UnEncrypted

I. Cravero, C. Dagnino, S. Remogna

Figure 2: Example 2, Approach 1. The graphs of (a) the exact solution, (b) the approximation computed with m = 7, n = 3, (c) the discrete L∞-norm error computed on Ψ.

Figure 3: Example 2, Approach 2.(a) Parameter domain Ω0, (b) physical domain Ω and (c) control points.

Then, we perform the same h-refinement of Approach 1. In this case the smoothness vector μ¯η has elements equal to one, while μ¯ξ has all of the elements equal to one except the element corresponding to s = 12 , that is equal to zero. We report the results in the third row of Table 2. In order to obtain a basis, according to Section 2, we remark that we have to neglect two inner B-splines either with C1 smoothness everywhere or with C0 smoothness only on the boundary of their support, because in this case the domain Ω is subdivided into two subdomains.

In Figs. 4(a)÷(c) we give the graphs of the exact solution, the approximation computed

Quadratic B-splines on criss-cross triangulations for elliptic problems

with m = 7, n = 3 and the discrete L∞-norm error computed on Ψ.

Figure 4: Example 2, Approach 2. The graphs of (a) the exact solution, (b) the approximation computed with m = 7, n = 3, (c) the discrete L∞-norm error computed on Ψ.

Table 2: Example 2. Error in L2-norm versus interval number per side.

In [11] the authors solve the same problem, by considering the two above approaches, but they use a method based on biquadratic tensor product B-splines. If we analyse our results and theirs, we can conclude that the two methods are comparable.

Example 3

In this example we consider the Poisson’s problem in the domain shown in Fig. 5(b)

where ΓN is given by the two segments with endpoints (-4,0), (0,0) and (-2,4), (2,4), respectively and ΓD is given by the two parabolic sections with endpoints (-4,0), (-2,4) and (0,0),

5(a) we show the parameter domain Ω , with the associated knot sequences and, in Fig. 5(b), the corresponding physical domain Ω, with the control points.

I. Cravero, C. Dagnino, S. Remogna

Figure 5: Example 3.(a) Parameter domain Ω0, (b) physical domain Ω and (c) control points.

Then, we perform h-refinement, considering m, n = 1, 3, 7, 15, 31 and, in Table 3, we report the results. In Figs. 6(a) ÷ (c) we give the graphs of the exact solution, the approximation computed with m = n = 7 and the discrete L∞-norm error computed on Ψ.

Table 3: Example 3. Error in L2-norm versus interval number per side.

Figure 6: Example 3. The graphs of (a) the exact solution, (b) the approximation computed with m = n = 7, (c) the discrete L∞-norm error computed on Ψ.

References

[1]P. Costantini, C. Manni, F. Pelosi, M. L. Sampoli, Quasi-interpolation in isogeometric analysis based on generalized B-splines, Comput. Aided Geom. Design 27

(2010) 656–668.

Quadratic B-splines on criss-cross triangulations for elliptic problems

[2] C. Dagnino, P. Lamberti, S. Remogna, BB-coe cients of unequally smooth quadratic B-splines on non uniform criss-cross triangulations, Quaderni Scientifici del Dipartimento di Matematica, Universit`a di Torino, 24 (2008) http://hdl.handle.net/2318/434

[3]C. Dagnino, P. Lamberti, S. Remogna, On unequally smooth bivariate quadratic spline spaces, Computational and Mathematical Methods in Science and Engineering ( J. Vigo Aguiar Ed.), (2009) 350–359.

[4]C. de Falco, A. Reali, R. Vazquez´ , GeoPDEs: a research tool for Isogeometric Analysis of PDEs, Adv. Eng. Softw. 42 (2011) 1020–1034.

[5]T. J. R. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric Analysis. Toward integration of CAD and FEA, WILEY, 2009.

[6]J. N. Lyness, R. Cools, A Survey of Numerical Cubature over Triangles, Mathematics and Computer Science Division, Argonne National Laboratory III 1994.

[7]A. Quarteroni, Numerical Models for Di erential Problems, Modeling, Simulation & Applications, Springer, 2009.

[8]P. Sablonniere` , Quadratic B-splines on non-uniform criss-cross triangulations of bounded rectangular domains of the plane, Report IRMAR 03-14, University of Rennes (2003)

[9]P. Sablonniere` , Quadratic spline quasi-interpolants on bounded domains of Rd, d = 1, 2, 3, Rend. Sem. Mat. Univ. Pol. Torino 61 (2003) 229–238.

