Dealing with Boundary Artifacts in MCMC-Based Deconvolution

Transcription

1 Dealing with Boundary Artifacts in MCMC-Based Deconvolution Johnathan M. Bardsley Department of Mathematical Sciences, University of Montana, Missoula, Montana. Aaron Luttman National Security Technologies, LLC, Las Vegas, Nevada. Abstract Many numerical methods for deconvolution problems are designed to take advantage of the computational efficiency of spectral methods, but classical approaches to spectral techniques require particular conditions be applied uniformly across all boundaries of the signal. These boundary conditions traditionally periodic, Dirichlet, Neumann, or related are essentially methods for generating data values outside the domain of the signal, but they often lack physical motivation and can result in artifacts in the reconstruction near the boundary. In this work we present a data-driven technique for computing boundary values by solving a regularized and well-posed form of the deconvolution problem on an extended domain. Further, a Bayesian framework is constructed for the deconvolution, and we present a Markov chain Monte Carlo method for sampling from the posterior distribution. There are several advantages to this approach, including that still takes advantage of the efficiency of spectral methods, that it allows the boundaries of the signal to This work was done by National Security Technologies, LLC, under Contract No. DE-AC52-6NA25946 with the U.S. Department of Energy and supported by the Site Directed Research and Development program. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. DOE/NV/25946???? addresses: bardsleyj@mso.umt.edu (Johnathan M. Bardsley), LuttmaAB@nv.doe.gov (Aaron Luttman) Preprint submitted to Linear Algebra and Its Applications September 3, 13

2 be treated in a non-uniform manner thereby reducing artifacts and that the sampling scheme gives a natural method for quantifying uncertainties in the reconstruction. Keywords: imaging, deconvolution, inverse problems, boundary conditions, Bayesian methods, Markov chain Monte Carlo MSC: 15A29, 65F22, 65C5, 65C6, 94A8 1. Introduction Many applications in image and signal processing are modeled using convolution: b(s) = a(s t)x(t)dt, s Ω, (1) Ω where b corresponds to the measured data, defined on Ω R d, which in image processing is commonly referred to as the field of view (FOV); x corresponds to the unknown object, which is to be recovered and is defined on the extended domain Ω containing Ω; and a is the known convolution kernel, also called the system response or point spread function (PSF). In this manuscript, the dimension d is either 1 or 2. For a typical kernel a, the values of b(s) for s near the boundary of Ω, Ω, will depend upon values of x(t) for t / Ω but near Ω. Thus in order for (1) to be accurate in general, Ω must contain Ω as well as the spatial locations t / Ω for which x(t) has influence on b(s) for s Ω. Nonetheless, it is standard practice to make assumptions about the values of x(t) for t / Ω so that Ω = Ω in (1). These assumptions are called boundary conditions in the imaging literature [7]. They are made for computational reasons, as they allow for the use of highly efficient spectral methods, but they also result in unrealistic artifacts in the estimates of x(t), for t Ω near Ω, when they are inaccurate. We describe a few of the most common boundary conditions and their numerical advantages in Section 2, but in each case a single condition and its associated assumptions are taken uniformly across the entire boundary of the signal. The assumptions often aren t accurate, with the result that boundary artifacts appear in the computed estimates of x. In this work, we present a simple and computationally efficient alternative for constructing data-driven boundary conditions by solving the deconvolution problem on an extended domain, based on work developed in [4]. The approach has the 2

3 advantage that no explicit assumptions are made about the values of x(t) for t Ω, hence minimizing boundary artifacts, but this comes at the expense of solving a severely underdetermined inverse problem. We overcome this by assuming a prior on x and sampling from the corresponding posterior distribution. In Section 3, we develop a Markov chain Monte Carlo method for the sampling and for using the results to quantify uncertainties in the signal reconstruction. In Section 4, the method is demonstrated on applications in 1D and 2D. In both cases, the applications come from image deblurring. The 1D example uses real data used for density calibrations in quantitative X-ray imaging. The accuracy of such calibrations depends fundamentally on deblurring the measured images, and the ability to quantify uncertainties associated with the deblurring process are necessary to subsequently estimate errors in the density calibrations. In our 2D example, as is usually the case in imaging, the image boundaries vary significantly in intensity and uniform boundary conditions are unnatural. A data-driven approach to dealing with the deblur near the boundaries can show great advantages. In both cases, enough samples are computed to adequately describe the posterior distributions and estimate uncertainties in the reconstructions. 2. Boundary Conditions and Structured Matrix Computations In practice, signals are measured at discrete points, and these measurements contain random errors. It is, therefore, typical to work with a numerically discretized version of (1) containing an additive error term: b = Ax + η, (2) where x R n and b R m are discretizations of x, b; A is the real-valued m n matrix that arises from approximating integration by a quadrature rule; and η N (, λ 1 I), which is to say that η is an independent and identically distributed (iid) Gaussian random vector with mean zero and variance λ 1 (or precision λ) across all pixels. To look in more detail into the structure of (2), we first consider 1D convolution, which after discretization of (1), is given by b i = i+k j=i k a i j x j = k a j x j+i, for i = 1,..., m, (3) j= k 3

