Key words. Boundary conditions, fast transforms, matrix algebras and Toeplitz matrices, Tikhonov regularization, regularizing iterative methods.

Transcription

1 REGULARIZATION OF IMAGE RESTORATION PROBLEMS WITH ANTI-REFLECTIVE BOUNDARY CONDITIONS MARCO DONATELLI, CLAUDIO ESTATICO, AND STEFANO SERRA-CAPIZZANO Abstract. Anti-reflective boundary conditions have been introduced recently in connection with fast deblurring algorithms: in the noise free case, it has been shown that they reduce substantially artifacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, reflective) and lead to O(n 2 log(n)) where n 2 is the size of the image. Here we consider the role of the noise. In dimension 1, we proposed a successful approach called re-blurring which is close to the Tikhonov regularization technique: more specifically, the normal equations product A T A is replaced by A 2 where A is the blurring operator. The aim of this paper is to extend to higher dimension the computational and the theoretical analysis performed in one dimension for signals. A wide set of numerical experiments concerning 2D images confirms the effectiveness of our proposal. Key words. Boundary conditions, fast transforms, matrix algebras and Toeplitz matrices, Tikhonov regularization, regularizing iterative methods. AMS subject classifications. 65F10, 65F15, 65Y Introduction. We consider the classical de-blurring problem of noisy and blurred images with space invariant point spread functions (PSFs). More precisely, the mathematical model of the continuous image blurring and noising is described by the following integral equation (see e.g [1]) g(x, y) = h(x θ, y ξ) f(θ, ξ) dθdξ + ν(x, y), (x, y) [0, 1] 2. R 2 related to a Fredholm operator of first kind with shift-invariant kernel. Here f is the (true) input object, h is the shift-invariant integral kernel of the operator (the continuous PSF), ν is the noise which arises in the process, and g is the observed image. Given the blurred and noisy image g, the image restoration problem is to recover a suitable approximation of the input object f. The previous equation is discretized by rectangle formulae over a uniform grid with step-size H (not very accurate discretization schemes are required since the object f is in general only piecewise regular). As a consequence, we obtain a system whose i-th equation is given by g(i) = j Z 2 f(j)h i j + ν(i), i Z 2, (1.1) where for s Z 2, the mask h s represents the (discrete) blurring operator (the discrete PSF), ν(s) is the noise contribution and g(s) is the blurred and noisy observed object. Here g(i) = g(h(i 1)), h i = h(h(i 1)), ν(i) = ν(h(i 1)) and f(i) represents an approximation of f(h(i 1)), for i Z 2. In analogy to the continuous setting, given h and some statistical knowledge of ν, the problem is to recover the unknown true Dipartimento di Fisica e Matematica, Università dell Insubria - Sede di Como, Via Valleggio 11, Como, Italy (marco.donatelli@uninsubria.it). Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, Genova, Italy (estatico@dima.unige.it). Dipartimento di Fisica e Matematica, Università dell Insubria - Sede di Como, Via Valleggio 11, Como, Italy (stefano.serrac@uninsubria.it, serra@mail.dm.unipi.it). 1

2 image f(s) (in the window of observation described by s {1,..., n} 2, H = (n 1) 1 ) from the knowledge of g(s) in the same window. It is clear that the system described by (1.1) is under-determined since we have n 2 equations and (n + m 1) 2 unknowns involved when the support of the PSF is a square of size m 2. In general if the support of the PSF contains more than one point then the number of unknowns exceeds the number of equations by at least n. In order to take care of this problem we can use two approaches: 1) a least square solution, 2) the use of appropriate boundary conditions (linear or affine relations between the unknowns outside the window of observation and the unknowns inside the windows of observation). The case 1) is disregarded by practitioners since it leads to several artifacts that often spoil the quality of the reconstructed object [1, 7]; the second solution in 2) is used and among the boundary conditions (BCs) we count zero Dirichlet, periodic, Neumann (also called symmetric or reflective), and anti-reflective. However the second solution has some unwanted side-effects as well, since some artifacts named ringing effects appear close to the borders of the reconstructed image: in this direction an interesting classification is provided. Zero Dirichlet BCs [1] impose an artificial discontinuity at the borders and so they can lead to serious ringing effects; the resulting structure is block Toeplitz with Toeplitz blocks (BTTB) so that it is not completely easy to handle from a linear algebra viewpoint: the multiplication by a vector can be done in O(n 2 log(n)) complex operations (see e.g. [4]) by using two-level fast Fourier transforms (FFTs) while the solution of an associated linear system is extremely costly in general (see e.g. [9, 13]). Periodic BCs [1] again impose an artificial discontinuity at the borders and therefore the related ringing effect are still not negligible in general; the resulting structure is block circulant with circulant blocks (BCCB) so that both solution of a linear system and matrix vector product can by achieved in O(n 2 log(n)) complex operations by FFTs. Neumann or symmetric or reflective BCs [11] preserve the continuity of the image but not the continuity of its normal derivative: as a consequence the ringing effect are sensibly reduced (by one order of magnitude); moreover the resulting structure is block Toeplitz + Hankel with Toeplitz + Hankel blocks so that the product is again possible by two-level FFTs while the solution of a linear system can by obtained in O(n 2 log(n)) real operations by two-level fast cosine transforms if the PSF is doubly symmetric. Anti-reflective BCs [12] preserve the continuity of the image and the continuity of its normal derivative as well: as a consequence the ringing effects are negligible with respect to the other BCs (see [12, 5]); the associated structure is more involved since it is block Toeplitz + Hankel with Toeplitz + Hankel blocks plus a structured low rank matrix: despite its apparent complicate structure, the product is again possible by two-level FFTs while the solution of a linear system can be obtained in O(n 2 log(n)) real operations by two-level fast sine transforms if the PSF is doubly symmetric [12, 5]. In the noise-free case, when considering generic images, we observed that the new boundary conditions are much more effective than previous ones (zero Dirichlet, periodic and reflective) in terms of reconstruction quality (see [12] and especially [5]). In the case of signals, when there is noise, we modified [6] the classical regularization 2

