Color Image Processing
This research is using quaternion (hypercomplex) algebra to study the definitions and properties of linear vector methods for processing color images.
The pixels of a color image can be thought of as vector quantities. For example, a typical encoding of a color image consists of the red, green and blue (RGB) color components, therefore each pixel can be identified as a 3 dimensional vector. Hence an entire image is an array of vectors. Naturally one would expect design engineers to treat the image as a vector field. But in practice they are instead treated as three separate red, green and blue images. This allows the reuse of grayscale image filters in a repeated fashion on the three color component images with the filtered results recombined to create a filtered color image.
This, however, does not do justice to the fact that the information in one color component is usually correlated to the information in the other components. To account for, and exploit, this correlation all three color components should be treated in a holistic fashion. This holistic, or unified, approach to color image processing is the focus of our research.
The common tools of signal processing, such as convolution, correlation, transfer functions and their equivalent spectral transforms have all been extended into multi-dimensional data--but only in an iterated, component-wise, fashion. Very few tools, which treat the multi-dimensional data itself as an entity, have been developed in this area. This leads to the larger issue of developing a unifying mathematical framework within which to design these new tools.
This is where quaternion numbers enter the picture. At the core of most signal processing algorithms one finds the extensive use of complex functions and operators. Complex numbers, being two dimensional in nature, cannot be used in processing signals of more dimensions without either iterating over a set of complex variables, or folding one dimension into another. Hypercomplex numbers are, however, of higher dimensions. In particular, quaternions are a four dimensional extension to the standard complex number. This means it should be well suited for 3-dimensional color images. One should in theory be able to define usable hyper-complex spectral transformations. Discovering these transformations and defining suitable extensions to standard convolution, correlation and transfer functions is the focus of our research.
To date, this work has defined hypercomplex Fourier transforms, correlation and convolution. Using these definitions several linear vector filters have been developed, including filters which perform color-dependent averaging and color-sensitive edge detection.
My publications page contains the references to work in this area published to date.