To conclude, we demonstrate an application for 2-D non-separable recursive filters. This is the method of *linear predictive coding*. Let us consider a 2-D signal *x*(*u*, ν) with a bounded support and centered, for example, a gray level image with *M* rows and *N* columns (we show such an image below), such that the summation below, which has a finite number of non-null terms, is null:

If the above conditions do not apply, we must subtract the constant μ/(*MN*) from each sample of *x* in order to have a centered signal.

It is important to reduce the redundancy of information between neighboring pixels. We can do this with a linear filtering operation: we look for a causal FIR filter, of transfer function:

where the order *m* and *n* are arbitrarily bounded, which, affected by the signal *x*(*u*, ν), give as an output signal *y*(*u*, ν), of minimal energy:

The filter being of finite impulse response and the input signal being of bounded support, the non-null terms are of finite number in the sum of equation (11.43).

To calculate this optimum filter, we can proceed in the following way. We write A = [*a _{h,k}*] (0 ≤

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