Approximation by Nonlinear Multivariate Sampling Kantorovich Type

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Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing a

Danilo Costarelli & Gianluca Vinti a

b

Dipartimento di Matematica , Università degli Studi Roma Tre , Roma , Italy

b

Dipartimento di Matematica e Informatica , Università degli Studi di Perugia , Perugia , Italy Accepted author version posted online: 13 Feb 2013.Published online: 17 Jul 2013.

To cite this article: Danilo Costarelli & Gianluca Vinti (2013) Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing, Numerical Functional Analysis and Optimization, 34:8, 819-844, DOI: 10.1080/01630563.2013.767833 To link to this article: http://dx.doi.org/10.1080/01630563.2013.767833

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Numerical Functional Analysis and Optimization, 34(8):819–844, 2013 Copyright © Taylor & Francis Group, LLC ISSN: 0163-0563 print/1532-2467 online DOI: 10.1080/01630563.2013.767833

APPROXIMATION BY NONLINEAR MULTIVARIATE SAMPLING KANTOROVICH TYPE OPERATORS AND APPLICATIONS TO IMAGE PROCESSING

Downloaded by [Danilo Costarelli] at 00:01 19 July 2013

Danilo Costarelli1 and Gianluca Vinti2 1

Dipartimento di Matematica, Università degli Studi Roma Tre, Roma, Italy Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy

2

In this article, we study a nonlinear version of the sampling Kantorovich type operators in a multivariate setting and we show applications to image processing. By means of the above operators, we are able to reconstruct continuous and uniformly continuous signals/images (functions). Moreover, we study the modular convergence of these operators in the setting of Orlicz spaces L (n ) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L p (n )-spaces, L log L(n )-spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and image processing applications are included. Keywords Exponential spaces; Image processing; Irregular sampling; L log L-spaces; Modular convergence; Orlicz spaces; Sampling Kantorovich operators. 2010 Mathematics Subject Classification 41A35; 46E30; 47A58; 47B38; 94A12.

1. INTRODUCTION In [17], the theory of linear multivariate sampling Kantorovich operators has been treated together with applications to image processing. It becomes interesting, not only from a mathematical point of view, but also in image processing, to have at disposal a theory for nonlinear sampling Kantorovich operators; indeed, in some istances, in order to reconstruct signals or images, one need to use a nonlinear process. To achieve our goal, the theory of nonlinear operators becomes crucial. Received 4 November 2011; Accepted 26 December 2012. Address correspondence to Gianluca Vinti, Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1 06123 Perugia, Italy; E-mail: [email protected] 819

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D. Costarelli and G. Vinti


The family of nonlinear sampling Kantorovich operators, here considered, is of the form n w (Sw f )(x) := (x ∈ n ), wx − tk , f (u)du (I) w A k R n k k∈ where f : n → is a locally integrable function such that the above series is convergent for every x ∈ n . Here : n × → is a kernel function satisfying suitable properties, tk = (tk1 , tk2 , , tkn ) is a vector where (tki )ki ∈ , i = 1, 2, , n is a sequence of real numbers with some properties (nonuniform sampling scheme). Moreover tk2 tk2 +1 tkn tkn +1 tk1 tk1 +1 w Rk := , × ··· × , × , (w > 0), w w w w w w and Ak := k1 · k2 · · · · · kn with ki := tki +1 − tki > 0, i = 1, 2, , n. The linear sampling Kantorovich type operators were first introduced in [1] in the univariate case and subsequently extended in [38] in the nonlinear univariate case. This kind of operator, such as those defined in (I), represents an averaged version in Kantorovich-sense of the generalized sampling operators introduced by Butzer and his school at Aachen in the 1980s (see, e.g., [4, 5, 8–11, 13–16, 28, 33, 36, 37]). The generalized sampling operators are often used in signal processing as an interesting algorithm to approximate signals; for further information about the classical theory of sampling operators and its extensions, the reader can see [7, 18, 21–24, 34]. One of the advantages, with respect to the Butzer’s operators, given by the above operators of the form (I) is that, instead of the sampling values f (k/w), one has an average of f in a n-dimensional parallelepiped containing k/w (also in the more general case of tk in place of k, obtaining an irregular sampling, so enlarging the range of applications). This fact allows us to reduce the “time-jitter” error that occur in signal processing when the sampling values cannot be matched exactly at the “node” tk /w but in a neighborhood of it. Practically, more informations are usually known around a point than precisely at that point. In this article, we give a pointwise and uniform approximation result (Theorem 4.1) for Sw f toward f when w tends to infinity. Moreover, in order to cover the case of discontinuous signals or images, we develop an approximation theory for the above family (I) in the setting of Orlicz spaces. The main convergence result (Theorem 4.5) in this istance (Orlicz spaces) is a modular convergence theorem (the modular convergence is the most natural one in this setting) which is based on a Luxemburg-norm convergence theorem in Cc (n ) (Theorem 4.3), on a modular density