4 or in the matrix-vector notation of (2), a b k a a k 1 b 2. = b m a k a a k a k a a k a k a a k x k+1. x 1. x ṃ.. (4) x m+k In this formulation, the discrete PSF {a j } k j= k is assumed to be known, either by modeling or direct measurement. Since, like b, it can be measured, it has its own predetermined size (2k+1) 1; we do not assume that it is symmetric about a unless otherwise stated. It is evident from (4) that the values of b i for i near 1 and m will depend upon the elements (x k+1,..., x ) and (x m+1,..., x m+k ), respectively, all of which lie outside of the FOV. Note that (4) has the form of (2) with n = m + 2k and hence is an underdetermined problem. As stated above, the standard approach for dealing with this issue is to make assumptions about the values of x outside of the FOV based on a priori knowledge or by relating the values to those within the FOV. Periodic boundary conditions correspond to (x k+1,..., x ) = (x m k+1,..., x m ) and (x m+1,..., x m+k ) = (x 1,..., x k ); Neumann boundary conditions correspond to a reflection of the signal about the boundaries, i.e.,(x k+1,..., x ) = (x k,..., x 1 ) and (x m+1,..., x m+k ) = (x m,..., x m k+1 ); and a zero (or Dirichlet) boundary condition corresponds to the assumption that (x k+1,..., x ) = (x m+1,..., x m+k ) =. In each case, the resulting linear system (2) becomes m m with x = (x 1,..., x m ), which is the unknown restricted to the FOV. One of the primary reasons for choosing one of the above boundary conditions is that there are efficient spectral methods that can be used to solve the resulting deconvolution problem. Periodic boundary conditions result in a circulant matrix A that can be diagonalized by the Discrete Fourier Transform (DFT) [7, 15]. With Neumann boundary conditions, the resulting matrix A has Toeplitz-plus-Hankel structure, and, if the convolution kernel a is symmetric, i.e., a i = a i, then A can be diagonalized by the discrete cosine transform (DCT) [11, Theorem 3.2]. Dirichlet boundary conditions give a Toeplitz matrix A [7, 15], and this can be embedded in a circulant matrix that can be diagonalized by the DFT. 4

5 In the two-dimensional (2D) case, b and x are obtained by columnstacking the M M array B and the N N array X, denoted b = vec(b) and x = vec(x), and A is determined by the K K convolution kernel a = {a ij } K i,j= K. The discretization of (1) in this setting yields the 2D discrete convolution equation b r,s = r+k s+k i=r K j=s K a r i,s j x ij, for r, s = 1,..., M. (5) Note then that X is N N with N = M + 2K. After stacking the columns of the two dimensional arrays defined by (5), the resulting system of linear equations is of the form (2) with n = N 2 and m = M 2, which is once again under-determined. (Here we have assumed that B and X are square, for the simplicity of notation, but all of the theory and algorithms apply to non-square signals as well, with the appropriate bookkeeping of indices.) As in the 1D case, boundary conditions can be used to turn (5) into an M 2 M 2 system of equations. The 2D versions of the zero, periodic, and Neumann boundary conditions, are represented, respectively, by the following extensions of X [7]: X, X X X X X X X X X, and X vh X h X vh X v X X v X vh X h X vh, where X v is the image that results from flipping X across its central vertical axis; X h is the image that results from flipping X across its central horizontal axis; and X vh is the image that results from flipping X across its vertical then horizontal axes. In all three cases, the central X corresponds to the unknowns within the FOV, and again the primary advantage of these assumptions is that there are efficient spectral methods, involving the 2D discrete Fourier and cosine transforms, that can be used to solve the corresponding deconvolution problems. The drawbacks to these boundary conditions are that the associated efficient computational methods are only applicable if the condition is applied uniformly to all boundaries and that these particular boundary conditions often lack physical motivation. Thus we must balance the gain of the computational efficiency against the losses of flexibility in how the boundaries are treated. 5

