Image Anal Stereol 2011;30:135-142 Original Research Paper MINKOWSKI-ADDITIVE MULTIMEASURES, MONOTONICITY AND SELF-SIMILARITY Davide La Torre1 and Franklin Mendivil2 1 Department of Economics, Business and Statistics, University of Milan, Italy; 2Department of Mathematics and Statistics, Acadia University, Wolfville, Nova Scotia, Canada. e-mail: davide.latorre@unimi.it, franklin.mendivil@acadiau.ca (Accepted August 31, 2011) ABSTRACT We discuss the main properties of positive multimeasures and we show how to define a notion of self-similarity based on a generalized Markov operator. Keywords: cone-positive multimeasure, fractal transforms, positive multimeasures, self-similarity. INTRODUCTION In the first part of this paper we introduce two different notions of positive multimeasures, namely positive multimeasures and cone-positive multimeasures, and we analyze some mononicity properties of these classes of multimeasures. In the second part we introduce a definition of self-similar multimeasure based on a Markov operator. There are many applications of set functions to different areas such as mathematical economics, decision theory and social sciences; for this reason, many variations on and extensions of measures have been provided in the literature including, for instance, subadditive and superadditive set functions, submeasures, null-additive set functions and so on. The notion of multimeasure (Vind, 1964; Debreu and Schmeidler, 1967; Artstein, 1972) is one of these possible generalizations which consider set-valued set functions instead of set functions. Some practical motivations behind this kind of mathematical structures can be found, for instance, in mathematical economics, when coalitions are considered as primitive economic units (Vind, 1964; Debreu and Schmeidler, 1967) or in the theory of capacities (Choquet, 1953). There has been a wealth of work on multimeasures and generalized measures, both theoretical and in applications (Choquet, 1953; Vind, 1964; Debreu and Schmeidler, 1967; Brooks, 1968; Artstein, 1972; Hildenbrand, 1974; Drewnokski, 1976; Pap, 1991; Kan, 1992; Alo etal., 1980; Guo and Zhang, 2004). Our purpose in this paper is to provide a useful class of multimeasures for modelling of images and information derivable from images. Images are often modeled as functions or measures and implicitly these are most often positive. In the case of measures, restricting to positive measures greatly simplifies the technical details as spaces of positive measures have nicer properties than spaces of signed measures (one example is the relationship between bounded variation of a positive measure ß and the boundedness of the total mass, ß(^), of ß). Thus we examine two different classes of positive multimeasures. Many images exhibit approximate self-similarity and this structure has proven very useful in applications in image compression, representation and analysis (Hutchinson, 1981; Barnsley and Demko, 1985; Barnsley etal., 1985; Barnsley, 1989; Forte etal., 1999; Forte and Vrscay, 1999; lacus and La Torre, 2005a;b; Kunze et al., 2007; 2008; 2012; La Torre and Mendivil, 2008; 2009; La Torre etal., 2009). For instance, in fractal image coding based on Iterated Function Systems (IFS) and their generalization, the self-similar attractor is defined in terms of a compact set and a positive measure supported on it. The positive measure is the unique fixed point of a Markov operator defined on a suitable space of positive measures. The main idea behind this paper is to extend this approach to the case of multimeasures, by defining an appropriate complete space of