Kantorovich Operators for Image Processing

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result for continuous functions with compact support in Orlicz spaces (see [2, 3]) and on a modular inequality for the nonlinear operators (I) (Theorem 4.4). From one side, the general setting of Orlicz spaces allows us to apply our theory to the particular cases of L p (n )-spaces, L log L(n ) Zygmund-type spaces (very important, e.g., in the theory of partial differential equations), exponential spaces (used for embedding theorems between Sobolev spaces), and others, in fact obtaining a unifying approach for the study of approximation problems in several concrete instances (see Section 5). On the other side, the setting of Orlicz space allows us to cover the case of not-necessarily continuous signals/images which, from the point of view of the applications, is very important. As a matter of fact, the problem of recostructing an image at the discontinuity points means to be able to describe in detail the edge or the countours of the image where jumps of gray levels occur; note that in the latter situation we have hight contrast areas. In the last part of the article, besides concrete examples of operators with particular kernels (e.g. Fejér’s and B-spline’s kernels) showing the versatility of our approximation approach, we apply our theory to image processing (see Section 6). Here we show, using MATLAB programming language, and by means of our approach based on nonlinear sampling Kantorovich operators, how we are able to reconstruct, for example, the well-known image of “Lena” (150 × 150 pixel resolution) and even to enhance it with increased pixel resolution (300 × 300). 2. PRELIMINARY NOTIONS In this article, we will denote by n and n the sets of elements k = (k1 , , kn ), where ki belongs, respectively, to and , for each i = n the usual 1, , n; n is defined analogously. We will 2consider on 2 1/2 Eucledean norm ·2 , defined by u 2 = (u1 + · · · + un ) , where u = (u1 , , un ), ui ∈ , for every i = 1, , n. Moreover, B(x, r ) ⊂ n will represents the closed ball of center x∈ n and set radius r > 0, that is, the n of all the elements u ∈ n such that x − u 2 ≤ r . Finally, u · v = i=1 ui vi and u = (u1 , , un ) will represent, respectively, the scalar product of u, v ∈ n and the product of ∈ and u ∈ n . We denote by C (n ) (resp. C 0 (n )) the space of all uniformly continuous and bounded (resp. continuous and bounded) functions f : n → endowed with the norm f ∞ := supu∈n f (u) . Cc (n ) is the subspace of C (n ) consisting of functions with compact support and M (n ) is the space of all (Lebesgue) measurable functions. We now recall some basic properties concerning Orlicz spaces.

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A function : 0+ −→ 0+ such that is continuous and non decreasing on 0+ with (0) = 0, (u) > 0 for every u > 0 and limu→∞ (u) = +∞, is called a -function. Let now consider a fixed -function . The functional I : M (n ) → [0, +∞] defined by I [f ] := ( f (x) )dx (f ∈ M (n )), n

is a modular on M (n ) (see e.g. [3, 29, 30]), which generates the Orlicz space


L (n ) := f ∈ M (n ) : I [f ] < +∞ for some > 0 L (n ) is a vector space and E (n ) := f ∈ L (n ) : I [f ] < +∞ for every > 0 , is the vector subspace of all finite elements of L (n ). In general, E (n ) is a proper subspace of L (n ) and these two spaces coincide if and only if satisfies the so called 2 -condition: (2 )

∃M > 0 : (2u) ≤ M (u)

(u ∈ 0+ );

moreover Cc (n ) ⊆ E (n ). In L (n ) we can introduce a concept of convergence, called “modular convergence”. We will say that a net of functions (fw )w>0 ⊂ L (n ) is modularly convergent to a function f ∈ L (n ), if limw→+∞ I [(fw − f )] = 0, for some > 0 ([30]). Furthermore, an F-norm on L (n ), called Luxemburg norm, is given by f := inf > 0 : I [f /] ≤ , which defines on L (n ) a strong notion of convergence. It is easy to show that a net of functions (fw )w>0 ⊂ L (n ) converges with respect to the Luxemburg norm to f ∈ L (n ) (i.e., limw→+∞ fw − f = 0), if and only if limw→+∞ I [(fw − f )] = 0, for every > 0. It is clear that, the norm convergence implies the modular convergence and the converse implication is true if and only if satisfies the 2 -condition. The modular convergence induces a topology (modular topology) on the space L (n ) and Cc (n ) is modularly dense in L (n ) (see [2, 29]). Some interesting examples of Orlicz spaces are L p (n )-spaces, L log L(n )-spaces and the exponential spaces, respectively generated by the convex -functions (u) = u p , 1 ≤ p < +∞, , (u) = u ln (e + u),


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≥ 1, > 0 and = e u − 1, > 0, for u ∈ 0+ . The -functions and , satisfy the 2 -condition while ( > 0) are examples of -functions that do not satisfy the 2 -condition. For further details concerning Orlicz space, see the following monographs [3, 25–27, 29, 31, 32].