[10]H. Speleers, C. Manni, F. Pelosi, M. L. Sampoli, Isogeometric analysis with Powell-Sabin splines for advection-di usion-reaction problems, Comput. Methods Appl. Mech. Engrg. 221-222 (2012) 132–148.

[11]A. V. Vuong, Ch. Heinrich, B. Simeon, ISOGAT: a 2D tutorial MATLAB code for Isogeometric Analysis, Comput. Aided Geom. Design 27 (2010) 644–655.

[12]R. H. Wang, Multivariate Spline Functions and Their Application, Science Press, Beijing/New York, Kluwer Academic Publishers, Dordrecht/Boston/London, 2001.

[13]R. H. Wang C. J. Li, A kind of multivariate NURBS surfaces, J. Comp. Math. 22

(2004) 137–144.

[14]G. von Winckel, Matlab procedure triquad, http://www.mathworks.com/matlabcentral/fileexchange/9230-gaussian-quadrature- for-triangles.

Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Image filtering with generalized fractional integrals

E. Cuesta1, A. Duran1, M. Kirane2 and S. A. Malik2

1 Department of Applied Mathematics, University of Valladolid, Spain

2 Laboratoire de Math´ematiques, Image et Applications, University of La Rochelle, France

emails: eduardo@mat.uva.es, angel@mac.uva.es, mokhtar.kirane@univ-lr.fr, salman.malik@univ-lr.fr

Abstract

In this work a new PDE technique based on fractional calculus for image restoration is proposed. The fractional order parameter, which controls the di usion from classical Gaussian filtering to the absence of smoothing, is chosen, according to the variation of the gradient of the image at each pixel, by using new convolution kernels in the corresponding Volterra equations. The new implementation is described and some numerical experiments are shown.

Key words: Image processing, Fractional integrals and derivatives, Volterra equations, Convolution quadrature methods.

MSC 2000: 44A35, 44K05, 45D05, 65R20, 68U10, 94A08.

1Introduction

This paper presents a numerical technique for image filtering based on the fractional partial di erential equation model

where ∂tα stands for the fractional time derivative of order 1 < α < 2 in the sense of Riemann–Liouville, Ω is a domain in R2 with boundary ∂Ω, ∂/∂η stands for the outward

Image filtering with generalized fractional integrals

normal derivative and u0 is the noisy image from which the objective is to restore the original (ideal) image. Thus u(t, x) stands for the restored image at the time level t. The range of the parameter α implies that (1) interpolates the linear parabolic heat equation (α = 1) and the linear hyperbolic wave equation (α = 2), which suggests a suitable adaptation of the model to image filtering processes, in order to handle the di usion following some dependence of α on the image gradient variation. The purpose of this work, based on the idea of applying (1) for each pixel of the image following the behaviour of the corresponding gradient [6], is to introduce an improvement of the selection of the viscosity parameter α, which is updated at every time step of the simulation. This leads to a better adaptation of the model (1) to the evolution of the image gradient.

Among the di erent methods presented in the literature for image processing, PDEs based models play a relevant role (see [1, 19] and references therein). In the particular case of restoration, use of the linear heat equation, α = 1 in (1), lead to several investigations, aiming to control the di usion process. They include:

1.Di usion models with edge stopping functions [12], which incorporates a gradientdependent di usion coe cient to control the process and avoid blurring e ect in edges and corners provided by the classical di usion given by the heat equation.

2.Variational numerical algorithms, based on the minimization of the total variation of the image subject to constraints involving statistical parameters of the noise [15].

3.Neighborhood filters [9, 20], consisting of local averaging, with di erent techniques, to smooth the image noise

4.Anisotropic approach [3, 18], where most of the previous models (and the fractional PDE based ones below) are enhanced to incorporate non canonical directions of diffusion.

As an alternative to the nonlinear models described above, fractional calculus approach for image filtering has been considered in e. g. [2, 5]. In the last reference which, along with [6], is based on, the heat equation is generalized to models of the form (1). The fractional parameter α (1, 2) plays the role of a viscosity term with limiting values that make (1) interpolate the linear parabolic heat equation and the linear hyperbolic wave equation. This makes the selection of α being a particularly interesting task to control the image di usion. On the other hand, in [6] the authors propose a refinement to handle the di usion in a not uniform way over the whole image. This consists of applying (1) with a possibly di erent value of α for each single pixel of the image. The corresponding Volterra equation is wellposed for all t > 0 (which is not guaranteed in some nonlinear models) and the numerical results are e cient and competitive.

