Polar Fourier Transformation and
Invertible Orientation Bundles

Processing Directional Features in Images

Markus A. van Almsick, Remco Duits and Bart M. ter Haar Romeny

Technische Universiteit Eindhoven

Abstract

Local image features are subject to scaling and rotation. Due to their fixed locality they are not subject to translation. A Fourier transformation in Cartesian coordinates expands a function in \(e\^\(i\ ω\&⇀\ x\&⇀\)\), which are eigenfunctions of translation. In polar coordinates, however, a Fourier transformation expands a function into  (e^(i m φ)) (Subscript[f, mn])[r], a basis that is an eigenbasis of rotation and thus much more appropriate for the study of local image features. We therefore implemented the Fourier transformation for polar coordinates in our add-on MathVisionTools and utilize it to examine directional properties in biomedical images.

One obtains the theory of orientation bundles by simply substituting the dilation group in wavelet theory by the SO(2) rotation group. Instead of applying wavelets at different scale, one looks at filters with different orientations. Orientation bundles have proven to be very practical when manipulating directional features in images.