LIE ANALYSIS SYMBOL

# LeftInvariantDerivatives LeftInvariantDerivatives[ObjPositionOrientationData, {σSpatial, σAngular}, derivativeIndex]
Computes the AderivativeIndex derivative(s) from ObjPositionOrientationData using Gaussian derivatives at specified scales {class -> TI} ;; XMLElement[span, {class -> special-character Sigma}, {RawXML[&#963;]}] ;; XMLElement[sub, {}, {Spatial}] and {class -> TI} ;; XMLElement[span, {class -> special-character Sigma}, {RawXML[&#963;]}] ;; XMLElement[sub, {}, {Angular}].

LeftInvariantDerivatives[ObjPositionOrientationData, derivativeIndex, options]
Computes the AderivativeIndex derivative(s) from ObjPositionOrientationData using a finite difference method which can be specified by using options.

## DetailsDetails

• LeftInvariantDerivatives computes the Left Invariant derivatives from an orientation score. The left-invariant frames are defined as
A1:= cos θ x + sin θ y,
A2:=-sin θ x + cos θ y,
A3:= θ,
thus with A1 the forward direction (aligned with the x-axis at θ=0), A2 spatial and perpendicular to A1 and A3 the derivative in angular direction.
• Internally the function makes use of the GaussianFilter to compute the Gaussian derivatives.
• The symmetry of odd-spatial-derivatives is taken into account using the element "symmetry" in ObjPositionOrientationData as -π, which means that the full periodic orientation score is the stored orientation score (from 0 to π) plus its conjugate.
• If the input orientation score is an complex orientation score, than both the real and imaginary part of the orientation scores are processed.
• The output of the function is again (just as the input) an Orientation Score
• LeftInvariantDerivatives does take the commutators into account when computing the Gaussian derivatives. These are computed in a separable way, which has the consequence that the angular derivative should always come up front. E.g., internally the A3A1 is rewritten by using the commutator relation A3A1=A1A3+A2. For details see [Franken 2008, Ch. 5.2.2].
• Using Normal on the output of LeftInvariantDerivatives results in an Association.
• If no scales (σSpatial and σOrientation) are specified, a finite difference method is used.
• Options that are supplied when using the Gaussian derivatives implementation are ignored as they are only valid in case of the Finite Difference method.
• The following options can be specified:
•  "Method" "Central" Specifies scheme used for finite differences. Other possible values are "Forward" and "Backward".
• References:
[Franken 2006]: Franken, E. M. (2008). Ph.D. Thesis: Enhancement of crossing elongated structures in images. Eindhoven University of Technology. Eindhoven, The Netherlands.

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Computing the First order derivative A2U using Gaussian derivatives

 In:= In:= Out= Computing multiple left invariant derivatives at once using Gaussian derivatives

 In:= Out= Computing the First order derivative A2U using a Finite Differences

 In:= Out= ### Properties & Relations  (3)Properties & Relations  (3)

InverseOrientationScoreTransform CakeWaveletStack GaborWaveletStack OrientationScoreTransform OrientationScoreTransform3D LeftInvariantDerivatives3D