Symmetry Analysis for a Fourth-order Noise-reduction Partial Differential Equation
aa r X i v : . [ n li n . S I] A ug Symmetry Analysis for a Fourth-order Noise-reduction PartialDifferential Equation
Andronikos Paliathanasis ∗ Institute of Systems Science, Durban University of TechnologyPO Box 1334, Durban 4000, Republic of South AfricaInstituto de Ciencias F´ısicas y Matem´aticas,Universidad Austral de Chile, Valdivia, Chile
P.G.L. Leach
Institute of Systems Science, Durban University of TechnologyPO Box 1334, Durban 4000, Republic of South AfricaSchool of Mathematical Sciences, University of KwaZulu-NatalDurban, Republic of South Africa
August 17, 2020
Abstract
We apply the theory of Lie symmetries in order to study a fourth-order 1 + 2 evolutionary partialdifferential equation which has been proposed for the image processing noise reduction. In particular wedetermine the Lie point symmetries for the specific 1+2 partial differential equations and we apply theinvariant functions to determine similarity solutions. For the static solutions we observe that the reducedfourth-order ordinary differential equations are reduced to second-order ordinary differential equations whichare maximally symmetric. Finally, nonstatic closed-form solutions are also determined.Keywords: Lie symmetries, Noise reduction, Image processing
During the last three decades significant work has been done on the subject of noise reduction in image processingwith the use of partial differential equations (PDEs). By noise reduction or noise removal we mean the algorithmwhich has to be applied to a digital image to remove the wrong data information, or artificial imprints addedto the digital image by the sensors; while during this procedure it is important that the original information ∗ Email: [email protected] ∂u∂t = − E L ( f | ∆ u | ) , (1)where the right hand side of (1) follows from the variational principle of the Action Integral, S = R f ( | ∆ u | ) d x .In the case of a two-dimensional flat space this becomes S = Z f ( | u ,xx + u ,yy | ) dxdy. (2)Consequently, f is the Lagrange function, while a second requirement for function f is f ′ > . Now, in the simplest case for which f is a linear function, i.e. f ( | ∆ u | ) = √ u ,xx + u ,yy , equation (1) takesthe following form( u xxxx + 2 u xxyy + u yyyy ) ( u xx + u yy ) − h ( u xx + u yy ) ,x i − h ( u xx + u yy ) ,y i p | u ,xx + u ,yy | + u t = 0 . (3)That particular linear function, f , can be seen as a natural extension of the Lagrangian for the second-orderdifferential equation studied in [5]. While equation (3) is a fourth-order equation, there are similarities withthe second-order minimal surface equation. In particular, in the minimal surface equation the Action Integralinvolves the minimization of a surface and not of an arc length, as is the case with the equations of motion [11–13].In this work, the equation (3) follows from the minimization of a length which is defined by second-orderderivatives.In this work we study the latter nonlinear fourth-order PDE by applying Lie’s theory on the symmetries ofdifferential equations. Lie’s theory is one of the main mathematical tools for the determination of solutions fornonlinear differential equations. The existence of a Lie symmetry for a given differential equation is importantbecause invariant functions can be constructed to reduce the order of the differential equation or the numberof the dependent variables [14]. Furthermore, Lie symmetries can be used to perform group classifications fordifferential equations [15] or identify well-known systems for instance see [16–25] and references therein. Theapplications of Lie symmetries cover various areas of Mathematical physics from liquid films and travel wavesolutions [26–29], the theory of diffusion [30, 31] and optical physics [32]The application of Lie symmetries to the theory of noise reduction is not new. Recently in [33] a groupclassification analysis was performed for the nonlinear second-order PDE proposed by Rudin et al [5] and Liesymmetries were applied for the determination of conservation laws. In this work we approach the problemdifferently, by applying the Lie