multimeasures and a contractive Markov operator which will be used to introduce a notion of self-similarity in this class. The paper is organized as follows: next section reviews the definition of (Minkowski additive) multimeasures and introduces a metric space of multimeasures which is complete once an extension of the Monge-Kantorovich metric is given. The third section introduces a weak notion of positive multimeasure; for a multimeasure O to be positive it is enough that 0 belongs to O(A) for all measurable sets A. In the fourth section a stronger notion of positive multimeasure is provided; in this context, a multimeasure <ä> is positive if <ä>(A) C P, for all measurable sets A, where P is an ordering cone. Finally a definition of self-similarity based on a notion of generalized Markov operator and some examples are provided in the last section. ADDITIVE MULTIMEASURES Consider a nonempty set Q. and a cr-algebra on Q. A set-valued measure or multimeasure on {O.,^) is a function (for more on multimeasures see Artstein, 1972) <ä>: ^ ^ {/sT C M'" : /sT / 0} , which for any sequence of disjoint sets A, G SS satisfies \=1 ^ i=l The right side is the infinite Minkowski sum given by ^Ki = I ^ki: ki G Ki, ^\ki\ < +oo|. i y i i ) For A c M'" and ^ G R'" recall that supp(^,A) =sup{^-x:xGA} defines the support function. For a multimeasure <ä> let <ä>'?(ß) = sup{^-x: X G <ä>(ß)} = supp(^,<ä>(ß)). Then <ä>'^(-) is a signed measure with values in (-00,00]. Versions of the Radon-Nikodym theorem have been proved for multimeasures (Artstein, 1972). Let B[c(M'") be the space of all compact and convex subsets of R*". In the following we work with multimeasures defined on the Borel cr-algebra ^ of a complete metric space Q. taking values in Hc(M'"). Let Q,Ke Hc(M'") with Q CK. We define to be the set of all Minkowski aadditive multimeasures <ä> on Q which take values in Hc(M'") and such that 1. <ä>(fl) = ß 2. There is some set D G ]H[c(M'") and a G fl so that /(x) d<ä>(x) C D for all / G Lipi (Q) with f{a) = 0. 3. <ä>(A) C ^ for all A. Let = {x G M'" : ||x|| = 1} be the unit sphere in R'". We define the following metric on R*"), Äf(l,2) = supiMK,f)• peSl In this, diviiß, v) is the natural extension of the usual Monge-Kantorovich metric between two probability measures (see Kunze er a/., 2012). It can be proved that the space R*") , ^m) is a complete metric space (Kunze et al., 2012). POSITIVE MULTIMEASURES We say that a multifunction F : D C R" ^ R'" is positive if 0 G F{x) for all x G D. In a similar way one can define positive multimeasure; given a measurable space (Q,^) a multimeasure 0 : ^ ^ X is said to be positive if 0 G <ä>(A) for all A G Let us define be the subspace of positive multimeasures; using the completeness of ^Q^xi^,^'") it is easy to show that R*"), (Im) is a complete metric space. The following result is easy to prove. Proposition 1. If is a positive multimeasure then <ä>^ is a positive measure for all p G S^. Given two positive multimeasures <ä>: ^ X and »/A : ^ ^ X we say that <ä> is absolutely continuous with respect to Y if is absolutely continuous with respect to y'' for all p ^ S^ and in this case we write <ä> < »/A. If <ä> is absolutely continuous with respect to Y and »//(A) = {0} then <ä>(A) = {0}. The following result provides a list of some simple properties of positive multi-functions and positive multimeasures. Let F be a positive multi-function and <ä> be a positive multimeasure. Then: 1. For all / : fl ^ R we have f{x)F{x) is a positive multi-function. 2. For all p G R*", the real-valued function FP defined by FP{x) = supp(p,F(x)) is nonnegative. 