Fractals, fractal-based methods and applications. Dr. D. La Torre, Ph.D.

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1 Fractals, fractal-based methods and applications Dr. D. La Torre, Ph.D. Associate Professor, University of Milan, Milan, Italy Adjunct Professor, University of Guelph, Guelph, Canada Adjunct Professor, Laurentian University, Sudbury, Canada 1

2 Book Fractal-based methods in analysis (2012) - ISBN Springer (with H.Kunze, F.Mendivil, E.R.Vrscay). 2

3 Editorials Guest Editor (with H. Kunze) of the International Journal of Mathematical Modeling and Numerical Optimization ( Special Issue on: Fractals, Fractal-based Methods and Applications Guest Editor (with E.Bongiorno, A.Micheletti, G.Naldi) of the Journal Image Analysis and Stereology ( Special issue on: Shape and Size in Medicine, Biology, Material and Social Sciences Editor of the International Journal of Applied Nonlinear Science 3

4 Outline Preliminary properties Iterated Function Systems (IFS) for measure valued functions IFS on multifunctions Inverse problems using fractal-based methods Self-similar objects in economics 4

5 Preliminary properties The following results provide the basis for fractalbased approximation methods. Theorem. ( Banach Fixed Point Theorem ) Let (X, d) be a complete metric space. Also let T : X X be a contraction mapping with contraction factor c [0, 1), i.e., for all x, y X, d(t x, T y) cd(x, y). Then there exists a unique x X such that x = T x. Moreover, for any x X, d(t n x, x) 0 as n. Theorem. ( Collage Theorem ) Let (X, d) be a complete metric space and T : X X a contraction mapping with contraction factor c [0, 1). Then for any x X, d(x, x) 1 d(x, T x), (1) 1 c where x is the fixed point of T. 5

6 Theorem. ( Continuity of fixed points ) Let (X, d) be a complete metric space and T 1, T 2 be two contractive mappings with contraction factors c 1 and c 2 and fixed points x 1 and x 2, respectively. Then where d(x 1, x 2) 1 1 c d sup(t 1, T 2 ), (2) d sup (T 1, T 2 ) = sup x X and c = min{c 1, c 2 }. d(t 1 (x), T 2 (x)) (3) 6

7 Iterated Function Systems (IFS) (M.Barnsley and J.Hutchinson) (X, d) denotes a compact metric base space, typically [0, 1] n. Let w = {w 1,, w N } be a set of 1-1 contraction maps w i : X X, to be referred to as an N-map IFS. Let c i [0, 1) denote the contraction factors of the w i and define c = max 1 i N c i. Now let H(X) denote the set of nonempty compact subsets of X and d H the Hausdorff metric, that is d H (A, B) = max{sup x A inf d(x, y), sup y B x B Then (H, h) is a complete metric space. inf d(x, y)} y A Associated with the IFS maps w i is a set-valued mapping ŵ : H(X) H(X) the action of which is defined to be ŵ(s) = N w i (S), i=1 S H(X), 7

8 where w i (S) := {w i (x), x S} is the image of S under w i, i = 1, 2,, N. Theorem. ŵ is a contraction mapping on (H(X), h): d H (w(a), w(b)) ch(a, B), A, B H(X). Corollary. There exists a unique set A H(X), such that w(a) = A, the so-called attractor of the IFS w. Moreover, for any S H(X), d H (w n (S), A) 0 as n. 8

9 IFS with Probabilities (IFSP) (M.Barnsley and J.Hutchinson) Let M(X) denote the set of Borel probability measures on X and d M the Monge-Kantorovich metric on this set: d M (µ, ν) = where [ sup f Lip 1 (X,R) X f(x)dµ X ] f(x)dν, Lip 1 (X, R) = {f : X R f(x 1 ) f(x 2 ) d(x 1, x 2 ), x 1, x 2 X}. We assume N i=1 p i = 1. Associated with this IFS with probabilities (IFSP) (w, p) is the so-called Markov operator, M : M(X) M(X), the action of which is S H(X). ν(s) = (Mµ)(S) = N i=1 p i µ(wi 1 (S)), 9

