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[σ]}] ;; XMLElement[sub, {}, {Spatial}] and {class -> TI} ;; XMLElement[span, {class -> special-character Sigma}, {RawXML[σ]}] ;; 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

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Computing multiple left invariant derivatives at once using Gaussian derivatives

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Computing the First order derivative A2U using a Finite Differences

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Scope  (2)Scope  (2)

Generalizations & Extensions  (1)Generalizations & Extensions  (1)

Options  (1)Options  (1)

Properties & Relations  (3)Properties & Relations  (3)

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