6 2.1. A Simple Alternative to Boundary Conditions Assumptions A simple alternative to enforcing boundary conditions on Ω was suggested in [4]. The basic idea in 1D is to solve the problem on the extended FOV represented in (4) by x = [x k+1,..., x, x 1,..., x m, x m+1, x m+k ] T. Rather than use a boundary condition to reduce the number of unknowns in x to m, i.e. those within the FOV, we instead zero pad the PSF as follows: â = [,...,, a }{{} k,..., a,..., a k,,..., ] }{{} m+2k. (m 1)/2 (m 1)/2 (Here we have assumed m is odd for the ease of notation, but, again, the results carry over, with appropriate modifications to indices, in the case that m is even.) Then, assuming a periodic boundary condition on x, we obtain an (m + 2k) (m + 2k) matrix Â. We then restrict Âx to the central m elements, i.e. those within the FOV, to obtain the model b = DÂx. (6) Then D is the m (m + 2k) matrix whose ith row is equal to row k + i of I m+2k, the (m + 2k) (m + 2k) identity matrix. Moreover, if restrict x to its central m elements, i.e. those within the FOV, the possible boundary artifacts are removed. Similar arguments in 2D also yield (6), except in that case  is diagonalizable by the 2D-DFT and the system is N 2 N 2, where N = M +2K. Note that multiplication by both D and  is extremely efficient and require low storage. However, (6) is underdetermined and cannot be solved directly using a spectral method. The focus of the remainder of this paper is on solving the under-determined linear system (6). A truncated iterative approach was taken in [4], where the Richardson-Lucy iteration was use, and in [14], where the Landweber iteration was used. Here instead we take a Bayesian approach and define a prior on x, which makes the resulting maximum a posteriori (MAP) estimation problem over-determined and well-posed. Moreover, the Bayesian formulation of the problem allows us to quantify uncertainties in our estimates both of x and of the regularization parameter by sampling from the posterior density function. 6

7 3. The Bayesian Solution of the Problem In this section, we formulate a Bayesian solution to (6). In order to simplify notation, we remove from  and add random noise to obtain b = DAx + η, (7) where in 1D, b, η R m, D R m n with n = m + 2k, and A R n n ; while in 2D b, η R M 2, D R M 2 N 2 with N = M +2K, and A R N 2 N 2. Then, given our assumption that η is an iid Gaussian random vector with variance λ 1, the probability density function for (7) is given by p(b x, λ) exp ( λ2 ) DAx b 2, (8) where denotes proportionality, and denotes the Euclidean norm. Computing the maximizer of the likelihood L(x b, λ) = p(b x, λ) is not a well-posed problem. The standard technique for overcoming this for inverse problems is regularization, which, in the context of Bayesian statistics, corresponds to the choice of the prior probability density function p(x δ), where δ > is another precision parameter. In our case, we use a Gaussian Markov random field (GMRF) to model the prior [2], which yields ( p(x δ) exp δ ) 2 xt Lx, (9) where the precision matrix δl is sparse, symmetric, and positive definite. We build L by assuming independent Gaussian increments (local differences), as in [2]. In 1D, we model local differences as follows: x i+1 x i N (, (w i δ) 1 ), i = 1,..., n, where a periodic boundary condition implies that x n+1 = x 1. This yields (see [2] for details) L = D T WD, where D is the forward difference derivative matrix with periodic boundary conditions, and W = diag(w 1,..., w n ). In 2D, we model local vertical and horizontal differences as follows: x i+1,j x i,j, x i,j+1 x i,j N (, (w ij δ) 1 ), i, j = 1,..., N, where, again, a periodic boundary condition is assumed. This yields (see [2] for details) L = D T v WD v + D h WD h, where D v = D I and D h = I D, 7

