LIE ANALYSIS TUTORIAL

# Enhancing Diffusion's on 2D Images

 LeftInvariantDerivatives[os, derivativeIndex] computes the left-invariant derivatives from os (ObjPositionOrientationData)using a finite differences method. LeftInvariantDerivatives[os,{σs,σo},derivativeIndex] computes the left-invariant derivatives from os using the Gaussian scales σs and σo OrientationScoreTensor[os,derivativeIndex] construct tensor derivativeIndex from os using the defined Gaussian σs and σo OrientationScoreTensor[os,{σs,σo},derivativeIndex] construct tensor derivativeIndex from os using a finite differences method OrientationScoreGaugeFrames[os,{σs,σo}] construct Gauge frames from os usins the Gaussian scales σs and σo OrientationScoreTransform[data] constructs an orientation score from data

Functions used for enhancing diffusion's.

 In:= Set example data
 In:= Generate some noise
 In:= Out= In:= ## Left Invariant Hypo-elliptic Diffusion

### Initialization

Construct orientation scores
 In:= Out= The effective parameters (based on the grid size)
 In:= Out= The diffusion constants relative to the ξ-normalized geometry
 In:= Out= The corresponding diffusion time (to reach the desired standard deviations)
 In:= ### Perform Diffusion in the Left-Invariant Frame

Function to compute left-invariant finite differences gradient
 In:= Perform diffusion with update step (u=u+dt*div(Mξ-1.D.Mξ-1.u)), note that the are due to the intrinsic geometry
 In:= Out= Show result
 In:= Out= ## Gauge Frames CEDOS

### Initialization

Settings
 In:= ### Compute Gauge Frames

Compute left-invariant Hessian
 In:= Out= Compute Gauge frames
 In:= Out= Perform ξ conversion to match
 In:= In:= Function for converting the tensor (in left-invariant frame)
 In:= Compute the tensor in the gauge frame and compute the orientation confidence (Laplacian in plane perpendicular to the main direction)
 In:= Out= ### Perform Diffusion along the Gauge Frames

Eq. (72) Franken Duits IJCV
 In:= In:= Helper function for computing the divergence of an orientation score object
 In:= Perform diffusion with TschumperléDeriche trace based update step (update via the Hessian)
 In:= Out= In:= Out= ## Left Invariant CEDOS

Left Invariant CEDOS
 In:= Out= Show results
 In:= Out= 