LIE ANALYSIS TUTORIAL

# Step By Step Gauge Frames SE(2)

In this tutorial we show how you can compute a data-adaptive (Gauge-)frame. First we show a step by step procedure for two different methods, then we show a short implementation that makes optimal use of the Lie Analysis package.

 OrientationScoreGaugeFrames[ObjPositionOrientationData, {σs,σo}, options] computes the data-adaptive frame from the ObjPositionOrientationData using the Gaussian scales σs and σo OrientationScoreTensor[ObjPositionOrientationData, {σs,σo},derivativeIndex] computes the Hessian or structure tensor from the ObjPositionOrientationData

Load package and some general functions and variables
 In:= In:= Out= In:= ## Step by Step Guide

### Method 1: Symmetric Product Hessian

Step 1: Compute orientation score  Step 2: Compute Hessian from orientation score  Step 3: Symmetrize the Hessian and make it dimensionless Step 4: Apply external regularization (smoothing the matrix field) Step 4: Compute eigenvectors Step 5: Compute curvature Step 5: Compute orientation confidence Show result  ### Method 2: Structure Tensor

Step 1: Compute orientation score  Step 2: Compute Structure Tensor from orientation score  Step 3: Symmetrize the Hessian and make it dimensionless Step 4: Apply external regularization (smoothing the matrix field) Step 4: Compute eigenvectors Step 5: Compute curvature Show result  ## Short Implementation

A minimal Lie Analysis setup    ## Plotting the Exponential Curves

Functions for the group product and exponential map Function for extracting the group element that maximizes the absolute value of the orientation score (optimal θ for each x,y) Extract the group elements and tangent vectors   Plot the results (projected to the xy plane)  • Fast Marching
• Processing on 2D Images
• Working With Objects