Spectrum of Fractal Interpolation Functions

Transcription

1 Spectrum of Fractal Interpolation Functions ikolaos Vasiloglou, Member, IEEE and Petros Maragos, Fellow, IEEE June 8, Abstract In this paper we compute the Fourier spectrum of the Fractal Interpolation Functions FIFs as introduced by Michael Barnsley We show that there is an analytical way to compute them In this paper we attempt to solve the inverse problem of FIF by using the spectrum results in a sequence of a sets that converges in the attractor of the operator A, which satisfies the condition A = W(A) = n= w n(a) Any system that uses the Hutchinson operator in order to generate iteratively the attractor A is called Iterated Function System (IFS) Iterated Function Systems Original Translation & Scaling The affine transform performs translation stretching and rotation on a given set In the special case of two dimensions the affine transform on a set S in -D space is described by the equation: [ ][ ] [ ] a b x e w(x, y) = + c d y f where (x, y) S The effects of an affine transform on a set are depicted in fig The union of affine transformations is called the Hutchinson operator: W = n= w n For a specified metric the distance h(a, B) between two sets A, B can be defined Under certain conditions [] the Hutchinson operator is contractive, h(w(a), W(B)) sh(a, B), s < Successive iterations with Hutchinson operator on a random set Submitted: Sep This research work was supported by the Greek Secretariat for Research and Technology and by the European Union under the program EΠET-98 with Grant # 98ΓT6, when both authors where with with the Department of Electrical & Computer Engineering, ational Technical University of Athens ikolaos Vasiloglou is now with Analytics 5 LLC Atlanta Georgia USA Petros Maragos is with the Department of Electrical & Computer Engineering, ational Technical University of Athens, Zografou, 577 Athens, Greece nvasil@ieeeorg,maragos@csntuagr Translation & Rotation Scaling & Reflection Figure : An affine transform can translate rotate and flip a -dimensional shape Fractal Interpolation Functions Fractal Interpolation Functions (FIF) is a special case of the -dimensional IFS and maintain all their characteristics FIF attractors are continuous functions that can be used to model continuous signals FIF interpolate a given set of + points (x n, y n ), n =, x < x < x < x < < x

2 The FIF that interpolates the above set is comprised of affine maps: [ ] [ ][ ] [ ] x an x en w n = +, n =,, y d n y f n c n satisfies the conditions of (), has a unique fixed point, the function F C, (TF)(x) = F(x), x [x, x ] The necessary condition is that the Interpolation Function passes from the + initial points, [ ] [ ] x xn w n = y y n [ ] [ ] x xn and w n = y y n n =,, () We call order of the FIF, the number of the affine maps The conditions provide 4 equations for 5 parameters, so d n, the vertical scaling factor is chosen to be the free parameter If we solve the above equations for a n, c n, e n, f n in terms of d n, we find :, The following restrictions guarantee the contractivity of T operator: a n < d n < Barnsley s operator is very useful because the attractor of an FIF can be generated with the iterative application of T on an initial signal F() : F m+ = TF m, F = lim m T m (F ), F C a n = x n x n x x, () e n = x x n x x n x x, () c n = y n y n x x d n(y y ) x x, (4) f n = x y n x y n x x d n x y x y x x (5) Let the real numbers a n, c n, e n, f n be defined by (-5) Barnsley [] introduced the operator T for the class C of continuous functions, T : C C by F (x) F (x) Attractor and (TF)(x) = c n l n (x) + d n F(l n (x)) + f n x [x n, x n ], n =,,, l n : [x, x ] [x n, x n ] the invertible transformation The above operator: l n (x) = a n x + e n is a contraction mapping according to Hausdorff metrics, Figure : Formation of the FIF attractor after successive iterations Discrete FIF One of the most interesting properties of the FIF is that its attractor is independent of the initial signal (initiator) If the initiator is a continuous signal, for example the linear interpolation between the given interpolation points, all the instances throughout all the steps of the iterations will be continuous signals, fig On the other hand if the initiator is a single point the attractor formed after infinite iterations

