Spaces of Functions and Lebesgue Integration

Before embarking the functions of bounded variation, we first introduce two important concepts of spaces (sets) of functions: (a) these are the spaces ofcontinuous functions C^m(Ω) etc. in the space of C(Ω) etc. (b) the Lebesgue spacesL^p etc. whose pth powers are integrable in an open set ofR orRⁿ, shown in Fig. 5.4.

Continuous Functions

Roughly speaking, a continuous function of a single variable may be characterized as one whose graph is an uninterrupted curve. On the other hand, a function is discontinuous in that its graph has a break. Another type of discontinuous function is one that is unbounded at some point. Continuous functions are defined on a subset Ω ofRⁿ and can be categorized:

1( ) L Ω

2( ) L Ω ( )

L^∞ Ω ( )

CΩ ( ) CΩ

Figure 5.4: The relationship between theL^p spaces and spaces of continuous functions.

5 Data-Driven Regularization for Variational Image Restoration in the BV Space

1. The space C(Ω). For any domain Ω in Rⁿ, the collection of all continuous functions defined on Ω forms a set, or space, which is denoted by C(Ω), shown in Fig. 5.4. The space of functions that are continuous on the closest set ¯Ω = Ω∪Γ (Γ and its boundary Γ) is denoted byC(¯Γ) and byC[a, b] for functions on the closed interval [a, b].

2. The spaces C^m(Ω) and C^∞(Ω). Among all the continuous functions defined on a subset Ω of Rⁿ, some have the properties that their first derivatives and possibly some derivatives of higher order are also continuous. It is very important to identify such functions with their derivatives ( of order m) are continuous on Ω. That is, C^m(Ω) = {u : u, ∂u/∂x, ∂u/∂y, ..., ∂^mu/∂x^k∂y^m−k(k = 0, ..., m) are all continuous functions} for Ω ⊂ R² and so on. Clearly, the inclusions C^∞(Ω) ⊂ ... ⊂ C^m(Ω) ⊂ C^m−1(Ω) ⊂ ... ⊂ C⁰(Ω) = C(Ω) hold, so that C^m constitutes a gradation which permits continuous func-tions to be classified according to their degree of smoothness: for any function in C^m(Ω), the higher the value of m, the smoother the function.

3. Continuous functions on compact sets. It turns out that continuous functions defined on compact sets (closed and bounded sets inRⁿ) may be characterized necessarily on such bounded sets. A functionf defined on a set Ω inRⁿis said to be bounded if it is possible to find a number M >0 such that f(x)≤M for all x∈Ω. In other words, the function does not “blow up” anywhere. Continuous functions behave in a special way on compact sets, shown in the definition.

Definition 5.2.1.1 Let Ω be a bounded domain (that is, a bounded open, connected set) in Rⁿ, andf a continuous function defined on the compact set Ω. Then,¯ (a) f is bounded on Ω¯ and, furthermore, f achieves its supremum and infimum on Ω.¯ (b) f is uniformly continuous on Ω.¯

The definition shows the function has a maximum for a given point z = supf(Ω) = maxf(Ω) for all pointsx∈Ω. A similar interpretation applies with respect to the infimum.

4. Lipschitz continuous functions. A function f defined on a set Ω in Rⁿ is said to be Lipschitz continuous or Lipschitz, if there exists a constantL >0 such thatf(x)−f(y)≤ L|x−y|for all x, y∈Ω. The definition of Lipschitz continuity does not require that the derivative exists at every point. It is straightforward to show that every Lipschitz function is uniformly continuous, although the converse is not true. If Ω is a compact set, then every continuously differentiable function on Ω is Lipschitz.

Measures of Sets in Rⁿ

However, many functions in practical applications are not continuous, and cannot therefore be accommodated in one of the spacesC^m(Ω), such as discontinuities, unconnected edges in images.

A simple example is to use Heaviside step function, H(x) =

0 ifx≤0 1 ifx >0

Although these functions are not continuous, they do possess the important property that they are integrable. Our aim is to set up a space of functions that may be classified according to their

( 2) f x

( )1

f x

x1

x2

xk

Δ M¹ M₂

y1

y2

Figure 5.5: a|b. The basic idea behind (a) Riemann integration and (b) Lebesgue integration.

integrable powers, e.g., Rb

a|f(x)|^pdx, p≥1. This permits the introduction of the spaceL^p(a, b) or, generally,L^p(Ω). For such demands, the case of the spacesC^mis possible to obtain a precise idea of the degree of smoothness of a function by determining the largest value of m for which it belongs toC^m. The smoothness of two functions may then be compared by determining the largest numbers m of the spaces C^m of which they are members. In the same way, we will see that theL^pspaces are alsonestedin the sense ofL^p ⊂L^qfor the case in whichp > q. Therefore, we note that these spaces also provide a means of comparing functions during the period of the integrability.

