LIE ANALYSIS TUTORIAL

Step By Step Gauge Frames SE(2)

Step by Step GuidePlotting the Exponential Curves
Short Implementation

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

Functions for the data-adaptive frame.

Load package and some general functions and variables
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Step by Step Guide

Method 1: Symmetric Product Hessian

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

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

A minimal Lie Analysis setup
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Plotting the Exponential Curves

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