3. AC B imphes that <ä>(A) C (A) C <ä>(Q) for all A. 4. For all p G R*", the signed measure <ä>^ is a positive measure. 5. If is a positive measure then the multimeasure <ä> defined by <ä>(A) = f^F(x)dju(x) is a positive multimeasure. 6. If 0 < /(x) < g(x), then L/(x)d<ä>(x) C ^g(x)d<ä>(x). Proposition 2. Suppose that F is a multifunction, F{x) is a convex set for all x, ß is a positive measure and <ä> defined by 0{A)=J^F{x)dß{x) is a positive multimeasure. Then F is ji-a.e. a positive multifunction. Proof. Suppose that the conclusion is false. Then there is some set A C Q with ^{A) > 0 and 0 ^ F{x) for all X e A. Let pn be a countable dense sequence in For each x G A, there is some n G N so that supp{pn,F{x)) < 0. Thus, with A'^ = {x G A ■. supp{pn,F{x)) < -l/m} we have A= U n,meN and so there are n,m with > 0. However, then this means that 0> - >f supp(p„,F(x))dM(x) = Ja'H supp [pn, I F{x) dß (x) ) = supp(p„ M^n)), \ JK J and thus 0 ^ which contradicts the fact that <ä> is positive. Thus F must be a positive multifunction. □ Corollary 1. If ^ is a positive multimeasure which is absolutely continuous with respect to the positive measure ji, then the Radon-Nikodym derivative F{x) o/<ä> with respect to ß is a positive multifunction. We mention that in Proposition 2 it also works if is a signed measure and not just a positive measure. To see this, one just decomposes Q. into positive and negative sets and apply the same proof to each part. Proposition 3. Suppose that <ä> is a positive multimeasure and f is a real-valued function on Q. Then v defined by v{A)= JJ{x)d^{x) is a positive multimeasure. Proof First suppose that f{x) > 0 for all x. Then for any A we have Q< jJ{x)d^P{x) = /(x)d<ä>(x)^ =supp(p,v(A)) , for any p G S^. Thus 0 G v(A) and so v is positive in this case. If /(x) < 0 for all x, we just use the fact that supp(p,/(x)<ä>(A)) = supp(-p,|/(x)|<ä>(A)) and get the same result. Now, if / is a general real-valued function, let P = {x G Q : /(x) > 0} and A^ = {x : /(x) < 0}. Then A = {Af^N) U (AnP) and so v(A) = v(A n A^) + v(A n P). However, by the argument from the first part (where / is positive) we see that 0 G v(AnA^) and 0 G v(AnP) and so 0 G v(A) and thus v is a positive multimeasure. □ We now deal with type of monotone convergence for positive multimeasures. Proposition 4. Let <ä>„ be a sequence of positive multimeasures with <ä>„(A) C defined by <ä>(A) = U<ä>„(A) is also a positive multimeasure. Proof The only non-trivial property is countable additivity. Thus, suppose {A^} are disjoint sets in Notice that C i;^<ä>„+i(A^). Using this and the definition of countably additive, we see that C £ 'Lm^i^m) iff for all £ > 0, there exists some G N and points yi G <ä>(Am;) with m, < N so that IE/J/ - CI < Further, this happens iff for all £ > 0 there is some N eN and yi G where n, < N and m, < N so that | Y^iJi - CI < On the other hand, C G «J^Iu^Am) iff for all £ > 0 there is some z G U„<ä>„(UmAm) with |z- CI < £ which happens iff for all £ > 0 there is some N eN and a points yi G <ä>„; (A^^) with m < N and m, „ be a sequence of multimeasures with <ä>n(A) C defined by <ä>(A) = U<ä>„(A)=lim<ä>„(A) ^ n n is a positive multimeasure, where the limit is taken in the Hausdorjf distance. Iff is any \\-integrable nonnegative function on Q then f /(x)d<ä>„(x) / f /(x)d<ä>(x) , Jü. Jü. where again the convergence is in the Hausdorff metric. Proof. For any A G ^^ and p G M*", we see that / <ä>^(A) = supp(p,<ä>(A)) = supp p, |J<ä>„(A) V n = supsupp(p,<ä>„(A)) = limsupp(p,<ä>„(A)) n « = lim<ä>^(A) ^ is a well-defined measure and thus <ä> is well-defined as a multimeasure with compact and