DEBLURRING AND SPARSE UNMIXING OF HYPERSPECTRAL IMAGES USING MULTIPLE POINT SPREAD FUNCTIONS G = XM + N,

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1 DEBLURRING AND SPARSE UNMIXING OF HYPERSPECTRAL IMAGES USING MULTIPLE POINT SPREAD FUNCTIONS SEBASTIAN BERISHA, JAMES G NAGY, AND ROBERT J PLEMMONS Abstract This paper is concerned with deblurring and spectral analysis of ground-based astronomical images of space objects A numerical approach is provided for deblurring and sparse unmixing of ground-based hyperspectral images (HSI) of objects taken through atmospheric turbulence Hyperspectral imaging systems capture a 3D datacube (tensor) containing: 2D spatial information, and D spectral information at each spatial location Pixel intensities vary with wavelength bands providing a spectral trace of intensity values, and generating a spatial map of spectral variation (spectral signatures of materials) The deblurring and spectral unmixing problem is quite challenging since the point spread function (PSF) depends on the imaging system as well as the seeing conditions and is wavelength varying We show how to efficiently construct an optimal Kronecker product-based preconditioner, and provide numerical methods for estimating the multiple PSFs using spectral data from an isolated (guide) star for joint deblurring and sparse unmixing the HSI datasets in order to spectrally analyze the image objects The methods are illustrated with numerical experiments on a commonly used test example, a simulated HSI of the Hubble Space Telescope satellite Key words image deblurring, hyperspectral imaging, preconditioning, least squares, ADMM, Kronecker product AMS Subject Classifications: 65F2, 65F3 Introduction Information about the material composition of an object is contained most unequivocally in the spectral profiles of the brightness at the different surface pixels of the object Thus by acquiring the surface brightness distribution in narrow spectral channels, as in hyperspectral image (HSI) data cubes, and by performing spectral unmixing on such data cubes, one can infer material identities as functions of position on the object surface [] Spectral unmixing involves the computation of the fractional contribution of elementary spectra, called endmembers By assuming the measured spectrum of each mixed pixel in an HSI is a linear combination of spectral signatures of endmembers, the underlying image model can be formulated as a linear mixture of endmembers with nonnegative and sparse coefficients G = XM + N, where M R Nm Nw represents a spectral library containing N m spectral signatures of endmembers with N w spectral bands or wavelengths, G R Np Nw is the observed data matrix (each row contains the observed spectrum of a given pixel, and we use N p to denote the number of pixels in each image), and X R Np Nm contains the fractional abundances of endmembers (each column contains the fractional abundances of a given endmember) Here, we assume X is a sparse matrix and N R Np Nw is a matrix representing errors or noise affecting the measurements at each spectral Department of Mathematics and Computer Science, Emory University sberish@emoryedu Department of Mathematics and Computer Science, Emory University nagy@mathcsemoryedu Research supported in part by the AFOSR under grant FA , and by the US National Science Foundation under grant no DMS-5627 Department of Computer Science, Wake Forest University, Winston-Salem, NC, USA plemmons@wfuedu His work was supported by grant no FA from the US Air Force Office of Scientific Research and by contract no HM582--C- from the US National Geospatial- Intelligence Agency

