A generalized fractal transform for measure-valued images

Nonlinear Analysis 71 (2009) e1598–e1607

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A generalized fractal transform for measure-valued images Davide La Torre a,∗ , Edward R. Vrscay b a

Department of Economics, Business and Statistics, University of Milan, Italy

b

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

article

info

Keywords: Measure-valued images Multifunctions Self-similarity Fractal transforms Iterated function systems

abstract Fractal image coding generally seeks to express an image as a union of spatially-contracted and greyscale-modified copies of subsets of itself. Generally, images are represented as functions u(x) and the fractal coding method is conducted in the framework of Ł2 or Ł∞ . In this paper we formulate a method of fractal image coding on measure-valued images: At each point x, µ(x) is a probability measure over the range of allowed greyscale values. We construct a complete metric space (Y , dY ) of measure-valued images, µ : X → M (Rg ), where X is the base or pixel space and M (Rg ) is the set of probability measures supported on the greyscale range Rg . A generalized fractal transform M is formulated over the metric space (Y , dY ). Under suitable conditions, M : Y → Y is contractive, implying the existence of a unique fixed point measure-valued function µ ¯ = Mµ ¯. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Most practical as well as theoretical works in image processing and mathematical imaging consider images as real-valued functions, u : X → Rg , where X denotes the base space or pixel space over which the images are defined and Rg ⊂ R is a suitable greyscale space. A variety of function spaces F (X ) may be considered depending on the application. In standard image processing schemes, F (X ) is usually assumed to be Ł2 or Ł∞ . Fractal image coding [1–3] seeks to approximate an image function as a union of spatially-contracted and greyscalemodified copies of subsets of itself, i.e., u ≈ Tu, where T is the so-called fractal transform operator. Under suitable conditions, T is contractive over the appropriate image function space and therefore possesses a unique fixed point u¯ : X → Rg . From a consequence of Banach’s theorem, known in the fractal coding literature as the collage theorem, making the collage error ku − Tuk small can make the approximation error ku − u¯ k small. Earlier works (by La Torre, Mendivil and Vrscay, see [4,5]) consider a multifunction approach in which a real-valued image function is replaced by a multifunction taking compact and convex values. There are, however, situations in which it is useful to consider the greyscale value of an image u at a point x as a random variable that can assume a range of values Rg ⊂ R. One example is the characterization of the statistical properties of a class of images, e.g., MRI brain scans, for a particular application, say image compression. Another example is statistical image processing as applied to the problem of image restoration (denoising or deblurring). Of course, it is not enough to know the greyscale values that may be assumed by an image u at a point x: one must also have an idea of the probabilities (or frequencies) of these values. This is the underlying idea of this paper; we replace real-valued image functions by measurevalued image functions, i.e., µ : X → M (Rg ), where X is the base or pixel space and M (Rg ) is the set of probability measures supported on the greyscale range Rg . We present some numerical examples of our measure-valued image approach near the end of this paper.

∗

Corresponding author. E-mail addresses: [email protected] (D. La Torre), [email protected] (E.R. Vrscay).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.239

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The structure of this paper is as follows. For the benefit of readers who are not very familiar with fractal-based methods of coding, we present some basics of the method of ‘‘Iterated Function Systems’’ (IFS). It is necessary to present the formulation of IFS-based methods over both image and measure spaces, since the method of fractal transforms over the space (Y , dY ) presented in Sections 6 and 7 represents a kind of ‘‘fusion’’ of these two methods. In Section 6, we construct an appropriate complete metric space (Y , dY ) of measure-valued images µ : [0, 1]n → M (Rg ). In Section 7, we formulate a method of fractal transforms over the metric space (Y , dY ). The iterative nature of this cross-scale self-similarity method utilizes more of the mathematical structure of the space. Under suitable conditions, a fractal transform operator M : Y → Y is contractive, implying the existence of a unique fixed point measure µ ¯ = Mµ ¯ . We also show that the pointwise moments of this measure satisfy a set of recursion relations that can be viewed as generalizations of the relations satisfied by moments of invariant measures for iterated function systems with probabilities (IFSP).