3 techniques (which are necessary for handling the effect of the noise) in order to exploit the quality of the AR BCs reconstruction and the efficiency of the related numerical procedures: we called this idea re-blurring. In this paper we extend both the computational analysis and the proposal to higher dimensions with special attention to the 2D case of classical images. We start by considering the Tikhonov regularization [1] method: we take a penalty positive parameter µ (generally small and associated with the level of the noise), an auxiliary operator T chosen among the identity and low order differential operators (generally depending on the type of the noise) and we minimize the functional Af g µ T f 2 2. (1.2) Here A denotes the blurring operator with the chosen BCs and the operator T is corrected with the same BCs in order to simplify the calculation and to reduce the computational cost. We remark that the minimization of (1.2) is equivalent to the modified normal equations [1] given by [ A T A + µt T T ] f = A T g. (1.3) Furthermore, if T includes the same BCs as A, then there is a computational advantage at least in the periodic case since the whole matrix A T A + µt T T shows a circulant structure. The same can be stated unchanged for the reflective BCs when the blurring operator is doubly symmetric, while we have difficulties both in the zero Dirichlet and anti-reflective BCs: in the first case, since the Toeplitz class is not an algebra (A T A is no longer Toeplitz and its blocks are no longer Toeplitz) and, in the second case, since the low rank term spoils the whole structure. It is intuitive to understand that the penalty functional µ T f 2 2 controls the growth of those components of f related to the noise on the data g. If g is contaminated by noise, then generally f consists of fast oscillations and hence T f is large (especially when T is a differential operator). On the other hand, if the regularization parameter µ is over estimated, then f becomes smooth, thanks to the large contribution of µ T f 2 2, but the residual Af g can have large norm, since the computed f is far from the real image. We choose experimentally the optimal parameter and we observe that the reflective BCs are still sensibly better than the other two classical BCs (periodic and zero Dirichlet); their behavior becomes similar only for high noise levels as SNR [1, 5] (SNR denoting the signal-to-noise ratio) since the boundary effects becomes negligible with respect to the noise contribution and therefore it does not make any sense to choose sophisticate BCs. The same observation stands if we consider the conjugate gradient (CG) applied to the normal equations A T Af = A T g (1.4) with optimal choice of the step where stopping the computation (see [7]). A negative surprise is that the anti-reflective BCs give results that deteriorate rapidly when the SNR decreases: therefore for a wide range of SNR between 1 and 100 the quality of the reconstruction is slightly better than the periodic and zero Dirichlet BCs reconstruction but it is seriously worse when compared with the reflective BCs. In [6], we have given an explanation of the latter facts and we proposed a basic modification of the normal equation approach (both of equation (1.3) to be solved exactly and of equation (1.4) to be solved by CG with early termination): more precisely, it the term A T is replaced by A (the re-blurring), the anti-reflective (AR) 3

4 choice becomes again superior among the considered BCs even in presence of high levels of noise. In this paper we focus the discussion on higher dimensions and especially on the case of images. We prove that the new formulation overcomes the computational difficulties due to the role of the low rank term in the product A T A. Under the assumption of doubly symmetry of the PSF, the restored image with n 2 pixels can be computed in O(n 2 log(n)) real operations by using few fast sine transforms and the re-blurred equation [ A 2 + µt 2] f = Ag. In the d-dimensional case, the cost is still O(n d log(n)) where n d is the size of the restored object, provided that the PSF is again fully symmetric, that is, symmetric with respect to each one of the d axes. By a theoretical point of view, it is important to remark that such full symmetry of the PSF is fulfilled in a lot of models for optical applications. For instance, in most 2D astronomical imaging with optical lens [1], the theoretical PSF is circularly symmetric, and hence, doubly symmetric; in the multi-image deconvolution of some recent interferometric telescopes, the PSF is doubly symmetric too (see Fig. 1 of [2]). In 3D out-of-focus deconvolution microscopy [10], the PSF is again fully (and then triply) symmetric. Indeed, the best optics yield PSFs that are symmetrical in the Z axes and circularly symmetric with respect to the X Y plane. On the other hand, it is important to remark that in real applications some optical distortions may be present, which cause asymmetries in the PSF. Although most of these distortions may be identified and corrected by interchanging special optical lens, the experimental PSF may come out asymmetric anyway. In that case it should be noticed that often the quoted asymmetry is very weak in the sense that r A = A + A T 1 A A T is a small number. Indeed, since (A + A T )/2 is the symmetric part of A and (A A T )/2 is the anti-symmetric part of A, then r A gives a measure of the degree of symmetry of A: with r A < 10% we can say that A is weakly asymmetric, with r A = 0 we have perfect symmetry. In any case, the application of the proposed re-blurring approach (1.3) or (1.4) with anti-reflective BCs gives quite good results, not only in these real settings of weak asymmetry, but also in the case of strongly asymmetric PSFs (see Table 6, Subsections 3.2 and 4.1 in [6]). The paper is organized as follows: in Section 2 we analyze the algebraic properties of the linear systems arising from d dimensional AR BCs that allow one to reduce the computation to few j-dimensional fast discrete sine transforms of type I (DST I) with j d. In Section 3 we briefly discuss the re-blurring idea. In Section 4 we report 2D numerical experiments that confirm the effectiveness of the proposal by including asymmetric examples as well. Section 5 is devoted to discuss future directions of research. 2. Linear systems with AR BCs. Following [3], in Subsection 2.1, we firstly introduce the (scalar) τ class and the multi-level τ class related the one level and multilevel-level discrete sine transform matrices. Secondly, in Subsection 2.2, we briefly describe the AR BCs in the case of signals, images, and d-dimensional objects and we report the structure of the matrix A representing the blur operator when the AR BCs are enforced. Subsection 2.3 contains new results which generalize the corresponding one-dimensional analysis in [6]: in particular we prove the structure of algebra (closure under linear combinations and multiplication) of a special class of matrices that includes any polynomial p(a) of any AR matrix A. In addition we show how to solve a linear system with invertible coefficient matrix of the form p(a) + µs where S is penalty (differential-like operator) with the right AR BCs: notice that by setting p(a) = A 2, we can use these theoretical results for our reblurring technique. 4

