Sampling Kantorovich operators and their applications to ... - wseas

Recent Advances in Applied Mathematics, Modelling and Simulation

Sampling Kantorovich operators and their applications to approximation problems and to Digital Image Processing DANILO COSTARELLI Roma Tre University Dipartimento di Matematica e Fisica Largo San L. Murialdo 1, 00146 Roma ITALY [email protected]

GIANLUCA VINTI University of Perugia Dipartimento di Matematica e Informatica Via Vanvitelli 1, 06123 Perugia ITALY [email protected]

Abstract: Here we present an overview on the theory of multivariate sampling Kantorovich operators, with some new applications to digital image processing (D.I.P.). Results concerning the convergence for the family of the sampling Kantorovich operators are showed. Moreover, in order to obtain applications to D.I.P., biomedical images are reconstructed and enhanced. The possible advantages that can be achieved by the present D.I.P. technique are discussed in details, both from the mathematical and the medical point of view. Key–Words: Sampling Kantorovich operators, Orlicz spaces, irregular sampling, biomedical images, image processing, vascular apparatus, CT images.

1

Introduction

ple values f (k/w) by mean values of the function f computed on suitable intervals of IR containing k/w. The main advantages produced by the sampling Kantorovich operators (other than the possibility to approximate discontinuous signals) is that ”time-jitter” errors are reduced. Time-jitter errors occur when the node k can not be matched exactly (or tk if we consider an irregular sampling scheme, i.e., a general sequence of nodes in place of k ∈ ZZ). The sampling Kantorovich operators reduces time-jitter errors calculating the information in a neighborhood of a point rather than exactly at that point. A multivariate extension of the results proved in [3] has been obtained in [15] in order to apply the above theory to image processing. In the last years, the theory of sampling Kantorovich operators has been largely studied in the following papers [24, 16, 17, 21, 25, 26, 8, 9, 18]. In particular, in [8, 9] some civil engineering models have been developed for studying the behaviors of buildings under seismic action. These models are based on a study made by using thermorgraphic images of building enhanced by an algorithm derived from the theory of sampling Kantorovich operators.

Applications to Approximation and to Signal Theory by means of family of sampling operators have been largely studied since the eighties. The sampling Kantorovich operators defined in (1) of Section 2 have been introduced in [3], in order to reconstruct not necessarily continuous, one-dimensional, signals. The above operators are very useful in the sampling and signal theories. One of the most important results to the Sampling Theory is given by the well-known Wittaker-Kotelnikov-Shannon (WKS) sampling theorem, see e.g., [20, 2]. The WKS sampling theorem provides an exact reconstruction formula for bandlimited, finite energy functions. These two conditions represent rather restrictive assumptions that allow to reconstruct only smooth functions. In order to weaken the above assumptions, the generalized sampling operators ([23, 1, 4, 5]) have been introduced. The generalized sampling operators are very suitable to approximate (in some sense) a given continuous signal starting from a sequence of its sample values of the form f (k/w), k ∈ ZZ and w > 0. However, in the applications signals have often discontinuities. For instance, images represent typical examples of multivariate signals not necessarily continuous, since jumps of gray levels in the edges of figures can be represented by discontinuities. The sampling Kantorovich operators revealed to be very useful to reconstruct these families of signals, and they represent an L1 −version of the generalized sampling operators. Their definition is obtained replacing the sam-

ISBN: 978-960-474-398-8

Here, we present an overview on the theory of multivariate sampling Kantorovich operators. Moreover, some new applications to digital image processing (D.I.P.) are obtained also considering biomedical images. The pointwise and uniform convergence for the family (Sw f )w>0 is showed, in case of bounded continuous and uniformly continuous functions. Then, the theory of Orlicz spaces is recalled

