Fractal images defined by an iterated function system (IFS) are specified by a finite number of contractive affine transformations. In order to plot the attractor of an IFS on the screen of a digital computer, it is necessary to determine a bounding area for the attractor. Given a point on the plane, we obtain a formula for the radius of a circle centred on that point that contains the attractor of the IFS. We then describe an algorithm to find the point on the plane such that the bounding circle centred on that point has minimum radius

Edalat, A

Sharp, WN

While, RL

English

Spiral - Imperial College Digital Repository

1Bounding the Attractor of an IFSAbbas Edalat, David W.N. Sharp and R. Lyndon While1Imperial College Research Report DoC 96/5email: {ae, dwns}@doc.ic.ac.uk, lyndon@cs.uwa.edu.auApril 16, 1999Keywords: Algorithms, analysis of algorithms, iterated function systems, fractals.AbstractFractal images defined by an iterated function system (IFS) are specified by a finite number ofcontractive affine transformations.  In order to plot the attractor of an IFS on the screen of a digitalcomputer, it is necessary to determine a bounding area for the attractor.  Given a point on the plane, weobtain a formula for the radius of a circle centred on that point that contains the attractor of the IFS.We then describe an algorithm to find the point on the plane such that the bounding circle centred onthat point has minimum radius.1   IntroductionIterated function system (IFS) fractal images, as popularised by Barnsley[1, 2], are constructed fromsheared, reduced, rotated and displaced copies of themselves.  For example, the curly image in Figure 1is constructed from two transformed copies of itself: the blackened rightmost curl and the less blackremainder.  The blackened rightmost curl is a reduced, rotated and displaced copy of the whole image,produced by applying transformation T 1 .  The remainder of the image is produced by applyingtransformation T 2 , which shrinks the whole image and rotates it anti-clockwise.  The curly image is theattractor of the IFS specified by the transformations T 1  and T 2 .Various algorithms exist for plotting fractal images from their affine transformations[1, 4, 7, 8, 9, 10].However, in order to plot a fractal image on the screen of a digital computer, all of these algorithmsrequire advance knowledge of a bounding area inside which the image is known to lie.  In this paperwe derive a formula which expresses the dimensions of a bounding circle for the attractor of an IFS interms of the parameters of its transformations.  Following the approach of [5], in Section 3 we derive                                                       1On sabbatical at Imperial College: normally at the Dept. of Computer Science, The University of Western Australia.2an upper bound for the radius of a bounding circle centred at an arbitrary (given) point on the plane.  InSection 4 we describe an algorithm for determining the centre at which the radius of the bounding circleis minimum.  Section 5 describes some results and Section 6 concludes the paper.Bounding areaT 1T 2Figure 1  The fractal image curly and its transformations.2   DefinitionsAn IFS is specified by n  contractive, affine transformations T i , 1 ≤ i ≤ n .  Each transformation T i  hasthe form x'y'     =a i bici d i     xy     +eif i     .  Each T i  has a fixpoint x0y0      that is mapped to itself under thetransformation, i.e. x0y0     =a bc d     x0y0     +ef     .  Solving the equation defining x0y0      givesx0y0     =(bi f i + ei (1 − d i )) δ(ci ei + f i (1 − ai )) δ     where δ = (1 − a i )(1 − d i ) − bi ci .The attractor of an IFS can be obtained as follows[6].  Let D  be any disk with T i ( D ) ⊆ D , 1 ≤ i ≤ n .The attractor is given by∩m≥ 0 Tm( D )where T ( A) = ∪i =1nT i ( A) .The effect of T i  on two points ( x1 , y1 )  and ( x2 , y2 )  is to map them to two points ( x1' , y1' )  and( x2 ' , y 2 ' )  that are closer together than the original points.  The contractivity factor of T i  is the least3number s i  such that ( x1 ' −x 2 ' )2 + (y1 ' −y2 ' )2 ≤ s i ( x1 − x 2 )2 + ( y1 − y2 )2 .  It is well known that[see,for example, 3]s i = α +β + (α − β )2 + γ 2where α = (a i2 + ci2 ) 2 ,β = (b i2 + d i2 ) 2 ,γ = a ib i + cid i .3   A bounding circle for the attractor of an IFSA bounding circle for an IFS is a circle that contains the attractor of the IFS.  