10 Theorem. M is a contraction mapping on (M(X), d H ): µ, ν M(X). d M (Mµ, Mν) cd M (µ, ν), Corollary. There exists a unique measure µ M(X), the so-called invariant measure of the IFSP (w, p), such that µ = Mµ. Moreover, for any µ M(X), d M (M n µ, µ) 0 as n. There is an ongoing research program on the construction of appropriate IFS-type operators, or generalized fractal transforms, over various spaces, i.e., function spaces and distributions, vector-valued measures, integral transforms, wavelet transforms, multifunctions, multimeasures. ( 10

11 Barnsley s spleenwort fern attractor on [0, 1] 2 is produced via the following IFSP: f 1 (x, y) = f 2 (x, y) = f 3 (x, y) = f 4 (x, y) = ( ( ( ( ) ( ) ( ) x 0 +, p 1 = 0.01, y 0 ) ( ) ( ) x 0.4 +, p 2 = 0.07, y ) ( ) ( ) x y ) ( ) ( ), p 3 = 0.07, x y 0.18, p 4 =

12 The maple leaf attractor on [0, 1] 2 is produced via the following IFSP: m 1 (x, y) = m 2 (x, y) = m 3 (x, y) = m 4 (x, y) = ( ( ( ( ) ( ) ( ) x 0.1 +, p 1 = 0.25, y 0.04 ) ( ) ( ) x , p 2 = 0.25, y 0.4 ) ( ) ( ) x , p 3 = 0.25, y ) ( ) ( ) x , p 4 = y

13 IFS on measure valued functions (D.La Torre, E.Vrscay, E.Ebrahimi, M.Barnsley) Most practical as well as theoretical works in image processing and mathematical imaging consider images as real-valued functions u : X R, where X denotes the base or pixel space over which the images are defined. At a particular point x in a (for simplicity) greyscale image I, we observe a pixel with a well defined intensity value, which we denote as u(x). Various function spaces F(X) may be considered. However, there are situations in which it is useful to consider the greyscale value of an image u at a point x to be a random variable that can assume a range of values R g R, the greyscale space. 13

14 One example is statistical image processing as applied, for example, to the problem of image restoration. It is useful to consider not only the real values that may be assumed by image u at x but also the probabilities that these values (or intervals around them) are assumed. We consider images as represented by measure-valued mappings, µ : X M(R g ), where M is the set of probability measures supported on R g. We construct an appropriate complete metric space (Y, d Y ) of measure-valued images µ : [0, 1] n M(R g ). Under suitable conditions, a fractal transform operator T : Y Y is contractive, implying the existence of a unique fixed point measure µ = T µ. We also show that the pointwise moments of this measure satisfy a set recursion relations that can be viewed as generalizations of the relations satisfied by moments of invariant measures for iterated function systems with probabilities (IFSP). 14

15 In what follows X = [0, 1] n will denote the base space, i.e., the support of the images. R g R will denote a compact greyscale space of values that our images can assume at any x X. And B will denote the Borel σ algebra on R g with µ L the Lebesgue measure. Let M denote the set of all probability measures on R g endowed with the Monge-Kantorovich metric d H defined earlier. For a given P > 0, let M 1 M be a complete subspace of M such that d H (µ, ν) P for all µ, ν M 1. We now define Y = {µ(x) : X M 1, µ(x) is measurable} and consider on this space the following metric d Y (µ, ν) = d H (µ(x), ν(x))dµ L. Theorem. The space (Y, d Y ) is complete. X 15

16 We now list the ingredients for a fractal transform operator in the space Y. The reader will note that they form a kind of blending of IFSP on measures and IFSM on functions: A set of N one-to-one contraction affine maps w i : X X, w i (x) = s i x + a i, with the condition that N i=1 w i(x) = X, A set of N greyscale maps φ i : [0, 1] [0, 1], assumed to be Lipschitz, i.e., for each i, there exists a α i 0 such that φ i (t 1 ) φ i (t 2 ) α i t 1 t 2, t 1, t 2 [0, 1], (4) For each x X, a set of probabilities p i (x), i = 1,, N with the following properties: p i (x) are measurable p i (x) = 0 if x / w i (X) and N i p i (x) = 1 for all x X. 16

