Nonlinear Techniques for Color Image Processing - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 460

2 and the orthogonal eigenvectors determine the
corresponding variation directions

are the extremum of d F

η and ξ

1
2 arctan

2 g 12
g 11

ξ = η + 2 .

η = , (12.30)

− g 22

Based on the eigenvalues, different gradient norms leading to various PDE
schemes can be developed. 55 , 65 , 68 , 79 - 81

12.4 Noise Reduction Filters for Color Image Processing

Several nonlinear techniques for color image processing have been proposed
over the years. Among them are linear processing methods, whose mathemat-
ical simplicity and the existence of a unifying theory make their design and
implementation easy. However, not all filtering problems can be efficiently
solved using linear techniques. For example, conventional linear techniques
cannot cope with nonlinearities of the image formation model and fail to
preserve edges and image details.
To this end, nonlinear color image processing techniques are introduced.
Nonlinear techniques, to some extent, are able to suppress non-Gaussian noise
and preserve important image elements, such as edges, corners, and fine
details, and eliminate degradations occurring during image formation and
transmission through noisy channels.

12.4.1 Order-Statistics Filters

One of the most popular families of nonlinear filters for impulsive noise re-
moval are order-statistics filters. 4 , 82-86 , 136 These filters utilize algebraic order-
ing of a windowed set of data to compute the output signal.
The early approaches to color image processing usually comprised exten-
sions of the scalar filters to color images. Ordering of scalar data, such as the
values of pixels in gray-scale images, is well defined and has been extensively
studied. 4 However, the concept of input ordering, initially applied to scalar
quantities, is not easily extended to multichannel data, as there is no univer-
sal way to define ordering in vector spaces. A number of different ways to
order multivariate data have been proposed. These techniques are generally
classified into the following: 86 - 89

•

Marginal ordering (M-ordering), where the multivariate samples are
ordered independently along each dimension

•

Reduced or aggregated ordering (R-ordering), where each multivariate
observation is reduced to a scalar value according to a distance
metric