Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
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1 Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge Processing Rf Chouder Noureddine Benhmidouche Received 3 October 6 Abstrct We propose in this work to find explicit exct solutions clled trvelling profile solutions to nonliner diffusion eqution tht occurs in imge processing. ome of these explicit solutions re relted with the phenomenon of contour enhncement in imge processing. We present generliztion of the results obtined by Brenbltt to study the contour enhncement in imge processing for exponent rnge of prmeter enhncement. Introduction Mny models which use nonliner diffusion equtions re proposed in imge processing nd contour enhncement. The technique of using PDEs in the imge edge hs firstly been introduced by Peron nd Mlik 6]. They hve proved tht imge intensity flux cn led to n enhncement of imge edge if the flux is directed opposite to the imge intensity grdient. Different pproches to the edge enhncement problem were proposed by severl uthors ]. In prticulr Mlldi nd ethin 7] proposed model bsed on the differentil-geometric pproch this model leds (fter proper scling) to the following eqution for imge intensity: u t (x t) = ( + u y )u xx u x u y u xy + ( + u x )u yy ( + u x + u y) +γ () where u(x y t) is the imge intensity flux nd γ is positive constnt. This eqution represents generl movement by curvture flow. The symptotic tretment of these models shows the enhncement of the intensity contrsts by formtion of regions of lrge intensity grdients. In order to focus on the boundry lyer where lrge grdients concentrte further simplifictions of this model were proposed in ] nd led to unidimensionl version of (): u t = u (+γ) x u xx. () Lbortory for Pure nd Applied Mthemtics University of M sil Box 66 Ichbili M sil 8 Algeri Lbortory for Pure nd Applied Mthemtics University of M sil Box 66 Ichbili M sil 8 Algeri
2 Trvelling Profile olutions for Nonliner Degenerte Prbolic Eqution This eqution governing the evolution of the imge intensity in the boundry lyer. In the rnge γ > eqution () flls into the clss of degenerte prbolic equtions with degenercy t u x =. In this pper we first propose to find exct solutions to eqution () by introducing the trvelling profile solutions which hve been proposed by Benhmidouche et l. in 4 5] in seeking exct solutions for some nonliner prtil differentil equtions. We seek solutions in the following form: ( ) x b(t) u (x t) = c(t)f (3) (t) where the coefficients (t) b (t) c (t) re functions which depend on time t nd f is the bsic profile these functions re to be determined. On the other hnd we wnt to generlize the results proposed in ] nd we show tht such behvior cn be observed in the lrger exponent rnge γ > directly by using solutions (3). We note tht this problem hs been investigted by Vsquez nd Brenbltt in 3] but with complicted conjugte formultion. Exct olutions to Nonliner Diffusion Eqution We now construct exct solutions of eqution () nd give prescription for properties such s the symptotic behvior blow up etc. for some exct solutions. For tht we use the trvelling profile solution s specil form of exct solutions to nonliner prtil differentil equtions. Trvelling profile solutions of eqution () re represented in the following form: u (x t) = c (t)f() with = x b(t) (t) for x R nd t > (4) where the prmeters (t) c(t) b(t) nd the profile f should be determined. This form permits us reducing the PDE () to n ODE. ubstituting the form of solutions (4) into eqution () we obtin: c c f f b f = c γ f. (5) γ f (+γ) A simple seprtion of vribles rgument implies tht the following conditions must hold: c c = α c γ γ c γ = (6) γ b = λc γ γ with prmeters α λ R nd the bsic profile f must stisfy the following ordinry differentil eqution in : f (+γ) f = αf + f + λf. (7)
3 R. Chouder nd N. Benhmidouche 3 We hve to solve system (6) to find the coefficients c (t)(t) nd b(t). Indeed from (6) we hve: { c(t) = K (t) α b(t) = λ (t) + A (8) where K > A re integrtion constnts. Inserting (8) into (6) we get: (i) In the first cse we hve two time behviors of the coefficients c (t) (t) nd b(t). They re given by: ] (t) = K γ t + A c(t) = K K γ for < t < T nd < (9) with And with b(t) = λ (t) = c(t) = K b(t) = λ = t + A ] α ] K γ t + A + A ] K γ t + A T = A K +γ. ] K γ α t + A ] K γ t + A + A ( + γ)α + γ for < t < nd > () nd A > K > A re constnts. (ii) In the second cse (for = ) we hve: (t) = A exp( K γ t) c(t) = K A α exp(k γ αt) b(t) = λ A exp( K γ t) + A for < t < () with A > K > A re constnts. We hve three time behviors of coefficients c (t) (t) nd b(t); these behviors depend on prmeters of similrity α nd. 3 ome Explicit Exct olutions of Nonliner Diffusion Eqution We consider now two interesting prticulr cses ccording to the vlues of α nd λ nd we seek new exct solutions of () we show lso the symptotic behviors of the solutions obtined.