Given a point u = ( x , y) ,we shall obtain a formula for the radius of a bounding circle centred on u .  We start by considering onetransformation of the IFS and then consider the IFS as a whole.3.1   A bounding circle for one transformationConsider the effect of a transformation T i  on an arbitrary point p  that lies on or inside a circle Bicentred on u .  T i  maps u  to the point u i  and p  to the point p i , as shown in Figure 2.  We denote theradius of Bi  by RBi (u) .1.  Bi  is a bounding circle for T i  if p i  lies on or inside Bi , i.e. ifup i ≤ RB i (u)2.  By the triangular inequality, up i ≤ uu i + u i pi , and we haveup i ≤ RB i (u)   if  uu i + u i pi ≤ RB i (u )3.  As T i  is contractive, u i p i ≤ siup , and we haveuu i + u i pi ≤ RB i (u )   if  uu i + siup ≤ RB i (u )4.  As p  lies on or inside Bi , up ≤ RBi (u) , and we haveuu i + siup ≤ RB i (u )   if  uu i + si RB i (u ) ≤ RB i (u)Solving for RBi (u)  givesRBi (u) ≥uu i1 − siHence, Bi  is a bounding circle for T i  if RBi (u) ≥uu i1 − si.4Note that uu i  is easily calculated from the coordinates of u .  Clearly, RBi (u)  is minimum when the twosides of the inequality have the same value.  Figure 2 illustrates the construction.Biuu ipp iFigure 2  The circle Bi  is a bounding circle for T i  if RBi (u) ≥uu i1 − si.3.2   A bounding circle for an IFS with n  transformationsApplying the formula for RBi (u)  to each of the n  transformations individually gives n  concentriccircles centred on u .  Clearly, the largest of the concentric circles is a bounding circle B  for the IFS.Its radius is given byRB (u ) = max i =1n uu i1 − s i4   Choosing an optimal centre for the bounding circleThe quality of the upper bound given by the formula for RB (u )  depends acutely on the choice of u .We shall describe an algorithm for deriving the optimal choice of u  for a given IFS.  We start byconsidering an IFS with two transformations and then extend the algorithm to the general case.4.1   An optimal centre for an IFS with two transformationsWe wish to find u = ( x , y)  where RB (u )  is minimum.  Letzi ( x , y) = RB i2( x , y) =( a i x + b i y + ei − x )2 + ( ci x + d i y + f i − y)2(1 − si )25The value of zi ( x , y)  for a transformation T i  forms an elliptic cone as ( x , y )  is varied, with the vertexof the cone at the fixpoint of T i .  The cones for the two transformations defining curly are shown inFigure 3.Figure 3  The elliptic cones for the two transformations defining curlyfor zi ( x , y) ≤ 14500 .  The intersection of the two cones is clear.The two cones meet in a conic section.  The point ( x , y , z i ( x, y ))  in the intersection that gives theminimum value of RB ( x , y )  is the optimal centre for the bounding circle.  Using Lagrange’s multipliers,this is obtained by minimising the functiong( x , y,λ ) = z1 ( x, y ) + λ ( z1( x , y ) − z 2 ( x , y))with respect to x , y  and λ  and solving numerically for x  and y .4.2   An optimal centre for an IFS with n  transformations: Algorithm ΕIn the case of three transformations, the optimal centre lies either at the intersection of the three cones(if they meet), or at the minimum of one of the three pairwise intersections.  For each candidate centre( x , y ) , we calculate RB ( x , y ) , and the candidate with the smallest value is the optimal centre.This algorithm generalises to an IFS with more than three transformations in the obvious manner.  Forn  transformations, the optimal centre lies either at the intersection of any k  cones, 3 ≤ k ≤ n  (if they6meet), or at the minimum of one of the nC 2  pairwise intersections.  For each candidate centre ( x , y ) , wecalculate RB ( x , y ) , and the candidate with the smallest value is the optimal centre.  We call thisAlgorithm Ε .Note that if a set of k  cones, 3 ≤ k < n , does not meet, then no superset of that set meets.  AlgorithmΕ  is O (n C2+nC 3 ) , i.e. O (n3 )  in the best case and O ( nC ii =2n∑ ) , i.e. O (2n )  in the worst case, for ntransformations.  The worst case is encountered rarely.5   ResultsWe exercised Algorithm Ε  on a range of fractals to get an idea of the tightness of the bounding circles.We used Algorithm Ε  to generate a bounding circle for a fractal, then plotted the fractal anddetermined the tightest possible bounding circle using a search technique.We found that, for typical fractals, Algorithm Ε  generates bounding circles that are at most 70% largerthan the tightest possible.  