CASCADIC MULTILEVEL METHODS FOR FAST NONSYMMETRIC BLUR- AND NOISE-REMOVAL. Dedicated to Richard S. Varga on the occasion of his 80th birthday.

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1 CASCADIC MULTILEVEL METHODS FOR FAST NONSYMMETRIC BLUR- AND NOISE-REMOVAL S. MORIGI, L. REICHEL, AND F. SGALLARI Dedicated to Richard S. Varga on the occasion of his 80th birthday. Abstract. Image deburring is a discrete i-posed probem. This paper discusses cascadic mutieve methods designed for the restoration of images that have been contaminated by nonsymmetric bur and noise. Proongation is carried out by noninear edge-preserving and noise-reducing operators, whie restrictions are determined by weighted oca east-squares approximation. The restoration probem is on each eve soved by an iterative method, with the number of iterations determined by the discrepancy principe. The performance of severa iterative methods is compared. Computed exampes demonstrate the effectiveness of the image restoration methods proposed. The discrepancy principe requires that an estimate of the norm of the noise in the contaminated image be avaiabe. We iustrate how such an estimate can be computed with the aid of the noninear Perona-Maik diffusion equation. Key words. i-posed probem, deburring, mutieve method, reguarizing iterative method, edge-preserving proongation 1. Introduction. The restoration of images that have been contaminated by bur and noise continues to receive considerabe attention. The burring may be caused by object motion, caibration error of imaging devices, and random fuctuation of the medium, e.g., the atmosphere. The noise typicay stems from the measurement equipment or transmission errors. We are interested in restoring images that have been contaminated by both bur and noise. In particuar, we woud ike to recover edges accuratey. This paper considers the restoration of two-dimensiona gray-scae images. These images can be represented by a rea-vaued function defined on a rectanguar region Ω R 2 or by the discretization of such a function. Let the function f δ represent the avaiabe observed bur- and noise-contaminated image and et the function û represent the associated (unavaiabe) bur- and noise-free image that we woud ike to recover. These functions are assumed to be reated by the degradation mode (1.1) f δ (x) = h(x, y)û(y)dy + η δ (x), x Ω, Ω where η δ represents additive noise in the avaiabe data f δ ; see, e.g., Chan and Shen [8] for a discussion on image modes. In many appications, the integra is a convoution, i.e., h(x, y) = k(x y) for some function k. For instance, Gaussian kernes k(x) = c 1 exp( c 2 x 2 ) with suitabe positive constants c 1 and c 2 often are used to mode atmospheric bur. The kernes h or k commony are referred to as point spread functions (PSFs). In appications of interest, the kerne is smooth or piecewise smooth and, therefore, the integra operator is compact. It foows that the singuar vaues of the integra operator custer at the origin, eading to that the integra operator has no bounded inverse. Department of Mathematics-CIRAM, University of Boogna, Piazza Porta S. Donato 5, Boogna, Itay. E-mai: morigi@dm.unibo.it. Department of Mathematica Sciences, Kent State University, Kent, OH 44242, USA. E-mai: reiche@math.kent.edu. Department of Mathematics-CIRAM, University of Boogna, Via Saragozza 8, Boogna, Itay. E-mai: sgaari@dm.unibo.it. 1

2 We woud ike to determine an accurate approximation of û when the observed image f δ and the kerne h are known. Straightforward soution of the inear mode (1.2) h(x, y)u(y)dy = f δ (x), x Ω, Ω for u typicay does not provide a meaningfu approximation of the desired noise- and bur-free image û due to the noise η δ in the right-hand side f δ and the fact that the integra operator does not have a bounded inverse. The atter property makes the task of soving (1.2) an i-posed probem; see, e.g., Eng et a. [11] for anayses of i-posed probems and discussions on numerica methods for their soution. In order to be abe to determine a meaningfu approximation of û, we first repace (1.2) by a nearby probem, whose soution is ess sensitive to perturbations in the data f δ, and then sove the new probem so obtained. This repacement is commony referred to as reguarization. The possiby most popuar reguarization approach is due to Tikhonov. The foowing form of Tikhonov reguarization is we suited for image restoration, (1.3) min u { Ω ( ( ) } 2 1 h(x, y)u(y)dy f (x)) δ + α R(u(x)) dx, 2 Ω where α > 0 is a reguarization parameter and (1.4) R(u) = ψ( u 2 ) is a reguarization operator. Here ψ is a differentiabe monotonicay increasing function and u denotes the gradient of u; see, e.g., Rudin et a. [27] and Wek et a. [31] for discussions on this kind of reguarization operators. The Euer-Lagrange equation associated with (1.3), suppied with a gradient descent which yieds a minimizer of (1.3) as t, is given by ( ) u (1.5) (t, z) = h(x, z) h(x, y)u(t, y)dy f δ (x) dx + α D(u(t, z)) t Ω Ω for z Ω and t 0. We use the initia function u(0, z) = f δ (z), z Ω. Let Ω denote the boundary of Ω. The derivation of (1.5) is based on the Dirichet boundary condition u(0, z) = 0, z Ω. Other boundary conditions, such as Neumann conditions, aso can be used and may in some situations give more accurate restorations; see, e.g., [22, 31] for discussions. We aso refer to D as a reguarization operator. Image restoration methods based on the Euer-Lagrange equation require that the reguarization operator D, as we as vaues of the reguarization parameter α and a suitabe finite interva of integration [0, T] be chosen. The determination of suitabe vaues of α and T generay is not straightforward. From (1.4), we obtain (1.6) D(u) = div(g( u 2 ) u), g(t) = dψ(t)/dt. The function g is referred to as the diffusivity. For instance, the choice ψ(t) = t gives g(t) = 1 and D(u) = u, where denotes the Lapacian. This reguarization operator typicay yieds over-smoothed restored images. Tota variation-type and 2