3 will be a continuous signal, but signals instances of the FIF throughout the iterations will be discrete signals, fig Although during all the iterations the instances are discrete signals or strictly mathematically speaking finite countable sets, the attractor is a continuous signal, an infinite and uncountable set Instances of a the formation of an FIF from a discrete initiator are shown in fig Figure : Generation of a discrete FIF It is essential to show that a good choice of the initiator is the + given interpolation points (x n, y n ), n =, After the first iteration in each of the subintervals between the points, new points are generated fig Let these points be (x s, y s ), s =,,( ) otice that all these ++( ) = + points belong to the attractor of FIF This wouldn t be true if the initiator was a different set apart from the given interpolation points By repeating this procedure after m iterations we get an M(m)-point discrete sequence that is a sampling of the FIF s attractor, with sampling period T s = M If the initiator included any other irrelevant point, that would not be mapped to an attractor s point after a finite number of iterations It can be proved that if the initiator is not the set of the initial interpolation points, then the error of the formed sequence after the mth iteration from the attractor decreases and goes to zero as m The number of the attractor s samples is : ( ) = ( ) + ( ) = + M = M(m) = m + (6) The discrete signal formed after the mth iteration over the discrete initiator, is called discrete FIF : f[n] = F(x n ), n =,,,M (7) Before expanding Barnsley operator T introduced above for the discrete FIF it is necessary to make clear that according to the strict definition of Fractals the discrete FIFs are not fractal sets because they are finite Discrete FIFs must be considered as approximations of the continuous Since the Barnsley operator assumes infinite resolution, it doesn t apply in discrete signals So for discrete FIF the following modified operator is used: T f[n] = (d k f[ n e k ] +c k ( n e k a a ) + f n) (8) k= (u(n e k ) u(n e k )) The Symbol f[n] defines that within two successive samples of f[n], zeros have been interpolated, fig 4 The term f n denotes the parameter of FIF as defined in (5) and should not be confused with the discrete signal f[n] Similarly to the continuous case, a discrete FIF of m iterations can be constructed with the following procedure: f m+ [n] = T f m [n] (9) It is obvious that when m the discrete FIF becomes continuous otice that T depends on the number of iterations m More specifically the parameters of FIF as defined in (-5) depend on the x n The x n change value because of the upsampling

4 F = F = and the points are evenly spaced Then(-5) becomes, a n =, () e n = n, () c n = F n F n, () f n = F n () 5 The above equations show that the FIF parameters are decoupled from each other This is very impor- because they can be estimated independently tant Moreover it is clear that if a n parameter is estimated Figure 4: Barnsley operator for discrete signals The then all parameters can be found directly, except for initiator is upsampled and interpolated according to d n (9 Any FIF can be transformed to an equivalent FIF that satisfies the first two conditions without loosing its fundamental properties More specifically fig 6 The generation of an FIF can be represented in terms shows how a given FIF can be transformed so as to of a linear system as shown in fig 5 satisfy the conditions for the first and the last point It is convenient to use an auxiliary affine map w aux that will rotate, scale and translate the given one ĝ(ω) ˆf(ω) The w aux transform is invertible and does not affect the intrinsic parameters a n, d n ˆq(ω) 5 5 Figure 5: Block diagram of the FIF Genaration 5 5 Computation of FIF s Fourier Spectrum In order to simplify (-5), it is very convenient to adopt the following assumptions x =, x =, Figure 6: An example of FIF out of range [, ] Applying an affine transform we can tie it at points (, ) and (, ) 4

5 Let A = {(x, y) : F(x) = y}, be the original attractor and w n(a) = A (4) n= where w n, [ ] [ ] [ ] [ ] x an x en w n = +, n =,, G(x) = y d n y f n c n and u(x) = { x < x G(x) is the piecewise linear function between the interpolation points : n= c n ( x e n a +f n )(u(x e n ) u(x e n )) The new transformed attractor is A = {(x, y ), F (x) = y } The new points are connected with the initial [ x y ] = [ ] [ aaux x d aux y c aux ] [ ] eaux + f aux That means A = w aux A These new points belong to a new FIF Setting in (4) A = waux A and applying the map w aux, with w auxw n w n= aux (A ) = A, (5) [ waux = /a aux caux a auxd aux /d aux ] By setting f aux = e aux =, d aux =, the affine maps of the new FIF are : w n [ ] [ x y = ] a n (c n d n c aux + c aux a n )/a aux d n [ ] [ ] x a y + aux e n c aux e n + f n otice that the new FIF has the same order and the same d n parameters Spectrum of Continuous FIF The application of the above simplifications to the Barnsley operator T results in the following equation for the FIF: FOURIER SPECTRUM FREQUECY (Hz) Figure 7: An example of Q(Ω) Applying continuous fourier transform [5]: F(Ω) = + F(x) F(Ω) F(x)e iωx dx F(Ω) = G(Ω) + af(aω) d n e iωen, (7) n= G(Ω) = n n= ( e iω e iω n (iω) ( cn a + iω(f n cnen a )) + cn a n n e iω n n e iω iω ) We define the function : (8) F(x) = n= (d nf( x en a )+ +c n ( x en a ) + f n )(u(x e n ) u(x e n )) (6) Q(Ω) = a n= (n ) iω d n e = a d n+ e iωn (9) n= 5