3. THE NONLINEAR MULTIVARIATE SAMPLING KANTOROVICH OPERATORS In this section we introduce the class of operator we will discuss in this article. Let n = (tk )k∈n be a sequence defined by tk = (tk1 , , tkn ), where each (tki )ki ∈ , i = 1, , n is a sequence of real numbers with −∞< tki < tki+1 < +∞, limki →±∞ tki = ±∞, for every i = 1, , n and such that there exist , > 0 for which ≤ ki := tki +1 − tki ≤ , for every i = 1, , n. Moreover, we denote by tk2 tk2 +1 tk1 tk1 +1 tkn tkn +1 w Rk := , × , (w > 0), , × ··· × w w w w w w the n-dimensional parallelepipeds of n identified by the sequence n = (tk )k∈n . We note that the Lebesgue measure of Rkw is given by Ak /w n , where Ak := k1 · k2 · · · · · kn . Let now : n × → be a kernel function, that is, satisfies the following assumption: • (1) k → (wx − tk , u) ∈ 1 (n ), for every x ∈ n , u ∈ and w > 0; • (2) (x, 0) = 0, for every x ∈ n ; • (3) is an (L, )-Lipschitz kernel, that is, there exist a measurable function L : n → 0+ and a -function such that (wx, u) − (wx, v) ≤ L(wx)( u − v ), for every x ∈ n , u, v ∈ and w > 0; • (4) for every j ∈ + and w > 0, 1 w (wx − tk , u) − 1 → 0 j (x) := sup 1/j ≤ u ≤j u k∈n as w → +∞, uniformly with respect to x ∈ n . If the function satisfies (i), i = 1, , 4, we will write ∈ . Moreover, we assume that the function L satisfies the following assumptions: • (L1) L ∈ L 1 (n ) and is bounded in a neighborhood of 0 ∈ n ;

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• (L2) k → L(u − tk ) ∈ 1 (n ), for every u ∈ n ; • (L3) there exists a number > 0 such that m,n (L) := sup L(u − tk ) u − tk 2 < +∞ u∈n k∈n

If the function L satisfies (Li), i = 1, 2, 3, we will write L ∈ . We can now introduce the following family of nonlinear operators. Definition 3.1. (Sw )w>0 by


(Sw f

Let ∈ and L ∈ . We define a family of operators

wn )(x) := wx − tk , Ak k∈n

f (u)du Rkw

(x ∈ n ),

where f : n → is a locally integrable function such that the series is convergent for each x ∈ n . Sw f (w > 0), will be called the nonlinear multivariate sampling Kantorovich operator applied to f . The above series represent the version of the generalized sampling series, where instead of the sampling values f (tk /w) one has an average of f in a n-dimensional parallelepiped Rkw ⊂ n . This approach is very useful in signal processing and allows us to solve the problem of the so-called “time-jitter errors.” We now recall the following lemma. Lemma 3.2. Let L ∈ . We have (i) m0,n (L) := supu∈n (ii) For every > 0

k∈n

lim

w→+∞

L(u − tk ) < +∞;

L(wx − tk ) = 0,

wx−tk 2 > w

uniformly with respect to x ∈ n . (iii) For every > 0 and > 0 there exists a constant M > 0 such that w n L(wx − tk )dx < , x 2 >M for sufficiently large w > 0 and tk such that tk 2 ≤ w. For a proof of Lemma 3.2, see [17].


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Remark 3.3.


(a) If f ∈ L ∞ (n ), by (i) of Lemma 3.2 and conditions (2) and (3), we obtain that (Sw f )w>0 are well defined. In fact n w (S f )(x) ≤ wx − t , f (u) du k w w A k Rk k∈n wn f (u) du L(wx − tk ) ≤ w A k R n k k∈ ≤ m0,n (L)(f ∞ ) < +∞, for every x ∈ n and w > 0, that is, Sw maps L ∞ (n ) into itself. (b) Instead of assuming that the function L is bounded in a neighborhood of 0 ∈ n and that m,n (L) < +∞, one can directly assume that for L the properties (i) and (ii) of Lemma 3.2 hold. (c) In the particular case of the equally spaced sequence tk = k = on L and the (k1 , k2 , , kn ), k ∈ n , the boundedness assumptions hypothesis (L3) can be replaced by supu∈n k∈n L(u − k) < +∞, where the convergence of the series is uniform on compact sets. Indeed, in this case the second part of Lemma 3.2 is an easy consequence of the property

lim

R →∞

L(u − k) = 0,

u−k 2 >R

uniformly with respect to u ∈ n (see [1, 10, 17]). 4. APPROXIMATION THEOREMS Now, we can prove the following. Theorem 4.1. Let f ∈ C 0 (n ). Then, for every x ∈ n , lim (Sw f )(x) = f (x)

w→+∞

In particular lim Sw f − f ∞ = 0,

w→+∞

for every f ∈ C (n ).