3 Perona-Maik reguarization are designed to reduce over-smoothing. Perona-Maik reguarization is obtained by choosing the diffusivity (1.7) g(s) = s/ρ, where ρ is a positive constant; see [24]. Tota variation-type reguarization is commented on beow. The present paper extends the mutieve image restoration methods described in [20] in severa ways. We aow nonsymmetric PSFs. This requires the use of different iterative methods than in [20], and a carefu comparison of the performance of these methods is provided. The restriction operators are defined by soving oca weighted east-squares probems. The purpose of the weights is to avoid smearing of edges. Computed exampes show that restriction by soving oca east-squares probems gives more faithfu restorations than restriction by oca averaging. The atter approach to restriction is appied in [20]. The proongation operators in our mutieve method are defined by piecewise inear proongation foowed by integration of the noninear Perona-Maik diffusion equation, determined by (1.6) with g given by (1.7), for a few time steps. These noninear proongation operators are compared to noninear proongation operators defined via tota variation-type reguarization in [20] and found to give restorations of somewhat higher accuracy. For theoretica purpose, we aso discuss proongation operators based on the inearized Perona-Maik diffusion equation. These inear proongation operators yied inear image restoration methods, which we show to be a reguarization methods in a we-defined sense. The quaity of the restored images obtained with the inearized proongation operators is amost as high as when noninear proongation operators, based on the noninear Perona-Maik diffusion equation, are used. Finay, we expore the possibiity of estimating the noise-eve in the avaiabe corrupted image by integrating the Perona-Maik diffusion equation a few time steps. We concude this section with discussions on the discretizations of the inear integra equation (1.2) and of the Euer-Lagrange equation (1.5), and comment on the computationa effort required for their soution. Discretization of (1.2) yieds the inear system of equations (1.8) Au = b δ, A R n n, u, b δ R n, where A is a discrete burring operator and b δ represents the avaiabe bur- and noise-contaminated image. Throughout this paper, the matrix A is assumed to be nonsymmetric, because otherwise we can empoy the methods discussed in [20]. Since the integra operator in (1.2) has many singuar vaues of different orders of magnitude cose to the origin, so does the matrix A. The matrix therefore is of i-determined rank and numericay singuar. Linear systems of equations with a matrix of this kind often are referred to as inear discrete i-posed probems. In image restoration appications, the right-hand side b δ of (1.8), rather than the right-hand side f δ of (1.2), is avaiabe. Each entry of b δ represents a pixe vaue. Moreover, a measured discrete PSF, from which A is constructed, rather than the PSF h, may be known. The matrix A typicay is very arge and therefore cannot be factored. Let û aso denote a discrete approximation of the desired continuous soution of (1.1). Then the vector ˆb = Aû (1.9) 3

4 represents a burred, but noise-free, image. Thus, (1.9) is a consistent inear system of equations for û. Our task is to determine an approximate soution of (1.8) that accuratey approximates û. Note that due to the error (1.10) e δ = b δ ˆb in b δ and the severe i-conditioning of A, the minima-norm east-squares soution of (1.8) generay is not a usefu approximation of û. We refer to the error e δ as noise. Note that in image restoration appications, it suffices to determine the discrete image û. A simpe approach to determining an approximation of the desired image û is to appy a few steps of an iterative method to the inear system of equations (1.8) with initia iterate zero. Anayses of many cassica iterative methods are provided by Varga [29]. Most iterative methods appied in the present paper are described by Saad [28]. Specificay, we consider the conjugate gradient method appied to the norma equations (LSQR), the generaized minima residua (GMRES) iterative method, as we as the cosey reated range restricted GMRES (RRGMRES) iterative method. The iteration number may be considered a discrete reguarization parameter. It is important not to carry out too many iterations in order to avoid severe error propagation. This approach of determining a restored image is referred to as reguarization by truncated iteration; see [2, 4, 5, 6, 11, 14, 18, 19] for discussions. For many image restoration probems, LSQR and RRGMRES, and for some probems aso GMRES, yied reasonabe resuts within ony a few iterations. The appication of these methods therefore typicay is quite inexpensive. However, due to cut-off of high frequencies, a of these iterative methods may introduce artifacts, such as ringing and fai to recover edges accuratey; see [7, Exampe 4.3] for an iustrative exampe with LSQR and GMRES. Computed exampes of Section 5 show that this shortcoming can be reduced by appying these iterative schemes in cascadic mutieve methods with suitabe restriction and proongation operators. We turn to the discretization of the Euer-Lagrange equation (1.5) with D determined by (1.6) and (1.7). Space discretization can be carried out in severa ways. For instance, finite difference or finite voume discretizations in space yied (1.11) du dt = (α L(u) AT A)u + A T b δ, where L(u)u is a discretization of the reguarization operator D(u) in the right-hand side of (1.5). Thus, L(u) is a discrete noninear diffusion operator. Expicit time stepping methods for the integration of (1.11) are expensive, because due to the CFL stabiity condition very many time steps have to be carried out; see [31] and [20, Exampe 4.2]. Semi-impicit time stepping methods aow onger time steps than expicit integration methods, but are not unconditionay stabe due to the operator A T A. Each time step requires the soution of a arge inear systems of equations. Therefore, semi-impicit integration methods aso are expensive. We concude that noninear modes based on (1.5)-(1.7) and (1.11) can provide denoising and deburring of good quaity; however, the soution of (1.11) by expicit or semi-impicit time stepping schemes is computationay expensive. Furthermore, as aready pointed out, the choices of appropriate vaues of the reguarization parameter α and the ength of the time interva T are not easy. A computed exampe with a semi-impicit integration method is presented in Section 5. The present paper describes new cascadic mutieve methods that share the computationa efficiency of truncated iteration for the soution of inear discrete i-posed 4