5 2.1. The τ algebra. Let Q be the n dimensional discrete sine transform matrix of type I (see [3]) with entries ( ) 2 jiπ [Q] i,j = n + 1 sin, i, j = 1,..., n. (2.1) n + 1 It is known that the matrix Q is orthogonal and symmetric (Q = Q T and Q 2 = I). For any n dimensional real vector v, the matrix vector multiplication Qv can be computed in O(n log(n)) real operations by the algorithm DST I. In the multidimensional case, setting Q (d) = Q Q (d times) and by considering v R nd, the vector Q (d) v can be computed in O(n d log(n)) real operations by the DST I procedure of level d. Let τ (d) be the space of all the d-level matrices that can be diagonalized by Q (d) : τ (d) = {Q (d) DQ (d) : D is a real diagonal matrix of size n d }. (2.2) Let X = Q (d) DQ (d) τ (d), then Q (d) X = DQ (d). Consequently we have Q (d) Xe 1 = DQ (d) e 1 with e t denoting the t-th vector of the canonical basis of R nd, that is, the eigenvalues [D] i,i of X are given by D i,i = [Q(d) (Xe 1)] i [Q (d) e 1 ] i, i = 1,..., n d. Therefore the eigenvalues of X can be obtained by means of a d-level DST I of the first column of X and, in addition, any matrix in τ (d) is uniquely determined by its first column. Now we report a characterization of the τ (d) class which is important for analyzing the structures of the AR BCs matrices: we start with the case of d = 1. Let us define the downshift of any vector v = (v 0,..., v n 1 ) T as σ(v) = (v 1, v 2,..., v n 1, 0) T. According to a Matlab like notation, we define T (v) as the n-by-n symmetric Toeplitz matrix whose first column is v and H(x, y) as the n-by-n Hankel matrix whose first and last column are x and y respectively. Every matrix of the class (2.2) for d = 1 can be written as (see Bini, Capovani [3]) T (v) H(σ 2 (v), Jσ 2 (v)), J = H(e n, e 1 ), (2.3) where v = (v 0,..., v n 1 ) T R n and the matrix J is the permutation flip matrix with J s,t = 1 if s + t = n + 1 and zero otherwise. This means that scalar τ structures are special instances of Toeplitz plus Hankel matrices. Now for d > 1, the description of τ (d) can be given recursively. More in detail, every τ (d) matrix is represented as (2.3) where every entry v j is a matrix of size n d 1 belonging to the algebra τ (d 1) and where the matrix J is replaced by J I n d 1. Finally, we stress that the class τ (d) will be indicated explicitly by τ m (d) when the corresponding matrix size m d is not clear from the context Structural properties. We have already mentioned that, in the generic case, periodic and zero (Dirichlet) BCs introduce a discontinuity in the signal while the Neumann reflective BCs preserve the continuity of the signal but introduce a discontinuity in the derivative (the normal derivative when d = 2). Our approach is to use an anti-reflection: in this way, at the boundaries, instead of having a mirrorlike symmetry around the vertical axis (reflective BCs), we impose a global symmetry around the boundary points. If (x 1, f 1 ) is the left boundary point and the (x n, f n ) is the right one, then the external points (x 1 j, f 1 j ) and (x n+j, f n+j ), j 1, are 5

6 computed as function of the internal points according to the rules f 1 j f 1 = (f j+1 f 1 ) and f n+j f n = (f n j f n ). Therefore if the support of the blurring function is m then we have f 1 j = f 1 (f j+1 f 1 ) = 2f 1 f j+1, for all j = 1,..., m, f n+j = f n (f n j f n ) = 2f n f n j, for all j = 1,..., m. Following the analysis given in [12], if h is symmetric i.e. h j = h j for every j, the precise structure of the AR matrix is A = z 1 + a z 2 + a a m z m + a m 1  z m + a m 1 a m.. 0 z 2 + a z 1 + a 0 where A 1,1 = A n,n = 1, z j = 2 m k=j h k, a j = h j,  has size n 2 and (2.4)  = T (u) H(σ 2 (u), Jσ 2 (u)), (2.5) with u = (h 0, h 1,..., h m, 0,..., 0) T, J s,t = 1 if s + t = n 1 and zero otherwise. According to the brief discussion in the Subsection 2.1 and in particular equation (2.3), relation (2.5) implies that  τ (1) n 2. Information concerning the structure and the eigenvalues of the blurring matrix A are contained in the following two statements which can be easily proven via (2.4) and (2.5). For all i we have: 1.  can be diagonalized by Q and  = P AP T with P = 0 0. I. 0 0 R (n 2) n (2.6) where I is the identity matrix of size n 2, 2. The n eigenvalues of A are given by 1 with multiplicity two and by the n 2 eigenvalues of Â. The anti-reflective BCs have a natural extension in two dimensions. In particular, for a blurring operator represented by an m-by-m PSF, the point f 1 j,t, 1 j m, 1 t n, is represented by 2f 1,t f j+1,t. Analogously, for 1 j m, 1 s, t n, we have f s,1 j = 2f s,1 f s,j+1, f n+j,t = 2f n,t f n j,t and f s,n+j = 2f s,n f s,n j. It is important to notice that when both indices lie outside the range {1,..., n} (this happens close to the 4 corners of the given image), then these 2D AR BCs reduce to the one-dimensional case by making anti-reflection around the x axis and then around the y axis separately. For instance, concerning the corner (1, 1), we set f 1 j,1 l = 4f 1,1 2f 1,l+1 2f j+1,1 + f j+1,l+1, for 1 j, l m (the idea around the other corners is similar). Regarding the two-dimensional blurring, it is worth mentioning the following generalization of equations (2.4), (2.5), and Statements 1. and 2. 6

