Nonlinear structure-enhancing filtering using plane-wave prediction

Lower-upper-middle filter

The lower-upper-middle (LUM) filter is a nonlinear filter that is simple to define and yet effective for noise attenuation in non-stationary signal processing (Hardie and Boncelet, 1993). It has two parameters, one for smoothing and the other for sharpening. A general class of LUM filters includes LUM smoothers and LUM sharpeners as special cases (Appendix B).

By manipulating the parameters $k$ (for smoothing) and $l$ (for sharpening), the lower-upper-middle (LUM) filter takes on a variety of characteristics. We found that $k=l=(N-1)/2$ works well for our synthetic and field data examples. In the synthetic example, special smoothing and sharpening parameters, $k=l=7$, were chosen to balance the ability between noise attenuation and fault protection. After applying the LUM filter on the synthetic noisy image, we obtain the image shown in Figure 3d. Comparing with Figure 3b, the LUM filter displays the similar-quality result as Gaussian similarity-mean filter. However, the LUM filter is somewhat easier to control than the similarity-mean filter.

The standard median filter with filter-window length $15$ is compared to the lower-upper-middle (LUM) filter. After applying the median filter on the prediction direction of Figure 1d, the result is shown in Figure 3c. When comparing with the mean filter (Figure 3a), the median filter has a better fault-protection ability but weaker noise-attenuation result. However, it still makes edges of some faults ambiguous. The LUM filter uses smoothing and sharpening parameters to limit the smoothing characteristics of the standard median filter. Therefore, it strikes a reasonable balance between structure enhancement and fault protection. Figure 4c and 4d show the difference between the noisy image (Figure 1b) and structure-enhancing results with the standard median filter (Figure 3c) and the LUM filter (Figure 3d). While coherent events can be preserved by either of the two filters, the median filter has a better result of fault protection than the mean filter (Figure 4c) and the LUM filter can further reduce the fault damage of the standard median filter (Figure 4d).

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Figure 4. Difference between Figure 1b and structure-enhancing results (Figure 3). Standard mean filtering (a), similarity-mean filtering (b), standard median filtering (c), and lower-upper-middle (LUM) filtering (d).

The key steps of our method are illustrated schematically in Figure 5.

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Figure 5. Schematic illustration of the proposed workflow.

Nonlinear structure-enhancing filtering using plane-wave prediction