8 with D the derivative matrix used in the 1D case, denoting the Kronecker product, and W = diag(vec ( {w ij } N i,j=1). Bayes Theorem then states that the posterior probability density function p(x b, λ, δ) can be expressed as p(x b, λ, δ) p(b x, λ)p(x δ) ( exp λ 2 DAx b 2 δ ) 2 xt Lx. (1) Maximizing (1) with respect to x yields the MAP estimator. We can equivalently minimize the negative-log of (1), given by ln p(x b, λ, δ) 1 2 DAx b 2 + α 2 xt Lx, = 1 [ ] [ ] DA b 2 2 (αl) 1/2 x. (11) where α = δ/λ is the r egularization parameter from classical inverse problems [15]. Note that (11) shows that the MAP estimator is the least squares solution of an over-determined linear system, of dimension (m + n) n in 1D and (M 2 + N 2 ) N 2 in 2D Sampling from the Posterior Density Function In order to compute an estimate for the maximizer of (1), we assume Gamma distributed hyper-priors on λ and δ and use the sampling scheme of [1]. Thus p(λ) λ α λ 1 exp( β λ λ), (12) p(δ) δ α δ 1 exp( β δ δ), (13) with α λ = α δ = 1, and β λ = β δ = 1 4, which have mean and variance α/β = 1 4 and α/β 2 = 1 8, respectively. Note that α = 1 yields exponentially distributed hyper-priors, however we present the full Gamma hyper-prior here because it is conjugate to the Gaussian distribution, making sampling straightforward, and other choices for α and β may be advantageous in other situations. Given the large variance values, the hyper-priors should have a negligible effect on the sampled values for λ and δ. 8

9 The posterior probability density then has the form p(x, λ, δ b) p(b x, λ)p(λ) p(x δ)p(δ) = λ n/2+α ( λ 1 δ (n 1)/2+αδ 1 exp λ 2 DAx b 2 δ ) 2 xt Lx β λ λ β δ δ. (14) The prior and hyper-priors were chosen because they are conjugate densities [6], by which we mean that the full conditional densities have the same form as the corresponding prior/hyper-prior; specifically, note that x λ, δ, b N ( (λa T D T DA + δl) 1 λa T D T b, (λa T D T DA + δl) 1),(15) λ x, δ, b Γ (m/2 + α λ, 12 ) DAx b 2 + β λ, (16) δ x, λ, b Γ ((n 1)/2 + α δ, 12 ) xt Lx + β δ, (17) where N and Γ denote Gaussian and Gamma distributions, respectively. The power in (15)-(17) lies in the fact that samples from these three distributions can be computed using standard statistical software, and a Gibbsian approach can be applied to (15)-(17) yielding the following MCMC method of [1] for sampling from (14): MCMC Method for Sampling from p(x, δ, λ b).. Initialize δ, and λ, and set k = ; 1. Compute x k N ( (λ k A T D T DA + δ k L) 1 λ k A T D T b, (λ k A T D T DA + δ k L) 1) ; 2. Compute λ k+1 Γ (m/2 + α λ, 12 ) DAxk b 2 + β λ δ k+1 Γ ((n 1)/2 + α δ, 12 ) (xk ) T Lx k + β δ ; 3. Set k = k + 1 and return to Step 1. The computational bottleneck in this MCMC method is Step 1. However, we can efficiently compute the sample x k by first generating new random data 9