7 Theorem 2.1. [12] Let the blurring function (PSF) h be quadrantally symmetric, i.e., h i,j = h i, j for all i, j. Then the blurring matrix A, with anti-reflecting boundary conditions, has the form A = z 1 + a z 2 + a 1 0. a m z m + a m 1 A z m + a m 1 a m. 0 z 2 + a z 1 + a 0 (2.7) where each symbol a j and z j denotes a block of size n having exactly the same structure as in (2.4), the matrix A has size n 2 with blocks of dimension n and where A = (P I)A(P T I) with P defined as in (2.6), the external structure of A is (block) τ (1) n 2 matrix and in addition (P P ) A (P T P T ) τ (2) n 2. Finally, each element of the blocks in (2.7) has a computable (in O(n 2 ) operations) expression which depends only on the entries h i,j of the blurring function. Now we are ready for generalizing the result to the d dimensional setting. We use again anti-reflection with respect to every axis separately. We set f i = 2f b f i, (2.8) where i = (i 1,..., i d ) is a d-index such that at least one component (let us say the k-th entry, k {1,..., d}) is of the form i k = 1 j k or i k = n + j k with j k 1, and b = (b 1,..., b d ), i = (i 1,..., i d ) are d-indices such that b s = i s = i s s k, { 1 if ik = 1 j b k = k, n if i k = n + j k, { i 1 + jk if i k = k = 1 j k, n j k if i k = n + j k. For j = (j 1,..., j d ), let r(j) denote the integer number #{s {1,..., d} : j s / {1,..., n}}. Clearly r(j) {0,..., d}. By (2.8) we easily infer that for i such that r(i) > 0 we have r(b) = r(i ) = r(i) 1. Now the use of anti-reflection separately means that we apply (2.8) recursively to f b and f i if r(b) r(i ) > 0. We terminate the recursion when the new indices in the righthand-side are such that their function r( ) vanishes (i.e. the corresponding d-indices determine points inside the d-dimensional window of observation). The resulting matrix is described as follows. Theorem 2.2. Let the d-dimensional blurring function (PSF) h be quadrantally symmetric, i.e., h i = h i where i = (i 1,..., i d ) is a d-index and i = ( i 1,..., i d ). 7

8 Then the blurring matrix A, with anti-reflecting boundary conditions, shows the form A = z 1 + a z 2 + a 1 0. a m z m + a m 1 A z m + a m 1 a m. 0 z 2 + a z 1 + a 0 (2.9) where each symbol a j and z j denotes a block of size n d 1 and represents a matrix of the form (2.9) where d is replaced by d 1. the matrix A has size n 2 with blocks of dimension n d 1 and where A = (P I I)A(P }{{} T I I) with P defined as in (2.6), }{{} d 1 d 1 the external structure of A is a (block) τ (1) n 2 matrix and P (d) A (P (d) ) T τ (d) n 2 where P (d) = P P. } {{ } d Finally, each element of the blocks in (2.9) has a computable (in O(n d ) operations) expression which depends only on the entries h i of the blurring function Fast O(n d log(n)) solution of AR BCs linear systems. For d positive integer number, we define the classes of matrices S d as follows. For d = 1, M S 1 if M = α v ˆM w β, (2.10) with α, β R, v, w R n 2, and ˆM τ (1) n 2. For d > 1, M S d if M = α v M w β, (2.11) with α, β S d 1, v, w R (n 2)nd 1 n d 1, v = (v j ) n 2 j=1, w = (w j) n 2 j=1, v j, w j S d 1 and M = ( ) Mi,j n 2 having i,j=1 external τ (1) n 2 structure; every block Mi,j of M belongs to S d 1. Notice that in the previous block matrices the presence of blanks indicates zeros or block of zeros of appropriate dimensions. The following preliminary result shows an important link between the classes S d and τ (d). Lemma 2.3. For every matrix M S d with M defined according to equation (2.11), we have P (d) M [P (d)] T = (I n 2 P (d 1)) ( [ M I n 2 P (d 1)] ) T τ (d) 8

