Estimation of Parameters

We will consider the estimation of the parameters of two-dimensional, causal AR models. For the case of Gaussian signals, we can use the well known Yule-Walker equations that uses the autocorrelation function of the signal. However, we wish to match the bispectrum of the signal to that of the AR model and hence we can not use the Yule-Walker equations directly.

In this case the matrix equation to be solved can be derived form the Brillinger-Rosenblatt summation formula [5,8]. The resulting algorithm is known as the Q-slice algorithm in the one-dimensional case. We have extended this to the two-dimensional case. We need to solve for the
AR parameters denoted by a ( i )  from the following equation

Here N refers to the neighbour set ( in our example N = {(-1,0), (0,-1), (-1,-1)} and we have 3 parameters). c3x is the third order cumulant of x and  the equation is valid for any  t1 that belongs to N and  t2   for all t2 = (t2(1), t2(2)) such that t2(1) >=0,  t2(2) >=0  and t2 is not equal to (0,0).
 

Results of Parameter Estimation

We used an AR model with the following parameters to generate a texture of size 64 x 64. A Rayleigh input noise distribution was used.The resulting texture is shown below.


 

        a(-1, -1)         -0.9686
        a(-1,  0)           0.9704
        a(0,  -1)           0.9735

 


 


Using this generated texture, we estimated the AR parameters using the algorithm explained above. The extracted parameters are given below.


 
  
        a(-1, -1)     -0.9946
        a(-1,  0)      0.9746
        a(0,  -1)      0.9880
It is clear that the estimates are very close to the actual parameters. Also, a new texture synthesized using the extracted parameters is given below. As can be seen, the two  textures seem to have similar properties.


 

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last updated on 5th May 2000.