Codebook design and k-means algorithm:- 

Definition:- In vector quantisation we need to determine the reconstruction levels and corresponding cells C_i.  A list of reconstruction levels is called a "codebook" (CB). If there are L-reconstruction levels in the list it is referred to as an L-level codebook.

The CB is needed at the transmitter to quantise a source vector to one of the L-reconstruction levels and at the receiver to determine the reconstruction level from the received codeword. This implies that the same CB should be known both to transmitter and receiver.

·  Design of codebook:

Unlike scalar quantisation, reconstruction levels in VQ are vectors and cell boundaries are no longer points but lines.

The optimum determination of  and C_i depends on the error criterion used. The often used mean square error (MSE) criterion is expressed as average distortion:

with

is a distance/norm measure.

Optimal design of codebook is  highly non linear problem. Attempts to solve this needs the following two conditions.

Condition 1: For  vector  to be quantized to one of the reconstruction levels, the optimal quantiser must choose the for which has the smallest distortion between and .

This implies,  if and only if ,

	(3.4.4)

for

Condition 2: Each reconstruction level must minimize the average distortion D in C_i, that is, minimise wrt

The level that satisfies condition  2 is called the centroid of C_i.. If condition (1) is not satisfied, the average distortion D can be reduced by imposing condition (1). If condition (2) is not satisfied, then average distortion D can be reduced by imposing condition (2).The above conditions are necessary for optimal solution.

It is to be noted that condition (1) satisfies a rule for quantising without explicit use of C_i. In other words, the distortion measure together with all 's for , specifies all the cells C_i for  .Similarly, condition (2) shows a way to determine from C_i and

The two conditions show that given , and C_i are not independent.

That is, specifies C_i and C_i specifies .

Only reconstruction levels alone are sufficient in the codebook. Explicit information about cells does not have to be stored.