A PROBABILISTIC POWER DOMAIN ALGORITHM FOR FRACTAL IMAGE DECODING

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1 Stochastics and Dynamics, Vol. 2, o. 2 (2002) 6 73 c World Scientiic Publishing Company A PROBABILISTIC POWER DOMAI ALGORITHM FOR FRACTAL IMAGE DECODIG V. DRAKOPOULOS Department o Inormatics and Telecommunications, Theoretical Inormatics, University o Athens, Panepistimioupolis 57 84, Athens, Hellas vasilios@di.uoa.gr A. KAKOS and. IKOLAOU Division o Inormation Systems, Bank o Greece, 34 Mesogeion Ave., 52 32, Cholargos, Hellas Received 2 January 2002 Revised April 2002 A new algorithm, called herein the random power domain algorithm, is discussed; it generates the image corresponding to an iterated unction system with probabilities, a technique used in ractal image decoding. A simple complexity analysis or the algorithm is also derived. Keywords: Fractals; random algorithm; image comparison; iterated unction systems; power domain. AMS Subject Classiication: 28A80, 94A08, 68Q0, 68Q55. Introduction A number o algorithms have been proposed or rendering the invariant measure supported by the attractor o an (hyperbolic) iterated unction system, orifs or short, on the plane. In what ollows, however, a competitive alternative will be described and implemented as a consequence o the introduction o domain theory in dynamical systems, measures and ractals [2]. It uses the probabilistic power domain as its computational model and generates the invariant measure o an IFS with probabilities. The proposed algorithm, ater comparing with the most commonly used probabilistic algorithms or rendering the invariant measure supported by the attractor, namely the Random Iteration Algorithm (RIA, see []) and the Random Adaptive Cut Algorithm (RACA, see [4]), shows to be, under all known circumstances, substantially aster than both the others and much more eicient than the RIA in terms o memory requirements. Since the RIA chooses the points o the attractor at random, it will unavoidably redraw some points more than once. Figure shows a demography o the dendrite shown in Fig. 7(e). The dendrite is put on a 6

2 62 V. Drakopoulos, A. Kakos &. ikolaou (a) (b) (c) Fig.. Distribution o an orbit rendering a dendrite displayed in vertical bars using (a) the RIA, (b) the RACA and (c) the proposed algorithm. square lattice on R 2 with small mesh size; the number o points rom the dendrite in each little box o that lattice is measured and represented by a vertical bar, thus visualizing the distribution. Apparently, the border o the dendrite is visited most requently, while the other points seem to be avoided most oten. This substantial waste o time is the main reason why the proposed algorithm, which produces no more than the necessary points, is that aster. Furthermore, many o the points o the attractor will never be plotted, due to the randomness o the RIA; to make matters worse, the attractors produced by two successive runnings o the algorithm will not be exactly the same. The RIA theoretically suggests that its corresponding iterated procedure should be perormed or an ininite number o times beore the actual attractor is produced. In the words o Monro and Dudbridge in [5] There is no deinite stopping criterion or the RIA. On the contrary, the RACA uses a criterion to terminate the recursive descent o the tree o transormations and the idea behind it can be served as a basis or a magniication algorithm. The main advantage o our algorithm is that it encapsulates an economical and enhanced stopping criterion as opposed to the RACA; roughly speaking, it terminates (in inite time) automatically and quickly on any digitized screen without needing to ix a number o iterations in advance. Moreover, or a given discrimination capability o the computer screen, the proposed algorithm has a determined upper and lower bound or the number o computations required beore the best possible attractor or the given resolution is constructed. The exact number o computations, however, cannot be speciied analytically in a closed ormula, but can be very easily computed with the aid o a computer. An analytic description o the algorithm as well as a digest o the theoretical undamentals, on which its model is based, ollows. However, an extended abstract or the theory can be ound in [3], where power domains are discussed along with IFS. A general reerence oriented toward computational theory is [6]. 2. The Probabilistic Power Domain First o all a link between measure theory and domain theory is needed. The link is established in [2]; however, or a deep understanding o the algorithm and its implementation that ollows, we shall make an account o the most undamental conclusions that appear in that paper. We are especially interested in Borel measures, which have the signiicant property that the measure o any Borel set is determined by the topologically important