invariants to perform the reduction process for the PDE (3). The importance ofthe application of Lie point symmetries on noise reduction PDEs is not only that closed-form solutions can bederived, but Lie symmetries can be applied also to recognize patterns in an image as proposed by Big¨un [34,35].The plan of the paper is as follows.In Section 2 the basic properties and definitions of Lie point symmetries of differential equations are pre-sented. Moreover, we present the Lie symmetries for equation (3) and that of the time-independent case. The ∆ denotes the Laplace operator. Let function Φ describe the map of an one-parameter point transformation such as Φ (cid:0) u (cid:0) x i (cid:1)(cid:1) = u (cid:0) x i (cid:1) withinfinitesimal transformation t ′ = t i + εξ (cid:0) t, x i , u (cid:1) (4) x i ′ = x i + εξ i (cid:0) t, x i , u (cid:1) (5) u ′ = u + εη (cid:0) t, x i , u (cid:1) (6)and generator Γ = ∂t ′ ∂ε ∂ t + ∂x ′ ∂ε ∂ x + ∂u∂ε ∂ u , (7)where ε is the parameter of smallness and x i = ( x, y ).Let u (cid:0) x i (cid:1) be a solution of the PDE H ( u, u ,t , u ,x ... ) = 0; (8)then under the map Φ, function u ′ (cid:0) x i ′ (cid:1) = Φ (cid:0) u (cid:0) x i (cid:1)(cid:1) is a solution for the latter differential equation if and onlyif the differential equation is also invariant under the action of the map, Φ, i.e.Φ ( H ( u, u ,t , u ,x ... )) = 0 . (9)When the latter expression is true, the generator Γ is called a Lie point symmetry for the differential equation.It is straightforward to prove that this condition becomesΓ [ n ] ( H ) = 0 , (10)in which Γ [ n ] describes the n th prolongation/extension of the symmetry vector in the jet-space of variables, (cid:8) t, x i , u, u ,i , u ,ij , ... (cid:9) .The importance of the existence of a Lie symmetry for a given PDE is that from the associated Lagrange’ssystem, dtξ t = dx i ξ i = duη , (11)zeroth-order invariants, U [0] (cid:0) t, x i , u (cid:1) are able to be determined which can be used to reduce the number of theindependent variables of the differential equation and lead to the construction of similarity solutions. Last butnot least, the admitted symmetries of a given differential equation constitute a Lie algebra. Consider now u ( t, x, y ) = u ( x, y ). Then equation (3) is simplified as follows.The resulting differential equation is of fourth-order and it is given as( u xxxx + 2 u xxyy + u yyyy ) ( u xx + u yy ) − h ( u xx + u yy ) ,x i − h ( u xx + u yy ) ,y i = 0 . (12)3 , ] X X X X X X − X X X X X X X − X X X − X − X , ] Y Y Y Y Y Y Y − Y Y Y Y Y Y Y − Y Y − Y Y − Y − Y − Y Y Y Y ∞ Lie pointsymmetries. X = ∂ x , X = ∂ y , X = y∂ x − x∂ x , X = u∂ u X = x∂ x + y∂ y , X ∞ = Ψ ( x, y ) ∂ u with ∆Ψ = 0 . The Lie Brackets of the admitted Lie symmetries are presented in Table 1, from which it is clear that theadmitted Lie algebra is { A ⊕ s A } in the Morozov-Mubarakzyanov Classification Scheme [36–39]. The vectorfields { X , X , X } form the E Lie algebra, more specifically they are the isometries of the two-dimensionalEuclidean space, while the set of vectors { X , X , X , X } form the Homothetic algebra for the two-dimensionalEuclidean space. For the time-dependent equation (3) the admitted Lie point symmetries are calculated to be Y = ∂ x , Y = ∂ y , Y = y∂ x − x∂ x , Y = t∂ t − u∂ u ,Y = 5 t∂ t + x∂ x + y∂ y , Y = ∂ t , Y ∞ = ¯Ψ ( x, y ) ∂ u with ∆ ¯Ψ = 0 . The Lie Brackets of the Lie point symmetries are presented in Table 2; from which we infer that the admittedLie algebra is { A ⊕ s A } ⊕ s A , .The table of Lie Brackets is 4 , ] Z Z Z Z Z − Z Z Z Z Z Z Z − Z Z − Z − Z { Y , Y } , while also the Y symmetry follows because thedifferential equation is invariant under time translations. Before we proceed with the application of the Lie invariants, we determine the Lie point symmetries of thefollowing differential equation ,( u xxxx + 2 u xxyy + u yyyy ) ( u xx + u yy ) − h ( u xx + u yy ) ,x i − h ( u xx + u yy ) ,y i + λ q | u ,xx + u ,yy | u = 0 (13)which is the time-indepedent equation (3) with a linear source λu . In the following Section we see how thisequation it can follow from (3) for a specific value of the parameter λ .The admitted Lie