10 ˆb N (b, λ 1 I) and ĉ N (, (δl) 1 ) and then by solving the least squares problem [ ] x k λ 1/2 2 k (DAx ˆb) = arg min x δ 1/2 k L 1/2 (x ĉ) { λk = arg min x 2 DAx ˆb 2 + δ } k 2 L1/2 (x ĉ) 2. (18) To see that x k has the correct probability density, note that the solution of the normal equations for (18) is given by x k = (λ k A T D T DA + δ k L) 1 (λ k A T D T ˆb + δk L 1/2 ĉ) = (λ k A T D T DA + δ k L) 1 λ k A T D T b + w, (19) where w N(, (λ k A T D T DA + δ k L) 1 ), which agrees with the distribution in Step 1. The solution of (18) (or equivalently, (19)), and hence the sample in Step 1, is computed directly in 1D, whereas in 2D the preconditioned conjugate gradient (PCG) algorithm is used with the preconditioner M = λ k A T A + δ k L, which can be diagonalized by the 2D-DFT in our examples. For determining convergence of the MCMC chain, we use the approach described in [1, 6], which monitors a statistic ˆR. Once ˆR is sufficiently near 1 for all sampled parameters, the samples from the last half of each of the MCMC chains are treated as samples from the target distribution. In [6], an ˆR threshold of 1.1 is deemed acceptable. For more detail on this convergence diagnostic, see [1, 6]. Finally, we note that the MCMC method yields sample distributions for x, λ and δ. Thus we can obtain an estimates of x, λ and δ by computing, for example, the sample mean or median, and we emphasize that a classical regularization parameter selection is not needed in this formulation. Moreover, the samples also provide a means of quantifying uncertainty in the estimates of x, λ, and δ. For example, we can compute sample variance or create histograms of sampled parameters, both of which are done in the numerical experiments that follow Poisson Noise Another noise distribution that commonly appears in imaging applications is the Poisson model. In this case, rather than (2), we have the statistical model b = Poisson(Ax + γ), () 1

11 where γ is the m 1 vector of background counts or the so-called dark field and is assumed known. The probability density function for () is given by ( ) n 2 p(b x) exp {([Ax] j + γ j ) + b j ln([ax] j + γ j )}, (21) j=1 where denotes proportionality, and this can be approximated by ln p(b x) 1 2 C 1/2 (DAx (b γ)) 2 + O( h 3 2, h 2 2 k 2, h 2 k 2 2, k 2 2), where denotes equal up to an additive constant, C = diag(b), h = x x true, and k = b DAx true [3]. Thus, we use the approximate likelihood function as follows: ( p(b x) exp 1 C 1/2 (DAx (b γ)) ) 2. (22) 2 Using this likelihood function leads to the MAP estimation problem { 1 x δ = arg min x 2 C 1/2 (DAx (b γ)) 2 + δ } 2 xt Lx. (23) Notice that the noise precision λ is not present in this case. This is due to the fact that [C] ii = b i provides an approximate of the variance of the data at the ith pixel. Thus only α = δ must be estimated, which can be done as above using, for example, the discrepancy principle [15]. Assuming a gamma hyper-prior for δ, as above, leads to the posterior density function p(x, δ b) p(b x) p(x δ)p(δ) = δ (n 1)/2+αδ 1 (24) ( exp 1 2 C 1/2 (DAx (b γ)) 2 δ ) 2 xt Lx β δ δ. Note that only x and δ need to be sampled in this MCMC method, which can be derived as in the Gaussian case. MCMC Method for Sampling from p(x, δ b). 11

12 . Initialize δ and set k = ; 1. Compute ˆb N (b, C), ĉ N (, (δl) 1 ), and then x k = arg min x { 1 2 C 1/2 (DAx (ˆb γ)) 2 + δ k 2 L1/2 (x ĉ) 2 }. 2. Compute δ k+1 Γ ((n 1)/2 + α δ, 12 ) (xk ) T Lx k + β δ ; 3. Set k = k + 1 and return to Step 1. Once again, in 1D, we compute x k in Step 1 directly, whereas in 2D we use conjugate gradient (CG) iterations. Because of the presence of C in the optimization problem, an efficient preconditioner for CG does not exist, however C can be viewed as a preconditioner of sorts, as it seems to accelerate the convergence of CG. 4. Numerical Experiments In this section we present the results obtained by applying our framework to both real and synthetic examples in 1D and 2D One Dimensional Examples We begin with a comparison of the results of our method to the results obtained using classical boundary conditions, calculated on synthetic data. This is followed by an example of real data from X-ray radiography, and the results of the sampling with an anisotropic prior and the data-driven boundary conditions are presented One-Dimensional Example for Comparing Boundary Conditions We begin with a synthetic 1D deconvolution problem in order to illustrate how the various boundary conditions work. Consider the 1D convolution model b(s) = 1 A(s s )x(s )ds, 21/1 s /1, 12