5 probems (1.8) with the edge-preserving property of noninear modes (1.5). The mutieve methods proceed from coarser to finer eves and are based on reguarization by truncated iteration on each eve. Proongation of a coarse-eve approximation of u to a finer grid is carried out by noninear edge-preserving and noise-reducing operators. Restrictions are computed by a oca weighted east-squares method that preserves structures in the image. For many image restoration probems, the mutieve methods demand fewer matrix-vector product evauations on the finest eve than the corresponding 1-eve truncated iterative methods and often determine restorations of higher quaity. In fact, the quaity of restorations obtained with mutieve methods can be higher than restorations computed by much more expensive methods for the soution of noninear modes (1.5); see Section 5 as we as [20, Exampe 4.2]. This paper is organized as foows. Section 2 introduces the mutieve framework, reviews reevant properties of the iterative methods mentioned, and discusses a stopping criterion for the iterations on each eve based on the discrepancy principe. This criterion requires that an estimate of the norm of the noise in the avaiabe contaminated image b δ be avaiabe. Section 3 defines the restriction operators appied to the avaiabe noise- and bur-contaminated images, and shows a noise-reducing property, Section 4 describes the proongation operators used and Section 5 presents numerica image restoration exampes, which iustrate the performance of mutieve methods based on the LSQR, GMRES, and RRGMRES iterative methods. Section 5 aso describes how an estimate of the norm of the noise in the data, b δ, can be computed using the Perona-Maik noninear diffusion equation. It foows that the mutieve methods of the present paper aso can be appied when no estimate for the norm of the noise in b δ is expicity known. Concuding remarks can be found in Section 6. The appication of mutieve methods to the soution of i-posed probems is not new. Severa soution methods for i-posed probems appy a mutieve scheme to the reguarized Tikhonov equations; see, e.g., Jacobsen et a. [17], Hucke and Staudacher [16], Zhu and Chen [32], and references therein. These methods reduce the computationa effort, but do not improve the quaity of the computed approximate soution, when compared with a standard iterative method appied on the finest eve ony. Mutieve methods that are appied directy to the equations (1.2) or (1.8), and reguarize by truncated iteration, are described in [9, 10, 20, 25]. However, the methods [9, 10] are not reguarization methods; see Section 2 for a discussion on the atter. The scheme in [20] is for symmetric probems and the approach in [25] is not taiored to image restoration. The present paper is the first one to discuss mutieve methods for the soution of nonsymmetric image restoration probems of the form (1.8). 2. Cascadic mutieve methods. This section first describes the 1-eve standard iterative methods used in the mutieve methods, discusses termination of the iterations by the discrepancy principe, and then presents the mutieve methods. The initia approximate soution, u 0, is assumed to be zero for a iterative methods. Introduce for v = [v (1), v (2),...,v (n) ] T R n the weighted east-squares norm (2.1) v = ( 1 n n i=1 ) 1/2 (v (i)) 2. GMRES is one of the most popuar iterative methods for the soution of argescae inear we-posed probems; see [28, Chapter 6] for a thorough treatment of this method. The jth iterate, u j, determined by GMRES appied to the soution of (1.8) 5

6 soves the minimization probem Au j b δ = min Au b δ, u j K j (A, b δ ), u K j(a,b δ ) where K j (A, b δ ) = span{b δ, Ab δ,...,a j 1 b δ } is a Kryov subspace. The computation of u j requires the evauation of j matrix-vector products with the matrix A. The iterates u j determined by the cosey reated RRGMRES method appied to the soution of (1.8) satisfy Au j b δ = min Au b δ, u j K j (A, Ab δ ). u K j(a,ab δ ) Thus, RRGMRES and GMRES differ in the Kryov subspaces used; a iterates of RRGMRES ive in the range of A. For many inear discrete i-posed probems RRGM- RES determines better approximations of the desired soution û of (1.9) than GMRES; see Section 5 for some computed exampes, as we as [5] for a discussion. When A is symmetric, RRGMRES reduces to MR-II, an iterative method proposed and anayzed by Hanke [14, Chapter 6] for the soution of inear i-posed probems with a symmetric indefinite operator A. The computation of u j requires j + 1 matrix-vector product evauations with A. LSQR is an impementation of the conjugate gradient method appied to the norma equations associated with (1.8); see [1, 23] for detais. The jth iterate, u j, determined by LSQR satisfies Au j b δ = min Au b δ, u j K j (A A, A b δ ), u K j(a A,A Ab δ ) where A denotes the adjoint of A. The computation of u j requires the evauation of j matrix-vector products with both A and A, giving a tota of 2j matrix-vector product evauations. A the above iterative methods minimize the norm of residua error b δ Au j over a Kryov subspace. Since the Kryov subspaces are nested, we have Au j+1 b δ Au j b δ for a the iterative methods considered. This property makes it natura to terminate the iterations by the discrepancy principe, defined beow. Let δ denote the norm of the error (1.10) in the right-hand side of (1.8), i.e., (2.2) δ = e δ, and assume that a fairy accurate estimate of δ is known. This aows us to use the discrepancy principe to decide how many steps to carry out with the iterative methods. In Exampe 5.6 of Section 5, we discuss an approach to compute an estimate of δ for probems for which such an estimate is not expicity avaiabe. Definition (Discrepancy Principe). Let γ > 1 be a fixed constant and et δ be given by (2.2). The vector u is said to satisfy the discrepancy principe if b δ Au γδ. We terminate the iterations as soon as an iterate u j that satisfies the discrepancy principe has been computed. Stopping Rue 2.1. Let δ and γ be the same as in the Discrepancy Principe. Terminate the iterations when for the first time (2.3) b δ Au j γδ. 6

7 Denote the resuting stopping index by j(δ). Note that, in genera, j(δ) increases as δ decreases with γ kept fixed. An iterative method equipped with this stopping rue is said to be a reguarization method if the computed iterates u j(δ) satisfy (2.4) im sup u j(δ) û = 0, δց0 e δ δ where û is the minima-norm soution of (1.9). Hanke [14, Theorem 6.15] and Nemirovskii [21] have shown that LSQR is a reguarization method in a Hibert space setting. Cavetti et a. [6] show that GMRES is a reguarization method under more stringent conditions. The atter proof carries over to RRGMRES. In particuar, a the iterative methods mentioned satisfy (2.3) in finite dimension in the absence of breakdown of GMRES and RRGMRES. How to circumvent breakdown is discussed in [26]; see aso beow for comments on the atter. We are in a position to present mutieve methods based on the iterative methods described above. A termination criterion simiar to Stopping Rue 2.1 is appied on each eve. Let W 1 W 2 W be a sequence of nested subspaces of R n of dimensions dim(w i ) = n i with n 1 < n 2 <... < n = n. We refer to the subspaces W i as eves, with W 1 being the coarsest and W = R n the finest eve. Each eve is furnished with a weighted east-squares norm; eve W i has a norm of the form (2.1) with n repaced by n i. This section discusses mutieve methods for image restoration for which we et (2.5) n i 1 = 1 4 n i, 1 < i. For notationa simpicity, we assume the image to be square, i.e., the image is represented on eve i by a square array of pixes p (j,k) i, 1 j, k n i ; here j and k denote the horizonta and vertica coordinates, respectivey. Let A i R ni ni be the representation of the burring operator A on eve W i. The matrix A i is determined by discretization of the integra operator (1.2) simiary as A. This defines impicity the restriction operator R i : R n W i, such that (2.6) A i = R i AR i. Some discretizations require that the adjoint Ri be repaced by another operator Q i : W i R n ; however, this is not the case for the computed exampes of Section 5. We define R = I. The choice of restriction operators R i for determining the restrictions A i of A is in our experience ess crucia for achieving high-quaity restorations than the choice of restriction operators R (ω) i : R n W i for reducing the avaiabe bur- and noisecontaminated image represented by the right-hand side b δ in (1.8); thus, we et (2.7) b δ i = R (ω) i b δ, 1 i <, where the R (ω) i are determined by repeated oca weighted east-squares approximation, inspired by a staircasing -reducing scheme recenty proposed by Buades et a. 7