6 otice that the above function is the discrete time fourier transform of the discrete sequence {d, d, d }, so we deduce that Q(Ω kπ) = Q((k + )π Ω), k =,, (fig 7) FOURIER SPECTRUM (db) FOURIER SPECTRUM (db) FOURIER SPECTRUM (db) FREQUECY (Hz) FREQUECY (Hz) FREQUECY (Hz) Figure 8: Left column: Fractal interpolation between points (, ), (5, ), (5, 4), (75, 5), (, ) after,,5 iterations right column: Corresponding spectrums The Fourier spectrum satisfies the following equation: F(Ω) = Q(Ω)F(aΩ) + G(Ω) () Through Barsley Operator in frequency domain the Fourier spectrum can be computed iteratively, fig 8 F m+ (Ω) = Q(Ω)F m (aω) + G(Ω) () F (Ω) = After infinite iterations: i F(Ω) = G(Ωa i ) Q(Ωa j ) () i= j= Spectrum of Discrete FIF From the () it is obvious that the FIF signals are not band-limited As a result the spectrum of a discrete FIF is aliased Assume T s = M has been chosen as the sampling period, then the Discrete Time Fourier Transform (DTFT) [5] is : ˆf(ω) = M f[n]e iωn, ω = ΩT s n= As in continuous case for the computation of the DTFT the T operator is used in frequency domain: ˆf m+ (ω) = ˆf m (aω) d k ae iωep + ĝ m (ω) () p= Posing the analogy between the continuous and the discrete case we define the q function : ˆq(ω) = ˆq m (ω) = a d p e iωep p= d p e iωep, p= and e p = p (M ), so ˆq(ω) = p= d p+e iωp (M ) p =,,, = p= d p+e ( iωp)(m m i) i=, p =,,, (4) ˆf m+ (ω) = ˆq m (ω) ˆf m (aω) + ĝ m (ω) (5) After m iterations m i ˆf m (ω) = ĝ m i (ωa i ) ˆq i j+ (ωa j ) (6) i= j= The Discrete Fourier Transform can be computed after keeping the frequency values between and π and sampling the spectrum at ω =, π M, π M,, (M )π M, fig 9 In the continuous case Q(Ω) has period π In the discrete case where ω [, π] the ˆq(ω) function has period π M 6

7 FOURIER SPECTRUM (db) FREQUECY (Hz) Figure 9: 5-point FIF attractor with d n = 74, 8, 77, 85, 88 and the spectrum after the 5th iteration of linear equations ˆq() ˆq[ω ] e iω e ( )iω ˆq[ω ] = e iω e ( )iω ˆq[ω ] 5 5 e iω R e ( )iω R d d d d (7) FIF parameter estimation using spectrum FIF modelling of signals has been proposed by Mazel [], using information of signal in time domain In this section the spectrum of signal is used to estimate its parameters, provided the FIF order is known Given the signal f[n], n =,,,M, let be FIF s estimated order We expect its spectrum to satisfy the following equation: (b) (a) (c) ˆq[ω k ] = ˆf m+ [ω k ] ĝ[ω k ], ˆf m [aω k ] (d) (e) aω k =, π M, π π,, (M ) M M All values that zero denominator are excluded otice that f m+ [n] and f m [n] signals must have the same length Considering also that f m+ [] = f m [] and f m+ [M(m + )] = f m [M(m)], where M(m), M(m + ) their lengths In order to equalize their lengths it is necessary to interpolate with zeros f m [n] Because of the term ˆf m [aω k ] the computation of FFT must be on kπ M, k =,,M This is done by padding the signal with M M zeros The d n are determined by the solution of the system Figure : (a) FIF without noise In (b),(c) FIF contaminated with noise SR = and SR = In (d),(e) reconstructed FIF 4 Results The above algorithm was tested in FIF signals contaminated with white noise and the parameters extracted were very close to the real, fig The main advantage of the method is the use of FFT It is well known that FFT is quite simple and easily implemented The first disadvantage of the method is that 7

8 it cannot find the FIF order In the above experiments we tried for =,, until the estimated ˆq function satisfied the periodicity and symmetry conditions mentioned earlier The second and most important problem is that it is very sensitive in the window effect of the FFT It is known that although a part of FIF signal is self affine it is not an FIF Trying to model it as an FIF results in wrong estimations In the example of fig, it is evident that the period of the ˆq function has been expanded Having chosen the order, the period of ˆq is known π M In the right plot of fig it is evident that the period of ˆq is much higher than the expected But although the spectrum is not the best method for solving the inverse problem it can be used as powerful analysis tool As shown above it can reveal the FIF nature of a signal and it can also help in the prediction of a missing part of a time series, given that it belongs to class of FIF [4] H DI Abarbanel, Analysis of Observed Chaotic Data, Springer Verlag, ew York 996 [5] A VOppenheim, R W Schafer, Deiscrete-Time Signal Processing, Prentice Hall, FOURIER TRASFORM 5 5 FOURIER TRASFORM ORMALIZED FREQUECY (ω/π) x ORMALIZED FREQUECY (ω/π) x 4 Figure : Left : ˆq(ω) Right : ˆq(ω) of the corrupted FIF References [] B B Mandelbrot, The Fractal Geometry of ature WH Freeman and Company, ew York, 98 [] M F Barnsley, Fractals Everywhere, Academic Press, 99 [] D S Mazel, M H Hayes, Using Iterated Function Systems to Model Discrete Sequences, IEEE Transactions on Signal Processing, Vol 4 (7), pp 74-74, July 99 8

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10 FOURIER SPECTRUM (db) FREQUECY (Hz)

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