17 The action of the fractal transform operator T : Y Y defined by the above is as follows: For a v Y and any subset S [0, 1], ν(x)(s) = (T µ(x))(s) = N i=1 p i (x)µ(w 1 i (x))(φ 1 (S)), where summation is performed only over those i for which wi 1 (x) is defined, i.e., x w i (X). i Y, Theorem. Let p i = sup x X p i (x). Then for µ 1, µ 2 ( n d Y (T µ 1, T µ 2 ) s i α i p i ) d Y (µ 1, µ 2 ). i=1 17

18 We show that the moments of measures in the space (Y, d Y ) also satisfy recursion relations when the greyscale maps φ i are affine. We now consider the local or x-dependent moments of a measure µ(x) Y, defined as follows, g n (x) = R g s n dµ x (s), m = 0, 1, 2,. where we use the notation µ x = µ(x) in the Lebesgue integral for simplicity. g 0 (x) = 1 for x X. We now derive the relations between the moments of a measure µ Y and the moments of ν = T µ where T is the fractal transform operator defined before. Let h n denote the moments of ν = T µ. Computing, we have h n (x) = R g N i=1 p i (x)[φ i (s)] n d(µ w 1 i (x) )(s) 18

19 For affine greyscale maps of the form φ(s) = α i s+ β i, we have h n (x) = [ n N j=0 i=1 p i (x)c nj α j i βm j i ] g j (wi 1 (x)) where c nj = ( n j ). One may compare the above result for the IFSP case. The place-dependent moments h n (x) are related to the moments g n evaluated at the preimages wi 1 (x). 19

20 IFS on multifunctions (D.La Torre, F.Mendivil, E.Vrscay) We now formulate some IFS-type fractal transform operators on the space of set-valued mappings over closed and bounded intervals of R n. We first consider a couple of metrics over these spaces and then establish the Lipschitz constants of the fractal transforms in these metrics. We introduce some IFS operators of the space of multifunctions. We recall that a setvalued mappings or multifunction F : X Y is a function from X to the power set 2 Y. We recall that the graph of F is the following subset of X Y graph F = {(x, y) X Y : y F (x)}. If F (x) is a closed, compact or convex we say that F is closed, compact or convex valued, respectively. Let (X, B, µ) be a finite measure space; a multifunction F : X Y is said to be measurable if for each open O Y we have F 1 (O) = {x X : F (x) O } B 20

21 A function f : X Y is a selection of F if f(x) F (x), x X. In the following we will suppose that Y is compact and F (x) is compact for each x X. Define F = {F : X H(Y )}. We place the following two metrics on F; the first is d (F, G) = sup x X d H (F (x), G(x)) and the second (here µ is a finite measure on X and p 1) d p (F, G) = ( X d H (F (x), G(x)) p dµ(x)) 1/p. Proposition. The space (F, d ) is a complete metric space. Proposition. Let Y be a compact interval of R and suppose that F (x) is convex for each x X and for all F F. Suppose that all F F are measurable. Then (F, d p ) is a complete metric space. 21

22 Let w i : X X be maps on X and φ i : H(Y ) H(Y ) with Lipschitz constants K i. Define T : F F by T (F )(x) = i φ i (F (wi 1 (x))). Proposition. If K = max i K i < 1, then T is contractive in d. Proposition. Assume that dµ(w i (x)) s i dµ(x) where s i 0. Then ( d p (T (F ), T (G)) i K p i s i) 1/p d p (F, G). 22

23 Define the operator T : F F by T (F )(x) = i p i (x)φ i (F (wi 1 (x))) where the sum depends on x and is over those i so that x w i (X). We require that the functions p i satisfy that i p i(x) = 1 The idea is to average the contributions of the various components in the areas where there is overlap. Proposition. We have [ d (T (F ), T (G)) sup x ] p i (x)k i d (F, G). i Proposition. Let p i = sup x p i (w i (x)) and s i 0 be such that dµ(w i (x)) s i dµ(x). Then we have ( ) 1/p d p (T (F ), T (G)) C(n) K p i sp i pp i d p (F, G). i 23