3 A Probabilistic Power Domain Algorithm or Fractal Image Decoding 63 open sets or, alternatively, the compact sets; such a Borel measure is called regular. More ormally, a Borel measure μ on a locally compact Hausdor space is called regular, i, or all Borel subsets B o, wehave μ(b) =in{μ(o) B O, O open} =sup{μ(f ) B F, F compact}. It can be proved that, i is a locally compact, second countable Hausdor space as in our case, every open set will be σ-compact (it is a countable union o compact sets) and every Borel measure is regular. The computational model o the new algorithm or the construction o the invariant measure o an IFS with probabilities is based on the probabilistic power domain. Hence the ollowing deinitions are essential or the understanding o the proposed algorithm. The lattice o open sets o a topological space is denoted by Ω(). Deinition. An (e)valuation on a topological space is a map ν :Ω() [0, ) which satisies: (i) ν(u)+ν(v )=ν(u V )+ν(u V ) (ii) ν( ) = 0, and (iii) U V ν(u) ν(v ). A continuous (e)valuation is an (e)valuation such that whenever A Ω() isa directed set (with respect to ) oopensetso, then ( ) ν O =supν(o). O A O A Deinition 2. The Probabilistic Power Domain P o a topological space consists o the set o continuous (e)valuations ν on with ν() and is ordered as ollows: μ ν i and only i or all open sets O o, μ(o) ν(o). Deinition 3. For any b, thepoint (e)valuation based at b is the (e)valuation n b :Ω() [0, ) deined by {, i b O, n b (O) = 0, otherwise. Any directed set (μ i ) i I o (e)valuations has a least upper bound (l.u.b.) given by i μ i = μ, whereoro Ω() wehaveμ(o) =sup i I μ i (O); hence (P, ) is a directed complete partial order (d.c.p.o.), with bottom element (which is the P : (O) =0,O Ω()) It is easy to prove that any inite linear combination n i= r in bi o unit point (e)valuations n bi with constant coeicients r i [0, ), i =, 2,,n is also an (e)valuation on. What is really interesting about (e)valuations is that, i is an ω-continuous d.c.p.o., then P is an ω-continuous d.c.p.o. as well; moreover any (e)valuation

4 64 V. Drakopoulos, A. Kakos &. ikolaou on an ω-continuous d.c.p.o. is the directed l.u.b. o linear combinations o point (e)valuations and it extends uniquely to a measure. Hence (e)valuations and measures are in act the same on ω-continuous d.c.p.o s. The set o all positive Borel measures μ on a locally compact, second countable Hausdor space with μ(), denoted by M(), can be partially ordered by putting μ ν i and only i or all Borel sets B, μ(b) ν(b). With this ordering M() becomes a d.c.p.o. 3. A Computational Model For any Hausdor metric space the upper space U consists o all nonempty compact subsets o (see Fig.2), that is U = { C C compact}. This space has a topology, called the upper topology, whose base is the collection a = {C U C a}, where a Ω() isanopenseto. This means that, or any open subset a o, the collection o all compact nonempty subsets o that are included in a orms an element o the base or the upper topology. This topology is T 0 ; the specialisation ordering u o U is the superset inclusion, i.e. A u B a Ω()[A a B a] A B. Under this ordering ( u )thespace(u, ) becomes a d.c.p.o., which means that every directed set has a l.u.b. The l.u.b. o a directed set o compact subsets is their intersection; the elements o are maximal elements o U. The singleton map s: U with s(x) ={x} can be used or embedding onto the set o maximal elements o U. I U is ω-continuous, then we can identiy (e)valuations and measures on U, or a continuous (e)valuation μ on U extends uniquely to a Borel measure on U and a measure on U is a continuous (e)valuation. The support o the invariant measure o an IFS has the very important property that it contains exactly the { x } x O O U =[0, ] [0, ] =[0, ] [0, ] Fig. 2. The upper space or the unit square.

