Multi-scale and multi-orientation medical image analysis

Inspired by multi-scale and multi-orientation mechanisms recognized in the first stages of our visual system, this chapter gives a tutorial overview of the basic principles. Images are discrete, measured data. The optimal aperture for an observation with as little artefacts as possible, is derived from first principles and leads to a Gaussian profile. The size of the aperture is a free parameter, the scale.Convolution with the derivative of the Gaussian to any order gives regularized derivatives, enabling a robust differential geometry approach to image analysis. Features, invariant to orthogonal coordinate transformations, are derived by the introduction of gauge coordinates. The multi-scale image stack (the deep structure) contains a hierarchy of the data and is exploited by edge focusing, retrieval by manipulations of the singularities (top-points) in this space, and multi-scale watershed segmentation. Expanding the notion of convolution to group-convolutions, rotations can be added, leading to orientation scores. These scores are exploited for line enhancement, denoising of crossing structures, and contextual operators. The richness of the extra dimensions leads to a data explosion, but all operations can be done in parallel, as our visual system does.