4 4 Trvelling Profile olutions for Nonliner Degenerte Prbolic Eqution 3. Cse: α = R + λ R A prticulr cse in the discussion now is the cse where α = R + λ R. This corresponds to the following form of solutions: u (x t) = c f() with = x b(t) () (t) with interesting ppliction of these solutions in imge processing generlizing the results proposed in ]. To nlyze the process of contour enhncement in imge processing nd resolving the free boundry problem formulted with eqution () Brenbltt ] hs introduced the intermedite symptotic solution : u (x t) = c f() with = x x (t + t ) γ for constnt c > nd x t fixed the profile f is n incresing function to be determined. He proves tht the rte of enhncement depends on the hypotheses concerning the imge intensity flow (the prmeter γ). The mthemticl problem ws investigted by Brenbltt ] nd consists in solving this eqution with suitble boundry dt nmely u = on the left-hnd side of the contour nd u = on the right-hnd side nd initil conditions u(x ) = u (x) stisfying < u < nd u > in n intervl I = ( b) nd constnt vlues otherwise zero to the left to the right joining the levels f = t finite distnce = c < to the level f = t = c. At these levels when tken t finite distnce = ±c the grdients re infinite. In this model the phenomenon of grdient enhncement tkes plce for ll γ > : The sptil grdient of the solutions u x increses with time nd its support shrinks 3]. Indeed the scling implies tht: u x (x t) = c (t + t ) /γ f () with = x x (t + t ) γ which shows tht the solution is concentrted in n incresingly nrrower strip = {(x t) : x x c(t + t ) /γ } with grdients tht diverge like t /γ s t. With the form of solutions () we cn generlize these results. Indeed if we replce this form of solutions in () we obtin: b f f = c γ A seprtion of vribles rgument implies tht: f. (3) γ f (+γ) { = c γ γ b = λc γ γ (4) where λ re rbitrry constnts nd γ >.
5 R. Chouder nd N. Benhmidouche 5 The eqution for the profile f (3) becomes: ( + λ) df ( ) (+γ) df d = d f d d. (5) According to different vlue of prmeter γ we discuss the solutions of (4) nd (5). The cse: γ ] ; {} : The resolution of system (4) is not difficult indeed from (4) we hve: If we replce (6) in (4) we get ( γ (t) = ( b(t) = λ nd ( (t) = ( b(t) = λ γ c (+γ) γ c (+γ) c (+γ) γ c (+γ) t + A ) γ b(t) = λ (t) + A. (6) t + A ) γ t + A ) γ + A t + A ) γ + A for < t < (7) for < t < T = A c (+γ) (8) γ where > nd A > A re constnts. The eqution for the profile f (5) cn be written s: ( ) (+γ) df d f = ( + λ) where > nd λ R d d fter integrtion we obtin ( ) (+γ) df = ( + γ) d nd nother integrtion gives for ] df d = (+γ) C +γ + γ λ + C + λ + k with k constnt ( ) ] (+γ) + λ C (9) λ C. For = x b(t) (t) this reltion suggests free-boundry problem for determintion of the imge intensity evolution in the boundry lyer l(t) nd r(t) such tht l(t) = b(t) λ + C (t) x b(t) λ C (t) = r(t)
