# Multichannel Data Ordering Schemes

### color vector marginal reduced

**Definition:** Multichannel data ordering schemes are used for color image filtering and enhancements in order to produce better and more efficient results.

Probably the most popular family of nonlinear filters is the one based on the concept of robust order-statistics. In univariate (scalar) data analysis, it is sufficient to detect any outliers in the data in terms of their extremeness relative to an assumed basic model and then employ a robust accommodation method of inference. For multivariate data however, an additional step in the process is required, namely the adoption of the appropriate sub-ordering principle, such as marginal, conditional, partial, and reduced ordering as the basis for expressing extremeness of observations.

Using marginal ordering the vector’s components are ordered along each dimension independently. Since the marginal ordering approach often produces output vectors which differ from the set of vectorial inputs, application of marginal ordering to natural color images often results in color artifacts. In conditional ordering the vector samples are ordered based on the marginal ordering of one component. Similarly to marginal ordering, conditional ordering fails to take into consideration the true vectorial nature of the color input. Using partial ordering the samples are partitioned into smaller groups which are then ordered. Since partial ordering is difficult to perform in more than two dimensions, it is not appropriate for three-component signals such as RGB color images. Finally, in reduced (or aggregated) ordering each vector is reduced to a scalar representative and then the vectorial inputs are ordered in coincidence with the ranked scalars. The reduced ordering scheme is the most attractive and widely used in color image processing since it relies an overall ranking of the original set of input samples and the output is selected from the same set.

To order the color vectors **x** 1 , **x** 2 ,**x** *N* , located inside the supporting window, the reduced ordering based vector filters use the aggregated distances or the aggregated similarities

associated with the vectorial input **x** *i* , for *i* = 1,2,*N* . By ordering the scalar values *D* 1 , *D* 2 ,*D N* , the ordered sequence of scalar values *D* (1) = *D* (2) == *D* ( *i* ) == *D* ( *N* ) , for *D* ( *i* ) { *D* 1 , *D* 2 ,*D N* } and *i* = 1,2,*N* , implies the same ordering of the corresponding vectors **x** *i* as follows: **x** (1) = **x** (2) = = **x** ( *i* ) == **x** ( *N* ).

Many noise removal techniques utilize the lowest ranked sample **x** ( *1* ) , referred to lowest vector order-statistics, as the output. This selection is due to the fact that: i) the lowest ranked vector **x** (1) which is associated with the minimum aggregated distances *D* (1) is the most typical sample for the vectorial set **x** 1 , **x** 2 ,**x** *N* , and that ii) vectors that diverge greatly from the data population usually appear in higher ranks of the ordered sequence. Therefore, the lowest ranked vector **x** (1) can also be used in vector color image zoomers to minimize the processing error and produce sharply looking images. On the other hand, vector edge detectors often utilize the lowest and the uppermost ranked vectors to determine edge discontinuities in color images.

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