256


in order to introduce the general setting in which we study a reconstruction theorem for not necessarily continuous functions by means of sampling Kantorovich operators. Orlicz spaces are functional spaces including, as particular case, the Lp -spaces. Several examples of multivariate kernels for the operators here considered are shown (see e.g., [15, 11, 12, 13, 10, 14]). For what concerns the applications discussed in the paper, the algorithm based on the theory of sampling Kantorovich operators for image reconstruction and enhancement is described in detail. A concrete examples is showed by processing a portion of a CT (computer tomography) image depicting the aorta artery. A detailed discussion concerning the possible advantages that can be derived by means of the algorithm for image reconstruction and enhancement based on sampling Kantorovich operators has been done. In particular, we discuss the possible advantages from the point of view of medical diagnosis in case of some pathology of the vascular apparatus.

symbols Rkw denote the sets of the form:

2

3

Approximation of continuous functions

We first introduce some notations. In what follows we denote by tk = (tk1 , . . . , tkn ) ∈ IRn vectors where each (tki )ki ∈ZZ is a strictly increasing sequence of real numbers, such that δ ≤ ∆ki := tki+1 − tki ≤ ∆, ki ∈ ZZ, for some δ, ∆ > 0, and limki →±∞ tki = ±∞, for every i = 1, . . . , n. A function χ : IRn → IR is said to be a kernel if the following requirements are satisfied: (χ1) χ ∈ L1 (IRn ) and is locally bounded on the origin; (χ2) For every u ∈ IRn ,

P

k∈ZZ n

χ(u − tk ) = 1;

(χ3) For some β > 0, X χ(u − tk ) ·ku−tk kβ < +∞,

mβ,Πn (χ) := sup

2

u∈IRn k∈ZZ n

where k · k2 denotes the usual Euclidean norm in IRn . The multivariate sampling Kantorovich operators (Sw )w>0 are defined by: "

(Sw f )(x) :=

X

χ(wx − tk )

k∈ZZ n

nZ

w Ak

Rkw

#

f (u) du , (1)

for every x ∈ IRn , where f : IRn → IR is a locally integrable functions such that the above series is convergent at any point, and χ is a kernel function. The ISBN: 978-960-474-398-8

Rkw

tkn tkn +1 tk1 tk1 +1 × ... × , := , , w w w w

w > 0 and Ak := ∆k1 · · · ∆kn . We now investigate the convergence properties of the above operators in (1) when one deals with continuous signals, see e.g., [15]. Theorem 1 Let f : IRn → IR be a continuous and bounded function. Then, for every x ∈ IRn , lim (Sw f )(x) = f (x).

w→+∞

In particular, for f ∈ C(IRn ) (C(IRn ) denotes the space of uniformly continuous and bounded functions on IRn ), we have lim kSw f − f k∞ = 0,

w→+∞

where k · k∞ denotes the usual sup-norm.

Approximation of not necessarily continuous functions

We first recall some basic notions concerning Orlicz + spaces. Let ϕ : IR+ 0 → IR0 be a ϕ-function (see, e.g., [19, 6]), i.e., ϕ is continuous, non-decreasing, ϕ(0) = 0, ϕ(u) > 0 for u > 0, and limu→+∞ ϕ(u) = +∞. ϕ ϕ We R define the modular functionalnI , by: I [f ] := n IRn ϕ(|f (x)|) dx, for f ∈ M (IR ), where M (IR ) denotes the space of Lebesgue measurable functions on IRn . Then, the Orlicz space generated by ϕ is defined by: Lϕ (IRn ) := {f ∈ M (IRn ) : I ϕ [λf ] < +∞, for some λ > 0} . Now, we can introduce in Lϕ (IRn ) a notion of convergence, called modular convergence, the most natural in this setting, see e.g., [19, 6, 24]. We will say that a family of functions (fw )w>0 in Lϕ (IRn ) is modularly convergent to f ∈ Lϕ (IRn ) if there exists λ > 0 such that: lim I ϕ [λ(fw − f )] = 0.