2 2 S BERISHA, J NAGY AND R PLEMMONS band or wavelength; see, eg [, 4] If we assume that the data at each wavelength has been degraded by a blurring operator, H, then the problem takes the form G = HXM + N Given a spectral library of the endmembers, M, and assuming that we have computed a priori the parameters defining the blurring operator H, the goal becomes to compute the nonnegative and sparse matrix of fractional abundances, X A major challenge for deblurring hyperspectral images is that of estimating the overall blurring operator H, taking into account the fact that the blurring operator point spread function (PSF) can vary over the images in the HSI datacube That is, the blurring can be wavelength dependent and depend on the imaging system diffraction blur as well as the effects of atmospheric turbulence blur on the arriving wavefront, see eg, [5, 9,, 2] We point especially to recent development of the hyperspectral Multi Unit Spectroscopic Explorer (MUSE) system installed on the Very Large Telescope (VLT) being deployed by the European Southern Observatory (ESO) at the Paranal Observatory in Chile The MUSE system will collect up to 4, bands, and research is ongoing to develop methods for estimation of the wavelength dependent PSFs for deblurring the resulting HSI datacube for ground-based astrophysical observations, see eg [9, 2] In particular, Soules, et al [] consider the restoration of hyperspectral astronomical data with spectrally varying blur, but assume that the spectrally varying PSF has already been provided by other means, and defer the PSF estimation to a later time In this paper we consider the estimation of a Moffat function parameterization of the PSF as a function of wavelength and derive a numerical scheme for deblurring and unmixing the HSI datacube using the estimated PSFs Moffat function parameterizations capture the diffraction blur of the telescope system as well as the blur resulting from imaging through atmospheric turbulence, and have been used quite effectively by astronomers for estimating PSFs; see, for example, the survey by Soulez et al [] This paper is outlined as follows In Section 2 we review a numerical approach for deblurring and sparse unmixing of HSI datacubes for the special case of a homogeneous PSF across the wavelength, based on work by Zhao et al [4] Section 3 concerns estimating the wavelength dependent PSFs to model the blurring effects of imaging through the atmosphere, and application of a numerical scheme for this multiple PSF case using a preconditioned alternating direction method of multipliers We show how to efficiently construct an optimal Kronecker product-based preconditioner to accelerate convergence In order to illustrate the use of our method, some numerical experiments on a commonly used test example, a simulated hyperspectral image of the Hubble Space Telescope satellite, are reported in Section 4 Some concluding comments are provided in Section 5 2 Numerical Scheme for the Single PSF Case In this section we describe and expand upon the numerical scheme used in [4] for solving the hyperspectral image deblurring and unmixing problem by using the Alternating Direction Method of Multipliers (ADMM) in the single PSF case Here, it is assumed that the blurring operator H is defined by a single PSF and each column of XM is blurred by the same H The authors in [4] have presented a total variation (TV) regularization method for solving the deblurring and sparse hyperspectral unmixing problem, which takes the form min X 2 HXM G 2 F + µ X + µ 2 T V (X)

3 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 3 where H R Np Np is a blurring matrix constructed from a single Gaussian function assuming periodic boundary conditions, and µ, µ 2 are two regularization terms used to control the importance of the sparsity and the total variation terms Numerical schemes for both isotropic and anisotropic total variation are presented in [4] For isotropic TV, the above problem can be rewritten as subject to min N 2 HXM p G 2 F + µ V + µ 2 N m i= j= W ij 2 (2) D h X = W (), D v X = W (2), V = X, V K = {V R Np Nm, V }, where W i,j = [ ] W () i,j, W (2) i,j R 2, W () i,j = D i,hx j, W (2) i,j = D i,vx j, i N p, j N m Here, the matrices D h and D v represent the first order difference matrices in the horizontal and vertical direction, respectively The authors in [4] solve the above problem using an alternating direction method The problem in (2) can be decoupled by letting f (X) = 2 HXM G 2 F and N p f 2 (Z) = X K (V ) + µ 2 N m i= j= W ij 2 + µ V where Z = W () W (2) V, X K = { if V K otherwise The constraints are expressed as BX + CZ = D h D v I Np N p X I 3Np 3N p W () W (2) = 3Np N m, V where we use I and to denote, respectively, an identity matrix and matrix of all zeros (subscripts on these matrices define their dimensions, and may be omitted later in the paper, if dimensions are clear from the context) Furthermore, by attaching the Lagrange multiplier to the linear constraints the augmented Lagrangian function of (2) is written as L(X, Z, Λ) = f (X) + f 2 (Z)+ < Λ, BX + CZ > + β 2 BX + CZ 2 F,

4 4 S BERISHA, J NAGY AND R PLEMMONS where Λ () Λ = Λ (2) R 3Np Nm, Λ (3) β > is the penalty parameter for violating the linear constraints, and <, > is the sum of the entries of the Hadamard product With this formulation the hyperspectral unmixing and deblurring problem is solved using an alternating direction method consisting of solving 3 subproblems at each iteration k: Step : X k+ arg min L(X, Z k, Λ k ) Step 2: Z k+ arg min L(X k+, Z, Λ k ) Step 3: Λ k+ Λ k + β(bx k+ + CZ k+ ) The X-subproblem, or Step, consists of solving X k+ arg min{ X 2 HXM G 2 F + < Λ, BX + CZ > + β 2 BX + CZ 2 F } (22) The above subproblem is the solution of the classical Sylvester matrix equation H T HXMM T + βb T BX = H T GM T βb T CZ k B T Λ k (23) Similar alternating minimization schemes have been used for solving the hyperspectral unmixing problem in [3] and [6] However, the key step of the alternating minimization scheme presented in [4] consists in transforming the matrix equation (23) to a linear system which has a closed-form solution In particular the authors in [4] reformulate the Sylvester matrix equation in (23) as where (MM T H T H + βi B T B)x = ĝ, x = vec(x) = vec( [ x x n ] ) = x x n, x i = ith column of X, and similarly, ĝ = vec(h T GM T βb T CZ k B T Λ k ) Let M = UΣV T be the singular value decomposition of M, and let H = F ΓF and B T B = F ΨF be the Fourier decomposition of H and B T B, respectively Here, we assume spatially invariant blur with periodic boundary conditions The above linear system takes the form (U F )(ΣΣ T Γ 2 + I Ψ 2 )(U T F )x = ĝ and thus a direct solution is given by x = (U F )(ΣΣ T Γ 2 + I Ψ 2 ) (U T F )ĝ For a detailed description of the solution of Steps 2 and 3 of the alternating minimization approach see [4]