24 The idea of this section is that to each pixel of an image is associated an interval which measures the error in the value for that pixel. In this situation, therefore, we restrict our set-valued functions so that they only take closed intervals as values. We also need to restrict the φ i maps so that they map intervals to intervals. Thus, we shall consider X = [0, 1] n for n = 1 or 2 and Y = [a, b]. For each x, let β(x) H be an interval in Y. Then we define T : F F by T (F )(x) = β(x) + i p i (x) α i F (wi 1 (x)) where α i R. 24

25 The inverse problem can be formulated as follows: Given a multifunction F F, find a contractive operator T : F F that admits a unique fixed point F F such that d (F, F ) is small enough. It is in general a very difficult task to find such operators. A tremendous simplification is provided by the Collage Theorem. The inverse problem then becomes one of finding a contractive operator that maps the target multifunction F as close to itself as possible. Corollary. Under the assumptions of the Collage Theorem we have the following inequality d (F, T F ) p i sup max{a i (x), Āi(x)} i x X where A i (x) = min F (x) min(β(x)+α i F (wi 1 )(x)), Ā i (x) = max F (x) max(β(x)+α i F (wi 1 (x))) and p i = sup x X p i (w i (x)). Corollary. Under the assumptions of the Collage Theorem we have the following inequality d p (F, T F ) p min F min T F p p+ max F max T F p p 25

26 Inverse problems using fractal-based methods (H.Kunze, D.La Torre, E.Vrscay) We analyze a different technique, based on the method of collage distances for contraction maps, to solve inverse problems arising in theory of DEs with initial and boundary conditions. We now recall a method of solving inverse problems for differential equations using fixed point theory for contractive operators. A number of inverse problems may be viewed in terms of the approximation of a target element x in a complete metric space (X, d) by the fixed point x of a contraction mapping T : X X. In practice, from a family of contraction mappings T λ, λ Λ R n, one wishes to find the parameter λ for which the approximation error d(x, x λ ) is as small as possible. 26

27 Thanks to a simple consequence of Banach s fixed point theorem known as the Collage Theorem, most practical methods of solving the inverse problem for fixed point equations seek to find an operator T for which the collage distance d(x, T x) is as small as possible. One now seeks a contraction mapping T λ that minimizes the so-called collage error d(x, T λ x). In other words, a mapping that sends the target x as close as possible to itself. This is the essence of the method of collage coding which has been the basis of most, if not all, fractal image coding and compression methods. Many problems in the parameter estimation literature for differential equations can be formulated in such a collage coding framework 27

28 Differential equations with initial conditions as ẋ = f(t, x), x(0) = x 0 Picard integral operator associated with the initial value problem (T x)(t) = x 0 + t 0 f(s, x(s)) ds. We assume classical conditions to get the solution of this problem Let {φ i } be a basis of functions in L 2 and consider the first n elements of this basis, that is, n f a (s, x) = a i φ i (s, x). i=1 Each vector of coefficients a = (a 1,, a n ) R n then defines a Picard operator T a. 28

29 Given a target solution x(t), we now seek to minimize the collage distance x T a x 2. The square of the collage distance becomes δ δ (a) 2 = x T a x 2 2 = t n x(t) a i φ i (s, x(s))ds 0 i=1 and the inverse problem can be formulated as min λ P (λ) The minimization may be performed by means of classical minimization methods. 2 dt 29

30 Given a Hilbert space H, a bilinear functional a : H H R and a nonzero linear functional φ : H R, many boundary value problems can be reduced to the solution of the following variational problem: Find u H such that a(u, v) = φ(v) (5) for all v V. Theorem. (Lax-Milgram representation) Let H be a Hilbert space and φ be a bounded and nonzero linear functional. Suppose that a is a bilinear form on H H which satisfies the following: There exists a constant M > 0 such that a(u, v) M u v for all u, v H, There exists a constant m > 0 such that a(u, u) m u 2 for all u H. Then there is a unique vector u φ(v) = a(u, v) for all v H. H such that 30

31 Suppose that for a given Hilbert space H, we have a target element u H and a family of bilinear functionals a λ. By Lax-Milgram theorem, there exists a unique vector u λ such that φ(v) = a λ (u λ, v) for all v H. We would like to determine if there exists a value of the parameter λ such that u λ = u or, more realistically, u λ u is sufficiently small. 31