8 [3]. The precise definition of the R (ω) i and a discussion of their noise-reducing property are provided in Section 3. We et R (ω) = I. The choice of proongation operators from eve i 1 to eve i is important for the performance of the mutieve method. We appy noninear proongation operators P i : W i 1 W i, 1 < i, defined by piecewise inear interpoation foowed by integration of the Perona-Maik equation over a short time interva; see Section 4 for detais on the atter. These operators are designed to be noise-reducing and edge-preserving. For comparison, we aso consider inear proongation operators L i : W i 1 W i, 1 < i, defined by piecewise inear interpoation, and inear proongation operators L i : W i 1 W i, 1 < i, obtained by inearization of the Perona-Maik diffusion equation. Computed exampes reported in Section 5 show the noninear proongation operators P i to yied more accurate restorations than the inear operators L i and L i. The mutieve methods of the present paper are cascadic, i.e., they first determine an approximate soution of A 1 u = b δ 1 in W 1 using one of the iterative methods GMRES, RRGMRES, or LSQR. We refer to the iterative method as IM in Agorithm 2.2 beow. The iterations with this method are terminated as soon as an iterate that satisfies a stopping rue reated to the discrepancy principe has been determined. This iterate is mapped from W 1 into W 2 by the proongation P 2. A correction of this mapped iterate in W 2 is computed by IM. Again, the iterations are terminated by a stopping rue reated to the discrepancy principe. The approximate soution in W 2 determined in this manner is mapped into W 3 by P 3. The computations are continued in this fashion unti an approximation of û has been determined in W = R n. The atter approximation is post-processed by oca weighted east-squares approximation, as suggested by Buades et a. [3], in order to reduce staircasing. This smoothing step is denoted by the operator S : W W in Agorithm 2.2, which outines the computations for the mutieve methods. Agorithm 2.2. Mutieve Agorithm Input: A, b δ, δ, 1 (number of eves); Output: approximate soution u W of (1.8); Determine A i and b δ i from (2.6) and (2.7), respectivey, for 1 i ; u 0 := 0; for i := 1, 2,..., do u i,0 := P i u i 1 ; u i,mi := IM(A i, b δ i A iu i,0 ); Correction step: u i := u i,0 + u i,mi ; endfor u := S u ; In the agorithm u i,mi := IM(A i, b δ i A iu i,0 ) denotes the computation of the approximate soution u i,mi of (2.8) A i z i = b δ i A iu i,0 by m i iterations with one of the iterative methods GMRES, RRGMRES, or LSQR, using the initia iterate u i,0 = 0. We remark that the structure of Agorithm 2.2 is the same as that of [20, Agorithm 2.2]; however, the iterative methods, as we as the proongation, restriction, and smoothing operators appied in the agorithms differ. Cascadic mutieve methods reated to those of Agorithm 2.2 aso are discussed in [25]. Again, the methods differ in the choices of proongation and restriction operators. 8

9 Moreover, no smoothing operator S is appied in [25]. Let (2.9) ˆbi = R (ω) i ˆb, 1 i, i.e., ˆb i is the (unavaiabe) representation of the burred but noise-free image ˆb on eve i, with ˆb = ˆb. Our stopping rue on each eve is based on the assumption that there are constants c i independent of δ, such that (2.10) b δ i ˆb i c i δ, 1 i, where δ satisfies (2.2). The noise-reducing property of the restriction operators R (ω) i, see (3.9) in Section 3, suggests the choice (2.11) c i = 1 3 c i+1, 1 i <, c = γ, where γ is the same as in (2.3). Stopping Rue 2.3. Let δ and the c i be the same as in (2.10) and denote the iterates determined by the IM iterative method appied to the soution of (2.8) with initia iterate u i,0 = 0 by u i,m, m = 1, 2,.... Terminate the iteration on eve i as soon as an iterate u i,mi that satisfies b i A i u i,0 A i u i,mi c i δ has been determined, where m i = m i (δ) denotes the termination index. Since images are represented by pixes, image restoration probems ive in finite dimensiona spaces. It is quite straightforward to show that the LSQR-based mutieve method defined by Agorithm 2.2, with inear proongation operators P i and S = I, is a reguarization method for finite dimensiona probems. The P i may, for instance, be the operators L i defined by piecewise inear interpoation. Aternativey, we may define the P i by time integration of a inearized Perona-Maik diffusion equation; see Section 4 for more detais on the atter. In Theorem 2.4 and beow, R(M) and N(M) denote the range and nu space, respectivey, of the matrix M. Theorem 2.4. Let, for 1 i, the equation (2.12) A i u = ˆb i be consistent with minima-norm soution û i. In particuar, û = û. Let the proongation operator P i in Agorithm 2.2 be inear for a i and et S = I in the ast step of Agorithm 2.2. Assume that (2.13) R(P i ) R(A i ), 1 < i. Let the LSQR-based Agorithm 2.2 determine approximate soutions of the norma equations associated with the inear systems (2.8) for 1 i. Assume that the right-hand sides of the norma equations are generic, i.e., they do not beong to an invariant subspace of ow dimension. Let the errors in the right-hand sides restricted to W i be of the form (2.14) b δ i ˆb i = c i δ w i, 1 i, where w i W i is a unit vector independent of δ, and the c i are the positive constants in (2.10). Terminate the LSQR-iterations in Agorithm 2.2 on each eve according to 9