w→+∞

Now, by means of a modular estimate for the operators (1) and using a density result, we may state the following modular convergence theorem, see [15]. Theorem 2 Let ϕ be a convex ϕ-function. For every f ∈ Lϕ (IRn ), there exists a λ > 0 such that: lim I ϕ [λ(Sw f − f )] = 0.

w→+∞

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Now, we apply the above theory in Lp -spaces, which represent important examples of Orlicz spaces, very useful in applications, e.g., in applications to signal and image processing. Choosing ϕ(u) = up , 1 ≤ p < +∞, we obtain Lϕ (IRn ) = Lp (IRn ) and I ϕ [f ] = kf kpp , where k · kp is the usual Lp -norm. Then, from Theorem 2, we obtain the following corollary.

Moreover, as further example we mention the Jackson-type kernels, defined by

Jk (x) = ck sinc2k

x , 2kπα

x ∈ IR,

with k ∈ IN , α ≥ 1, where the normalization coefficients ck are given by

Corollary 3 For every f ∈ Lp (IRn ), 1 ≤ p < +∞,

Z

ck := lim kSw f − f kp = 0.

2k

sinc IR

u 2kπα

−1

du

,

w→+∞

see e.g., [7], where the sinc-function is defined by: The above corollary allow us to reconstruct Lp signals, in Lp -sense, therefore not necessarily continuous. Other examples of Orlicz spaces for which the theory of sampling Kantorovich operators can be applied, are given by the Zygmund spaces (or interpolation spaces) generated by the ϕ-function ϕ(u) = uα logβ (u + e), u ≥ 0, with α ≥ 1, β > 0, and by γ the exponential spaces, generated by ϕ(u) = eu − 1, u ≥ 0, with γ > 0; see e.g., [6, 3, 15, 25, 21].

4

(

sinc(x) :=

5

I(x, y) :=

s−1

, +

where the function (x)+ := max {x, 0} denotes the positive part of x ∈ IR, see e.g., [7, 15, 11, 12, 13, 17, 10, 14, 18]. The central B-splines of order s ∈ IN , satisfy (χj), j = 1, 2, 3, for n = 1, and in particular (χ2) is satisfied for every positive β, see e.g., [15, 16]. ISBN: 978-960-474-398-8

aij · 1ij (x, y) ((x, y) ∈ IR2 ),

where 1ij (x, y), i, j = 1, 2, ..., m, are defined by 1ij (x, y) = 1, for (x, y) ∈ (i − 1, i] × (j − 1, j] and 1ij (x, y) = 0 otherwise. In this way I(x, y) maps every (i, j) to the corresponding aij . The family of bivariate sampling Kantorovich operators (Sw I)w>0 (for some kernel χ) approximate I in Lp -sense (see Corollary 3) and can be used to reconstruct and to enhance the original image. To obtain a new image (matrix), we sample Sw I (for some w > 0) with a fixed sampling rate. Clearly, we can sample Sw I considering different sampling rates and this is possible since we know Sw I analytically in all its domain. If the sampling rate is chosen higher than the original one, a new image with an increased resolution with respect to the original image, can be obtained. The above procedure has been implemented by using MATLAB, in order to obtain an algorithm based on the multivariate sampling Kantorovich theory. Now, we show some practical reconstruction and enhancement of a biomedical image that could be successfully applied to perform accurate diagnosis concerning some pathology of the vascular apparatus. First of all, we begin showing in Fig. 1 a CT image without contrast medium, depicting the aorta artery. Our region of interest (ROI) in Fig. 2 is delimited

Lemma 4 allows us to construct multivariate kernels of the product type starting from univariate ones. For this reason, now we list some well-known examples of one-dimensional kernels satisfying the assumptions of Lemma 4. Examples of kernels with compact support are furnished by the well-known central B-spline of order s ∈ IN , defined by: s +x−i 2

m X m X i=1 j=1

Lemma 4 Let χi : IR → IR, i = 1, . . . , n, be one-dimensional kernels satisfying assumptions (χj), jQ = 1, 2, 3, for n = 1. Then, setting: χ(x) := n n i=1 χi (xi ), x ∈ IR , we have that χ is a multivariate kernel and it satisfies conditions (χj), j = 1, 2, 3.