5 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 5 3 Numerical Scheme for the Multiple PSF Case In this section we provide the problem formulation for the general case where each column of the matrix XM is generally blurred by a different blurring operator In particular, the deblurring and hyperspectral unmixing problem with multiple PSFs takes the form H H 2 XMe XMe 2 = g g 2 (3) H Nw XMe Nw g Nw where e i R Nw is the i th unit vector, g i is the i th column of the observed matrix G, and each blurring matrix H i is defined by different PSFs that vary with wavelength For example, in astronomical imaging, a blurring operator H for a particular wavelength, λ, might be accurately modeled with a circular Moffat function PSF(α, α, α 2, λ) = α 2 ( + i2 + j 2 ) α2 π(α + α λ) (α + α λ) 2 (32) Moffat functions are widely used to paramaterize PSFs in astronomical imaging The parameters α, α and α 2 are the Moffat function shape parameters for the associated PSF which are to be estimated from the data, see, eg [9] Notice that problem (3) can be rewritten as H XMe H 2 XMe 2 g = g 2 H Nw XMe Nw g Nw By utilizing Kronecker product properties the above equation can be reformulated as (e T M T H )x g (e T 2 M T H 2 )x g 2 (e T N w M T H Nw )x = where x = vec(x) Thus, the multiple PSF hyperspectral image deblurring and unmixing problem can be formulated as Hx = g g Nw, where H = m T H m T 2 H 2 m T N w H Nw g, x = vec(x), g = g 2 g Nw Hence, the X-subproblem (22) for the multiple PSF formulation takes the form { } X k+ arg min X 2 Hx g 2 F + < Λ, BX + CZ > +β 2 BX + CZ 2 F

6 6 S BERISHA, J NAGY AND R PLEMMONS That is, using Kronecker product properties and by applying the vectorizing operator, vec( ), X k+ is a solution of the { min x 2 Hx g 2 F +vec(λ)t ((I B) x + vec (CZ)) + β } 2 (I B) x + vec(cz) 2 F, where x = vec(x) Now, if we set the gradient of the augmented Lagrangian for the X-subproblem to then we obtain H T Hx + β ( I B T B ) x = H T g β ( I B T ) vec(cz k ) vec(b T Λ k ) The above equation can be rewritten as (H T H + βi B T B)x = ĝ, (33) where ĝ = H T g β ( I B ) T vec(cz k ) vec(b T Λ k ) Notice that m T H H T H = [ ] m H T m 2 H2 T m Nw HN T m T 2 H 2 w m T N w H Nw = m m T H T H + m 2 m T 2 H T 2 H m Nw m T N w H T N w H Nw Using the decompositions H T i H i = F Γ 2 i F and BT B = F Ψ 2 F equation (33) takes the form (m m T F Γ 2 F +m 2 m T 2 F Γ 2 2F + +m Nw m T N w F Γ 2 N w F +I F Ψ 2 F )x = ĝ Thus, the X-subproblem to be solved for the multiple PSF case is (I F )(m m T Γ 2 + m 2 m T 2 Γ m Nw m T N w Γ 2 N w + I Ψ 2 )(I F )x = ĝ Notice that the middle part of the coefficient matrix in the above linear system is not diagonal as in the single PSF case, and thus there is not an explicit solution of the X-subproblem for multiple PFSs However, the coefficient matrix for multiple PSFs is symmetric positive definite (spd) and thus we use the conjugate gradient method to solve the above subproblem Construction of a preconditioner is described next 4 Conjugate Gradient Preconditioner The X-subproblem involves the coefficient matrix (I F )(m m T Γ 2 + m 2 m T 2 Γ m Nw m T N w Γ 2 N w + I Ψ 2 )(I F ) Our goal is to approximate m m T Γ m Nw m T N w Γ 2 N w by one Kronecker product A D, where D is a diagonal matrix, and A is spd If we can find such an approximation, then we can compute A = UΣU T, and (I F )(m m T Γ 2 + m 2 m T 2 Γ m Nw m T N w Γ 2 N w + I Ψ 2 )(I F ) (I F )(A D + I Ψ 2 )(I F ) = (I F )(UΣU T D + I Ψ 2 )(I F ) = (U F )(Σ D + I Ψ 2 )(U T F )