9 with P (t) = P P (t times tensor product), P (0) = 1, and P being the matrix given in (2.6). Proof First we observe that P (d) M [P (d)] T = (I n 2 P (d 1)) ( [ M I n 2 P (d 1)] ) T (2.12) holds owing to the structure of P. For d = 1 the thesis is trivially true since, by equations (2.10) (2.11), we have M = ˆM τ (1) and I n 2 P (d 1) = I n 2 P (0) = I n 2. For d > 1 we use induction. By (2.12), we have ( [ P (d) M P (d)] ) T [ = P (d 1) (M ) i,j P (d 1)] T (2.13) i,j with (M ) i,j S d 1 by definition of the class S d. Consequently, by the inductive assumption it follows that every block P (d 1) (M ) i,j [ P (d 1) ] T belongs to the algebra τ (d 1). Moreover M has external τ (1) structure by definition of the class S d and therefore relation (2.13) implies that the matrix P (d) M [ P (d)] T possesses external τ (1) structure as well. In conclusion P (d) M [ P (d)] T has external τ (1) structure and every block lies in the class τ (d 1), which is equivalent to the relationship P (d) M [ P (d)] T τ (d). Thanks to the latter Lemma and from (2.10) and (2.11), it is evident that the AR BCs matrices described in (2.4), (2.7), and (2.9) belongs to the classes S d with appropriate choice of d 1. Now we are ready for giving some algebraic and computational properties of the class S d. Theorem 2.4. Let f = [f j ] n j=1 and g = [g j] n j=1 be two vectors with real entries and size n d. The following facts hold true: (i) every linear system Mf = g with M S d can be solved in O(n d log(n)) arithmetic (real) operations whenever M is an invertible matrix; (ii) every matrix vector product g := Mf with matrix M S d costs O(n d log(n)) arithmetic (real) operations; (iii) the space S d is an algebra, i.e., it is closed under linear combinations, product and inversion. Proof (by induction) (i) For d = 1 the first part of the theorem has been proved in [6], Theorem 2.1. For d > 1 we assume that the desired claims (i), (ii), and (iii) hold for every j d 1. From a direct inspection of M, it follows that the determinant of M is given by det(α)det(β)det( ˆM): therefore the invertibility of M is equivalent to claim that α, β, and ˆM are invertible matrices. The solution of the given linear system is obtained as follows: 1) solve the systems αf 1 = g 1, βf n = g n ; 2) compute c = [g] n 1 j=2 vf 1 wf n ; 3) y = [f] n 1 j=2 is computed by solving the system M y = c. The first step consists of solving two invertible linear systems with coefficient matrices α and β both belonging to S d 1 : therefore the inductive assumption, part (i), tells us that the first step costs O(n d 1 log(n)) arithmetic operations. In the second step we perform 2(n 2) matrix vector product with matrices v j and w j, belonging to S d 1, for j = 1,..., n 2, and 2(n 2)n d 1 additions: 9

10 the related global cost is dominated by O(n d log(n)) arithmetic operations again by the inductive assumption (part(ii)). The argument for step 3) is a little bit more involved. The matrix M has external τ (1) n 2 structure and every block lies in S d 1. Therefore we can perform a block diagonalization of M by using DST I transforms of size n 2. More precisely we have M [ ] = Q n 2 I n d 1 diagj=1,...,n 2 D j Qn 2 I n d 1 with D j S d 1, because they are obtained as linear combinations of the blocks of M and, by the inductive assumption, part (iii), S d 1 is an algebra (and therefore a vector space). As a consequence, by performing 3n d 1 DST I of size n 2 we reduce the computation of the vector y to the solution of n 2 systems with coefficient matrices D j, j = 1,..., n 2. Since the cost of one DST I of size n 2 is O(n log(n)) arithmetic operations and since D j S d 1, we conclude that the cost of the whole step 3) is given by O(n d log(n)) arithmetic operations. (ii) For d = 1 the second part of the theorem has been proved in [6], Theorem 2.1. For d > 1 we assume that the desired claims (i), (ii), and (iii) hold for every j d 1. The desired matrix vector product is given by 1) compute g 1 = αf 1, g n = βf n ; 2) compute [g] n 1 j=2 = M [f] n 1 j=2 + vf 1 + wf n. By following the same considerations as in the previous item we easily see that both steps 1) and 2) cost O(n d log(n)) arithmetic operations. (iii) For d = 1 the third part of the theorem has been proved in [6], Theorem 2.1. For d > 1 we assume that the desired claim (iii) holds for every j d 1. We observe that the closure under (real) linear combinations is trivial since S d is a (real) linear space by definition. If M 1 and M 2 belong to S d then its product M 1 M 2 can be written as with α 1 α 2 v 1 α 2 + M 1 v 2 M 1 M 2 w 1 β 2 + M 1 w 1 β 1 β 2 M j = α j v j M j w j β j with all the blocks ( ) Mj, s, t = 1,..., n 2, α s,t j, v j, w j, β j, j = 1, 2, belong to S d 1. Since S d 1 is an algebra, by the inductive step it follows that α 1 α 2, β 1 β 2 and every block of the matrices v 1 α 2, M1 v 2, M1 M2, w 1 β 2, M1 w 1 are in the same algebra S d 1. Finally, since M1 and M2 have external τ (1) n 2 structure, it is evident that by performing a block diagonalization it follows that M1 M2 has also external τ (1) n 2 structure and therefore the claim is proven. The O(n d log(n)) algorithm proposed in the point (i) of Theorem 2.4 can be used for solving any linear system of the form [p(a)+µs]x = b where the coefficient matrix p(a) + µs is invertible, where A is the AR BCs matrix and S is any differential-like operator with AR BCs. 10,