point symmetries are four and they are Z = ∂ x , Z = ∂ y , Z = y∂ x − x∂ y Z = x∂ x + y∂ y + 10 u∂ u . with Lie Brackets given in Table 3. Easily we can infer that the Lie symmetries form the Lie algebra { A ⊕ s A } ⊕ s A .We proceed with the application of the Lie symmetry vectors for equation (3). The nonlinear PDE is an 1 + 2 fourth-order equation. We continue our analysis by eliminating the time-derivative. For that, we start by performing reductions with the use of the vector fields (a) Y and (b) Y . ∂ t The application of the autonomous symmetry Y to equation (3) provides that the solution, u , is static, i.e., u ( t, x, y ) = u ( x, y ). Consequently the resulting equation is that of expression (12). Now we can continue withthe reduction process by using the Lie symmetries X I . .1.1 Travelling-wave solution Consider the vector field X + αX . Then the resulting Lie invariants are ζ = y − αx , u = w ( ζ ). We consider ζ to be the new indepedent variable and w the new dependent variable. Thus equation (12) is simplified to thefollowing fourth-order ordinary differential equation (ODE) (cid:0) a + 1 (cid:1) d wdζ d wdζ − (cid:18) d wdζ (cid:19) ! = 0 . (14)For this equation easily we can calculate the admitted Lie point symmetries which are¯ X = ∂ w , ¯ X = ∂ ζ , ¯ X = w∂ w , X = ζ∂ ζ and X = ζ∂ w which form the Lie algebra { A ⊕ s A } ⊕ s A , .While we could continue with the application of Lie point symmetries to equation (14), it can easily beintegrated and we get w ( ζ ) = w ln ( ζ − ζ ) + w ( ζ − ζ ) , (15)where w , and ζ , are four constants of integration.Because the parameter α is not involved in the resulting ODE (14) the solution holds also for reductionprocess of (12) with the vector field X or X . Moreover, equation (14) can be written easily as the second-orderordinary differential equation 2
W d Wdζ − (cid:18) dWdζ (cid:19) = 0 , (16)where W ( ζ ) = d wdζ . We can easily see that it is maximally symmetric and admits eight Lie point symmetries,which form the sl (3 , R ) Lie algebra. Equation (16) can be easily linearised after the transformation W ( ζ ) = (cid:0) ¯ W ( ζ ) (cid:1) − [40–42].. Consider now the application of the Lie symmetry vector X , i.e. of the rotational symmetry. The invariantfunctions are calculated to be r = x + y and u = h ( r ). The resulting ODE is − d hdR (cid:18) d hdR + d hdR (cid:19) + (cid:18) d hdR (cid:19) + 3 (cid:18) d hdR (cid:19) = 0 (17)after we applied the transformation r = exp ( R ) in order to simplify the reduced equation.Equation (17) admits the four-dimensional Lie algebra, 2 A ⊕ s A , , which comprises the following symmetryvectors X ′ = ∂ R , X ′ = ∂ h , X ′ = h∂ h and X = R∂ h . The fourth-order ODE can be easily reduced to the second-order ODE − ρ (cid:18) d ρdR + dρdR (cid:19) + ρ + 3 (cid:18) dρdR (cid:19) = 0 (18)by applying the change of variable ρ = d hdR . 6urprisingly, equation (18) is maximally symmetric, that is, it admits as Lie point symmetries the elementsof the sl (3 , R ) Lie algebra in the representation ∂ R , ρ∂ ρ , e − R ρ ∂ ρ , Re − R ρ ∂ ρ , R∂ R + Rρ∂ ρ ,e ρ √ ρ ( ∂ R + ρ∂ ρ ) , e ρ √ ρ ( R∂ R + ρ ( R − ∂ ρ ) , R ∂ R + R ( ρR − ρ ) ∂ ρ . The generic solution of (18) is given to be ρ ( R ) = ρ e R (( R − R )) − , (19)where ρ , R are constants of integration. From the scaling symmetry X we determine the invariants θ = arctan (cid:0) yx (cid:1) and u = g ( θ ). The reduced ODE is2 (cid:18) d gdθ (cid:19) (cid:18) d gdθ (cid:19) − (cid:18) d gdθ (cid:19) − (cid:18) d gdθ (cid:19) = 0 , (20)which is invariant under the Lie symmetry vectors X ∗ = ∂ θ , X ∗ = ∂ g , X ∗ = g∂ g and X ∗ = θ∂ g .The generic solution of equation (20) easily can be determined. It is g ( θ ) = g ln (cos ( θ − θ )) + g ( θ − θ ) (21)in which g , and θ , are four constants of integration.However, what is important to mention is that equation (20) can be written again as a second-order ODEby applying the change of variable G = (cid:16) d gdθ (cid:17) . The resulting second-order ODE is maximally symmetric andunder the G = ¯ G − takes the simple form of the harmonic oscillator d ¯ Gdθ + ¯ G = 0 . (22) t∂ t − u∂ u We continue our analysis by applying the invariants