10 Stopping Rue 2.3 to obtain the iterates u i W i for 1 i. The mutieve method so defined is a reguarization method on each eve, i.e., (2.15) im u i û i = 0, 1 i. δց0 Proof. The proof of Theorem 2.4 in [20] is sufficienty genera to aow adaption to the present situation. The constants c i > 0 can be chosen arbitrariy. If we ony require (2.15) to hod for i =, then the requirements of the theorem can be reaxed. For instance, then we ony have to require that (2.13) hods for i =. Remark 2.1. The fact that an iterative method is a reguarization method in the sense that it satisfies (2.15) does not guarantee that it produces accurate restorations. However, methods that vioate (2.15) may yied poor restorations when the noise e δ in b δ is of sma norm δ; cf. (2.2). Resuts reated to Theorem 2.4 in infinite dimensions are discussed in [25, Theorem 3.2 and Coroary 3.1]. Donatei and Serra [9, 10] describe noncascadic mutieve methods, in which fu V - or W-cyces are carried out. These methods are in [10] shown not to be reguarization methods in the sense of the present paper; nevertheess, the computed exampes presented in [9, 10] ook nice. When (2.13) does not hod for the index i, the computed iterate u i may have a component in N(A i ) and, therefore, not converge to the minima-norm soution of (2.12) as δ ց 0. Numerica experience indicates that the condition (2.13) does not pose a practica probem; in a our experiments, Agorithm 2.2 computed iterates u i that are numericay orthogona to N(A i ) for a i. A matrix M is said to be range symmetric if R(M) = R(M ). Theorem 2.4 aso hods for RRGMRES-based mutieve methods when the matrices A i are range symmetric or nonsinguar. When A i is singuar and not range symmetric, the mutieve method may determine an approximation of û i that has a component in N(A i ). GMRES-based mutieve methods are ony guaranteed to determine approximations of û i orthogona to N(A i ) when A i is nonsinguar. Impementations of GMRES and RRGMRES for inear systems of equations with a singuar matrix are discussed in [26]. 3. Noise-reducing restriction operators. The noninear mode (1.5) based on the Perona-Maik reguarization operator defined by (1.6) with g given by (1.7) may determine restored images that suffer from so-caed staircasing, i.e., the computed images dispay arge fat regions separated by artificia boundaries, instead of a desired smooth surface. Recenty, Buades et a. [3] proposed post-processing of restored images by oca weighted east-squares approximation in order to reduce this effect. : W i W i 1 by soving these east-squares probems as foows. Let p i 1 denote the image on eve i 1 of size n i 1 n i 1 obtained by restriction of the image p i on eve i of size n i n i. Superscripts denote pixe We define the mappings M (ω) i denotes the pixe with coordinates (j, k) of p i. For each pixe p (j,k) i 1 of the image p i 1, we sove a weighted east-squares probem using a 3 3 window of neighboring pixes. This window and the coordinates of the pixes invoved coordinates, i.e., p (j,k) i are dispayed by Figure 3.1. Pixes p (j,k) i 1 of the coarser image p i 1 are shown with gray background. Introduce the weight function (3.1) ω (2j,2k) i (s, t) = exp ( ( κ p (2j+s,2k+t) i 10 ) ) 2 p (2j,2k) i,

11 (2j 1,2k 1) (2j 1,2k) (2j 1,2k+1) (2j,2k 1) (2j,2k) (2j,2k+1) (2j+1,2k 1) (2j+1,2k) (2j+1,2k+1) Fig Pixe mask used in the oca weighted east-squares computation. with κ a positive constant, and consider the oca weighted east-squares probem ( ) 2 min p (2j+s,2k+t) (2j,2k) (3.2) i (α 0 + α 1 s + α 2 t) ω i (s, t) α 0,α 1,α 2 s,t {0,±1} for the coefficients {α 0, α 1, α 2 } of the inear function. Let {ˆα 0, ˆα 1, ˆα 2 } denote the soution and define p (j,k) i 1 := ˆα 0. We sove the east-squares probems (3.2) for a 1 j, k n i 1. This defines M (ω) i. The number of arithmetic foating point operations required to sove (3.2) for a pixes in W i 1 is proportiona to n i 1 and therefore modest. The soution of each east-squares probem can be computed, e.g., by determining a modified QRdecomposition of a 9 3 matrix based on modified Househoder transformations; see Guiksson and Wedin [12] for detais on the atter. The modifications are required because of the weights in the east-squares probem. Appication of the mappings M (ω) k for k =, + 1,..., i + 1, in order, defines the weighted restriction operators R (ω) i : R n W i according to (3.3) R (ω) i = M (ω) i+1 M(ω) i+2...m(ω), 1 i <. Appying the mapping R (ω) i to the right-hand side of (1.8) yieds the vectors b δ i ; cf. (2.7). The restriction operators (3.3) typicay yied more faithfu restorations than simper restriction operators defined by oca averaging of neighboring pixe-vaues; see Exampe 5.2 of Section 5 for an iustration. 11

12 Since east-squares approximation impies smoothing and the noise e δ = eδ in b δ = bδ contains a significant high-frequency component, the restrictions b δ i of bδ, for i <, defined by (2.7) typicay contain ess noise than b δ. This is iustrated by the foowing resut. Proposition 3.1. Let κ = 0 in (3.1) and assume that the entries e (j,k), 1 j, k n, of the noise-vector e δ are uncorreated random variabes with zero mean. Let e δ 2 V denote the average variance of the entries, i.e., e δ 2 V = 1 n n j,k=1 Var(e (j,k) ). Assume that the image is periodic, so that boundary effects can be ignored, and et Then e δ 1 = R (ω) 1 eδ. e δ 1 V 4 9 eδ V, e δ 1 V 2 9 eδ V. Proof. For i =, the east-squares probem (3.2) with ω (2j,2k) = 1 for a j can be expressed as (3.4) with H = min Hα α R q(j,k) α = [α 0, α 1, α 2 ] T R 3, T R 9 3, q (j,k) = [p (2j,2k), p (2j+1,2k), p (2j 1,2k),..., p (2j 1,2k 1) ] T R 9. Denote the soution of (3.4) by ˆα = [ˆα 0, ˆα 1, ˆα 2 ] T. Since H has orthogona coumns, we obtain p (j,k) 1 = ˆα 0 = 1 p (2j+s,2k+t). 9 s,t {0,±1} The component e (j,k) 1 of the vector eδ 1 represents the noise in p(j,k) 1 and is the average of the errors e (2j+s,2k+t) in the nine pixes p (2j+s,2k+t), s, t {0, ±1}. Therefore, and it foows that Var(e (j,k) 1 ) = 1 81 s,t {0,±1} Var(e (2j+s,2k+t) ) (3.5) e δ 1 2 V = 1 n 1 Var(e (j,k) 1 n ) 1 j,k=1 12