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Proof. We first note that Sw f is well-defined since f is bounded. Let x ∈ n be fixed. Then we can write n w (S f )(x) − f (x) ≤ − (wx − t , f (u) du − t , f (x)) wx k k w w A k Rk k∈n (wx − tk , f (x)) − f (x) + k∈n

wn f (u) − f (x) du L(wx − tk ) ≤ w A k R k k∈n (wx − tk , f (x)) − f (x) := I1 + I2 + k∈n


By the continuity Now, we estimate I1 . Let > 0 be fixed. of f at x, there exists > 0 such that f (u) − f (x) < whenever u − x 2 ≤ (u ∈ n ). We can use to split I1 in two additional summands:   wn   f (u) − f (x) du + I1 =   L(wx − tk ) Ak Rkw w

w

wx−tk 2 ≤ 2 wx−tk 2 > 2 := I1,a + I1,b For every u ∈ Rkw ⊂ n , if wx − tk 2 ≤ w /2, we have 1 √ 1 u − x ≤ u − tk + tk − x ≤ n · + , 2 w 2 w w 2 2 and since

√ n·

I1,a ≤

w

<

2

for sufficiently large w > 0, we obtain

wx−tk 2 ≤w /2

wn L(wx − tk ) Ak

du Rkw

≤ m0,n (L) · (),

for sufficiently large w > 0. Moreover, I1,b ≤ 2 f ∞

L(wx − tk )

wx−tk 2 >w /2

and by the property (ii) of Lemma 3.2, we obtain that I1,b → 0 as w → +∞ (in particular, the convergence is uniform with respect to x ∈ n ).


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Now, we estimate I2 . Since f is bounded, for every > 0 there exists j ∈ + such that supu∈n |f (u)| ≤ j with 1/j < . Set Bj = u ∈ n : 0 < f (u) < 1/j , we can rewrite I2 as follows: I2 = (wx − tk , f (x)1Bj ) − f (x)1Bj k∈n + (wx − tk , f (x)1n \Bj ) − f (x)1n \Bj k∈n ≤ (wx − tk , f (x)1Bj ) − f (x)1Bj + f (x) jw (x) k∈n := I2,a + I2,b , taking into account that (wx − tk , 0) = 0, for every k ∈ n and where 1Bj and 1n \Bj are the characteristic functions of the measurable sets Bj and n \ Bj . It is clear that I2,b ≤ f ∞ jw (x) and using (4) it is easy to see that I2,b → 0 as w → +∞ (also in this case, the convergence is uniform with respect to x ∈ n ). For I2,a we obtain I2,a ≤ L(wx − tk ) f (x)1Bj + f (x)1Bj k∈n

≤ m0,n (L) · (1/j ) + 1/j ≤ m0,n (L) · () + Since is arbitrarily chosen, we have shown that I1 = I1,a + I1,b → 0

as w → +∞,

and I2 = I2,a + I2,b → 0 as w → +∞; follows. therefore, since (Sw f )(x) − f (x) ≤ I1 + I2 , the assertion By similar arguments we can prove that limw→+∞ Sw f − f ∞ = 0, if f ∈ C (n ). Remark 4.2. We can observe that, since for every f ∈ Cc (n ) there exists a positive constant such that suppf ⊂ B(0, ), we have f (u)du = 0, Rkw

for every tk B(0, w ), being Rkw ∩ B(0, ) = ∅.

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Then, the nonlinear multivariate sampling Kantorovich operators of f is well-defined and reduces to the finite sum wn (Sw f )(x) = f (u)du , wx − tk , Ak Rkw tk 2 ≤w

for every x ∈ n and w > 0. Let now be a convex -function. By Jensen’s inequality and Fubini-Tonelli theorem, we obtain   wn   I [(Sw f )] ≤  f (u) du  dx wx − tk , Ak Rkw n tk 2 ≤w

  wn  f (u) du  ≤  L wx − tk  dx Ak Rkw n t ≤w

k 2     ≤  L wx − tk f ∞  dx n tk 2 ≤w

( m0,n (L)(f )) ∞ ≤ L wx − tk dx m0,n (L) n tk 2 ≤w

( m0,n (L)(f )) ∞ L1 < +∞, = w n m0,n (L) tk 2 ≤w

for every > 0. This inequality shows that Sw f ∈ E (n ) ⊂ L (n ), for every f ∈ Cc (n ) and w > 0. In order to obtain the desired result of modular convergence in Orlicz spaces for the family of nonlinear multivariate Sampling Kantorovich operators, we first prove the following norm-convergence theorem. Theorem 4.3. Let be a convex -function. For every f ∈ Cc (n ) we have lim Sw f − f = 0 w→+∞

Proof. Let f ∈ Cc (n ). We must show that limw→+∞ I [(Sw f − f )] = 0 for every > 0 or equivalently that the family (((Sw f − f )))w>0 converges to zero in L 1 (n ), for every > 0. We will use the Vitali convergence theorem in L 1 (n ).

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By Theorem 4.1 and the continuity of , it is easy to see that lim Sw f − f ∞ = 0, w→+∞

for every > 0. Let now > 0 be fixed and , > 0 such that suppf ⊂ B(0, ) and > + . By Lemma 3.2 (iii), for such , > 0, there exists a constant M > 0 (we can assume M > ), such that w n L(wx − tk ) dx < ,


x2 >M

for every sufficiently large w > 0 and tk 2 ≤ w. Then, as made in Remark 4.2, by Jensen’s inequality and Fubini-Tonelli theorem, we have ( (Sw f )(x) ) dx x2 >M





w  f (u) du   L wx − tk  dx Ak Rkw x2 >M tk 2 ≤w

    ≤  L wx − tk f ∞  dx x2 >M tk 2 ≤w

( m0,n (L)(f )) ∞ w n L wx − tk dx ≤ n w m0,n (L) x2 >M tk 2 ≤w

( m0,n (L)(f ∞ )) 0 represents the number of terms of the above series in fact corresponding to the number of sets Rkw having nonempty intersection with B(0, ). We estimate G . For every w ≥ 1 we obtain n w n w n