7 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 7 4 Kronecker Product Approximation One, very simple approximation, can be obtained by replacing each Γ j by a single matrix, such as by an average of all diagonal matrices, Γ avg In this case we use A = MM T and D = Γ 2 avg However, it is difficult to determine the quality of using such a simple approach Therefore, we seek a different and possibly optimal approximation Consider m m T Γ m Nw m T N w Γ 2 N w = [ m T Γ ] m Γ m Nw Γ Nw where = CC T, C = [ ] m Γ m 2 Γ 2 m Nw Γ Nw m Γ m 2 Γ 2 m Nw Γ Nw m 2 Γ m 22 Γ 2 m 2Nw Γ Nw = m NmΓ m Nm2Γ 2 m NmN w Γ Nw m T N w Γ Nw Notice that if we can approximate C with C ˆM ˆΓ, where ˆΓ is diagonal, then CC T ˆM ˆM T ˆΓ 2 For example, one such simple approximation, as discussed above, is ˆM = M, ˆΓ = Γavg Instead of using this approach, we show how to find an optimal minimizes C = vec(m NmΓ ) T vec(m 2 Γ 2 ) T vec(m 22 Γ 2 ) T vec(m Nm2Γ 2 ) T vec(m Nw Γ Nw ) T vec(m 2Nw Γ Nw ) T vec(m NmN w Γ Nw ) T = C ˆM ˆΓ F ˆM and ˆΓ that Using ideas from Van Loan and Pitsianis [], we can find the approximation C ˆM ˆΓ by transforming C to tilde space; the optimal Kronecker product approximation of a matrix is obtained by using the optimal rank- approximation of the transformed matrix In our case, the tilde transformation of C is given by vec(m Γ ) T m vec(γ ) T vec(m 2 Γ ) T m 2 vec(γ ) T m Nmvec(Γ ) T m 2 vec(γ 2 ) T m 22 vec(γ 2 ) T m Nm2vec(Γ 2 ) T m Nw vec(γ Nw ) T m 2Nw vec(γ Nw ) T m NmN w vec(γ Nw ) T = m vec(γ ) T m 2 vec(γ 2 ) T m Nw vec(γ Nw ) T

8 8 S BERISHA, J NAGY AND R PLEMMONS m m 2 = m Nw vec(γ ) T vec(γ 2 ) T vec(γ Nw ) T where m ij is the ith entry in vector m j, for i =,, N m and j =,, N w To find the optimal Kronecker product approximation of C, first observe (see, eg, []) that, C ˆM ˆΓ F = C vec( ˆM)vec(ˆΓ) T 2, and thus the optimal Kronecker product approximation problem is equivalent to an optimal rank- approximation of C By the Eckhart-Young theorem (see, eg, [2]), the best rank- approximation is obtained from the largest singular value and corresponding singular vectors of C; that is, C σ ũ ṽ T The matrices ˆM and ˆΓ are then constructed from σ, ũ, and ṽ ; specifically, vec( ˆM) = σ ũ, vec(ˆγ) = σ ṽ 42 Computing the largest singular triplet of C The matrix C can be quite large, N m N w Np 2, so it is important to exploit its structure in order to efficiently compute the largest singular triplet; simply exploiting sparsity is not enough To describe how we do this, we first define matrices m vec(γ ) T m 2 M = and T = vec(γ 2 ) T, m Nm vec(γ Nw ) T so that C = MT Now notice that since each Γ j is an N p N p diagonal matrix, then at most N p columns of T are nonzero Thus, there is a permutation matrix P such that T P = [ T ] T = [ T ] P T, where is an N w ( N 2 p N p ) matrix of all zeros, and T = is an N w N p matrix, where γ j is a vector containing the diagonal elements of Γ j Next, observe that the structure of M allows us to efficiently compute a thin QR decomposition, γ T γ T 2 γ T N w M = QR,

