { 2 . 0 , 6 . 0 , 10 . 0 } what does it converge to? You will find you need to enunciate a tie-breaking rule, that is, if a training point is

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