convex values. Furthermore, 0 G <ä>n(A) C <ä>(A) C K for all n and A G Thus <ä> is a bounded positive multimeasure. Notice that we know lim<ä>^(A) = <ä>^(A) for all p G R*" and Ag ^ and that this trivially implies that for all simple functions g we have lim f g(x)d<ä>^(x) = f g(x)d<ä>^(x) « Jo. Jo. for each p. We know |<ä>n(A)| < |<ä>(A)| for all n and thus / is also |<ä>„|-integrable for any n. Since any selector for <ä>„ is also a selector for <ä>„+i and for <ä>, for all n we have JQ. /(x)d<ä>„(x)c f /(x)d<ä>„+i(x)c f /(x)d<ä>(x). Jq. Jq. Further, /^/(x)d<ä>(x) C R*" is compact and convex. To show the reverse inclusion, suppose that it isn't true. That is, suppose that there is some zG^/(x)d<ä>(x) but z^L:=U^/(x)d<ä>„(x). Then since L is compact and convex, there is some £ > 0 and a. p e S^ with z-p > supp(p,L) + e which implies that jjix)d^pix) = supp [p,jjix)d^nix) < supp (^p,^/(x)d<ä>(x)^ -e/2 = [ f(x)d<^(x)-e/2 Jq for any n. Now, there is an increasing sequence of simple functions gk which converge upwards to / on Q.. So, by the Monotone Convergence Theorem for finite positive measures we know there is some N so that k>N implies that f gk{x)d^P{x)> f f{x)d^P{x)-e/2. Jq Jq But then [ g,(x)d<ä>^(x)""°° [ gk{x)d^P{x) Jq <— Jq < [ /(x)d<ä>^(x)< [ f(x)d<^(x)-s/2 Jq Jq < f gk{x)d<^{x) , Jq which is a contradiction. □ CONE-POSITIVE MULTIMEASURES Let P C R*" be a pointed convex cone. P induces in R*" an order in the usual manner, that is a

0,yp e P} be the dual cone. The following properties are easily proved: 1. ACPiff{0} 0 and supp(-^,A) < 0. 5. A,B C P with A supp(-^,ß) for all q^P*. 6. A + C = B with C C Rl' and ^ G implies that supp(^,A) < supp(^,ß). We say that a multifunction F : D CW is cone-positive if F{x) CP for all x G D. In a similar way one can define cone-positive multimeasure; given a measurable space (Q,^), where be a sigma-algebra defined over Q, a multimeasure : ^ X is said to be cone-positive if <ä>(A) C P for all A G Let us define be the subspace of cone-positive multimeasures; by using the completeness of it is easy to show that is a complete metric space. We have the following properties: 1. if A C Z? then <ä>(A)

(ß), that is <ä> is monotone with respect to P 2. if <ä> is a cone-positive multimeasure and p e P* then <ä>^ is a positive measure and is a negative measure 3. if 0 < /(x) < g(x) for all x and <ä> is a cone-positive multimeasure, then 0 < Xi/d<ä>< Xigd<ä>, VA 4. if P is a cone-positive multifunction and is a positive measure then <ä>(A) = J F{x)dß defines a cone-positive multimeasure 5. if <ä> is a cone-positive multimeasure and is absolutely continuous with respect to a positive measure ß, then d<ä>/d;U is a cone-positive multifunction 6. if <ä> is cone-positive multimeasure and / is a positive function then v(A) = /^/(x)d<ä>(x) is a cone-positive multimeasure. Proposition 6. Let A,B (Z P, convex and compact. ThenA

supp{-p,B). Proof. Since A

supp{—p,B). Suppose the converse is not true. Then there exists b e B with b +P wd then a ^b for all a e A. So for aGA,3paGP* such that pa{a) > Pa{b), that is -Pa{b) < -Pa{a). But we assumed supp(-p,A) > supp(-p,B) so we have a contradiction. □ In particular A C P iff supp(-p,A) < 0 for all peP*. Proposition 7. Suppose that f^F(x)dju(x) CP far all A. Then F{x) C P ß-a.e. Proof. For all p ^P* and all A we have that jmpp{-p,F{x))d^{x) <0, that is supp{-p,F{x)) < 0 ß-a.e. and then F{x) C P ß-a.e. □ Given two cone-positive multimeasures and <ä>, we say that is absolutely continuous with respect to <ä> and we write <ä> ^ , iff <ä>(A) = {0} implies *F(A) = {0}. It can be easily proved that this is equivalent