6 6 Trvelling Profile olutions for Nonliner Degenerte Prbolic Eqution where (t) b(t) re given by (7) nd (8). If we replce b(t) given by (6) we obtin l(t) = A C (t) x A + C (t) = r(t). Here C( is n integrtion ) constnt. ( Further ) integrtion nd using the boundry conditions f = f = then we obtin λ+c λ C f () = ] (+γ) γ C +γ + γ +λ C dη η ] (+γ) nd C = ] (+γ) dη + γ η ] Thus the trvelling profile solutions ssumes the form: ] (+γ) γ C +γ u (x t) = c + γ The grdients re given by o by (9) we hve (+γ) x A C(t) u x (x t) = c (t) f (). u x (x t) = c ] (+γ) C +γ (t) + γ γ+ γ dη η ]. (+γ). ( ) ] (+γ) x A. C(t) Therefore the vlidity of the symptotic eqution () improves with time. Thus the grdients blow up like O(t γ ) s t nd like O((T t) γ ) s t T. The width of the trnsition region r(t) l(t) is equl to C (t) = C ( γ c (+γ) t + A ) γ which decreses with time. In this cse the phenomenon of grdient enhncement tkes plce: The sptil grdient of the solutions u x increses with time nd its support shrinks see figures nd. The cse: γ = : The eqution () becomes: u t = u x u xx. ()
7 R. Chouder nd N. Benhmidouche t= u(xt) t= Lter times x Figure : The evolution of the imge intensity distribution u(x t) for γ = the Beltrmi flow..9.8 t=.7 u(xt) t= Lter times x Figure : The evolution of the imge intensity distribution u(x t) for γ = /4. If we replce the solution () in eqution () we obtin f b f = c f f. () A seprtion of vribles rgument implies tht the following conditions must hold: { where λ re rbitrry constnts. The resolution of system () gives: { (t) = A e c t b(t) = λ A e c t + A The eqution for the profile f becomes: ( + λ) df d = = c () b = λc for < t <. (3) ( ) df d f d d (4) which cn be written s ( ) 3 df d f d d = ( + λ) where R + nd λ R
8 8 Trvelling Profile olutions for Nonliner Degenerte Prbolic Eqution fter integrtion we obtin df d = C ( ) ] + λ C ( where C > is n integrtion constnt. By using the boundry conditions f ( ) nd f = we obtin fter integrtion: for λ C f () = ( π ) π rcsin + λ + πc ; = π Cπ λ π Cπ λ π. Thus the trvelling profiles solutions ssumes the form ( u (x t) = c π rcsin π x A ) + ] C(t) for l(t) = b(t) λ + Cπ π where (t) nd b(t) re given by (3). We cn write: The grdients re given by (t) x b(t) λ Cπ (t) = r(t) π l(t) = A C π (t) x A + C (t) = r(t). π u x (x t) = c (t) f () = c C(t) ( π x A ) ]. C(t) λ+c The grdients blow up exponentilly s t. The width of the trnsition region r(t) l(t) is given by C π A e π t c which decreses with time. In this cse the phenomenon of edge enhncement tkes plce for this clss of equtions (γ = ) see figure Cse: α = R λ R Now we discuss nother cse so we ssume tht u (s t)ds = const. R ) =
9 R. Chouder nd N. Benhmidouche t= u(xy) t= Lter times x Figure 3: The evolution of the imge intensity distribution u(x t) for γ = the men curvture flow. We cn see tht from (4): ( ) s b (t) u (s t)ds = c (t)f ds = (t)c(t) f ()d = const R (t) R this implies R c(t) = const (t) (5) from reltion (8) nd (5) we hve α = this cse corresponds to physicl lw of conservtion of mss. In this cse new explicit solutions re obtined to eqution () which depends on the prmeter γ. The cse: γ ] ; {} : The eqution of profile f (7) becomes: After integrtion we obtin: f (+γ) f = ( + λ) f] where α λ R. (6) γ + f γ = ( + λ)f + k with k is constnt. If we put for exmple f () = f () = then k = nd we obtin: which implies tht Integrting once more we get γ + γ + f γ+ f γ = (γ + )( + λ)f f γ+ df = ((γ + )( λ)) γ+ d. (γ + ) γ γ+ γ+ = γ where K is n integrtion constnt. ] ( λ) γ γ+ + K for γ ; {}