5 A Probabilistic Power Domain Algorithm or Fractal Image Decoding 65 M () M() P U PU : β 0 : O 0 : : O 0 Fig. 3. The probabilistic power domain. points o the attractor. We will now extend the notion o the support in the case o (e)valuations. An (e)valuation μ PU is said to be supported in s(), i μ(u \ s()) = 0; i μ is supported in s(), then the support o μ is the set o points y s() such that μ() > 0 or all neighbourhoods U o y. WedenotebyS() theset that contains all the supported (e)valuations in s() (S() PU). Since the invariant measure o an IFS is a normalized Borel measure, we restrict our attention to normalized measures and normalized (e)valuations. We denote by M (), P U, S () the sets which contain the normalized Borel measures, the normalized (e)valuations and the normalized supported (e)valuations, respectively (see Fig. 3). PU and P U have exactly the same maximal elements and the elements o S () are maximal elements o PU. The maps e: M () S () withe(μ)(o) μ(s (O)) or any O Ω() and j : S () M () withj(ν)(b) ν(s(b)) or any Borel subset B o are well deined and continuous. Since j is the inverse o e, the restrictions o e and j give an isomorphism between M () ands (). Let us recall that PU is ω-continuous whenever U is ω-continuous and in this case any (e)valuation μ PU is the directed l.u.b. o linear combinations o point (e)valuations. I B U is a countable order basis o U, then { n } n r i n bi b i B,r i rational and r i i= is a countable order basis o PU. This observation provides us with an eective way o obtaining measures on as the directed l.u.b. o linear combinations o point (e)valuations on U. I is compact, then both U and P U have bottom elements; the bottom element o U is the space itsel, whereas the bottom element o P U is the unit point (e)valuation n based at U with {, i O U, n (O) = 0, otherwise. i=

6 66 V. Drakopoulos, A. Kakos &. ikolaou The above theoretical ramework can be used or the construction o the invariant measure o an IFS with probabilities as ollows. 4. The Random Power Domain Algorithm Let {;, 2,, ; p,p 2,,p } or, more briely, {; ; p } be an hyperbolic IFS with probabilities; the operator T : P U P U given by T (μ)(o) = p i μ(i (O)) i= is well deined and continuous, and thereore has a least ixed point; it can be proved (see [2]) that the least ixed point o T is supported in and its support is the unique attractor o the IFS. Since an element o S () is a maximal element o P U, we immediately deduce that an hyperbolic IFS with probabilities has a unique ixed point. I μ is the unique ixed point o T,thenj(μ) M () isthe unique invariant measure o the IFS. We correspond the (e)valuation n i i2 in to node i i2 in o the nonprobabilistic tree shown in Fig. 4. The nodes o the tree are point (e)valuations ordered as usual so that the construction o the invariant measure can be obtained by using the tree shown in Fig. 5. Then, the tree is traversed until the outcome o any node i i2 in becomes smaller than a pixel on the computer screen. Then a pruning occurs and the corresponding branch will produce no more children. This tree, however, cannot be employed in any eicient implementation. The solution will be the use o an equivalent tree (see Fig. 6) that can be traversed eiciently. We prove that this alternative tree is computationally equivalent to the original one. The two trees have exactly the same irst two levels. From the third level onwards, however, the dierences start. Even i the second level is identical in both trees, a dierent third level is produced. It is our intention to prove that this does not aect the theoretical basis and that the third level is computationally equivalent. I at some node o the second level s i s i2 2 <εthen holds, due to Fig. 4. The IFS tree.

7 A Probabilistic Power Domain Algorithm or Fractal Image Decoding 67 n p 2 n p p n p p n p 2 n n p 2p n p p 2 n... p n p n p 3 n p 3n p p 2 n p 2 p p n 2p n p p 2 n Fig. 5. The IFS tree with probabilities. n p 2 n p p n p p n p 2 n n p 2p n p p 2n... p n p n p 3 n p 2 p n p 2 p n p p 2 p 2 n p n Fig. 6. The action tree with probabilities. p 3 n the commutative property o the real numbers, s i2 s i 2 <εalso holds. Hence, in the case o the irst tree, s i s i2 2 < ε means that the children o i i2 ( i i2, i i2 2,, i i2 ) will not be produced and the pixel i i2 will be plotted. For this same tree, s i2 s i 2 <εmeans that the children o i2 i ( i2 i, i2 i 2,, i2 i ) will not be produced and the pixel i2 i will be plotted. On the other hand, or the second tree, the condition s i s i2 2 < ε means that the children o i i2 ( i2 i, i2 i 2,, i2 i ) will not be produced and the pixel i i2 will be plotted. Moreover, the condition s i2 s i 2 < ε, or the second tree, means that the children o i2 i ( i i2, i i2 2,, i i2 ) will not be produced and the pixel i2 i will be plotted. It is now more than obvious that the third level o the two trees are computationally equivalent. Using induction we can prove that the two trees are equivalent at all levels. Thus the two trees are, or our purposes, computationally equivalent. Our contribution, i.e. the discovery o the equivalent tree, has beneicially eected the implementation o the algorithm. ot a single redundant computation is perormed. On the other hand, the memory requirements have signiicantly dropped due to the introduction o this second tree.