11 A sketch of the algorithm in the 2D case. Theorem 2.4 leads to a precise sketch of a recursive O(n d log n) algorithm for computing the restored object. In the following, we give an explicit direct procedure for computing the 2D restored image by using the re-blurred equation. More precisely, by ordering the unknowns according to the scheme (1, 1), (1, 2),..., (1, n), (2, 1), (2, 2),..., (2, n),..., (n, 1), (n, 2),..., (n, n), we first compute the unknowns on the frontier of the domain (f i,j where either i or j belong to {1, n}), and then solve a linear system of size (n 2) 2 whose coefficient (2) matrix  belongs to the algebra τ n 2. That is,  = (P P )A(P T P T ) (2.14) with P defined as in (2.6). The precise details of the solution process may be outlined as follows: 1. First determine f 1,1, f 1,n, f n,1, and f n,n (the corresponding rows of A are zero except for the diagonal element); 2. update the right hand side and compute the unknowns f 1,2, f 1,3,..., f 1,n 1 and f n,2, f n,3,..., f n,n 1 by solving two separate τ (1) n 2 linear systems; 3. update the right hand side and compute the unknowns f 2,1, f 3,1,..., f n 1,1 and f 2,n, f 3,n,..., f n 1,n by solving two separate τ (1) n 2 linear systems; 4. update the right hand side and compute the remaining unknowns by solving a linear system with the τ (2) n 2 matrix  in (2.14). It is clear that the algorithm involves O(n 2 log n) arithmetic operations all due to fast (unilevel and two-level) discrete sine transforms of type I. 3. Regularization and re-blurring. When the observed signal (or image) is noise free, then there is a substantial gain of the reflective BCs with respect to the periodic or zero Dirichlet BCs and, analogously, there is a significant improvement of the AR BCs with regard to the reflective ones (see [12, 5]). Since the de-convolution problems is generally ill-posed (high frequency errors, i.e., noise components are greatly amplified) independently of the chosen BCs, it is evident that we have to regularize the problem. Two classical methods, i.e., Tikhonov regularization (see (1.3)) and iterative solvers (conjugate gradient [7] or Landweber method [1]) for normal equations (see (1.4)) can be used. We observe that in both the cases, the coefficient matrix is a shift of A T A and that the right hand side is given by A T g. The only BCs that are seriously spoiled by this approach are the AR BCs: more in detail, even in presence of a moderate noise, its precision becomes worse with respect to the reflective BCs and only slightly better than the other two BCs (see Table 3.1 in [6]). The reason relies upon the matrix A T (see [6]): we recall that h is symmetric and therefore, concerning the other BCs, the matrix A T is still a blurring operator since A T = A, while, in the case of the AR BCs matrix, A T cannot be interpreted as a blurring operator. A (normalized) blurring operator is characterized by nonnegative coefficients such that every row sum is equal to 1: in the case of A T with AR BCs the row sum of the first and of the last row can be substantially bigger than 1. This means that new observed signal A T g has artifacts at the borders and this reduces the quality of the reconstruction. Furthermore, the structure of the matrix A T A is also spoiled and, in the case of images (d = 2) we lose the O(n 2 log(n)) computational cost for solving a generic 11

12 system A T Ax = b. The reason of this negative fact is that A T A / S 2. More precisely, for A S 2, we have α a β A T A = b M c, γ d δ where a = (a j ) n 2 j=1, b = (b j) n 2 j=1, c = (c j) n 2 j=1, d = (d j) n 2 j=1, M = ( Mi,j ) n 2 i,j=1 with a j, b j, c j, d j, M i,j, α, β, γ, δ being expressible as the sum of a matrix belonging to S 1 and a matrix of rank 2. Therefore since M has external τ (1) n 2 structure, it follows that A T A can be written as the sum of a matrix in S 2 and a matrix of rank proportional to n. The cost of solving such a type of linear systems is proportional to n 3 by using e.g. Shermann-Morrison formulae (which by the way can be numerically unstable [8]). In higher dimension the situation worsens becouse in the d-dimensional setting the solution of the normal equation linear system is asymptotic to n 3(d 1), where n d is the matrix order. In order to overcome the problem due to the multiplication by A T (which arises only with the most precise AR BCs), we replace A T by A. We observe that in the other three cases (when the blurring coefficients are symmetric i.e. h j = h j for every j), the associated normal equation can be read as A 2 f = Ag. Therefore the observed image g is re-blurred. The re-blurring is the key of the success of the regularization techniques in (1.3) and (1.4): indeed also the noise is blurred and this makes the contribution of the noise less evident. To make clear the idea, we consider the case of A ill-conditioned and positive definite. Thus, instead of using the Tikhonov approach, we can consider the Riley approach which consists in solving Af = g by CG with early stopping or by taking the solution of [A + θs] f = g (3.1) with best parameter θ. If we set S and T as the identity, then the new formulation can be compared with the Tikhonov one in (1.3): all the numerical experiments uniformly show that: the solution of (1.3) with best parameter is always better than the solution of (3.1) with best parameter; the solution of (1.4) by CG with best termination is always better than the solution of Af = g by CG with best termination. The reason of that relies upon the fact that the re-blurring smoothes the noise in the right hand side. In conclusion we propose to replace (1.3) and (1.4) with the following: [ A 2 + µt 2] f = Ag (3.2) to be solved with the right choice of the parameter µ and A 2 f = Ag (3.3) to be solved by conjugate gradient (or Landweber) with early termination. Notice that when T is the identity and we consider zero Dirichlet, periodic or reflective BCs, 12

13 L. Eldén s PSF Experimental symmetric PSF Figure 4.1. Point Spread Functions under our assumptions of symmetry, the new proposal coincides with the classical ones. Therefore the novelty solely concerns the AR BCs. As we will see in the numerical experiments, with this modification, the AR BCs are still very convenient even in presence of noise. Notice that the cost of the solution of a linear system (3.2) is of the order n d log(n) thanks to Theorem 2.4 since both A 2 and T 2 (thus A 2 +µt 2 ) belong to the class S d defined in Section 2.3. The re-blurring idea can be generalized in many ways: for instance the re-blurred equation A 2 f = Ag can be replaced by A k+1 f = A k g with k > 1 even if the choice k = 1 seems to be practically the most convenient. 4. Numerical experiments. We test the following two PSFs. (I) Eldén PSF It is associated to the separable blurring h i,j = a i a j with { 4 a s = 51 k 0.15(sh) if r s < 9, 0 otherwise where k σ (t) is the Gaussian distribution with zero mean and standard deviation σ, and the mesh size h = 4/n. (II) Symmetrization of the experimental PSF It is related to an experimental PSF developed by US Air Force Phillips Laboratory, Lasers and Imaging Directorate, Kirtland Air Force Base, New Mexico. The following input (true) images f are of pixels. (1) Linear Function: z = 5x 2y + 10 with (x, y) equidistant in [ 5, 5] [ 5, 5]. (2) atan : z = 1 1+x 2 +y 2 with (x, y) equidistant in [ 5, 5] [ 5, 5]. (3) Face of actor (4) Blocks of flats We compute the blurred and noisy image g, by means of the convolution between the PSF matrix and the input data f. The noise ν is white, that is, Gaussian with zero mean. In the first test (Table 4.1, Figures ), the relative noise ν / Af is about 4%, i.e., SNR = 25. In the latter tests (Tables 4.2 and 4.3, Figures 4.9 and 4.10), we compare the restorations of the real image (3) for several levels of noise on the blurred image, ranging from SNR = 5 to the noiseless case SNR =. We consider only the internal portion of the blurred and noisy image g, and we try to recover the internal portion of the true input data f. 13