!

Digital image processing

Here, we apply the multivariate sampling Kantorovich operators in order to process digital images. A bidimensional gray scale image A (matrix) can be represented by a step function I belonging to Lp (IR2 ), 1 ≤ p < +∞. We define I by:

In the above theory an important role is played by the kernels χ. For the sake of simplicity, in what follows we consider only the case tk = k ∈ ZZ n of an uniform sampling scheme. The following lemma is useful in order to choice suitable kernels for the above multivariate operators.

s i

x 6= 0, x = 0.

For other useful examples of kernels, see also [7, 6, 11, 12, 13, 10, 14].

Concrete examples of kernels

s X 1 Ms (x) := (−1)i (s − 1)! i=0

sin(πx)/πx, 1,

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Figure 1: CT image without contrast medium. In the red

Figure 3: Fig.2 enhanced by the sampling Kantorovich

square is depicted the aorta artery.

operators S20 I based upon a bivariate Jackson-type kernel.

Figure 2: ROI of the CT image without contrast medium

ble resolution with respect to each spatial dimension (480 × 480 pixel). Even if, the contours of the lumen vessel are not yet clearly visible, our next step is to apply some algorithms of edge detection (such as those based on wavelet decompositions) to the above image in Fig. 3, in order to extract the contours which are of our interest and to allow the doctors to perform a better diagnosis. Obviously, we expect that the lumen of the vessel results visible in a better way in the image reconstructed by the sampling Kantorovich algorithm since, in general, Fig. 3 is more detailed than the image in Fig. 2.

of Fig. 1, depicting the aorta artery.

6 by the red square (240 × 240 pixel) on the CT image showed in Fig. 1. The main purpose in processing the biomedical image in Fig 2 is to emphasize the contours of the lumen in the vessel. The above detail is very important from a medical point of view, since allows the doctors to distinguish the thrombotic areas from the vessel lumen, and to perform an accurate diagnosis of the pathologies, such as aneurysms or stenosis, and to set up a therapeutic or surgery treatment. Obviously, the above problem could be solved by using contrast medium in CT investigations, but the contrast medium is quite invasive, and cannot be always exploitable. For this reason, having at disposal techniques of digital image processing (D.I.P.) we became able to perform, from the original CT images obtained without contrast medium, a better diagnosis. A first step in processing Fig. 2 is the reconstruction and the enhancement by the algorithm based on sampling Kantorovich operators. In particular, in Fig. 3 the image obtained by the operators Sw I, with w = 20 and based upon the bivariate Jackson-type kernel obtained by Lemma 4 using the univariate kernel J12 (x), x ∈ IR, with parameters k = 12 and α = 1 (see Section 4) is showed. At the present step, the final image is enhanced with respect to the original one, in the sense that it has been reconstructed with douISBN: 978-960-474-398-8

Conclusions

The implementation of the theory of sampling Kantorovich operators, performed by MATLAB revealed to be very useful for digital image reconstruction and enhancement. In particular it can be used as starting point of a series of D.I.P. algorithms that can be applied to biomedical images depicting arteries, in order to extract the contours of the vessel lumen from the vessel itself. The above procedure can results very useful from a medical point of view, in order to improve diagnosis of some pathologies of the vascular apparatus. The research will be performed with the collaboration of some doctors of the ”Department of Surgical and Biomedical Sciences” of the University of Perugia (Italy). Acknowledgements: The research was supported by the national research group GNAMPA of the Istituto Nazionale di Alta Matematica - INdAM (grant No. U 2014/000237). The authors would like to thank Prof. Enrico Cieri and Drs. Giacomo Isernia and Gioele Simonte for providing the CT image in Fig. 1. References: [1] L. Angeloni and G. Vinti, Rate of approximation for nonlinear integral operators with applications 259


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