Multidimensional lattice filter

Lattice filters are implicitly solving the problem of finding the inverse of the autocorrelation matrix of the signal involved in the normal equations of linear prediction. Therefore, as PEFs they are carrying information of the signal spectra. Multidimensional filters are similar and probably even more important for the real applications.

Extending lattice filters to two dimensions is not a trivial task. There have been several successful attempts of solving this problem (e.g. (Parker and Kayran, 1984)). If in the one-dimensional case there are two fields involved (forward and backward prediction errors), in 2D there are even more fields that need to be updated (4 in case of a 2x2 quarter-plane filter). Naturally, the reflection coefficients now become vectors depending on the filter size. Finding optimal values for them in this case involves a matrix inversion at every lattice stage 23#23 and at every sample 35#35. However, it was shown in (Kayran, 1996) that the problem of finding multidimensional lattice filter may be solved using one-dimensional lattice approach.

The first step is to locally order the 2D signal into a 1D array (Figure 3). In this case, ordering corresponding to an assymetric plane was used. After this, the first iteration of the algorithm is to combine all the neighboring pairs within this 1D vector to give the first estimate of the forward and backward errors at all the points. The second iteration combines the pair of points jumping across one sample, the third - jumping across two samples, etc. We iterate until the jump is equal to the whole filter length. To make this filter adaptive, the reflection coefficients are updated by the same Equation 4.

q-plane a-plane
q-plane,a-plane
Figure 3.
Ordering of 2D signal into 1D array: (a)-quarter-plane model, (b)-assymetric plane. Evaluating the prediction error at the 0-th sample (shown by black circle) involves previous samples (shown in white circles). Numbers correspond to ordering of the samples involved in calculations. NR
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The result of applying a 2D lattice filter of size 2x3 on the stacked data is shown in Figure 4. Two-dimensional prediction error filters destroy the correlation in two dimensions, which is why the continous reflections are suppressed. However, because the dip of the events is changing in space and time, crossing events are harder to predict. By changing the forgetting parameter it is possibe to control the adaptiveness of the filter and significantly remove the correlated (in space and time) events. Making this parameter even smaller than shown in the figure removes most of the events, making the output white noise. Increasing the filter size might also help in predicting more slopes, because it will be capturing more spatial information.

WGstack0 WGstack-2-0.99 WGstack-2-0.92
WGstack0,WGstack-2-0.99,WGstack-2-0.92
Figure 4.
2D lattice filter applied to raw stack (with filter size 9#9, 2#2-forgetting parameter): (a) - original stack, (b) - 10#10, (c) - 11#11. ER
[pdf] [pdf] [pdf] [png] [png] [png]


2018-06-10