In order to give such spaces a proper treatment, it is necessary to introduce the notion of Lebesgue measure. This in turn allows us to introduce the notion ofLebesgue integration, which is a generalization of the “standard” Riemann integration. Then we define the spaces L^p(Ω).

Lebesgue measure is an important measure method in the well-established measure theory in mathematics. In order to introduce the Lebesgue integral, we return first to the definition of the Riemann integral. The basic idea of the Riemann integral is to divide [a, b] into a finite numberN of subintervals, thekth subintervals having length ∆x_k, and the approximation area under the graph f is the sum of the forms: f(x1)∆x1 +f(x2)∆x2 +...+f(xN)∆xN, shown in Fig. 5.5(a). Thus, the Riemann integral is denoted by Rb

af(x)dx which is used widely and adequate for most purposes.

However, the Riemann integral suffers from certain deficiencies, e.g., it is unable to deal with the functionf(x) = 1, x is rational (while f(x) = 0, xis irrational) on the interval [0,1]. Secondly, in contrast to the Riemann integral, for a more general Lebesgue integral, the approximation to the integral of f can be progressively improved, not by further subdivisions of the domain, but by refining the approximation to f. The approximating functions that serve this purpose are indeed known as simple functions, and are defined to be functions that take on a finite number of values. Provided that with the subsets M_k on their constant values, the integral of f can be approximated by a sum of the form, y₁µ(M₁) +y₂µ(M₂) +...+y_Nµ(M_N), shown in Fig. 5.5(b), where µ(Mk) is a measure ofMk. The limit number N is a nice improvement for the approximation off. Therefore, in this measurable space Ω, the Lebesgue measure is defined to satisfy those four criteria for measurable sets: (1) Ω itself; (2) Ω−M, for M ∈ M; (3) all open sets in Ω; and (4) M1∪M2..., for any countable family {M₁, M2, ...,} of disjoint sets in M. Also, functions that are Riemann-integral are also Lebesgue-integral, and the two integrals coincide.

Lebesgue Integration and the Space L^p(Ω)

We say that a function defined on a measurable set Ω in R is measurable if the inverse image f⁻¹(M) of any measurable set M in R is itself measurable. Therefore, we can verify any

5 Data-Driven Regularization for Variational Image Restoration in the BV Space

= +

f f⁺ f⁻

Figure 5.6: a|b|c. The positive and negative parts of a function in the Lebesgue integral.

continuous function is measurable. Heaviside function is a measurable function. Since sums of measurable functions are measurable, we can conclude that every step function is measurable.

Based on the intuitively obvious character, the Lebesgue integral of a simple functionson Ω is defined by

Z

Ω

sdx=a₁µ(M₁∩Ω) +a₂µ(M₂∩Ω) +...+a_Nµ(M_N ∩Ω) (5.12) whereM_kare measurable and pairwise disjoint. To obtain the Lebesgue integral of a measurable functionf we first set up a sequence of nondecreasing simple functions that approximatef. Next, we evaluate the integrals of these simple functions and take the limit to obtain the integral of f. Of course, f is a nonnegative measurable function on Rⁿ with a nondecreasing sequence S of simple functions on Rⁿ such that lim

n→∞sn(x) = f(x) at all points x inRⁿ. Therefore, when f is a measurable function defined on a measurable set Ω and f is nonnegative on Ω, then the Lebesgue integral off over Ω is defined by

Z

Ω

f dx= lim

k→∞

Z

Ω

s_kdx (5.13)

where s_k are nondecreasing simple functions that approximate f. Indeed, for well-behaved functions, e.g., piecewise continuous functions: it is clear that the Lebesgue integral like the Riemann integral, amounts to the area under the graph of the function. However, there are Lebesgue-integrable functions which are not Riemann-integrable. To complete the theory of the Lebesgue integral, we extend the treatment to include functions that are not necessary nonnegative. Suppose that f is any measurable function. Then f may be decomposed into positive part f⁺(x) = f(x), if f(x) ≥ 0 (otherwise f(x) = 0) and negative part f⁻(x) = 0, if f(x) ≥ 0 (f⁻(x) = −f(x), otherwise) shown in Fig. 5.6. More concisely, we can write f⁺ = ¹₂(f +|f|) and f⁻ = ¹₂(|f| −f) so that f = f⁺−f⁻. It is possible to show that f⁺ and f⁻ are both measurable. The summable function R

Ωf (Lebesgue integral exists) can be decomposed as the sum of two nonnegative functions,

Z

Ω

f dx= Z

Ω

f⁺dx− Z

Ω

f⁻dx

Now we note that it is possible to haveR

Ωf dx= +∞ for a nonnegative function. This lemma is very useful in the function of bounded variation for image deblurring and denoising.