2. Direct and inverse problem for a fractal transform The action of a generalized fractal transform T : Y → Y on an element u of the complete metric space (Y , dY ) can be summarized in the following steps (see [6]):

• it produces a set of N spatially-contracted copies of u, • it modifies the values of these copies by means of a suitable range mapping, • it recombines them using an appropriate operator in order to get the element v ∈ Y , v = Tu. In all these cases, under appropriate conditions, the fractal transform T is a contraction and thus Banach’s fixed point theorem guarantees the existence of a unique fixed point u¯ = T u¯ . Theorem 1 (Banach). Let (Y , dY ) be a complete metric space. Also let T : Y → Y be a contraction mapping with contraction factor c ∈ [0, 1), i.e., for all u, v ∈ Y , dY (Tu, T v) ≤ cdY (u, v). Then there exists a unique u¯ ∈ Y such that u¯ = T u¯ . Moreover, for any v ∈ Y , dY (T n v, u¯ ) → 0 as n → ∞. The inverse problem is crucial for applications: given T : Y → Y and a ‘‘target’’ element u ∈ Y , we look for a contraction mapping T with fixed point u¯ such that dY (u, u¯ ) is as small as possible. In practical applications, however, it is difficult to construct solutions to this problem and one relies on the following simple consequence of Banach’s fixed point theorem, known as the ‘‘Collage Theorem’’. Theorem 2 (‘‘Collage Theorem’’ [7,1,13]). Let (Y , d) be a complete metric space and T : Y → Y a contraction mapping with contraction factor c ∈ [0, 1). Then for any u ∈ Y , dY (u, u¯ ) ≤

1 1−c

dY (u, Tu),

(1)

where u¯ is the fixed point of T . Instead of trying to minimize the error dY (u, u¯ ), one looks for a contraction mapping T that minimizes the collage error dY (u, Tu). The following result states a continuity result for the fixed points. Theorem 3 (‘‘Continuity of Fixed Points’’ [8]). Let (Y , dY ) be a complete metric space and T1 , T2 be two contractive mappings with contraction factors c1 and c2 and fixed points u¯ 1 and u¯ 2 , respectively. Then dY (¯u1 , u¯ 2 ) ≤

1

dY ,sup (T1 , T2 ),

(2)

dY ,sup (T1 , T2 ) = sup dY (T1 u, T2 u)

(3)

1−c

where

u∈Y

and c = min{c1 , c2 }. The next sections will be devoted to the introduction of classical fractal transforms for image analysis.

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3. Iterated function systems and invariant measures In what follows, (X , d) denotes a compact metric ‘‘base space’’, typically [0, 1]n . Let w = {w1 , . . . , wN } be a set of one-toone contraction maps wi : X → X , to be referred to as an N map IFS (see [9]). Let ci ∈ [0, 1) denote the contraction factors of the wi and define c = max1≤i≤N ci . Note that c ∈ [0, 1). Now let H (X ) denote the set of nonempty compact subsets of X and h the Hausdorff metric. Then (H , h) is a complete ˆ : H (X ) → H (X ) the action of which is metric space [10]. Associated with the IFS maps wi is a set-valued mapping w defined to be

ˆ (S ) = w

N [

wi (S ),

S ∈ H (X ),

(4)

i =1

where wi (S ) := {wi (x), x ∈ S } is the image of S under wi , i = 1, 2, . . . , N.

ˆ is a contraction mapping on (H (X ), h): Theorem 4 ([11]). w h(w(A), w(B)) ≤ ch(A, B),

A, B ∈ H (X ).

(5)

Corollary 1. 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 ), h(wn (S ), A) → 0 as n → ∞. Now let M (X ) denote the set of Borel probability measures on X and dH the Monge–Kantorovich metric on this set: dH (µ, ν) =

Z sup

f ∈Lip1 (X ,R)

f (x)dµ −

X



Z

f (x)dν ,

(6)

X

where Lip1 (X , R) = {f : X → R | |f (x1 ) − f (x2 )| ≤ d(x1 , x2 ), ∀x1 , x2 ∈ X }.