13 (3.6) (3.7) = 1 81n 1 4 n 1 81n 1 j,k=1 j,k=1 s,t {0,±1} n Var(e (j,k) ) = 16 81n Var(e (2j+s,2k+t) ) n j,k=1 Var(e (j,k) ) = eδ 2 V. The inequaity (3.7) is a consequence of the fact that some errors e (2j+s,2k+t) contribute to four errors e (j,k) 1. This can be seen from Figure 3.1. The center pixe in the figure is in W 1 and gets contributions from the four corner pixes on eve W. Moreover, no pixe in W contributes to more than four pixes in W 1. The equaity on the eft in (3.7) foows from (2.5). Every pixe in W contributes to at east one pixe in W 1. We therefore obtain anaogousy to (3.5)-(3.7) the inequaity e δ 1 2 V = 1 n 1 81n n 1 j,k=1 j,k=1 s,t {0,±1} n Var(e (j,k) ) = 4 81n Var(e (2j+s,2k+t) ) n j,k=1 Var(e (j,k) ) = 4 81 eδ 2 V. This estabishes the proposition. Coroary 3.2. Let the conditions of Proposition 3.1 hod and assume further that (3.8) Then Var(e (j,k) ) = η 2, 1 j, k n. e δ 1 V = 1 3 eδ V. Proof. The resut foows from (3.5)-(3.6). Approximating e δ k V by e δ k for k = 1,, we obtain from Coroary 3.2 that (3.9) e δ eδ. We wi use the factor 1/3 in our mutieve method; cf. (2.11). 4. Edge-preserving noninear proongation operators. We describe the noninear edge-preserving proongation operators P i used in the computed exampes. They have previousy been appied in [20], where further detais on their impementation are provided; see aso [30]. The proongation operators consist of two parts: first the approximate soution on eve W i 1 determined in the correction step of Agorithm 2.2 is mapped into W i by piecewise inear interpoation. This is carried out by the operator L i. The eement L i u i 1 in W i then is used as initia function for a discretization of the initia-boundary vaue probem for the Perona-Maik noninear diffusion equation (4.1) u t = div(g( u 2 ) u), 13

14 where g is the Perona-Maik diffusivity (1.7). Integration over a short time interva removes noise whie preserving rapid spacia transitions, such as edges. Discretization of (4.1) in space yieds the initia vaue probem (4.2) du dt = L(u)u, u W i t > 0, with initia function u 0 = L i u i 1. We set u to zero on the boundary. Here L(u)u is the same discretization of the Perona-Maik operator as in (1.11). The differentia equation (4.2) is designed to determine an eement in W i that has edges cose to those of u i 1 in W i 1. The matrix L(u) in (4.2) has, genericay, five nonvanishing entries in each row. This makes the evauation of L(u)u quite inexpensive. Integration is performed by carrying out about 10 time steps of size about 0.2 with an expicit method. The sma number of time steps avoids difficuties due to numerica instabiity and keeps the computationa work required for integration negigibe. We found it to be beneficia to appy more time steps the more noise-contaminated the avaiabe image. However, in our experience the exact choices of the number of time steps and of the size of the time steps is not crucia for the good performance of the mutieve methods. The proof of Theorem 2.4 requires the proongation operators to be inear. We therefore aso consider the discrete inearized Perona-Maik diffusion equation (4.3) du dt = L(u0 )u, where u 0 = L i u i 1. Integration over a short time interva with initia vaue u 0 defines the inear proongation operator L i. Exampe 5.2 of Section 5 shows the operators L i to yied restored images of amost the same quaity as the proongation operators P i. However, we do not advocate the use of the inearized operators L i instead of the noninear operators P i since integration of (4.2) is fast and easy. This concudes the description of the components of Agorithm 2.2. Thus, the restriction operators are defined as described in Section 3 and are based on a oca weighted east-squares approximation scheme described by Buades et a. [3]. The atter aso is used for the smoother S on the finest eve. Proongation is carried out as described in the present section. Our mutieve methods seek to reduce the bur by soving inear systems of equations on each eve, and to reduce the noise by integrating a discretization of the noninear diffusion equation (4.1). We remark that noise-reduction aso can be achieved by other means, such as by integrating a differentia equation based on the tota variation-norm. This is discussed in [20]. A comparison of the atter approach with the one of the present paper reported in [20] showed integration of (4.1) to yied restored images of sighty higher quaity. 5. Computed exampes. We iustrate the performance of Agorithm 2.2 appied to the restoration of two-dimensiona gray-scae images that have been contaminated by bur and noise. The images are represented by arrays of 8-bit pixes. The computed exampes compare cascadic mutieve methods based on the GMRES, RRGMRES, and LSQR iterative methods. Define the noise-eve (5.1) ν = eδ ˆb. 14

15 We assume that an accurate estimate of ν is avaiabe in a exampes of this section, except for Exampe 5.6, and therefore choose the parameter γ in (2.3) and (2.11) cose to unity; specificay, we et γ = Exampes 5.1 and 5.2 discuss restoration of images that have been contaminated by space-variant Gaussian bur and noise, whie Exampes 5.3 and 5.4 are concerned with images that are corrupted by motion bur and noise. In Exampe 5.5 we compare the mutieve method to the noninear mode (1.5). We iustrates how the noise-eve can be estimated by integrating the Perona-Maik diffusion equation over a short time interva in Exampe 5.6. Estimates so obtained are used in the stopping rue of Agorithm 2.2 and we compare the performance of the agorithm for exact and estimated noise-eves. The matrices A i defined by (2.6) do not have to be expicity stored; it suffices to define functions for the evauation of matrix-vector products with the A i. For the exampes of the present section, these products can be computed efficienty by using the structure of the A i ; see, e.g., [22, 15] for discussions. The matrices corresponding to the finest eve are numericay singuar in a exampes. The dispayed restored images provide a quaitative comparison of the performance of the proposed cascadic mutieve methods. The Peak Signa-to-Noise Ratio (PSNR), (5.2) PSNR(u, û) = 20 og u û db, where û is the bur- and noise-free image and u the restored image determined by Agorithm 2.2, provides a quantitative comparison. The norm u û is the Root Mean Squared Error (RMSE) of u û; cf. (2.1). The numerator 255 is the argest pixe-vaue that can be represented with 8 bits. A high PSNR-vaue indicates that the restoration is accurate; however, the PSNR-vaues are not aways in agreement with visua perception. The computations are carried out in MATLAB with about 16 significant decima digits. GMRES RRGMRES LSQR ν PSNR # iter PSNR # iter PSNR # iter Tabe 5.1 Exampe 5.1. Restoration of corrupted versions of the corner image. The tabe reports PSNRvaues of the restored images determined and the number of iterations (# iter) required for severa noise-eves ν and severa cascadic mutieve methods. The 1-eve methods are standard iterative schemes appied on the finest eve ony. The point spread function is nonsymmetric and Gaussian with band = 7 and σ 1 = 4, σ 2 = 1. 15

