Dealiasing prediction filters with a neural net

Dealiasing the prediction filter $P(\omega,s)$ is a process of identifying, for each frequency, L of its zeros that are also zeros of the prediction filter $\hat{P}(\omega,s)$. Let us call these L zeros of $P(\omega,s)$``genuine'' zeros and the rest (M-1)L zeros ``fake'' zeros. Comparing equations (13) with (11) shows that, for genuine zeros, $\phi_{m}$ in equation (13) vanishes; while for fake zeros, $\phi_{m}$ does not vanish. Hence, if we project all zeros of $P(\omega,s)$ onto the $\omega$-k_x plane, as shown in Figure , the location of each genuine zero as the function of $\omega$ follows a linear trajectory which, or the extension of which, passes through the origin. The location of each fake zero also follows a linear trajectory, but neither this trajectory nor its extension passes the origin. Equation (13) also shows we know that each genuine zero has (M-1) fake zeros associated with it. We can predict where the associated fake zeros are if we know the position of the genuine zero. From above observations, it is apparent that dealiasing the prediction filter is equivalent to a pattern recognition process that retains the genuine zeros and rejects fake zeros. Our method uses a neural net to accomplish this goal.

 

zind
Figure 1 The principle of dealiasing the prediction filter. Each black dot indicates the projection of a zero of the prediction filter onto the $\omega$-k plane: (a) before dealiasing; (b) after dealiasing.

To design a neural net for the problem, we must first find all zeros of the prediction filter $P(\omega,s)$ by scanning the spectrum of the filter over the s plane. Then we project these zeros onto the discretized $\omega$-k_x plane. We define the input zero-indicator array as follows:

$$v_{ij} = \left\{ \begin{array} {ll} 1 & \hbox{if a zero at $(\omega_i,k_j)$} \\ \\ 0 & \hbox{otherwise}\end{array}\right.$$ (14)

and, similarly, the output zero-indicator array is defined as follows:

$$u_{ij} = \left\{ \begin{array} {ll} 1 & \hbox{if a ``genuine... ...$(\omega_i,k_j)$} \\ \\ 0 & \hbox{otherwise}.\end{array}\right.$$ (15)

Now we can retain a genuine zero by setting its indicator to 1 and reject a fake zero by setting its indicator to . For each v_ij=1, we define two regions, the exhibition region C_ij and the inhibition region D_ij. The exhibition region contains the indicators that are on the linear trajectory passing through both position (i,j) and the origin, and that are in the neighborhood of position (i,j). The inhibition region contains the indicators that indicate the associated fake zeros. Next, we compute

$$U_{ij}={1 \over \vert C_{ij}\vert}\sum_{kl \in C_{ij}} v_{kl},$$ (16)

where |C_ij| is the number of indicators in C_ij. Then we use the winner-take-all (WTA) algorithm to determine the value of u_ij, as follows:

$$u_{ij}=\left\{ \begin{array} {ll} 1 & \hbox{if} \ U_{ij} \gt... ...and} \ v_{ij} = 1 \\ \\ 0 & \hbox{otherwise}.\end{array}\right.$$ (17)

This algorithm is applied iteratively, as shown in Figure . Figure  displays an example of dealiasing the zeros of the prediction filters computed from field data. In spite of a few mistakes that can be identified by the human eye, the result is satisfactory. Most of the genuine zeros are retained and the fake zeros removed. Noted that the convergence of the iterative WTA algorithm is not guaranteed. However, the algorithm will never diverge because of the discrete and finite nature of the problem. Generally speaking, after several iterations, the output of the algorithm will oscillate among a few solutions that are all close to the globally optimal solution.

 

neunet2
Figure 2 An illustration of the iterative winner-take-all (WTA) algorithm.

 

cwlneu
Figure 3 An example of dealiasing the prediction filter with a neural net: (a) input zero indicators, (b) output zero indicators.