Nonlinear structure-enhancing filtering using plane-wave prediction |

In this appendix, we review lower-upper-middle (LUM)
filters introduced by Hardie and Boncelet (1993). Consider a window function
containing a set of samples centered about the sample . We
assume to be odd. This set of observations will be denoted by
. The rank-ordered set can be written as

The estimate of the center sample will be denoted .

**Lower-upper-middle smoother**-

Lower-upper-middle (LUM) smoother is equivalent to center-weighted medians (Justusson, 1981). The output of the LUM smoother with parameter is given by

where .Thus, the output of the lower-upper-middle (LUM) smoother is if . If , then the output of the LUM smoother is . Otherwise the output of the LUM smoother is simply .

**Lower-upper-middle sharpener**-

We can define a value centered between the lower- and upper-order statistics, and . This midpoint or average, denoted , is given by

Then, the output of the lower-upper-middle (LUM) sharpener with parameter is given by

Thus, if , then is shifted outward to or according to which is closest to . Otherwise the sample is unmodified. By changing the parameter , various levels of sharpening can be achieved. In the case where , no sharpening occurs and the lower-upper-middle (LUM) sharpener is simply an identity filter. In the case where , a maximum amount of sharpening is achieved since is being shifted to one of the extreme-order statistics or .

**Lower-upper-middle filter**-

To obtain an enhancing filter that is robust and can reject outliers, the philosophies of the lower-upper-middle (LUM) smoother and lower-upper-middle (LUM) sharpener must be combined. This leads us to the general lower-upper-middle (LUM) filter. A direct definition is as follows:

where is the midpoint between and defined in equation B-3, and .

Nonlinear structure-enhancing filtering using plane-wave prediction |

2013-07-26