8 68 V. Drakopoulos, A. Kakos &. ikolaou I a branch is pruned, or example at node i i2 in, then the corresponding atomic measure or the pixel o the attractor will be p i p i2 p in, which is assigned by the corresponding point (e)valuation p i p i2 p in n i i2 in. However, two pruned branches might correspond to the same pixel. In this case, the measure contribution o all branches that end up at the same pixel should be summed up beore any rendering o the pixel takes places. In this way, a record o which branches correspond to the same pixel must be kept so that one is able to add up all individual contributions to determine the actual atomic measure o every screen pixel o the attractor. In this way a graphic representation o the invariant measure is achieved. Although the theory suggests that at depth n, T (n) μ 0 is an (e)valuation, μ 0 being any normalized (e)valuation, the unbalanced pruning o the tree might raise some objections regarding the soundness o our implementation. However, i d max is the level at which the last pruning takes place, then we can saely assume that our approximation to the invariant measure μ is T (dmax) μ 0, μ 0 being the uniorm measure on the screen. This means that we can make the equivalent theoretic assumption that all branches o the point (e)valuation tree o Figs. 5 and 6 are pruned at level d max. The ollowing argument justiies this assumption. For any branch o the tree which is pruned at an earlier stage d than d max we can insert its children and grandchildren eectively pruned in the tree diagram until the complete level d max is reached. Then our approximation to the invariant measure is still a sound one or, i at level d we had decided that the parent node represented less than a pixel on the screen, then its children would still be included in that pixel (due to the ordering o nodes); then their total measure contribution to the pixel would be equal to that o their parent at level d. This means that the sum o the atomic measures that all children have in total would be equal to what the node at level d had. Thus, this balanced pruning which we just proved to be equivalent to the unbalanced one we perorm justiies our argument that what we actual produce is a measure, in act T (dmax) μ The Algorithm and Its Complexity Hence the algorithm can be described roughly as ollows; we start rom the ixed point which corresponds to the transormation with the greatest probability and traverse the action tree until a parallelogram i i2 i becomes small enough so that it is actually a pixel on the computer screen. This happens when s i s i2 s i 2 <ε, ε =/M,whereM is the resolution o the screen, and in that case this branch has contributed p i p i2 p i to the measure o the pixel i i2 i. Ater having sum up all measure contributions or each pixel, we color the attractor pixels accordingly. In that way, a graphic representation o the invariant measure is produced. The actual algorithm in a orm o pseudo-code, which also provides a deinition or the equivalent tree o Fig. 6, has as ollows:

9 A Probabilistic Power Domain Algorithm or Fractal Image Decoding Start.. Compute all contractivity actors s i. 2a. Assign measure on the whole screen, i.e. μ([0, ] 2 )=. 2b. All atomic measures μ(x) opixelsx [0, ] 2 are initially set to zero. 3a. Call procedure measure ([x 0,x 0,,x 0 ], 0, [ 2, 2,, 2], [,,,]), or x 0 [0, ] 2. 3b. Plot all pixels x [0, ] 2 with a color proportional to their atomic measure μ(x). 4. End. where procedure measure([x,x 2,,x ],q,[d,d 2,,d ], [m,m 2,,m ]) { or i =,, do { x i = i (x q ) d i = s i d q m i = p i m q } or i =,, do { i (d i > /M ) then do call measure([x,x 2,,x ], i,[d,d 2,, d ], [m,m 2,,m ]) else do μ(x i ) = μ(x i ) + m i } } Having described the algorithm, we shall try to identiy the number o computations needed or the construction o the attractor. Since the contractivity actors o the aine transormations are known, we can ind an estimation or the depth o the tree. I s max and s min are the largest and the smallest contractivity actors, respectively, then the depth o the tree will lie between [ ln M d max = ln s max ] + and d min = [ ln M ln s min ] + that are the greatest and the smallest depth, respectively. This means that in the worst case we need 0( dmax ) = 0[( dmax+ )/( ) ], thus giving O( dmax ) while in the best case we need 0( dmin )= 0[( dmin+ )/( ) ] computations. This number is explained by the act that nine computations are needed or the calculation o the new point and only one computation is needed or the calculation o the quantity s i (s i2 s in 2) since at each step the quantity s i2 s in 2 has already been computed in previous nodes.