(7)

For 1 ≤ i ≤ N, let 0 < pi < 1 be a partition of unity associated with the IFS maps wi , so that i=1 pi = 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

PN

ν(S ) = (M µ)(S ) =

N X

pi µ(wi−1 (S )),

∀S ∈ H (X ).

(8)

i=1

(Here, wi−1 (S ) = {y ∈ X | wi (y) ∈ S }.) Theorem 5 ([11]). M is a contraction mapping on (M (X ), dH ): dH (M µ, M ν) ≤ cdH (µ, ν),

µ, ν ∈ M(X ).

(9)

Corollary 2. 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 ), dH (M n µ, µ) ¯ → 0 as n → ∞. The reader is referred to the book by Barnsley [1] for more detailed discussions as well as numerous examples. 4. Moment relations for measures and IFSP In applications, it is most convenient to employ affine IFS maps. In this case, the moments of the invariant measure µ ¯ of the Markov operator M satisfy a set of relations that allow them to be computed recursively [9,1]. We illustrate with the one-dimensional case, i.e., X = [0, 1]. The extension to higher dimensions is quite straightforward. The affine IFS maps will be denoted as follows,

wi (x) = si x + ai ,

i = 1, 2, . . . , N .

(10)

The moments of a probability measure µ ∈ M (X ) are defined as

Z

xn dµ,

gn = X

n = 0, 1, 2, . . . .

(11)

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By definition, g0 = 1. Now let ν = M µ. Then, from Eq. (8), the moments of ν are given by

Z

xn d(T µ)(x)

hn = X

Z

N X

n

=

x d X

pi µ ◦ wi

(x)

i=1

Z X N

=

! −1

pi xn d(µ ◦ wi−1 )(x)

X i =1

Z X N

=

pi [wi (x)]n dµ(x)

X i =1

Z X N

=

pi [si x + ai ]n dµ(x).

X i =1

Expansion of the binomial followed by an interchange of summation and integration yields the result hn =

" n   X N X n j

j=0

# j n −j pi si ai

gj .

(12)

i =1

If we let g = (g0 , g1 , . . .)T ,

h = (h0 , h1 , . . .)T ,

(13)

denote the (infinite) moment vectors of µ and ν , respectively, then the Markov operator M is seen to induce a linear mapping h = Ag, where A is represented by a lower triangular matrix. This was originally pointed out in [6]. In the special case that µ = µ ¯ = Mµ ¯ , the invariant measure of the IFSP, then hn = gn and Eq. (12) can be rearranged to yield 1−

N X

! pi sni

gn =

i =1

" n−1   X N X n j

j =0

# j n−j pi si ai

gj .

(14)

i=1

This result, originally derived in [9], shows that the moments gn of the invariant measure µ ¯ may be computed recursively, starting with g0 = 1. 5. IFS on function spaces For the moment, we consider a general function space F (X ) supported on X . The essential components of a fractal transform operator are as follows. (1) A set of N one-to-one contraction maps wi : X → X with the condition that ∪Ni=1 wi (X ) = X . (2) A set of associated greyscale maps φi : R → R that are assumed to be Lipschitz on R, i.e., for each φi there exists a Ki ≥ 0 such that

|φi (t1 ) − φi (t2 )| ≤ Ki |t1 − t2 |,

for all t1 , t2 ∈ R.

(15)

The above two sets of maps are said to comprise an ‘‘Iterated Function System with greyscale maps’’ (IFSM) (w, Φ ) [6]. For each x ∈ X , this IFSM produces one or more fractal components defined as gi (x) =



φi (u(wi−1 (x))), 0,

if x ∈ wi (X ), otherwise.

(16)

If several fractal components exist for an x ∈ X , then they are combined with an operation that is suitable for the space in which we are working (see [6] for more details and examples of the various function spaces that can be considered). The natural operation in Łp (X ) is the summation operation: For u ∈ Łp (X ),

v(x) = (Tu)(x) =

N X i=1

gi (x).