16 (a) (b) (c) (d) Fig Bur- and noise-free images used in the numerica experiments: (a) corner, pixes; (b) pepper, pixes; (c) izard, pixes; (d) Varga, pixes. Exampe 5.1. We compare image restorations determined by Agorithm 2.2 with the GMRES, RRGMRES, and LSQR iterative methods and noninear proongation. The number of iterations is determined by Stopping Rues 2.1 or 2.3. We determine restorations of contaminated versions of the origina pixe corner image dispayed in Figure 5.1(a). The contamination is caused by noise and spatiay variant Gaussian bur. The nonsymmetric burring matrix is given by A := I 1 (T 1 T 1 ) + I 2 (T 2 T 2 ) R n n, n = Here I 1 is the diagona matrix, whose first n/2 diagona entries are one, and the remaining entries zero, and I 2 := I I 1. The operator denotes the Kronecker product and the T are banded Toepitz matrices, which represent Gaussian bur in one space-dimension. Specificay, T = [t () jk ] n j,k=1 { t () jk := 1 σ 2π exp ( (j k)2 2σ 2 is defined by ), if j k band, 0, otherwise. 16

17 (a) (b) Fig Exampe 5.1. (a) Restoration determined by RRGMRES of bur- and noisecontaminated corner image with noise-eve ν = , and (b) edge-map for restoration. (a) (b) Fig Exampe 5.1. (a) Restoration determined by GMRES of bur- and noise-contaminated corner image with noise-eve ν = , and (b) edge-map for restoration. This modes the situation when the eft and right haves of the image are degraded by different Gaussian burs. The matrix A is nonsymmetric; its first n/2 rows are the made up of the first n/2 rows of T 1 T 1 and its ast n/2 rows are the ast n/2 rows of T 2 T 2. The parameter band determines the haf bandwidth of the Toepitz matrices appied. Enarging band increases the storage requirement, the arithmetic work required for the evauation of matrix-vector products with A, and to some extent the burring. In this exampes, we et band = 7 and σ 1 = 4, σ 2 = 1. We first iustrate properties of the GMRES, RRGMRES, and LSQR iterative methods when used as 1-eve methods. Throughout this section, 1-eve method refers to a basic iterative method appied on the finest eve ony without smoothing. We appy the methods to restore the contaminated image. The restoration deter- 17

18 mined by RRGMRES is shown in Figure 5.2(a); Figure 5.2(b) dispays the edge map for the restored image. This map is determined by the edge-detector of gimp, a pubic domain software too for image processing. 1 The restored image determined by LSQR and the associated edge map are visuay indistinguishabe from those obtained with RRGMRES; they therefore are not shown. Figure 5.3(a) shows the restoration determined by GMRES and Figure 5.3(b) depicts the edge-map for the restoration. Whie the restorations determined by RRGMRES and GMRES ook fairy simiar when dispayed in the size shown, the associated edge-maps differ; the edge-map for the restoration computed by GMRES shows ess sharpness and much more noise. PSNR-vaues for the restored images and the number of iterations required by 1-eve methods for severa noise-eves are reported in the first row of each horizonta bock of Tabe 5.1. The tabe shows 1-eve LSQR to determine the most accurate restorations for any noise-eve. This 1-eve method is the computationay most expensive one, because it requires the argest number of iterations and each iteration demands the evauation of two matrix-vector products; GMRES and RRGMRES ony require the evauation of one matrix-vector product per iteration. Tabe 5.1 shows -eve GMRES, for > 1, to give accurate restorations for noiseeves ν < with ess computationa effort than -eve LSQR. However, -eve RRGMRES and LSQR produce better restorations for high noise-eves ν The -eve GMRES method often requires fewer iterations than -eve RRGM- RES, but generay deivers restorations of worse quaity. Images restored by GMRES typicay maintain sharp edges, but suffer from propagated noise and some ringing near the borders. This is iustrated by the edge map of Figure 5.3(b) for 1-eve GM- RES. For sma noise-eves, the PSNR-vaues for images restored by -eve GMRES are high; however, restorations determined by -eve RRGMRES generay are more visuay peasing, even when they have ower PSNR-vaues. Tabe 5.1 shows restorations obtained by cascadic mutieve methods to be of higher quaity, as measured by the PSNR-vaues, than restorations computed by 1- eve methods. This is in agreement with visua perception. The coumns abeed # iter dispay the number of iterations on each eve. The -tuppets show, from eft to right, the number of iterations for increasing eve index. Thus, the eftmost entry shows the number of iterations on the coarsest eve and the rightmost entry the number of iterations on the finest eve. The number of required iterations can be seen to increase as the noise eve decreases, both for 1-eve and mutieve methods. Because of the noted poorer quaity of restorations determined by -eve GMRES, 1, when compared with restored images determined by -eve RRGMRES and -eve LSQR, we omit -eve GMRES methods in the remainder of this section. Exampe 5.2. We compare Agorithm 2.2 for piecewise inear and noninear proongation operators. Tabes report the performance of the agorithm when used with the iterative methods RRGMRES or LSQR. The agorithm is appied to bur- and noise-contaminated versions of the images shown in Figure 5.1(a) and (c). The coumns with header L i show resuts obtained with the piecewise inear proongation operators L i without smoothing in the ast step of the agorithm. Coumns abeed P i dispay resuts obtained when the noninear proongation operators P i are used and smoothing is appied in the ast step of the agorithm. Simiary, the coumns abeed L i in Tabes 5.2 and 5.3 show the performance of Agorithm 2.2 with the inearized proongation operators L i described in Section 4 and no smooth- 1 gimp is described and avaiabe at 18

19 L i P i Li band PSNR # iter PSNR # iter PSNR # iter Tabe 5.2 Exampe 5.2. Restoration of corrupted versions of the corner image by RRGMRES-based cascadic mutieve methods. The tabe reports PSNR-vaues of the restored images and the number of iterations (# iter) required for piecewise inear proongation operators (L i ), noninear proongation operators (P i ), and inearized proongation operators described in Section 4 ( L b i ). The burring operator A is nonsymmetric; it is determined by Gaussian PSFs defined by the parameters band and σ 1 = 4, σ 2 = 1. The noise eve is ν = L i P i Li band PSNR # iter PSNR # iter PSNR # iter Tabe 5.3 Exampe 5.2. This tabe differs from Tabe 5.2 ony in that LSQR-based cascadic mutieve methods are appied instead of RRGMRES-based mutieve methods. L i P i band PSNR # iter PSNR # iter Tabe 5.4 Exampe 5.2. Restoration of corrupted versions of the corner image by LSQR-based cascadic mutieve methods, using the averaging restriction operators for obtaining the vectors b δ i in (2.7). The tabe reports PSNR-vaues of the restored images and the number of iterations (# iter) required both for piecewise inear proongation operators (L i ) and noninear proongation operators (P i ). The images to be restored are the same as for Tabe