10 70 V. Drakopoulos, A. Kakos &. ikolaou The exact number o computations perormed beore the construction o the attractor is C( 2), where {, i x<ε C(x) = i= C(s ix)+0, otherwise, where is the number o aine transormations and ε =/M, M being the resolution o the screen. Although, it is extremely diicult to ind a closed ormula which gives the exact number o computations, the above mentioned recursive ormula can be used to obtain this number with the aid o a computer. (a) (b) (c) (d) (e) () (g) Fig. 7. The Sierpiński triangle, a dendrite, a spiral, a meander, a second dendrite, a ern lea and a maple lea Sierpinski Dendrite Spiral Meander Dendrite II Fern lea Maple lea Time (sec) RPDA Method RACA RIA Sierpinski Dendrite Spiral Meander Maple lea Fern lea Dendrite II IFS Fig. 8. The time results o the systematic comparison.

11 A Probabilistic Power Domain Algorithm or Fractal Image Decoding 7 6. Comparative Results The current implementation o our algorithm is written in Microsot Visual Basic 6.0. It is capable o drawing ractal images using the Random PDA, the RIA and the RACA, and displaying the depth o the action tree, the number o points used or rendering and the total runtime. The ractal images (M = 00) used or the comparisons o the various algorithms are illustrated in Fig. 7. The Random PDA was inally tested and rated by comparing the various ractal attractors produced by it versus the attractors produced using the RIA and the RACA. Time results are given in CPU seconds on a Pentium III PC with a 400 MHz CPU clock running Windows 98. As we have already stated, the proposed algorithm is more eicient than the RIA and the RACA; the systematic comparison we perormed gave the results shown in Fig. 8 and justiies our argument. 7. Conclusions Our algorithm is extremely eicient or the decoding o pictures which have been coded using some method based on IFS. This is explained by the act that natural images such as the ace o a person lack sel-similarity and hence a lot o aine transormations are employed or the coding; each o them, however, will have a very small contractivity actor and hence our tree will have small depth (0 or the maple lea, 3 or the ern lea, 45 or the spiral and 8 or the other igures); so, the construction o the attractor will terminate very quickly, since only a ew number o points are necessary (see Table ). Table. The number o pixels used or the systematic comparison. Pixels drawn RPDA RACA RIA Sierpiński Dendrite Spiral Meander Dendrite II Fern lea Maple lea One should also mention that without the use o the equivalent trees, it is doubtul i this perormance o the Random PDA could have ever been achieved. Any implementation that avoids their use is bound either to make use o huge amount o memory or unavoidably to resort to recomputations. In both cases, such an implementation becomes highly ineicient and the perormance o the algorithm would be substantially decreased.

12 72 V. Drakopoulos, A. Kakos &. ikolaou Appendix A transormation is aine, i it may be represented by a matrix A and translation t as (x) =Ax + t, or(ix R 2 ) ( ) ( )( ) ( ) x a b x d = +. y c s y e The code o isthe6-tuple(a, b, c, s, d, e), and the code o an IFS is a table whose rows are the codes o, 2,,. I we add one extra column with the corresponding probabilities p,p 2,,p, then we are talking about the code o an IFS with probabilities. We list the IFS codes (see Tables 2 8) or the examples discussed in the main text. Table 2. The IFS code or the Sierpiński triangle. a b c s d e p Table 3. The IFS code or a dendrite. a b c s d e p Table 4. The IFS code or a spiral. a b c s d e p Table 5. The IFS code or a second dendrite. a b c s d e p

13 A Probabilistic Power Domain Algorithm or Fractal Image Decoding 73 Table 6. The IFS code or a meander. a b c s d e p Table 7. The IFS code or a ern lea. a b c s d e p Table 8. The IFS code or a maple lea. a b c s d e p Acknowledgements The authors wish to thank the anonymous reviewer or all the helpul suggestions. Reerences. M. F. Barnsley, Fractals everywhere (Academic Press, 993), 2nd edition. 2. A. Edalat, Dynamical systems, measures and ractals via domain theory, Inorm. Comput. 20 (995) A. Edalat, Power domains and iterated unction systems, Inorm. Comput. 24 (996) D. Hepting, P. Prusinkiewicz and D. Saupe, Rendering methods or iterated unction systems, infractals in the Fundamental and Applied Sciences, eds. H. O. Peitgen, J. M. Henriques and L. F. Penedo (orth-holland, 99), pp D. M. Monro and F. Dudbridge, Rendering algorithms or deterministic ractals, IEEE Computer Graphics Appl. 5 (995) G. D. Plotkin, A note on inductive generalization, in Machine Intelligence, eds. B. Meltzer and D. Michie (orth-holland, 970), Vol. 5,