13 with a Gaussian convolution kernel A(s) = exp( s 2 /(2γ 2 ))/ πγ 2, γ >. Then, discretizing the integral using mid-point quadrature yields the matrix A defined by [A] ij = h exp ( ((i j)h) 2 /(2γ 2 ) ) / πγ 2, 1 i, j n, (25) where h = 1/n with n the number of grid points in [, 1]. We use n = 1 and then restrict to the central 8 grid points to obtain the vector representation, b, of b on [21/1, /1]. Thus the model has the form b = DAx, where D is the matrix obtained by extracting rows 21 through of the 1 1 identity matrix. The true image x used to generate the data is plotted on the upper-left in Figure 1, as is the data b generated using noise model (2) with the noise variance λ 1 chosen so that the noise strength is 2% that of the signal strength. 5 5 true image extended image BC Neumann BC periodic BC Dirichlet BC true image extended image BC Neumann BC periodic BC Dirichlet BC (a) true image extended image BC Neumann BC periodic BC Dirichlet BC (b) (c) (d) Figure 1: The one-dimensional image used to generate the data and blurred noisy data (a), and a comparison of classical boundary conditions to the method presented here (b). Images (c) and (d) show zoomed in views of the left and right boundaries, respectively. 13

14 On the upper-right in Figure 1 is a plot of the MAP reconstruction of x obtained by minimizing (11) and using the four different choices of boundary conditions. For regularization, we used L = D T D and a fixed regularization parameter α = δ/λ chosen by hand. The lower plots in Figure 1 are magnifications of the upper-right plot near the left and right boundary of the FOV, in the left and right plots respectively. Note that our approach is the only one that retains the model matrix DA; the others yield a different A based on the boundary conditions assumption. The results are as one would expect: the worse the boundary condition assumption for the specific example, the worse the boundary artifacts. Listed from worst to best (measured by nearness to the true image near the boundaries) for this example, the boundary conditions are: Dirichlet, periodic, Neumann, and our extended domain approach. Note that the extended domain boundary condition yields only slightly better results than the Neumann boundary condition, but only on the right hand side, where the Neumann (or reflective) boundary condition is not quite correct, however, the difference is quite small A One-dimensional Example in X-ray Radiography, Poisson Approximation Most applications of imaging are qualitative, in that the goal is to produce image reconstructions that look as good as possible. In the security sciences, however, pulsed power X-ray radiography is a quantitative imaging diagnostic, used to calculate the true locations of image features in 3D space or to calculate the densities of the objects being imaged [1, 16, 9, 13]. In these applications it is often possible to accurately determine the sources and magnitude of noise, and such images are often dominated by Poisson noise due to particle counting on the CCD. In this case, it is natural to use the Poisson formulation detailed in Section 3.2. Figure 2 (a) shows an image taken from a pulsed power X-ray radiography system, and the object in the scene is a so-called step wedge, which is used to determine the X-ray transmission of the system. The object consists of a single material, with steps of different thicknesses. Thicker regions of the object correspond to darker regions of the image. A single vertical crosssection is extracted from the image (or several image columns are averaged together), and the resulting curve must be de-blurred before the transmission can be accurately determined. (See [8] for a simulation-based approach to computing transmission curves. Experimental approaches for high energy X-ray systems are largely undocumented in the literature.) 14

15 25 δ, the prior precision 15 5 (a) (b) 1 lineout MCMC sample mean 95% credibility bounds (c) Figure 2: Image (a) shows the radiograph of a so-called step wedge used for object density reconstruction. The goal is distinguish among the darkest shades of gray. In (c), on the bottom, is shown a vertical lineout (cross section) from the image, with the mean of the deblurred samples and 95% credibility bands. Image (b) gives the histogram of precision parameter, δ, samples. 15