14 Full Image ( ) Image Eldén PSF Exp. PSF Figure 4.2. True and blurred (and noisy) images of the face of actor Full Image ( ) Image Eldén PSF Exp. PSF Figure 4.3. True and blurred (and noisy) images of Blocks 14

15 PSF Eldén PSF Experimental BCs Linear atan Actor Blocks Linear atan Actor Blocks Periodic Reflective AR Table 4.1 Best relative restoration errors within 100 iterations of CG method. Image Size: See Figures 4.1, 4.2, and 4.3 for the two PSFs and for some original and blurred images. We show the different reconstructions corresponding to the following choices of boundary conditions: Periodic Reflective Anti-Reflective We did not report the zero Dirichlet reconstructions since they do not perform differently when compared with the periodic setting. We use the conjugate gradient with optimal termination (within 100 iterations) on the system (3.3), i.e., we use A 2 in place of A T A as in the normal equation approach (1.4). The results of Table 4.1 show that the AR BCs are generally better than the other choices. The left side of the Table concerns the Eldén PSF, which is a blurring operator well clustered at the center (see Figure 4.1). In this case, the improvement due to the AR BCs with respect to the reflective one is not large; both of such two BCs are quite better than the periodic one, though (see in particular the first column of Table 4.1 and the related Figure 4.8). A remark has to be given for the result of the second column. Here the image is the atan function, which is symmetric with respect to the center of the domain. As expected in this special case, periodic and reflective extensions lead to very similar results. The right side of Table 4.1 is related to the experimental symmetric PSF. Here the restoration errors of the AR BCs are much better than the other two choices. In particular, we point out that in the case of the (real image) face of actor, the AR BCs error is half the periodic s one (see the third column). The reason of this high improvement is due to the shape of the experimental PSF, since, basically, the more the support of the PSF is large, the more the ringing effects become important. Concerning the restored images of Figures , a general observation has to be made: with the AR BCs the ringing effects are practically not visible while they can be observed in the periodic case and even in the reflective case. The organized structure of the error close to the left and right borders in the first two items of Figure 4.8 shows such a fact. Figure 4.4 and Figure 4.5 show some restorations related to the Eldén PSF. Regarding the image of the face of actor, the artifacts due to ringing effects are easily visible for periodic BCs (see the light and dark lines close to the borders). Similar effects can be observed with the block of flat of Figure 4.5 inside the two bars. Much lower ringing effects are still present with the reflective BCs (see, for instance, the bottom of the image at the center of Figure 4.4), while they are not visible with the AR BCs (see the right image of Figure 4.4). Figure 4.6 and Figure 4.7 concerns the experimental PSF, whose blurring effects are very high. In this case, restorations with periodic BCs are totally unsatisfactory. Moreover, the AR BCs allow one to recover more details than the reflective BCs. This is rather evident in Figure 4.6, where the AR BCs restoration is the very best one. 15

16 Periodic Reflective AR BCs Figure 4.4. Reconstructions for the face of actor with Eldén PSF Periodic Reflective AR BCs Figure 4.5. Reconstructions for Blocks with Eldén PSF Periodic Reflective AR BCs Figure 4.6. Reconstructions for the face of actor with Exp. PSF Periodic Reflective AR BCs Figure 4.7. Reconstructions for Blocks with Exp. PSF 16

17 Periodic Reflective AR BCs Figure 4.8. Errors for the Linear image with Elde n PSF (to note that the z axes is [ 30, 30] in the periodic case while it is [ 1, 1] for the other two BCs). SNR=25 SNR=50 Figure 4.9. Relative restoration error for first 100 iterations of CG with Elde n PSF. SNR=25 SNR=50 Figure Relative restoration error for first 100 iterations of CG with experimental PSF. Two examples of convergence histories, that is, the relative restoration errors at any iteration, are plotted in Figures 4.9 and The left sides of the figures concern the restoration of the face of actor with the same level of noise SNR=25 just as before. In addition, here we consider a lower level of noise, i.e., SNR=50, and we compare the different convergence. It should be stressed that the AR BCs give the best results, especially better with the experimental PSF of Figure In this latter case, the errors with AR BCs continue to decrease even after the first 100 iterations. On the contrary, the restorations with periodic BCs start to deteriorate after the very first iterations, and the restorations with reflective BCs after about iterations. It is interesting to note that the AR BCs curves are quite flat. This is a very useful 17