The space of Lebesgue integrationL^p(Ω) is defined in an open set Ω. Letpbe a real number with p >1. A functionf(x) defined on a subset Ω ofRⁿis said to belong toL^p(Ω), iff is measurable

and if the (Lebesgue) integralR

Ω|f(x)|^pdxexits, i.e., is finite. The casep= 2 is special in many ways, and is referred to as square-integrable. Therefore, every bounded continuous function defined on a bounded set Ω belongs to L^p. If we let p → ∞, then we may define the space L^∞(Ω) to be the space of all measurable functions on Ω that are bounded almost everywhere on Ω.

We note that although L^∞(Ω) ⊂ ... ⊂ L^p(Ω) ⊂ ... ⊂ L¹(Ω), the space C(Ω) of continuous functions is not a subset of any of the L^p spaces, shown in Fig. 5.4. For example, the function f(x) = x⁻¹ belongs to C(0,1) but not to L^∞(0,1) since it is not bounded. But the space of bounded continuous functions, equivalently, the spaceC( ¯Ω) of continuous functions defined on a compact set ( ¯Ω) is a subset of L^∞(Ω). Fig. 5.4 also shows schematically how the spacesC^m(Ω) and L^p(Ω) are related.

Distributions and Sobolev Spaces

Form an integer, 1≤p≤ ∞ and Ω⊂R, we define theSobolev space,

W^m,p(Ω)^def={f ∈L^p(Ω);D^αf ∈L^p(Ω), 0≤ |α| ≤m} (5.14) where forα={α₁, α2, ..., αn} ∈Nⁿ, we put the partial derivative

D^αf = ∂^|α|f

∂x^α₁¹∂x^α₂²...∂x^αnⁿ

and |α|=α1+α2+...+αn (5.15)

Thus if |α| = m, then D^αf denotes one of the mth derivatives of f. The space is a normed space when endowed with the Sobolev norm k · k_m,p, The Banach spacefor the norm becomes

kfk_m,p=



 X

0≤|α|≤m

|D^αf|^p_Lp(Ω)





1/p

, 1≤p <∞ (5.16)

and in the case p=∞ kfk_m,∞= max

0≤|α|≤m|D^αf|_L∞(Ω) (5.17)

In the particular case p= 2, we have

W^m,2(Ω) =H^m(Ω) (5.18)

where the Sobolev spaceH^m(Ω) has been defined by taking as a point of departure in the Hilbert spaceL²(Ω). The results concerning the spacesW^m,p(Ω) are analogous to that obtained for the space H^m(Ω). The definition of the spaces W^s,p(Ω) for non-integral values of s, can be given by interpolation between L^p(Ω) andW^m,p(Ω). Fig. 5.7 shows the relationship between Hilbert spaces and Banach spaces, and the others. The theory of Hilbert spaces forces the use of the Lebesgue integral. As we have discussed previously, the widely used Riemann integral is only valid under very restrictive assumptions, in contrast to the Lebesgue integral. The Riemann integral leads only to pre-Hilbert spaces for which the fundamental Cauchy criterion is not valid.

5 Data-Driven Regularization for Variational Image Restoration in the BV Space

`, `

\ ^ (L G L G2( ), ^{^}2( )) 1 l¹

2 2

(W G W( ), ( ))G Sobolev spaces Lebesgue spaces

Hilbert space Banach space The Cauchy criterion is valid

Linear space (αu+βv) Normed space

(norm ||u||=(( | ) )u u¹² Pre-Hilbert space

(inner product (u|v))

Figure 5.7: The relationship between Hilbert spaces and Banach spaces, and others. Each Hilbert space is a Banach space; the most important Hilbert spaces are from the Lebesgue spacesL₂(G), L^C₂(G) and the related Sobolev spacesW₂¹(G) andcW₂¹(G). Roughly speaking, the real Lebesgue spaceL₂(G) (resp.

the complex Lebesgue spaceL^C₂(G)) consists of all functions. The theory of Hilbert spaces forces the use of the Lebesgue integral.

1. Real Lebesgue space L₂(G) is applied in Fourier series, integral equations, and partial differential equations.

2. Sobolev spaces W₂¹(G) andWc₂¹(G) are applied mainly in Dirichlet principle and the cal-culus of variations.

3. Complex Lebesgue space L^c₂(R^N) is applied mainly in quantum mechanics and Fourier transformation.

First of all, there is a category of Sobolev spaces that are Hilbert spaces. In Hilbert spaces, an inner product (u|v) is defined, allowing us to introduce the fundamental notion of orthogonality.

The Sobolev spaces provide a very natural setting for boundary value problems. This Banach space is a complete normed space, while Hilbert space is complete inner product space. Since every inner product defines a norm, every Hilbert space is a Banach space. Second, it is possible to obtain quite general results regarding existence and uniqueness of solutions in a variational setting, using these spaces. A third advantage is that, like the space C^m(Ω), Sobolev spaces provide a means of characterizing the degree of smoothness of functions. Finally, perhaps most important, is the fact that approximate solution methods such as the Galerkin and finite element methods. These methods are most conveniently and correctly formulated in finite-dimensional subspaces of Sobolev spaces.

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