to require that is absolutely continuous with respect to <ä>^ for all p G P*. The following result is an analog of the fact that a positive measure has bounded total mass iff it has bounded variation. Proposition 8. Let ^ be a cone-positive multimeasure with convex and compact values in R'^. Suppose P is such that int{P*) / 0- Then there exists K ^ is a positive measure and so = <ä>^(Q\A) + <ä>^(A) and this imphes <ä>^(A) < that is supp(p,<ä>(A)) < supp(p,<ä>(Q)) for all p G P*, VA. Let M = suppgp» ||p||^iSupp(p,<ä>(Q)) < oo and K = G P : X • p < M}. Clearly, supp(p,<ä>(Q)) < M for all p G P*, \\p\\ = 1, so (A) C K, for all A. Thus we show K is compact. Suppose not, then there existx„ G K such that ||x„ || > n. Let p G int{P*), \\p\\ = 1 and £ > 0 such that B^ip) C P*. Further, let g„ G M"' be such that ||g„|| = 1 and gnXn = ||x„|| > n. Then p + £g„ G P* and so The following proposition 9 and corollary 2 can be easily proved. Proposition 9. Suppose that 0 < fn / f, where f is a bounded function. Suppose further that <ä>„ is a sequence of cone-positive multimeasures such that n(A) C <ä>„+i(A), VA, Vn with < M for all n andpGP* with \\p\\ = 1. Let^{A) := [jn^n{A). Then J fndn J fd'i> in the Hausdorff metric. Corollary 2. Let <ä>„ be a sequence of cone-positive multimeasures and suppose that there is a compact set K with „(A) C <ä>„+i(A), VA C K, Vn. Then <ä>(A) := is a cone-positive multimeasure and Ij^fd^n JAf^^f'^^ ^^^ positive function f and M A (Z K. SELF-SIMILAR MULTIMEASURES Iterated applications of an IFS Markov operator operating on a probability measure converge to a self-similar measure which is invariant under the action of the functions in the IFS. We now turn to the definition of self-similarity through an IFS Markov operator on these multimeasures but, to explain this, we give abrief review of the relevant construction. Let {0.,d) be a complete metric space and let ^ be the corresponding Borel c7-algebra. Let w; : Q ^ Q for / G {1,2,..., A^} be uniformly contractive. That is, there exists 0 < c < 1 such that d{wi{x),wi{y)) < cd{x,y) for each i and all x,y G Q.. Let {pi)f^i be a collection of probabilities such that Pi > 0 and Pi = This determines the IFS {wi,pi,i = I...N). The Markov operator associated with this IFS is an operator on probability measures ß over {Q., which is defined by Mß(B)=Y,Piß {wTHB)) (1) M >{p + egn)Xn = pXn + EgnXn > En-M which is a contradiction. Thus K is compact. □ for all Borel sets B G If ß is supported on B and the Wi{B) are mutually disjoint then the result of this operator is to assign probability p i to Wi{B), that is Mß{wi{B)) = pi. A second application of M assigns probability pipj to the set Wi{wj{B)), a third application assigns probability PiPjPk to the set Wi{wj{wk{B))), and so on. This recursively partitions a limit probability measure over Q. which is concentrated on the fractal set defined by the w;'s. Our intention is to generalize this process to multimeasures; the formulation here of an IFS type method over multimeasures represents recent results from an ongoing research programme concerning the construction of appropriate IFS type operators or generalized fractal transforms, including integral and wavelet transforms, over various function spaces, distribution spaces and spaces of generalized measures. Fix two compact and positive sets K, Q (we can consider either notion of positivity) and recall that is complete. Let w/ : Q ^ Q for i = \,2,... ,N he contractive with the Lipschitz factor for Wi being q. We also take linear functions 7] : M'" ^ M'" with l^i TiQ = Q and E/es TiK C K for all 5 C {1,2,... (the choice K = XQ for some X > 1 often works, but might be overly restrictive). We define the IFS operator (2) for all ß G A simple argument shows that M<ä> G whenever <ä> G (to see this, it is useful to note that each 7] is continuous with respect to the Hausdorff metric since 7] is linear, and thus Lipschitz). Theorem 1. For the IFS Markov operator defined above we have / \ dM (M<ä>i, M<ä>2) < ^ Q11 11 Um (i , 2) (3) V for all <ä>i, <ä>2 G Now, we see that the function f = Y.i\\Ti\\f ow i has Lipschitz factor at most Q||7]||. Thus, when we take the supremum over all q and all Lipschitz functions, we get that / \ iM(M<ä>i,M<ä>2)< Y.^^mw iM(l,2), V as was desired. □ Example 5.1. As illustrative example, we choose K = Q CW" to be the closed unit ball and Q = [0,1] with Wi{x) = x/2 + ill for i = 0,1. Further let po G (0,1) and Pi = I—Po and define Ti = pj. Then the invariant multimeasure for the IFS Markov operator TiHw^'iB)) (4) is the measure Qji where ji is the probability measure which is the invariant distribution for the standard IFS with maps {wo,wi} and probabilities {po,pi}. In this case, the multimeasure is rather simple, being the product of the scalar (probability) measure ß and the set Q. Fig. 1 shows two multimeasure attractors of IFS Markov operators and they have been obtained with the following parameters: po = 0.3, pi = 0.1 for both of them and, for the rectangular, one direction is 0.3 and 0.7 and the other direction is 0.7 and 0.3 The Wi are X12 +i/I, i = 0..l. Proof First we note that for linear T and convex A, we have sup supp(^, TA) = sup supp(r*^,A) qeSi qeSi < ||r*|| supsupp(^,A) = ||r|| supsupp(^,A) . qeSi qeSi Let C = max;Q be the contraction factor for the IFS. For a given fixed q^Si and Lipschitz /, we have f{x) d [supp(^, MOi (x)) - supp(^, (x) )] = / /Wd Ja (wr^x))) i d[supp(^,0i(y))-supp(^,02(3'))] • Fig. 1. Rectangular and circular positive multimeasures. Example 5.2. Let Q. = [0,1], wo(x) = x/3, wi{x) = x/3 + 2/3 (so co = ci = l/3) and To{x,y) = Tiix,y) = a 0 vo V^'y V l-a 0 0 a J ( x\ \y) (5) (6) with l/2 is a cone-positive multimeasure with = Q, then = Q and r is a cone-positive multimeasure. Thus r(A) C r\p*eP*,\\p*\\=i{^ : X/?* < M} with the same M. It is also trivial to see that if is positive in the sense that 0 G (A) for all A, then is also positive in this same sense. Thus M preserves both classes of positive multimeasures. Fig. 2. Lower and upper Lena images CONCLUSIONS At the end of this paper, it is worth spending few words to justify why the concept of positive multimeasure arises quite naturally in the context of fractal image coding. In fact, in the classical IFS coding an image F can be modeled as an LP function or as a positive probability measure. Practically speaking, these are two identical ways to look at a given image; when a function-based representation is used, at each pixel in the domain the function assigns the color corresponding to that pixel. However, when the color of pixel can not be assigned with precision, it might be preferable to use a measure-based approach which assigns the averaged color of a given block of pixels. From a mathematical point of view, there exists a straight correspondence between the above approaches; when a positive function is integrated with respect to the Lebesgue measure, this leads to a positive measure and vice versa, if the the positive measure is absolutely continuous with respect to the Lebesgue measure then, by Radon theorem, there exists a density. In other words these two approaches can be understood as the two faces of the same medal. 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