w n−i n n G≤ 2 +1 =2 = 2n i i=0 w n−1 +n + · · · + 1 = 2n w n · n n−1 1 1

+n · + ··· + n w w n n−1

n n +n + · · · + 1 =: w n · K , ≤w · 2

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(where [·] denotes the integer part). Thus,


x2 >M

( m0,n (L)(f ∞ )) ( (Sw f )(x) )dx < · · K =: · C , m0,n (L)

for every w ≥ 1. Therefore, for > 0 there exists a set E = B(0, M ) such that for every measurable set F , with F ∩ E = ∅, we have ( Sw f (x) − f (x) ) dx = ( Sw f (x) )dx F F ≤ ( Sw f (x) ) dx < · C x 2 >M Finally, let B ⊂ n be a measurable set with B < (f ∞ > 0), where := max (2m0,n (L)(f ∞ )), (2 f ∞ ) Using Remark 3.3, in correspondence to > 0 and for every w > 0, ( (Sw f )(x) − f (x) )dx B 1 1 ≤ (2 (Sw f )(x) )dx + (2 f (x) )dx 2 B 2 B 1 1 ≤ (2m0,n (L)(f ∞ )) dx + (2 f ∞ )dx 2 B 2 B ≤ dx = B < B

Therefore, it follows that the integrals ( (Sw f )(x) − f (x) ) dx (·)

are equi-absolutely continuous, for every > 0 and so the proof is complete. Now, to obtain a modular convergence result in L (n ) for Sw , we need a modular continuity property for our nonlinear operators. In [1] and [17], it is shown for the linear (resp. univariate and multivariate) version of the generalized sampling Kantorovich operators that Sw maps the Orlicz space L into itself. This property does not occur in the nonlinear univariate case (see, e.g., [38]) and hence the same happens for the nonlinear multivariate case. For our operators to be well defined in some Orlicz spaces, we need a growth condition (already used in [38] for


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the theory in one-dimensional setting) on the composition of the function , which generate the Orlicz space and the function of the (L, )Lipschitz condition. Let be a fixed -function. We suppose that there is a -function such that, for every ∈ (0, 1), there exists a constant C ∈ (0, 1) satisfying (H)

(C (u)) ≤ (u)

(u ∈ 0+ ),


where is the function of the (L, )-Lipschitz condition (see [3]). Now, we are ready to prove the following. Theorem 4.4. Let be a convex -function satisfying condition (H) with convex. Then, given f , g ∈ L (n ), there exist ∈ (0, 1) and a constant > 0 such that I [(Sw f − Sw g )] ≤

L1 I [(f n m0,n (L)

− g )]

(1)

In particular, for every w > 0, Sw : L (n ) → L (n ) Proof. Let f , g ∈ L (n ) and we estimate (Sw f )(x) − (Sw g )(x) for x ∈ n . (S f )(x) − (S g )(x) w w wn wn f (u)du − wx − tk , g (u)du ≤ wx − tk , w w A A k k Rk Rk k∈n wn f (u) − g (u) du L(wx − tk ) ≤ w A k R n k k∈ Now, since (f − g ) ∈ L (n ), there exists > 0 (we can take ∈ (0, 1)) such that I [(f − g )] < +∞. Then, we can choose > 0 such that m0,n (L) ≤ C (where C is the constant of condition (H)). Thus, applying twice Jensen’s inequality and using Fubini-Tonelli theorem, we obtain (Sw f )(x) − (Sw g )(x) dx I [(Sw f − Sw g )] = n   n w f (u) − g (u) du dx  L(wx − tk ) ≤ Ak Rkw n k∈n

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wn |f (u) − g (u)|du m0,n (L) ≤ m0,n (L) k∈n Ak Rkw L(wx − tk )dx · 1


n

L1 wn f (u) − g (u) du C ≤ n w m0,n (L) k∈n Ak Rkw L1 wn f (u) − g (u) du ≤ n w m0,n (L) k∈n Ak Rkw wn L1 ≤ n f (u) − g (u) du w m0,n (L) k∈n Ak Rkw ≤

L1 I [(f n m0,n (L)

− g )] < +∞,

and, therefore, (1) is proved. It is now obvious that for g ≡ 0, Sw maps L (n ) in L (n ).

We can finally prove the main theorem of this section. Theorem 4.5. Let be a convex -function satisfying condition (H) with convex. Moreover, let f ∈ L + (n ). Then there exists a constant > 0 such that lim I [(Sw f − f )] = 0

w→+∞

Proof. Let f ∈ L + (n ). By a density result (see, e.g., [2]), there exists > 0 (we can assume ∈ (0, 1)) such that for every > 0 there is a function g ∈ Cc (n ) such that I + [(f − g )] < . Let now > 0 such that C ≤ min 3m n (L) , 3 , where the constant C is that of condition (H). By 0, the properties of and Theorem 4.4, we have I [(Sw f − f )] ≤ I [3(Sw f − Sw g )] + I [3(Sw g − g )] + I [3(f − g )] L1 ≤ n I [(f − g )] + I [(Sw g − g )] + I [(f − g )] m0,n (L) ≤ I + [(f − g )] + I [(Sw g − g )] < + I [(Sw g − g )], where := max n mLn1 (L) , 1 . 0, The assertion now follows from Theorem 4.3.