(17)

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Theorem 6. Let (w, Φ ) be an IFSM as defined above, with spatial contractions wi and Lipschitz greyscale maps φi . Then for p ≥ 1 and u, v ∈ Łp (X ),

kTu − T vk ≤

N X

1/p

ci

Ki ku − vk.

(18)

i=1

PN

Corollary 3. If c =

1/p

i=1

u¯ (x) = (T u¯ )(x) =

ci

N X

Ki < 1, then T is contractive in Łp (X ) with fixed point u¯ ∈ Łp (X ). The fixed point equation,

φi (¯u(wi−1 ))(x),

(19)

i=1

indicates that u¯ is ‘‘self-similar’’, i.e., that it can be written as a sum of spatially-contracted and greyscale-modified copies of itself. It is also convenient to define IFSM operators with condensation functions. For example, given a set of IFS maps wi , associated constants αi and condensation function b(x), x ∈ X , define the action of the associated operator T as follows: For u ∈ Ł(X ),

v(x) = (Tu)(x) = b(x) +

N X

αi u(wi−1 (x)).

(20)

i=1

6. A complete metric space (Y , dY ) of measure-valued images We first set up our space of measure-valued images. In what follows, X = [0, 1]n will denote the ‘‘base space’’, i.e., the support of the images. Rg ⊂ R will denote a compact ‘‘greyscale space’’ of values that our images can assume at any x ∈ X . (The following discussion is easily extended to Rg ⊂ Rm to accommodate color images, etc.) And B will denote the Borel σ algebra on Rg with µL the Lebesgue measure. Let M denote the set of all Borel probability measures on Rg and dH the Monge–Kantorovich metric on this set. For a given M > 0, let M1 ⊂ M be a complete subspace of M such that dH (µ, ν) ≤ M for all µ, ν ∈ M1 . We now define Y = {µ(x) : X → M1 , µ(x) is measurable}

(21)

and consider on this space the following metric dY (µ, ν) =

Z

dH (µ(x), ν(x))dµL .

(22)

X

We observe that dY is well defined, since µ and ν are measurable functions, dH is bounded and so the function ξ (x) = dH (µ(x), ν(x)) is integrable on X . Theorem 7. The space (Y , dY ) is complete. Proof. It is trivial to prove that this is a metric when we consider that µ = ν if µ(x) = ν(x) a.e. x ∈ X . To prove the completeness we follow the trail of the proof of Theorem 1.2 in [12]. Let µn be a Cauchy sequence in Y . So for all  > 0 there exists n0 such that for all n, m ≥ n0 we have dY (µn , µm ) <  . Let  = 3−k so you can choose an increasing sequence nk such that dY (µn , µnk ) < 3−k for all n ≥ nk . So choosing n = nk+1 we have dY (µnk+1 , µnk ) < 3−k . Let Ak = {x ∈ X : dH (µnk+1 (x), µnk (x)) > 2−k }. Then µL (Ak )2−k ≤

R Ak

dH (µnk+1 (x), µnk (x))dµL ≤ 3−k so that µL (Ak ) ≤

! µL

[ k≥m

Ak

≤

(23)

X

µL (Ak ) ≤

k≥m


m

X  2 k k≥m

3

2 m 3 2 3

2 k . 3




Let A =

T∞ S m=1

k≥m

Ak . We observe that




=

1−


.

(24)

Therefore µL (A) ≤ 3 32 for all m, which implies that µL (A) = 0. Now for all x 6∈ X \ A there exists m0 (x) such that for all m ≥ m0 we have x ∈ 6 Am and so dH (µnm+1 (x), µnm (x)) < 2−m . This implies that µnm (x) is Cauchy for all x 6∈ X \ A and

D. La Torre, E.R. Vrscay / Nonlinear Analysis 71 (2009) e1598–e1607

e1603

so µnm (x) → µ(x) using the completeness of M . This also implies that µ : X → M is measurable, that is µ ∈ Y . To prove µn → µ in Y we have that dY (µnk , µ) =

Z

dH (µnk (x), µ(x))dµL

ZX =

lim dH (µnk (x), µni (x))dµL

X i→+∞

Z

dH (µnk (x), µni (x))dµL

≤ lim inf i→+∞

X

= lim inf dY (µnk , µni ) ≤ 3−k

(25)

i→+∞

for all k. So limk→+∞ dY (µnk , µ) = 0. Now we have dY (µn , µ) ≤ dY (µn , µnk ) + dY (µnk , µ) → 0 when k → +∞.