Considered here is an approach which goes more deeply into this topic. The selection of α on each pixel is modified. The new technique allows to modify this parameter during

E. Cuesta, A. Duran, M. Kirane, S. A. Malik

the simulation according to the dependence on the image gradient variation. The numerical resolution is carried out with convolution quadrature techniques from the new convolution kernels for the Volterra approach.

The paper is structured as follows. In Section 2 the approach explained in [6] is recalled, the discretization of our proposal for (1) is introduced and the corresponding implementation is described. The performance of the new procedure is shown in Section 3, with numerical experiments on some noisy images.

2Fractional models

2.1Volterra equation and discretization

We first briefly describe the pixel by pixel technique, introduced in [6] and that will be the starting point for our study. Problem (1) is first written in the integral form

supplemented with homogeneous Neumann boundary condition. Now (2) is discretized in space and time with a di erent value of α for each pixel of the image. On a uniform M × M pixel mesh of Ω, with mesh length h > 0, the Laplacian is approximated with second order central di erences, leading to a M2 × M2 pentadiagonal matrix h of a pattern shown in Figure 1

Figure 1: Sparsity pattern of the discretized Laplacian

Image filtering with generalized fractional integrals

Considered now is a set of values 1 < αj < 2, j = 1, 2, . . . , M2 and the semi-discrete

where u(t) stands for the M2 × 1 vector-image function at time t and at the pixel mesh, u0

1 < αj < 2, j = 1, 2, . . . , M .

Formula (4) is finally discretized in time by means of a convolution quadrature method. A brief description is as follows (see [4, 10, 11] for the necessary explanations and details). The convolution integral in (4) is written in the form

connecting −i∞ to +i∞ (where Bromwich inversion formula for K is applied) and Y (λ, t) stands for the solution of the ordinary di erential equation

Now the discretization of (7) provides numerical approximations of (4) via (6). The corresponding formulas are written in terms of the generating power series

where τ is the time-step of the discretization and δ(ξ) is the quotient of the generating polynomials of the underlying numerical method used in (6) which, in our case, will be the backward Euler formula [7]. Thus, δ(ξ) = 1 − ξ, and if un is an approximation to u(tn), for the time level tn = nτ, n ≥ 0, then (4) is approximated by the convolution quadrature method

j=0

E. Cuesta, A. Duran, M. Kirane, S. A. Malik

In practice, these matrix coe cients are computed in an e cient way by using FFT techniques [11].

2.2Description of the modification

The implementation in [6] establishes values of α, for each single pixel, in the first time level and they will be remained fixed during the rest of the computation. A more realistic approach would be to consider nonconstant fractional orders (viscosity parameter). To this

end, a coherent definition of

∂−α(t)g(t), t ≥ 0, (11)

where g : [0, +∞) → X is absolutely continuous on a Banach space X is required. To our knowledge, the first approach in this sense is given in [16], where

is adopted. Unfortunately, (12) lacks a convolution structure. A natural extension to the constant case would consists of replacing (3) by

as the convolution kernel and therefore

# t

∂−α(t)g(t) := k g(t) = k(t − s)g(s) ds. (14)

0

The main objection to this choice is that the Laplace transform of (13), which is necessary in (6) (as well as in many other procedures), cannot be explicitly computed. Finally, the most widely adopted definition (and which is our choice in this paper) for (11) is obtained from taking k(t) as the inverse Laplace transform of

Image filtering with generalized fractional integrals

where α¯(z) stands for the Laplace transform of α(t) (see [14]). Notice that if α(t) = α is a constant, then this and (13) coincide with (3). Some numerical experiments, carried out in this sense for a future work, show that the results for the last two choices of the kernel k(t) do not di er notably.

This time-dependent fractional order strategy is introduced in the formulation of [6], by considering di erent functions α(j)(t), j = 1, . . . , M2 for each single pixel, leading to a semidiscrete approximation (9) with a convolution kernel K(t) = I(t) · h where (5) is replaced by

In order to use (8), a computable expression of the Laplace transform of K is required. This can be obtained as follows. Note that the time discretization (9) only requires to have the values of the α(j) at the time levels 0 = t0 < t1 < t2 < . . . < tN = T . Then we can set

N

α(j)(t) = α(0j) + (α(mj) − α(mj)−1)U(t − tm), m=1

therefore the Laplace transform of each component of the matrix valued function I reads simply as

2.3Implementation

It is of interest to make some comments on the implementation. They essentially concern the automatic selection of the fractional parameter α, in order to improve the performance in the preservation of edges and corners, the noise filtering and the adaptation to the evolution of the di usion. The main requirements about the first two points [6] are summarized as follows :