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5. APPLICATIONS, EXAMPLES AND GRAPHICAL REPRESENTATIONS


We will now apply our convergence theorems to some special spaces. Let (u) = u p , 1 ≤ p < +∞, u ∈ 0+ and consider the Orlicz space generated by such . In this case, satisfies the 2 -condition, the modular p I coincides with ·p and L (n ) = L p (n ). Now, if we choose (u) = u (i.e. is strongly Lipschitz), we have that satisfies condition (H) with (u) = (u) = u p and C = . Moreover, we obtain the following proposition. Proposition 5.1. For every f ∈ L p (n ), 1 ≤ p < +∞, we have lim Sw f − f p = 0 w→+∞

Moreover, there holds S f ≤ −n/p m0,n (L)(p−1)/p L1/p f 1 w p p In order to weaken the choice of , we consider (u) = u q/p (1 ≤ q < p < +∞). In this case, condition (H) is satisfied with (u) = u q and C = q/p . We obtain the following. Proposition 5.2. Let 1 ≤ q < p < +∞ and , as above. Then q/p f , S f ≤ −n/p m0,n (L)(p−1)/p L1/p 1 w q p for every f ∈ L q (n ) and Sw : L q (n ) → L p (n ) is well defined. Moreover, for every f ∈ L p (n ) ∩ L q (n ) we have lim Sw f − f p = 0 w→+∞

Other important spaces in the applications are the “interpolation spaces” generated by the convex -functions , (u) = u log (e + u), u ≥ 0, ≥ 1, > 0 (L , (n ) = L log L(n )). The functions , satisfy the 2 -condition and the corresponding modular functional is now given by , f (x) log (e + f (x) )dx (f ∈ M (n )) I [f ] = n

For more information on interpolation spaces, see, for example, [6, 35]. Now, choosing (u) = u, we obtain for the interpolation spaces, as in the case of (u) = u p , that condition (H) is satisfied for (u) = , (u) and C = . Then we can write the following proposition.

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Proposition 5.3. For every f ∈ L log L(n ), with ≥ 1 and > 0, we have lim Sw f − f L log L = 0 w→+∞

Moreover,

(S f )(x) log (e + (S f )(x) ) dx w w n L1 f (x) log (e + m0,n (L) f (x) ) dx, ≤ n 1− m0,n (L) n


for every > 0, i.e., Sw : L log L(n ) → L log L(n ). Finally, we consider the case of the “exponential spaces" ([19, 20])

generated by the convex -functions (u) = e u − 1, > 0, u ∈ 0+ . The spaces L (n ) are examples of Orlicz spaces where the norm convergence and the modular convergence are not equivalent, since does not satisfy the 2 -condition. The modular functional generated by is given by

(exp( f (x) ) − 1)dx (f ∈ M (n )) I [f ] = n

If we set (u) = u, condition (H) is fulfilled for (u) = (u) and C = . We obtain the following proposition. Proposition 5.4. Let f ∈ L (n ). Then, (exp( (Sw f )(x) ) − 1)dx n L1 ≤ n (exp(m0,n (L) f (x) ) − 1)dx, m0,n (L) n for some > 0. In particular, Sw : L (n ) → L (n ), for every w > 0. Moreover, there exists > 0 such that lim (exp( (Sw f )(x) − f (x) ) − 1)dx = 0 w→+∞

n

It is clear that, taking (u) = u, one can furnish other estimates and convergence results for our operators in the spaces generated by the functions and . In order to give concrete examples of the above nonlinear multivariate sampling Kantorovich operators, the most natural way is that of considering kernel functions of the form (wx − tk , u) = L(wx − tk )gw (u),


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where (gw )w>0 , gw : → , is a family of functions satisfying gw (u) → u uniformly as w → +∞ and such that there exists a -function with gw (u) − gw (v) ≤ ( u − v ),


for every u, v ∈ and w > 0. Assumptions (1), (2), (3), (4) and (L1), (L2), (L3) result in the request that: (1) k → L(wx − tk ) ∈ 1 (n ), for every x ∈ n and w > 0, L ∈ L 1 (n ) is bounded in a neighborhood of 0 ∈ n and there exists a number > 0 such that L(wx − tk ) wx − tk 2 < +∞; m,n (L) := sup x∈n k∈n