10 Trvelling Profile olutions for Nonliner Degenerte Prbolic Eqution Then the solution of (6) is written under the form: f () = ] (γ + )(γ + ) γ+ ( ]γ+ ) γ+ γ+ γ+ ] γ ( γ+ λ)+ + K γ + for γ ] ; {}. Finlly n explicit exct solution to the nonliner degenerte prbolic eqution () is given s follows: ( u (x t) = c(t) (γ + )(γ + ) ]γ+ γ+ γ+ ( x b(t) ] γ ) ] γ+ γ+ γ+ λ) + + K γ (t) + (7) for γ ] ; {}. The coefficients c (t) (t) nd b(t) re given explicitly by: ] (t) = K γ t + A ] c(t) = K K γ t + A for < t < T nd < (8) ] b(t) = λ α K γ t + A + A with And (t) = c(t) = K b(t) = λ α ] K γ t + A T = A K +γ. ] K γ t + A ] K γ t + A + A for < t < nd > (9) where = ( + γ) > for γ The cse: γ = : ] ; {} nd A K > A re constnts. Clerly if we set γ = in (9) nd () the coefficients c (t) (t) nd b(t) re given by: (t) = K αt + A ] c(t) = K K b(t) = λ α αt + A K αt + A ] ] + A for < t < T nd = α < (3)
11 R. Chouder nd N. Benhmidouche with And (t) = K αt + A ] c(t) = K K b(t) = λ α αt + A K αt + A T = A K α. ] ] + A for < t < nd = α > (3) where A K > A re constnts. The eqution of profile f (7) becomes: f f = (α + λ) f] where α λ R. (3) Integrtion of (3) nd putting f () = f () = nd fter routine computtion yield f f = α + λ. Integrting once more we obtin: f = ln α + λ + K α where K is n integrtion constnt. Therefore the solution of (3) is written under the form: f () = ( α Returning to the vribles x nd t we obtin: ) ] ln α + λ + K. + ( ) u (x t) = c(t) α ln b(t) αx + λ (t) + K where c (t) (t) nd b(t) re given by (3) nd (3). + ] (33) 4 Conclusion In this work we hve presented some explicit exct solutions to nonliner degenerte prbolic eqution tht occurs in imge processing. We hve introduced generl form of self similr solutions clled trvelling profiles solutions. We hve lso given generl results obtined for free boundry problem investigted by Brenbltt ] to study the contour enhncement in imge processing. We hve obtined new results in the lrger exponent rnge of prmeter enhncement γ >.
12 Trvelling Profile olutions for Nonliner Degenerte Prbolic Eqution References ] L. Alvrez P. L. Lions nd J. M. Morel Imge selective smoothing nd edge detection by nonliner diffusion II IAM J. Numer. Anl. 9(3)(99) ] G. I. Brenbltt elf-similr intermedite symptotics for nonliner degenerte prbolic free-boundry problems tht occur in imge processing PNA 98(3)() ] G.I. Brenbltt nd J. L. Vsquez Nonliner diffusion nd imge contour enhncement Interfces Free Bound 6()(4) ] N. Benhmidouche Exct solutions to some nonliner PDEs trvelling profiles method Electronic Journl of Qulittive Theory of Differentil Eqution 5(8) 7. 5] N. Benhmidouche nd Y. Ariou New Method for Constructing Exct olutions to Nonliner PDEs Interntionl Journl of Nonliner cience 7(4)(9) ] P. Peron nd J. Mlik cle-spce nd edge detection using nisotropic diffusion IEEE Trnsctions on Pttern Anlysis nd Mchine Intelligence (7)(987) ] R. Mlldi nd J. A. ethin Imge Processing: Flows under Min/Mx curvture nd Men Curvture Grphicl Models nd Imge Processing 58()(996) 7 4.
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