18 SNR Periodic Reflective AR AR-corner (0% noise) Table 4.2 Best relative restoration errors within 100 iterations of CG method varying the SNR for the Eldén PSF. SNR Periodic Reflective AR AR-corner (0% noise) Table 4.3 Best relative restoration errors within 100 iterations of CG method varying the SNR for the Experimental PSF. feature since it yields that the estimation of the optimal number of iterations, which generally is a difficult task, can be done with low precision. Table 4.2 and 4.3 show the best relative restoration errors for some levels of noise on the data. The test image is again the face of actor. As already mentioned, the choice of the BCs is important mainly if the noise on the data is low, that is, for high values of SNR. In the first two rows of tables 4.2 and 4.3 the errors due to noise dominate the restoration process and therefore the choice of particular BCs is not significant. In the next rows, where the noise is lower, the choice of the BCs becomes crucial. In particular, the AR BCs improve a lot the quality of the restorations with respect to the others BCs (see, for instance, the case with SNR=50 with the experimental PSF on the right side). In the last column of both tables we show the results concerning the AR BCs with the antireflection around the corners proposed in [5]. In such a paper, it is shown that in the two-dimensional case we can use a different strategy at the corners (called choice a) in [5]): for instance the antireflection around the corner (1,1) is f 1 j,1 l = 2f 1,1 f j+1,l+1, 1 j, l m. From an experimental point of view, usually this choice works slightly better than the previous one (see tables 4.2 and 4.3), but unfortunately the corresponding coefficient matrix A does not belong to the class S 2 : indeed A can be seen as the matrix in (2.7) plus a sparse matrix R m having at most O(n) nonzero rows and columns (with reference to the d level case, O(n d 1 ) nonzero rows and columns have to be added to the matrix in (2.9)). Therefore we can not apply the fast algorithm described in Subsection 2.3. However, if we use an iterative method as solver, the matrix-vector product can be again computed in O(n d log(n)) arithmetic operations irrespectively of the support of the PSF. Finally, for the sake of completeness, we treat the asymmetric case since it is the most common in a variety of real applications. In analogy with the corner antireflection, the algorithm in Subsection 2.3 can not be used. On the other hand, the matrix 18

19 SNR Periodic Reflective AR-corner Figure Best relative restoration errors within 100 iterations of CG method for the non symmetric PSF. PSF P (2) A(P (2) ) T is a BTTB plus a block Hankel with Hankel blocks and therefore only O(n 2 log(n)) arithmetic operations are necessary in a matrix-vector multiplication with any of these matrices. Therefore the matrix-vector product with the full coefficient matrix A requires again O(n 2 log(n)) arithmetic operations and this property is essential when using an iterative solver. From a qualitative point of view, as we can see from the table in Figure 4.11, the AR BCs give again very good restoration, especially for low levels of noise as in the symmetric case considered in the previous tests. We remark that all the discussion concerning the quality of the reconstruction in the AR BCs is completely independent of the symmetry of the PSF and hence the proposal in this note can be used in a general context too. 5. Conclusions. In this paper we considered a basic modification (re-blurring) of the normal equation approach (both for Tikhonov regularized equation (1.3) to be solved exactly and for equation (1.4) to be solved by CG with early termination): by using this modification, the anti-reflective choice becomes again convenient among the considered boundary conditions even in presence of high level of noise. We have shown that the new formulation allows to overcome the computational difficulties due to the role of the low rank term in the product A T A of the blurring operator A: more precisely, the restored d-dimensional object can be computed in O(n d log(n)) real operations by using few fast j-level sine transforms with j d, if A is fully symmetric, that is, symmetric with respect to each one of the d axes. Although the analysis of the model covers the symmetric case in more detail, we have shown that the proposed re-blurring with AR boundary conditions gives good results even in the asymmetric setting. In particular, both in the symmetric and asymmetric cases, the proposed technique allows very high reduction of the artifacts (ringing effects) of the reconstructed images with respect to the Dirichlet BCs and the periodic BCs, and improvements even with respect to the reflective BCs; good filtering of the components of the data related to the noise; fast computation, i.e. O(n d log(n)) in d-dimensional settings, if we use iterative solvers where only fast matrix-vector products are required (refer to Theorem 2.4, item (ii)); easier choice of the regularization parameter in connection with iterative solvers (in other words it is easier to determine a quasi-optimal iteration where to stop). Under these considerations, it should be favorable to extend the technique to real applications such as LBT or 3D microscopy. Future works will be addressed to this 19

20 study, since the properties of the PSF (which is close to be symmetric) and the characteristic of the images (which are nonzero nearby the boarders) seems to fit quite well with the proposed AR BCs re-blurring. Acknowledgements The work of all the authors was partially supported by MIUR, grant number REFERENCES [1] M. Bertero and P. Boccacci 1998 Introduction to inverse problems in imaging Inst. of Physics Publ. Bristol and Philadelphia, London, UK. [2] M. Bertero and P. Boccacci 2000 Image restoration for Large Binocular Telescope (LBT) Astron. Astrophys. Suppl. Ser. 147 pp [3] D. Bini and M. Capovani 1983 Spectral and computational properties of band symmetric Toeplitz matrices Linear Algebra Appl. 52/53 pp [4] R. H. Chan and M. Ng 1996 Conjugate gradient methods for Toeplitz systems SIAM Rev. 38 pp [5] M. Donatelli, C. Estatico, J. Nagy, L. Perrone, and S. Serra Capizzano 2003 Anti-reflective boundary conditions and fast 2D deblurring models Proceeding to SPIE s 48th Annual Meeting, San Diego, CA USA, F. Luk Ed, 5205 pp [6] M. Donatelli and S. Serra Capizzano 2003, Anti-reflective boundary conditions and reblurring. Inverse Problems, in press. [7] H. Engl, M. Hanke, and A. Neubauer 1996 Regularization of Inverse Problems Kluwer Academic Publishers, Dordrecht, The Netherlands. [8] G. Golub and C. Van Loan 1983 Matrix Computations The Johns Hopkins University Press, Baltimore. [9] N. Kalouptsidis, G. Carayannis, and D. Manolakis 1984 Fast algorithms for block Toeplitz matrices with Toeplitz entries Signal Process. 6 pp [10] J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello 1999 Three-Dimensional Imaging by Deconvolution Microscopy METHODS 19 pp [11] M. Ng, R. H. Chan, and W. C. Tang 1999 A fast algorithm for deblurring models with Neumann boundary conditions SIAM J. Sci. Comput. 21 pp [12] S. Serra Capizzano 2003 A note on anti-reflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput pp [13] S. Serra Capizzano and E. Tyrtyshnikov 1999 Any circulant-like preconditioner for multilevel matrices is not superlinear, SIAM J. Matrix Anal. Appl., 22-1, pp