(2) gw (0) = 0, for every w > 0; (3) for every j ∈ + , there holds gw (u) w j (x) := sup L(wx − tk ) − 1 → 0 u k∈n 1/j ≤ u ≤j as w → +∞, uniformly with respect to x ∈ n . If gw (u) ≡ u for every w > 0, the operators Sw f are linear and the convergence results obtained in this article reduce to the known result of [17]. In general, it is not so easy to verify if a function L : n → 0+ satisfies the conditions 1 and 3. Therefore may be useful to construct examples of functions L using the following simple procedure based on univariate functions satisfying suitable properties (see, e.g., [10, 17]). For the sake of simplicity, in what follows we will consider the uniform sequence tk = k, k ∈ n . Let now L1 , L2 , , Ln ∈ L 1 () such that Li (u − k) < +∞, (i = 1, 2, , n) m0, (Li ) := sup u∈ k∈

where = (k)k∈ and the convergenceof the series is uniform on compact sets of . Moreover, we assume that k∈ Li (u − k) = 1, for every u ∈ and i = 1, 2, , n. n Setting L(u) := i=1 Li (ui ), we obtain that L ∈ L 1 (n ) since

n

L(u)du =

n !

n i=1

Li (ui )du1 · · · dun =

n ! i=1

Li (ui )dui < +∞,

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and m0,n (L) = sup

u∈n k∈n

L(u − k) =

n !

m0, (Li ) < +∞,

i=1

and the convergence is uniform on compact subsets of n . Further,


k∈n

L(u − k) =

n !

Li (ui − ki ) = 1,

(2)

i=1 ki ∈

for every u ∈ n . Thus by Remark 3.3 (c), we can say that L satisfy condition 1, while by (2) we obtain condition 3, taking into account that gw (u) → u uniformly as w → +∞. In the following we will take (gw )w>0 of the form: " u 1−1/w , 0 < u < 1, gw (u) = (3) u, otherwise, for every u ∈ and w > 0. A first prototypical example of nonlinear multivariate sampling Kantorovich operator of the above type is based on the multivariate Fejér’s n kernel n (x) := i=1 F (xi ) (see, e.g., Figure 1 for n = 2), where F is the well-known (univariate) Fejér’s kernel x 1 2 , F (x) := sinc 2 2 (see, e.g., [12]) and where the sinc-function is defined by   sin(x) , x ∈ \ 0 , sinc(x) := x 1, x = 0 n We have that k∈n n (u − k) = 1, for every u ∈ , n is bounded, summable and satisfies all the others required conditions (see [10] and

FIGURE 1 Bivariate Fejér’s kernel 2 .

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FIGURE 2

Graph of the function f .


[17]). In this case (uniform sampling), we can assume 1n (wx − k, u) := n (wx − k)gw (u), n

and the nonlinear multivariate Sampling Kantorovich operators Sw1 f take now the form wn 1n n (wx − k)gw f (u)du , (Sw f )(x) = Ak Rkw k∈n for every x ∈ n and w > 0.

2

For example, we apply now our nonlinear operators Sw1 f (where 2 (x, y) = F (x)F (y) is the two-dimentional Fejér’s kernel) to the particular discontinuous function f ∈ L p (2 ), for every 1 ≤ p < +∞, defined by (Figure 2).   −1 ≤ x ≤ 1 and − 1 ≤ y ≤ 1, 3, (4) f (x, y) = 6  , otherwise  2 2 x +y The nonlinear (two-dimensional) sampling Kantorovich operators for the function f defined in (4) in case of w = 5 and w = 10 are given in Figure 3 (in an octant of the plane).

FIGURE 3

2

The function f (black) with the nonlinear (two-dimensional) sampling Kantorovich 2

operator S5 1 f (gray) and S101 f (gray), respectively.

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The graphs show how the operators approximate the function f ; the two-dimensional Fejér’s kernel does not have compact support over 2 , hence in order to evaluate the sampling ' series one needs to compute an infinite number of mean values w 2 R w f (u1 , u2 ) du1 du2 at any point


k

(x, y) ∈ 2 . Then the sampling series must be truncated and this leads to truncation errors. In general, this problem does not arise for the operators Sw f having kernels with compact support. In this case, the infinite sum reduces to a finite one. Examples of kernels with compact support can be constructed by using the multivariate B-splines of order k ∈ + , defined by nk (x) := n n i=1 Mk (xi ), x ∈ , where Mk are the well-known (univariate) B-spline, given by k−1 k k 1 i k (−1) , +x −i Mk (x) := i (k − 1)! i=0 2 + x ∈ (where the function (x)+ = max x, 0 is the positive part of x). Now, we set ˜ kn (wx − k, u) := nk (wx − k)gw (u), x ∈ n and w > 0. It is shown in [10] and [17] that the kernels nk satisfy assumption 1, and it is also easy to show that 3 holds, for every k ∈ + and so ˜ kn are kernel functions. In this case, the nonlinear multivariate sampling Kantorovich operators ˜ n Swk f take the form n w ˜ kn (Sw f )(x) = nk (wx − k)gw f (u)du , Ak Rkw k∈n for every x ∈ n and w > 0.

˜ 2

Let’s now consider the particular case of Sw3 f , applied to the above function f defined in (4). First, we recall that the B-spline M3 is given by  3 1   − x 2, x ≤ ,  4 2  2  1 3 1 3 M3 (x) := − x < x ≤ , ,  2 2 2 2     3 0, x > , 2 for x ∈ , and let 23 (x, y) = M3 (x)M3 (y) (see Figure 4) be the twodimentional B-spline of order 3.