Some examples are illustrated in Figures 4(a)—(d).  The outer circle in eachfigure is derived using Algorithm Ε  and the inner circle is the tightest possible bounding circle.Numerical results are given in Table 1 and the parameters for the fractals are given in Appendix A.0012412400124124Figure 4(a)  Bounding circles for curly. Figure 4(b)  Bounding circles for fern.70012412400124124Figure 4(c)  Bounding circles for square. Figure 4(d)  Bounding circles for square rotated π / 4 .Radius CentreCurlyFernSquareSquare + π/4(70.2, 61.7)Algorithm Ε Search56.476.5 (61.2, 71.5)51.451.4(62.0, 62.0)(62.0, 62.0)5155Centre(73, 60)(58, 51)(62, 62)(62, 62)5248Diff.11%39%0%7%Radius ( ±1)Table 1  Bounding circles for the fractals illustrated.The square fractal is instructive.  The basic square is specified by four transformations, each onemapping the square to one of its quadrants.  None of the transformations rotate the square andAlgorithm Ε  is able to derive the tightest possible bounding circle exactly (within measuring error).  Ifwe introduce a rotation of π / 4  to each transformation, we get the attractor shown in Figure 4(d),which clearly occupies a smaller area than the original.  However, Algorithm Ε  derives the samebounding circle: in fact it derives the same bounding circle whatever rotation is introduced.6   ConclusionsPlotting a fractal image from its transformations on the screen of a digital computer requires advanceknowledge of a bounding area for the image.  We have derived a formula that expresses the radius of abounding circle for a fractal in terms of the parameters of the transformations specifying the fractal anddescribed an algorithm for finding the point at which a bounding circle should be centred to minimise its8radius.  Experiments indicate that this algorithm gives reasonable results for typical fractals.  This willimprove the effectiveness of algorithms that plot IFS fractal images.AcknowledgementsWe would like to thank Peter Potts for his help with Mathematica.References[1] BARNSLEY, M.F.: “Fractals Everywhere”, Boston: Academic Press, 1988.[2] BARNSLEY, M.F. and SLOAN, A.D.: “A Better Way to Compress Images”, Byte Magazine,January 1988.[3] EDALAT, A.: “Power Domain Algorithms for Fractal Image Decompression”, Dept. ofComputing Internal Report DoC 93/44, Imperial College, 1993.[4] EDALAT, A.: “Power Domains and Iterated Function Systems”, Information andComputation, Vol. 124, pp. 182-97, 1996.[5] EDALAT, A., SHARP, D.W.N. and WHILE, R.L.: “An Upper Bound on the Area Occupiedby a Fractal”, Proc. ICASSP '95, Detroit, Michigan, Vol. 4, pp. 2443-6, May 1995.[6] FALCONER, K.: “Fractal Geometry: Mathematical Foundations and Applications”, Wiley,Chichester, 1990.[7] HEPTING, D., PRUSINKIEWICZ, P. and SAUPE, D.: “Rendering Methods for IteratedFunction Systems”, Proc. IFIP Fractals, 1990.[8] MONRO, D.M., DUDBRIDGE, F. and WILSON, A.: “Deterministic Rendering of Self-affineFractals”, IEE Colloquium on the Application of Fractal Techniques in Image Processing, IEEPublications, December 1990.[9] SHARP, D.W.N. and CRIPPS, M.D.: “Parallel Algorithms That Solve Problems byCommunication”, Proc. 3rd IEEE Symposium on Parallel and Distributed Processing, Dallas,Texas, pp. 87-94, December 1991.[10] SHARP, D.W.N. and WHILE, R.L.: “Pattern Recognition Using Fractals”, Proc. Int. Conf. onParallel Processing, Illinois, Vol. III, pp. 82-9, August 1993.Appendix AA.1   CurlyCurly is specified by two transformations with the following parameters.a b c d e fT 1 -0.18 0.18 -0.18 -0.18 113.46 90.52T 2 0.89 0.33 -0.33 0.89 -13.64 27.28A.2   FernFern is specified by four transformations with the following parameters.a b c d e f9T 1 0.85 0.09 -0.02 0.85 0.68 3.54T 2 0.00 0.00 0.00 0.16 67.64 86.34T 3 0.20 -0.60 0.10 0.22 115.66 60.37T 4 -0.15 0.64 0.11 0.24 11.50 66.8910A.3   SquareSquare is specified by four transformations.  With a rotation of θ , they have the following parameters.a b c d e fT 1cos θ2−sin θ2sin θ2cos θ231( 2 − cos θ + sin θ ) 31( 2 − cos θ − sin θ )T 2cos θ2−sin θ2sin θ2cos θ231( 2 − cos θ + sin θ ) 31( 4 − 2 − cos θ − sin θ )T 3cos θ2−sin θ2sin θ2cos θ231( 4 − 2 − cos θ + sin θ ) 31( 2 − cos θ − sin θ )T 4cos θ2−sin θ2sin θ2cos θ231( 4 − 2 − cos θ + sin θ ) 31( 4 − 2 − cos θ − sin θ )

Bounding the Attractor of an IFS

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Bounding the Attractor of an IFS

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