32 Theorem. (Generalized Collage Theorem) Suppose that a λ (u, v) : F H H R is a family of bilinear forms for all λ F and φ : H R is a given nonzero linear functional. Let u λ denote the solution of the equation a λ (u, v) = φ(v) for all v H as guaranteed by the Lax-Milgram theorem. Given a target element u H then where F (λ) = u u λ 1 m λ F (λ), sup a λ (ū, v) φ(v). v H, v =1 In order to ensure that the approximation u λ is close to a target element u H, we can, by the Generalized Collage Theorem, try to make the term F (λ)/m λ as close to zero as possible. If inf λ F m λ m > 0 then the inverse problem can be reduced to the minimization of the function F (λ) on the space F, that is, inf λ F F (λ). 32

33 Example. Consider the following one-dimensional steady-state diffusion equation d ( κ(x) du ) = f(x), 0 < x < 1, (6) dx dx u(0) = 0, u(1) = 0, (7) The inverse problem is: given u(x) and f(x) on [0, 1], determine an approximation of κ(x). As a specific experiment, let f(x) = 8x and κ true (x) = 2x+1. In which case the solution is u true (x) = x x 2. noise ε κ collage d 2 (κ collage, κ true ) x x x x

34 Self-similar objects in economics (D.La Torre, S.Marsiglio, F.Privileggi) We study an optimal growth model under uncertainty in which the social planner seeks to maximize the representative household s infinite discounted sum of instantaneous utility functions which are assumed to be logarithmic subject to the laws of motion of physical, k t, and human, h t, capital. At each time t, the planner chooses consumption, c t, and the share of human capital, u t, to allocate into production of a unique homogeneous consumption good which uses a Cobb-Douglas technology that combines physical and human capital. 34

35 The social planner problem solves: V (k 0, h 0, z 0, η 0 ) = max {c t,u t } E 0 β t ln c t t=0 subject to k t+1 = z t kt α (u t h t ) 1 α c t h t+1 = η t [(1 u t ) h t ] φ k 0 > 0, h 0 > 0, z 0 {q 1, q 2, 1}, η 0 {r, 1} where E 0 denotes expectation at time t = 0, 0 < β < 1 is the discount factor, k t and h t denote physical and human capital at time t, 0 < α < 1 and 0 < φ < 1. The Bellman equation reads as: V (k t, h t, z t, η t ) = max c t,u t [ln c t + βe t V (k t+1, h t+1, z t+1, η t+1 )]. 35

36 The solution V (k, h, z, η) of the Bellman equation is given by: V (k, h, z, η) = θ + θ k ln k + θ h ln h + θ z ln z + θ η ln η, where θ k, θ h, θ z, θ η and θ are constants. The optimal policy rules for consumption and share of human capital allocated to physical production are respectively given by: c t = (1 αβ) (1 βφ) 1 α z t k α t h 1 α t u t = 1 βφ, while physical and human capital follow the (optimal) dynamics defined by: k t+1 = αβ (1 βφ) 1 α z t k α t h 1 α t h t+1 = (βφ) φ η t h φ t 36

37 Assume that α φ and let [ ] α φ r = exp 1 α (2 ln q 2 ln q 1 ) Then the one-to-one logarithmic transformation (k t, h t ) (x t, y t ) defined by: { x t = ρ a ln k t + ρ b ln h t + ρ c y t = ρ d ln h t + ρ e, with ρ a, ρ b, ρ c, ρ d, and ρ e constants, defines a contractive linear IFSP which is composed of the three maps w 1, w 2, w 3 : R 2 R 2 given by: (x t+1, y t+1 ) = (αx t, φy t ) (x t+1, y t+1 ) = (αx t + (1 α) /2, φy t + (1 φ)) (x t+1, y t+1 ) = (αx t + (1 α), φy t ). with probabilities p 1, p 2, and p 3. It converges to an invariant distribution supported on a (generalized) Sierpinski gasket with vertices (0, 0), (1/2, 1) and (1, 0). 37