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FIGURE 4

FIGURE 5

˜ 2

Bivariate B-spline kernel 23 .

The function f (black) with the nonlinear (two-dimensional) sampling Kantorovich ˜ 2

operator S5 3 f (gray) and S103 f (gray), respectively.

The nonlinear (two-dimensional) sampling Kantorovich operators generated by 23 , for the function f defined in (4) in case of w = 5 and w = 10, are plotted in Figure 5. The graphs show how the operators approximate the function f . Moreover, if we compare the graphs of the nonlinear bivariate sampling Kantorovich operators generated by the spline kernels and the Fejér’s kernels, it is clear that the approximation by the series based on 23 is considerably better than the approximation obtained by the series ˜ 2

based on 2 (see, e.g., Figure 6). This means that applying the series Sw3 f , a reasonably good approximation can be achieved by taking into account 2

fewer mean values of f than those used for the series Sw1 f . One can also use, instead of M3 , linear combination of univariate Bspline of different degree, such as L1 (x) := 4M3 (x) − 3M4 (x),

˜ 2

L2 (x) := 5M4 (x) − 4M5 (x)

2

˜ 2

(x ∈ ),

2

FIGURE 6 f (black) with S5 3 f (dark gray), S5 1 f (gray) and S103 f (dark gray) and S101 f (gray), respectively.

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or linear combinations of translates of B-splines, e.g., L3 (x) :=

5 1 M3 (x) − M3 (x + 1) + M3 (x − 1) 4 8

(x ∈ ),

in order to construct examples of multivariate kernels improving the rate of approximation.


6. APPLICATION TO IMAGE PROCESSING The theory of sampling operators is used in signal processing to reconstruct and approximate signals. In particular, the theory of multivariate sampling operators can be applied to image approximation (image processing), since images are well-known examples of multivariate signals. A static image can be represented by a two-dimensional function (signal), while a digital (static) image is a discrete signal and it can be represented by a two-dimensional matrix. In order to apply the theory of the nonlinear multivariate sampling Kantorovich operators to the image processing, the first step is to represent digital images (i.e. the matrices) with suitable functions, as made in [17] for the case of linear multivariate sampling Kantorovich operators. Every square matrix A = (aij )ij , i, j = 1, 2, , m, can be modeled as a step function I = I (x, y) (gray level image function), with compact support, belonging to L p (2 ), for every 1 ≤ p < +∞. The most natural way to define I is to consider I (x, y) :=

m m

aij · 1ij (x, y)

((x, y) ∈ 2 ),

i=1 j =1

where 1ij (x, y), i, j = 1, 2, , m, are the characteristics functions of the sets (i − 1, i] × (j − 1, j ] (i.e. 1ij (x, y) = 1, for (x, y) ∈ (i − 1, i] × (j − 1, j ] and 1ij (x, y) = 0 otherwise). Note that the above function I (x, y) is defined in such a way that to every pixel (i, j ) it is associated the corresponding grey level aij . Then, we can now consider the family of nonlinear bivariate sampling Kantorovich operators (Sw I )w>0 (for some kernel ) that is an approximating net of I in L p -sense, according to the convergence results obtained in Section 4. Now, in order to obtain a new image (matrix) that approximates the original one, it is sufficient to sample the operators Sw I (for some w > 0) with a fixed sampling rate. In particular, we can reconstruct the approximating images (matrices) taking into consideration different sampling rates and this is possible since we know the analytic expression of Sw I .

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FIGURE 7 “Lena.” Original image (150 × 150 pixel resolution).

If the sampling rate is chosen higher than the original sampling rate, one can get a new image that has an higher resolution than the original one’s, that is, it is built with a larger number of pixel compared to the original image (image enhancement). Now, we consider the following practical example, showing the application of the theory to the image in Figure 7 (150 × 150 pixel resolution). In Figures 8 and 9 we approximate the image in Figure 7 by means 2

of the nonlinear two-dimensional sampling Kantorovich operators Sw1 and ˜ 2

Sw3 , for w = 5 and w = 10, where 12 and ˜ 32 are the bivariate Fejér’s kernel and the bivariate B-spline kernel of order 3, respectively. The sampling rate in the images of Figures 8 and 9 is the same of the original one.

FIGURE 8

2

2

“Lena.” Approximation by S5 1 and S101 (150 × 150 pixel resolution).

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FIGURE 9

˜ 2

˜ 2

“Lena.” Approximation by S5 3 and S103 (150 × 150 pixel resolution).

2

2

˜ 2

˜ 2

FIGURE 10 resolution).

“Lena.” Approximation by S5 1 and S101 with increased resolution (300 × 300 pixel

FIGURE 11 resolution).

“Lena.” Approximation by S5 3 and S103 with increased resolution (300 × 300 pixel

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In Figures 10 and 11 we approximate the image in Figure 7 by means 2

˜ 2

of the nonlinear bivariate sampling Kantorovich operators Sw1 and Sw3 , for w = 5 and w = 10; here we take into account a number of sample values bigger than those of the image in Figures 8 and 9 (300 × 300 pixel resolution), so obtaining enhanced images. ACKNOWLEDGMENTS


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