# Image Processing via Scores on Lie Groups

We transform each image into a higher dimensional ’orientation score’, which attributes a complete distribution of orientations to each position in the image. On such a score new classes of analysis techniques can be performed before the stable inverse transformation is applied. The orientation score can be regarded as just a single entity in a larger Lie group theoretical framework, e.g. when dealing with multi-scale and multi-frequency processing. Our method relies on the combination of 5 principles:

• the image data are coherently transformed to a score, which is a complex-valued function on a higher dimensional space G beyond position space,
• the image data will be stably reconstructed from the higher dimensional space to ensure we do not spoil data-evidence before processing takes place in G,
• only the coherent features in G are amplified by means of contextual enhancement via left-invariant evolutions (PDE’s) on the score,
• optimal curves are extracted in G via geometrical control theory based on the enhanced scores,
• processing of multiple features (crossing lines, crossing textures, occlusions) do not involve an ad-hoc classification of complex structures (such as crossings).

The space  $G = \rm I\!R^d$ $\rtimes$ $T$  is typically a Lie group which is a semi-direct product of $\rm I\!R^d$ with another Lie group $T$ associated to the local features of interest.

# Lie Analysis Package Release 1.0 Containing Lots of new Features

Using our Mathematica package LieAnalysis, it is easy to create Orientation Scores for the analysis of your 2D or 3D images. The package is currently actively in development, containing new features in each update.