Covariant fractional extension of the modified Laplace-operator used in 3D-shape recovery

Covariant fractional extension of the modified Laplace-operator used in 3D-shape recovery ⋆ Richard Herrmann ∗ ∗GigaHedron, Berliner Ring 80, D-63303 Dreieich (e-mail: herrmann@gigahedron.com) Abstract: Extending the Liouville-Caputo definition of a fractional derivative to a nonlocal covariant generalization of arbitrary bound operators acting on multidimensional Riemannian spaces an appropriate approach for the 3D shape recovery of aperture afflicted 2D slide sequences is proposed. We demonstrate, that the step from a local to a nonlocal algorithm yields an order of magnitude in accuracy and by using the specific fractional approach an additional factor 2 in accuracy of the derived results. Keywords: Fractional calculus, computer graphics, image processing, shape recovery, confocal microscopy, modified Laplacian. 1. INTRODUCTION From a historical point of view, fractional calculus provides us with a set of axioms and methods to extend the concept of a derivative operator from integer order n to arbitrary order α, where α is a real or complex value. dn dxn →dα dxα (1) In the sense of (1) fractional calculus has been frequently applied in the area of image processing, see e.g. [Falzon (1994)], [Ortigueira (2003)], [Sparavigna (2009)]. Alternatively we may consider fractional calculus as a specific prescription to extend the definition of a local operator to the nonlocal case. In this lecture, we will present a covariant, multidimensional generalization of the fractional derivative definition, which may be applied to any bound operator on the Riemannian space. As a first application, we will propose a specific non-local extension of the modified local Laplace-operator, which is widely used in problems of image processing. We will especially compare the local to the nonlocal approach for 3D-shape recovery from a set of 2D aperture afflicted slide sequences, which may be obtained e.g. in confo- cal microscopy or autofocus algorithms [Zernike (1935)], [Spencer (1982)]. It will be shown, that a major improvement of results is achieved for the nonlocal version of the modified Laplace- operator. 2. THE GENERALIZED FRACTIONAL DERIVATIVE We will propose a reinterpretation of the fractional cal- culus as a specific procedure for a non-local extension ⋆ of arbitrary local operators. For that purpose, we start with the Liouville definition of the left and right fractional integral [Liouville (1832)]: LIα f(x)=        (Iα +f)(x)= 1 Γ(α) Z x −∞ dξ (x −ξ)α−1f(ξ) (Iα −f)(x)= 1 Γ(α) Z ∞ x dξ (ξ −x)α−1f(ξ) (2) With a slight modification of the fractional parameter α = 1 −a, where α is in the interval 0 ≤α ≤1. Consequently for the limiting case α = 0 I+ and I−both coincide with the unit-operator and for α = 1 I+ and I− both correspond to the standard integral operator. I+ and I−may be combined to define a regularized Liouville integral [Herrmann (2011)]: Iα f(x) = (1 2(Iα + + Iα −)f)(x) (3) = 1 Γ(α) Z ∞ 0 dξ ξα−1 f(x + ξ) + f(x −ξ) 2 (4) = 1 Γ(α) Z ∞ 0 dξ ξα−1ˆs(ξ)f(x) (5) where we have introduced the symmetric shift-operator: ˆs(ξ)f(x) = f(x + ξ) + f(x −ξ) 2 (6) The regularized fractional Liouville-Caputo derivative may now be defined as: arXiv:1111.1311v1 [cs.CV] 5 Nov 2011 ∂α x f(x) = Iα∂xf(x) (7) =  1 Γ(α) Z ∞ 0 dξ ξα−1ˆs(ξ)  ∂xf(x) (8) = 1 Γ(α) Z ∞ 0 dξ ξα−1 f ′(x + ξ) + f ′(x −ξ) 2 (9) = 1 −α Γ(α) Z ∞ 0 dξ ξα−1 f(x + ξ) −f(x −ξ) 2ξ (10) with the abbreviation ∂xf(x) = f ′(x). This definition of a fractional derivative coincides with Feller’s [Feller (1952)] definition F∂x(θ) for the special case θ = 1. We may interpret Iα as a non-localization operator, which is applied to the local derivative operator to determine a specific non-local extension of the same operator. There- fore the fractional extension of the derivative operator is separated into a sequential application of the standard derivative followed by a non-localization operation. The classical interpretation of a fractional integral is changed from the inverse operation of a fractional derivative to a more general interpretation of a non-localization proce- dure, which may be easily interpreted in the area of image processing as a blur effect. This is a conceptual new approach, since it may be easily extended to other operators e.g. higher order derivatives or space dependent operators, e.g. for ∂2 x we obtain: (∂2 x)α f(x) = Iα∂2 xf(x) (11) =  1 Γ(α) Z ∞ 0 dξ ξα−1ˆs(ξ)  ∂2 xf(x) (12) = 1 Γ(α) Z ∞ 0 dξ ξα−1 f ′′(x + ξ) + f ′′(x −ξ) 2 (13) = 1 −α Γ(α) Z ∞ 0 dξ ξα−1 f ′(x + ξ) −f ′(x −ξ) 2ξ (14) = 2 −α 2Γ(α) Z ∞ 0 dξ ξα−1 f(x + ξ) −2f(x) + f(x −ξ) ξ2 (15) which is nothing else but the Riesz [Riesz (1949)] definition of a fractional derivative. Therefore we define the following fractional extension of a local operator local ˆO to the non-local case nonlocal ˆOα f(x) = Iα local ˆOf(x) (16) as the covariant generalization of the Liouville-Caputo fractional derivative to arbitrary operators on R. This definition may be easily extended to the multidimen- sional case, interpreting t

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