(26)



7. A fractal transform operator M on (Y , dY ) In this section, we construct and analyze a fractal transform operator M on the space (Y , dY ) of measure-valued functions. 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 IFS-based methods on measures (IFSP) and functions (IFSM). For simplicity, we assume that X = [0, 1]. The extension to [0, 1]n is straightforward. (1) A set of N one-to-one contraction affine maps wi : X → X , wi (x) = si x + ai , with the condition that ∪Ni=1 wi (X ) = X . (2) A set of N greyscale maps φi : [0, 1] → [0, 1], assumed to be Lipschitz, i.e., for each i, there exists a αi ≥ 0 such that

|φi (t1 ) − φi (t2 )| ≤ αi |t1 − t2 |,

∀t1 , t2 ∈ [0, 1].

(27)

(3) For each x ∈ X , a set of probabilities pi (x), i = 1, . . . , N with the following properties: (1) pi (x) are measurable (2) p i (x) = 0 if x 6∈ wi (X ) and P N (3) i pi (x) = 1 for all x ∈ X . The action of the fractal transform operator M : Y → Y defined by the above is as follows: For a µ ∈ Y and any subset S ⊂ [0, 1],

ν(x)(S ) = (M µ(x))(S ) =

N X

pi (x)µ(wi−1 (x))(φi−1 (S )).

(28)

i =1

Theorem 8. Let pi = supx∈X pi (x). Then for µ1 , µ2 ∈ Y , dY (M µ1 , M µ2 ) ≤

n X

! |si |αi pi dY (µ1 , µ2 ).

(29)

i =1

Proof. Computing, we have dY (M µ1 , M µ2 ) =

Z

dH (M µ1 (x), M µ2 (x))dµL

X N X

Z =

dH X

≤

pi (x)µ1 (wi (x)) ◦ φi , −1

i =1

Z X n

−1

N X

! pi (x)µ2 (wi (x)) ◦ φi

i=1

pi (x)dH (µ1 (wi−1 (x)) ◦ φi−1 , µ2 (wi−1 (x)) ◦ φi−1 )dµL

X i =1

≤

Z X n

αi pi (x)dH (µ1 (wi−1 (x)), µ2 (wi−1 (x)))dµL

X i =1 n X

Z ≤ X

=

i=1

! |si |αi pi dH (µ1 (x), µ2 (x))dµL

i=1

n X

−1

! |si |αi pi dY (µ1 , µ2 ). 

−1

d µL

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Corollary 4. Let pi = supx∈X pi (x). Then M is a contraction on (Y , dY ) if n X

|si |αi pi < 1.

(30)

i=1

Consequently there exists a measure-valued mapping µ ¯ ∈ Y , such that µ ¯ = Mµ ¯. Examples: (1) The fractal transform M defined by the following two-IFS-map system on X = [0, 1]:

w1 (x) = w2 (x) =

1 2 1 2

x,

φ1 ( t ) =

x+

1 2

,

1 2

t,

φ2 ( t ) =

1 2

1

t+

2

.

The sets w1 (X ) and w2 (X ) overlap at the single point x =



p1 (x) = 1, p1 (x) = 0,

p2 (x) = 1 x ∈

 

 

1

p1

= p2

2

1

p2 (x) = 0 x ∈ 0,

1

=

2

1 2



1 2



2

1 2

so we let

,



,1 ,

.

It is easy to confirm that M is contractive. Its fixed point µ ¯ is given by

µ( ¯ x) = δ(t − x),

x ∈ [0, 1],

(31)

where δ(s) denotes the ‘‘Dirac delta function’’ at s ∈ [0, 1]. (2) A ‘‘perturbation’’ of the fractal transform M in Example 1 that is produced by adding the following IFS and associated greyscale maps:

w3 (x) =

1 2

x,

φ3 ( t ) =

1 2

t + 0.1.