20 L i band PSNR # iter PSNR # iter Tabe 5.5 Exampe 5.2. Restoration of corrupted versions of the izard image by RRGMRES-based cascadic mutieve methods. The tabe reports PSNR-vaues of the restored images and the number of iterations (# iter) required for piecewise inear proongation operators (L i ) and noninear proongation operators (P i ). The burring operator A is nonsymmetric; it is determined by Gaussian PSFs defined by the parameters band and σ 1 = 4, σ 2 = 1. The noise eve is ν = P i L i band PSNR # iter PSNR # iter Tabe 5.6 Exampe 5.2. This tabe differs from Tabe 5.4 ony in that LSQR-based cascadic mutieve methods are appied instead of RRGMRES-based mutieve methods. P i ing in the ast step of the agorithm. The computations reported in the coumns abeed L i and L i are covered by Theorem 2.4, whie the resuts reported in the coumns abeed P i are not. The tabes show cascadic mutieve methods to determine restorations of higher quaity than 1-eve methods, and noninear proongation foowed by smoothing to yied better restorations than inear proongation. The restoration quaity is measured in terms of PSNR-vaues. The tabes aso show the inearized proongation operators L i to produce restorations with amost the same PSNR-vaues as the non-inear proongation operators P i with smoothing. We aso compare the weighted east-squares restriction operators in (2.7) with the averaging restriction operators used in [20]. The averaging restriction operator from W i to W i 1 repaces groups of four adjacent pixes in the image represented by b δ i by one pixe, whose vaue is the average of the vaues of the four pixes it repaces. Tabe 5.4 reports resuts obtained with an LSQR-based cascadic mutieve method using averaging restriction operators. Comparison of Tabes 5.3 and 5.4 shows that the PSNR-vaues reported in the former are higher. This depends on that the weighted east-squares restriction operators (2.7) hande image discontinuities better. Finay, 20

21 (a) (b) Fig Exampe 5.3. izard and pepper images perturbed by motion bur with width= 15, θ = 10, and noise-eve ν = (a) (b) Fig Exampe 5.3. (a) Restoration of the corrupted izard image determined by the 3-eve LSQR-based mutieve method, and (b) restoration obtained by 1-eve LSQR. Tabes dispay resuts for the izard image; they are anaogous to Tabes The arithmetic work required by the cascadic mutieve methods is dominated by the matrix-vector product evauations on the finest eve. Their number therefore can be used to estimate the arithmetic effort required by the cascadic mutieve methods. The tabes of this exampe show the number of matrix-vector product evauations on the finest eve often to be smaer for -eve methods, with > 1, than for the corresponding 1-eve method. However, the significant advantage of cascadic mutieve methods over the corresponding 1-eve iterative methods is the higher quaity of the restorations determined. Exampe 5.3. We appy Agorithm 2.2 to the restoration of the images shown in Fig.5.1(b) and (c) that have been contaminated by inear motion bur and noise. The 21

22 (a) (b) Fig Exampe 5.3. (a) Restoration of the corrupted pepper image determined by the 3-eve LSQR-based mutieve method, and (b) restoration obtained by 1-eve LSQR. RRGMRES LSQR PSNR # iter PSNR # iter Tabe 5.7 Exampe 5.3. Restoration of the burred and noisy izard image shown in Figure 5.4(a). The tabe shows PSNR-vaues of the restored images and the number of iterations (# iter) required by RRGMRES- and LSQR-based cascadic mutieve methods. RRGMRES LSQR PSNR # iter PSNR # iter Tabe 5.8 Exampe 5.3. Restoration of the burred and noisy pepper image shown in Figure 5.4(b). The tabe shows PSNR-vaues of the restored images and the number of iterations (# iter) required by RRGMRES- and LSQR-based cascadic mutieve methods. PSF is represented by a ine segment of ength r pixes in the direction of the motion. The ange θ (in degrees) specifies the direction; it is measured counter-cockwise from the positive x-axis. The PSF takes on the vaue r 1 on this segment and vanishes esewhere. We refer to the parameter r as the width. The arger the width, the more i-conditioned the matrix A, and the more difficut the restoration task. Figure 5.4 dispays izard and pepper images that have been contaminated by motion bur defined by width= 15 and θ = 10, and by noise of eve ν = The izard image is representative of back and white images with we-defined edges, whie the pepper image is a gray-scae photograph with smoothed edges. Figure 5.5(a) shows the restoration of the corrupted izard image of Figure 5.4(a) 22

23 (a) (b) (c) Fig Exampe 5.4. (a) Image perturbed by motion bur and noise of eve ν = (b) Restoration determined by 1-eve Agorithm 2.2, (c) Restoration determined by Agorithm 2.2 with 2 eves using LSQR. (a) (b) (c) Fig Exampe 5.5. (a) Image perturbed by Gaussian nonsymmetric bur and noise of eve ν = (b) Restoration determined by the noninear PDE-mode (1.5) using a semi-impicit discretization. (c) Restoration determined by Agorithm 2.2 with 2 eves using LSQR. determined by Agorithm 2.2 based on LSQR, using 3 eves, noninear proongation, and smoothing. The image obtained by 1-eve LSQR is shown in Figure 5.5(b). Visua comparison shows the 3-eve LSQR-based cascadic mutieve method to give the most peasing restoration. This is in agreement with the PSNR-vaues reported in Tabe 5.7. Anaogousy, Figure 5.6(a) provides a visua comparison of restorations of the pepper image determined by the 3-eve LSQR-based cascadic mutieve method with noninear proongation operators and smoothing. The restoration determined by 1-eve LSQR is shown by Figure 5.6(b). Tabe 5.8 compares the PSNR-vaues for computed restorations of the pepper image. The tabe confirms our earier observation that LSQR-based cascadic mutieve methods determine better restorations than RRGMRES-based cascadic mutieve methods when the noise-eve of the contaminated image is high. Exampe 5.4. We determine restorations of a contaminated version of the origina pixe Varga image dispayed in Figure 5.1(d). The image is degraded by inear motion bur defined by width = 15 and θ = 10, and by noise of eve ν = Figure 5.7(a) dispays the contaminated image. The restoration obtained by 1-eve LSQR is shown in Figure 5.7(b). The PSNR-vaue for the restored image is This method requires 4 iterations to satisfy the discrepancy principe. Since the avaiabe image has ow resoution, we appy Agorithm 2.2 with ony 2 eves. The agorithm carries out 1 and 3 LSQR-iterations on the coarse and fine eves, respectivey, and yieds a restored image with PSNR= The atter image is shown in Figure 5.7(c). Exampe 5.5. This exampe iustrates the performance of the non-inear ap- 23