16 Given the nature of the data, it is undesirable to use the isotropic smoothness prior L = D T D. Hence we first compute the MAP reconstruction x δ defined by (23) with L = D T D and δ chosen using the discrepancy principle [15], and then use x δ to define an anisotropic smoothness prior as follows: ( ) L = D T 1 WD, but with W = diag (Dx DP ) This prior precision matrix is designed to pick up the edge information, since it allows for large increments (or local differences) where the derivative of the signal is large. The reconstruction shown in Figure 2 (b) is the mean of the samples computed using the MCMC scheme in Section 3.2. While the mean of the samples does not pick up the edge information clearly, the 95% credibility bands do show the step geometry of the object. Further, there are no discernible boundary artifacts in the reconstruction, due to the approach presented here. Figure 2 (c) shows the histogram of the samples of the precision (or regularization) parameter δ. The discrepancy principle value of the parameter was.49; the mean of the samples is.8 with an approximately single peaked distribution Two-Dimensional Examples Finally, we consider a two dimensional image deconvolution test case, in which the mathematical model is of the form b(s, t) = 3/2 3/2 1/2 1/2 a(s s, t t )x(s, t )ds dt, s, t 1. We discretize using mid-point quadrature on a uniform computational grid over [-1/2,3/2] [-1/2,3/2]. Then the vector x is while b is , since b is defined over [, 1] [, 1]. This yields a system of linear equations b = DAx; note that D is the discretization of the indicator function on [, 1] [, 1]. We assume that x is a periodic function on [-1/2,3/2] [-1/2,3/2], so that A has block circulant with circulant block structure and can be diagonalized by the 2D-DFT for low storage requirements and efficient matrix-vector multiplication Gaussian Noise, Synthetic Data Test Case The data b is generated using (7) with the noise variance λ 1 chosen so that the noise strength is 2% that of the signal strength. In order to 16

17 Figure 3: On the left is the two-dimensional image used to generate the data, and on the right is the blurred data corrupted by Gaussian noise. obtain the noise-free data DAx, we begin with an extended true image, compute 2D discrete-convolution (5) assuming periodic BCs, and then restrict to the central sub-image to obtain DAx. The central region of the image used to generate the data and the data b are shown in Figure 3. We reconstruct the image by sampling from the posterior density function p(x, λ, δ b) defined in (14) using the MCMC method described in Section 3.1. We computed 4 parallel MCMC chains and reached an R value of 1.5 when the length of the chains was 1.The initial values for the chains were δ =.1 and λ = 5. We plot the mean of the sampled images, with negative values set to zero, as the reconstruction on the upper-left in Figure 4; it had a relative error of.2. We also reconstruct x using MAP estimation with α = δ/λ = chosen using GCV and plot it in the upper-right in Figure 4; it had a relative error of x α x true / x true =.985. From the samples for λ and δ, on the lower-left in Figure 4, we plot histograms for λ, δ, and the regularization parameter α = δ/λ, which has a 95% credibility interval [ , ]. Note that the noise precision used to generate the data, λ = is contained in the 95% credibility interval for the λ samples: [3.139, 3.29]. And finally, we plot the pixel-wise standard deviation of the samples for x in the lower-right in Figure Poisson Noise, Synthetic Data Test Case Next, we use the same synthetic data example, but with the Poisson noise model (). We use the Gaussian approximation (22) so that the required 17

18 δ, the prior precision λ, the noise precision x α=δ/λ, the regularization parameter x Figure 4: Reconstructions, Gaussian Test Case. On the upper-left is the sample mean of the x samples computed using the MCMC method. On the upper-right is the MAP reconstruction with α chosen using GCV. On the lower-left are the histograms of the α and δ samples, as well as of the corresponding α = λ/δ. And finally, on the lower right is the pixel-wise standard deviation computed from the samples for x. 18

19 Figure 5: On the left is the two-dimensional image used to generate the data, and on the right is the blurred data corrupted by Poisson noise. large-scale minimization tasks are all quadratic, and hence only require an application of CG. The data b in the Poisson case corresponds to counts, so we increase the intensity of the true image from the previous example. The noise-free data DAx is obtained as in the Gaussian case. The central region of the image used to generate the data and the data b are shown in Figure 5. We reconstruct the image by sampling from the posterior density function p(x, δ b) defined in (14) using the MCMC method described in Section 3.2. We computed 4 parallel MCMC chains and reached an R value of 1.5 when the length of the chains was 13.The initial values for the chains were δ =.1. We plot the mean of the sampled images, with negative values set to zero, as the reconstruction on the upper-left in Figure 6; it had a relative error of We also reconstruct x using MAP estimation with δ = chosen using GCV. The reconstruction is given in the upperright in Figure 6 and it had a relative error of On the lower-left in Figure 6 we plot a histogram for the samples of δ, which has a 95% credibility interval [ , ]. And finally, we plot the pixel-wise standard deviation of the samples for x in the lower-right in Figure 6, which is smaller in the regions of the true image of lower intensity. 5. Conclusions We have presented an MCMC sampling scheme with a data-driven approach to dealing with boundary artifacts in deconvolution problems. The 19

20 δ, the prior precision x Figure 6: Reconstructions, Poisson Test Case. On the upper-left is the sample mean of the x samples computed using the MCMC method. On the upper-right is the MAP reconstruction with α chosen using GCV. On the lower-left are the histograms of the α and δ samples, as well as of the corresponding α/δ. And finally, on the lower right is the pixel-wise standard deviation computed from the samples for x.