to equation (3) of the symmetry vector Y . The invariantfunctions are x, y and u = u ( x,y ) t . The resulting PDE is equation (13) for the specific value λ = − Reduction with the use of the symmetry vectors Z + αZ provides the fourth-order ODE2 (cid:18) d wdζ (cid:19) (cid:18) d wdζ (cid:19) − (cid:18) d wdζ (cid:19) − s ( α + 1) − d wdζ w = 0 (23)in which ζ = y − αx and w = w ( ζ ).Equation (23) is invariant only under the two-dimensional Lie algebra A , with as elements the vector fields¯ Z = ∂ ζ , ¯ Z = ζ∂ ζ + 10 w∂ w . Z and it is w ( ζ ) = w ζ for w = w ( α ).The application of the two symmetries ¯ Z , ¯ Z to (23) leads to a second-order equation with no Lie pointsymmetries. From the vector field Z we get the invariants R = ln (cid:0) x + y (cid:1) and u = h ( r ). The reduced ODE is − d hdR (cid:18) d hdR + d hdR (cid:19) + (cid:18) d hdR (cid:19) + 3 (cid:18) d hdR (cid:19) − s(cid:18) d hdR (cid:19) he R = 0 , (24)which admits only one Lie point symmetry, namely Z ′ = ∂ R + 5 h∂ h . Application of this to (24) leads to a third-order ODE with no Lie point symmetries. A special solution of(24) can be found by applying the invariant of Z ′ . It is h ( R ) = h e R , where h is an imaginary number. The application of the scaling symmetry, Z , leads to a fourth-order ODE with only one symmetry, the vectorfield Z ∗ = Z the application of which reduces the equation to a third-order equation without Lie pointsymmetries. For convenience of the presentation we do not write the fourth-order or the third-order equations. In this work, we have focused on the application of Lie’s theory for the determination of invariant one-parameterpoint transformations for a fourth-order 1 + 2 evolution equation which has been proposed for the study of noisereduction in image-processing theory. The model of our consideration is a higher-order generalization of themodel proposed by Rudin et al [5] and it has been proposed later by You and Kaveh [10].We have performed the symmetry classification for the following kind of solutions: (i) time-dependentsolution, (ii) stationary solution with no source and (iii) stationary solution with linear source. The laterclassification has been useful in order to continue the reduction process on the 1 + 2 evolution equation to afourth-order ODE.The line point symmetries have been applied in order to perform second-reductions that they provide sim-ilarity solutions which belong to family of: travelling-wave solutions, radial solutions and scaling solutions.Surprisingly, in the case of static solutions all the second-reductions reduce to fourth-order ODEs which canbe written as linear second-order ODEs. That it is not the case of nonstatic solutions for which we were ableto reduce the fourth-order ODEs to third-order ODEs with the application of Lie symmetries. However, somespecific closed-form solutions were determined.It is important to mention the existence of the two symmetries u∂ u and Ψ ( x, y ) ∂ u which indicates linearity,even if the PDE (3) is always nonlinear. These two symmetries are related with the solution of the two-dimensional Laplace equation which is the common factor in (3). However, these solutions are not acceptablefor our consideration. For that reason we have not applied these two symmetries in the reduction process. On8he other hand, from the similarity solutions we determined with the use of the Lie symmetries we show thatnew solutions exist.The results of this work indicates that the theory of symmetries of differential equations can be play animportant role for the study of the nonlinear PDEs in the image-processing theory. Therefore, conservationlaws can be also determined by using Noether’s theorem or by using other approaches. Such a work is still inprogress and will be published elsewhere. Acknowledgements
PGLL Thanks the Durban University of Technology, the University of KwaZulu-Natal and the NationalResearch Foundation of South Africa for support. The authors thank Suranaree University of Technology,Sergey Meleshko and Eckart Schulz for the the hospitality provided while the bulk of this work undertaken.
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