The sets w1 (X ) and w3 (X ) overlap over the entire subinterval [0, 21 ] so we let p1 (x) = p3 (x) =

1 2

,

p1 (x) = p3 (x) = 0,

  p1

1 2



p2 (x) = 0 x ∈ 0, p2 (x) = 1 x ∈

  = p2

1 2

  = p3

1

2

=

1 3



1 2

1 2



,

 ,1 ,

.

Once again, it is easy to confirm that M is contractive. Its fixed point µ( ¯ x) is sketched in Fig. 1. The darkness of a point is proportional to the measure µ( ¯ S1 , S2 ) of the region in [0, 1]2 represented by the point. Note that the overlapping of the w1 and w3 maps over [0, 12 ] is responsible for the self-similar ‘‘splitting’’ of the measures µ( ¯ x) over this interval, since φ3 produces an upward shift in the greyscale direction. Since w2 (x) maps the support [0, 1] of the entire measure-valued function onto [ 12 , 1], the self-similarity of the measure over [0, 21 ] is carried over to [ 12 , 1].

8. Moment relations induced by the fractal transform operator In this section we show that the moments of measures in the space (Y , dY ) also satisfy recursion relations when the greyscale maps φi are affine. We now consider the local or x-dependent moments of a measure µ(x) ∈ Y , defined as follows, gn (x) =

Z

sn dµx (s),

m = 0, 1, 2, . . . .

(32)

Rg

where we use the notation µx = µ(x) in the Lebesgue integral for simplicity. By definition, g0 (x) = 1 for x ∈ X . Obviously the functions gm are measurable on X (since µ(x) are measurable) and bounded so that gm ∈ Ł1 (X , L). We now derive the relations between the moments of a measure µ ∈ Y and the moments of ν = M µ where M is the fractal transform operator defined in Eq. (28).

D. La Torre, E.R. Vrscay / Nonlinear Analysis 71 (2009) e1598–e1607

e1605

Fig. 1. A sketch of the invariant measure µ( ¯ x) for the three-IFS-map fractal transform in Example 2, x ∈ X = [0, 1], y ∈ Rg = [0, 1].

Let hn denote the moments of ν = M µ. Computing, we have hn (x) =

Z

sn d(M µ)x (s) Rg

Z

N X

n

=

s d Rg

i

pi (x)sn d(µw−1 (x) ◦ φi−1 )(s) i

Rg i=1 N X

Z =

(s)

pi (x)µw−1 (x) ◦ φi

i=1 N X

Z =

! −1

pi (x)[φi (s)]n d(µw−1 (x) )(s). i

Rg i=1

For affine greyscale maps of the form φ(s) = αi s + βi , we have hn (x) =

N X

Z

pi (x)(αi + sβi )n d(µw−1 (x) )(s) i

Rg i=1

=

N X n X

pi (x)

=

N X n X

n −j

cnj (αi s)j βi n −j

j

pi (x)cnj αi βi

Z

i

sj d(µw−1 (x) )(s)

i=1 j=0

Rg

" n N X X

#

j =0

d(µw−1 (x) )(s)

Rg

i=1 j=0

=

Z

m−j

j

pi (x)cnj αi βi

i

gj (wi−1 (x)),

(33)

i=1

where

  cnj =

n . j

(34)

The reader may compare the above result to that of Eq. (12) for the IFSP case. The place-dependent moments hn (x) are related to the moments gn evaluated at the preimages wi−1 (x). And it is the greyscale φ(s) maps that now ‘‘mix’’ the measures, as opposed to the spatial IFS maps wi (x) in Eq. (12). In the special case that µ = µ ¯ = Mµ ¯ , the fixed point of M, then hn (x) = gn (x) and we have g n ( x) =

" n N X X j =0

i =1

# j i

n−j i

pi (x)cnj α β

gj (wi−1 (x)).