21 method samples the unknown image x defined on an extended domain and then restricts to the field of view (FOV) of the imaging instrument, thus removing any boundary artifacts. The approach retains computational efficiency by assuming a periodic boundary condition on the extended domain. The resulting model is no longer diagonalizable by a fast transform, but efficient iterative methods nonetheless exist for its solution, and direct methods can be used in 1D cases. The MCMC method samples the unknown object x, as well as the noise precision λ and prior precision δ, making regularization parameter selection unnecessary. Moreover, in the Poisson noise case, we use a Gaussian approximation of the negative-log Poisson likelihood to extend our framework. In this case, there is no λ parameter and hence we only sample x and δ. The prior is defined by a Gaussian Markov random field (GMRF), which in all but one of the numerical experiments corresponds to the standard scaled negative-laplacian precision matrix δl. In the one experiment involving real data, however, an anisotropic smoothness prior was needed in order to obtain good results. In each of the numerical experiments presented, the sampling scheme worked well and boundary artifacts were negligible in the associated reconstructions. 6. Acknowledgements The authors would like to thank the U.S. Department of Energy Nevada Radiography Working Group for providing the X-ray radiography data. 7. References [1] J. M. Bardsley, MCMC-Based Image Reconstruction with Uncertainty Quantification, SIAM J. Sci. Comput., vol. 34, no. 3, 12, pp. A1316- A1332. [2] J. M. Bardsley, Gaussian Markov Random Field Priors for Inverse Problems, Inverse Problems and Imaging, vol. 7, no. 2, 13, pp [3] J. M. Bardsley and J. Goldes, Regularization Parameter Selection Methods for Ill-Posed Poisson Maximum Likelihood Estimation, Inverse Problems, 25 (9)

22 [4] M. Bertero and P. Boccacci, A simple method for the reduction of boundary effects in the Richardson-Lucy approach to image deconvolution, Astronomy and Astrophysics, 437 (5), pp [5] J. Besag, Spatial Interaction and the Statistical Analysis of Lattice Systems, Journal of the Royal Statistical Society, Series B, 36(2) (1974), pp [6] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, Second Edition, Chapman & Hall/CRC, Texts in Statistical Science, 4. [7] P. C. Hansen, J. Nagy, and D. O Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, Philadelphia, 6. [8] G. Hoff, S. F. Firmino, R. M. Papaleo, M. T. de Vilhena, Estimating Transmission Curves of Primary X-Ray Beams Used in Diagnostic Radiology, IEEE Transactions on Nuclear Science, 59(2) (12), pp , DOI:1.119/TNS [9] T. A. Kelley and D. M. Stupin, Radiographic Least Squares Fitting Technique Accurately Measures Dimensions and X-Ray Attenuation, Review of Progress in Quantitative Nondestructive Evaluation, (1998), pp [1] D. Mosher, R. Commisso, G. Cooperstein, S. Stephanakis, S. Swanekamp, and F. Young,Rod-Pinch X-Radiography for Diagnosis of Material Response, American Physical Society, 42nd Annual Meeting of the APS Division of Plasma Physics combined with the 1th International Congress on Plasma Physics, October 23-27, Quebec City, Canada Meeting ID: DPP, abstract #PO2.1. [11] M. K. Ng, R. H. Chan and W. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21(3) (1999), pp [12] H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, Chapman and Hall/CRC, 5. 22

23 [13] J. Smith,R. Carlson, et al., Cygnus Dual Beam Radiography Source, Pulsed Power Conference, 5 IEEE, pp.334,337, June 5, doi: 1.119/PPC [14] R. Vio, J. Bardsley, M. Donatelli, and W. Wamsteker, Dealing with edge effects in least-squares image deconvolution problems, Astronomy and Astrophysics, 442 (5), pp [15] C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2. [16] R. L. Whitman, H. M. Hanson, and K. A. Mueller, Image Analysis for Dynamic Weapons Systems, Los Alamos Report LALP-85-15,