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Polar Fourier Transformation and Invertible Orientation Bundles: Processing Directional Features in Images

Markus van Almsick
Organization: Technische Universiteit Eindhoven
Department: Department of Biomedical Engineering
URL: http://www.wolfram.com/services/training/almsick.html
Remco Duits
Bart M. ter Haar Romeny
Organization: Eindhoven University of Technology
Department: Department of Biomedical Engineering

2004 Wolfram Technology Conference
Conference location

Champaign IL

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*OverVector[ω]*OverVector[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.

*Applied Mathematics > Visualization
*Mathematics > Calculus and Analysis > Special Functions

Image rotation
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MarkusBartOrientation.nb (366.2 KB) - Abstract of talk [for Mathematica 5.0]