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In other words, the moments gn (x) satisfy recursion relations that involve moments of all orders up to n evaluated at preimages wi−1 (x). Note that this does not yield a rearrangement, analogous to Eq. (14), which will permit a simple recursive computation of the moments gn (x). Nevertheless, the moment functions can be computed recursively, as we now show. Firstly, note that for the particular case n = 1, the moment function g1 (x) is a solution of the fixed point equation g1 (x) =

N X

pi (x)[αi g1 (wi−1 (x)) + βi ].

(35)

i=1

(Note that g1 (x) is the expectation value of µ(x).) But this implies that g1 is the unique fixed point in Ł1 (X ) of the contractive place-dependent IFSM operator defined by N X

(Mu)(x) =

pi (x)[αi u(wi−1 (x)) + βi ].

(36)

i=1

This provides a method for computing g1 – at least approximately – by means of Banach’s Theorem. Starting at any u0 ∈ Ł1 the sequence M n u = M (M n−1 )u0 converges to g1 as n → +∞. Higher order moments can be computed in a similar recursive manner. To illustrate, consider the case n = 2. From Eq. (34), the moment g2 (x) satisfies the fixed point equation g2 (x) =

N X

pi (x)[αi2 g2 (wi−1 (x)) + 2αi βi g1 (x) + βi2 ].

(37)

i=1

In other words, g2 is the fixed point of a contractive IFSM operator, cf. Eq. (20), with condensation function b(x) =

N X

pi (x)[2αi βi g1 (x) + βi2 ].

(38)

i=1

From a knowledge of g1 , the moment function g2 may be computed via iteration. The process may now be iterated to produce g3 , etc. Finally, note that g1 and g2 determine the pointwise variance σ 2 (x) of the measure µ( ¯ x) :

σ (x) = 2

Z

(s − g1 (s))2 dµ ¯ x (s) Rg

= g2 (x) − [g1 (x)]2 .

(39)

9. Concluding remarks In this paper we have constructed an appropriate complete metric space (Y , dY ) of measure-valued images µ : [0, 1]n →

M([0, 1]) and a method of fractal transforms was formulated over this space. Under suitable conditions, the fractal transform

operator M : Y → Y is contractive, implying the existence of a unique fixed point measure µ ¯ = Mµ ¯ . The pointwise moments of this measure were also shown to satisfy a set of recursion relations. We plan to explore further uses of this measure-valued formalism in characterizing image self-similarity in a future paper as well as in other nonlocal image processing algorithms. Acknowledgements This work was started during a research visit by DLT to the Department of Applied Mathematics of the University of Waterloo. DLT thanks ERV for this opportunity. This work has also been supported in part by a Discovery Grant (ERV) from the Natural Sciences and Engineering Research Council of Canada (NSERC). References [1] [2] [3] [4] [5]

M.F. Barnsley, Fractals Everywhere, Academic Press, New York, 1989. Y. Fisher, Fractal Image Compression, Theory and Application, Springer Verlag, NY, 1995. N. Lu, Fractal Imaging, Academic Press, NY, 2003. D. La Torre, F. Mendivil, Iterated function systems on multifunctions and inverse problems, J. Math. Anal. Appl. 340 (2) (2008) 1469–1479. D. La Torre, F. Mendivil, E.R. Vrscay, Iterated function systems on multifunctions, in: G. Aletti et al. (Eds.), Math Everywhere – Deterministic and Stochastic Modelling in Biomedicine, Economics and Industry, Springer, pp. 125–138. [6] B. Forte, E.R. Vrscay, Theory of generalized fractal transforms, in: Y. Fisher (Ed.), Fractal Image Encoding and Analysis, in: NATO ASI Series F, vol. 159, Springer Verlag, New York, 1998. [7] M.F. Barnsley, V. Ervin, D. Hardin, J. Lancaster, Solution of an inverse problem for fractals and other sets, Proc. Natl. Acad. Sci. USA 83 (1985) 1975–1977.

D. La Torre, E.R. Vrscay / Nonlinear Analysis 71 (2009) e1598–e1607

e1607

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