9 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 9 where R = diag( m 2, m 2 2, m Nw 2 ) Thus, we can now rewrite C as C = MT = QR [ T ] P T = Q [ R T ] P T Notice that RT is an N w N p matrix, which is relatively small compared to the N m N w Np 2 matrix C In addition, because Q and P are orthogonal matrices, to compute the largest singular triplet of C, we need only compute the largest singular triplet of RT That is, if we denote the largest singular triplet of RT as (u t, σ t, v t ), and the largest singular triplet of C as (ũ, σ, ṽ ), then [ ] vt σ = σ t, ũ = Qu t and ṽ = P, where is an (N 2 p N p ) vector of all zeros It is also important to notice that the zero structure of ṽ implies that if vec(ˆγ) = σ ṽ, then ˆΓ is a diagonal matrix precisely what we need for our preconditioner 43 Approximation Quality of the Preconditioner In this subsection, we consider the approximation quality of the preconditioner It is difficult to give a precise analytical result on the quality of the approximation, but it is possible to get a rough bound on C ˆM ˆΓ 2 F The size of this norm depends on how the PSFs vary with wavelength Specifically, if ˆM ˆΓ is the matrix that minimizes the Frobenius norm, then C ˆM ˆΓ 2 F C M Γ avg 2 F N m N w = m ij 2 E j 2 F i= j= N w N m E j 2 F, where E j = Γ j Γ avg, and the last inequality results because, without loss of generality, we can assume m ij (see Fig 53) Thus, if Γ Γ Nw (that is, the PSFs are approximately equal), then we obtain a small approximation error with our preconditioner Because of the high nonlinearity of the PSFs, we have not been able to refine this bound any further However, we can say that it is known that all the wavelengthdependent exact PSFs for hyperspectral imaging through atmospheric turbulence are scaled versions of a base PSF associated with the optical path difference function [8] for the arriving light wavefront phase Also, the blurring effects of the PSFs become essentially much less at longer wavelengths, and there the scaling factor begins to approach one, so that the PSFs become almost identical, see eg, [5] The wavelength where this begins to occur is of course problem dependent and depends upon the level of turbulence We note that, when evaluating the quality of preconditioners, it is often useful to look experimentally at the singular values of the original matrix and the preconditioned system However, this can only be done for very small problems Figure 4 j=

10 S BERISHA, J NAGY AND R PLEMMONS shows the singular values for preconditioned and non-preconditioned systems We used PSFs of size and vary the number of wavelengths from 9 to 99 with a nm step size Note that the singular values of the preconditioned system tend to cluster more towards compared to the non-preconditioned system We also noticed that the singular values of the preconditioned system show a tendency to move away from The behavior of the singular values is similar as the number of wavelengths varies Even though there is not a tight clustering of the singular values around, as one would expect from an extremely effective preconditioner, our numerical results show that our Kronecker product-based preconditioner significantly reduces the number of necessary iterations for the convergence of the conjugate gradient method Fig 4 Singular values for the non-preconditoned (left) and preconditioned (right) systems with varying number of wavelengths 5 Numerical results In this section we test the proposed numerical scheme for the deblurring and unmixing model using single and multiple PSFs Simulated hyperspectral data are used to evaluate the proposed method We compare the multiple PSF approach with the single PSF approach The quality of the estimated fractional abundances of endmembers is evaluated by using the relative error defined by: X true X 2 X true 2, where X true is the matrix of true fractional abundances of endmembers and X is the computed fractional abundances of endmembers by the proposed method In all experiments, we used the circular Moffat functions defined in equation (32) to model the PSFs, with α = 242, α =, and α 2 = 266 We consider a simulated hyperspectral image of the Hubble Space Telescope, which is also used for testing in [4] Similar data was also used in [7] and [3] The signatures cover a band of spectra from 4nm to 25nm We use 99 evenly distributed sampling points, leading to a hyperspectral datacube of size Six materials typical to those associated with satellites are used, Hubble aluminum, Hubble glue, Hubble honeycomb top, Hubble honeycomb side, solar cell, and rubber edge The synthetic map of the satellite image is shown in Figure 5 The hyperspectral datacube is blurred by multiple circular Moffat point spread functions (see Figure 52), ie each column of G is blurred by a different circular Moffat function corresponding to a particular wavelength Note that as the wavelength, λ, increases there is less blurring present in those columns compared to the blurring in the columns

11 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF observed at shorter wavelengths, as expected eg [4] Gaussian white noise in the level of 3dB is also added to the datacube In the experiments for the numerical scheme with multiple PSFs we use all the PSFs for the reconstruction of the fractional abundances In the single PSF scheme we use an average of all the PSFs for reconstruction That is, the columns of G are blurred with different PSFs in both cases and we use one PSF representing the average of all PSFs for reconstruction in the single PSF numerical scheme The plot of relative errors is shown in Figure 54 One can observe that the use of multiple PSFs provides lower relative reconstruction errors compared to the relative errors obtained by the single PSF case It is a known fact that the blurring varies with different wavelengths Figure 54 shows that by taking this fact into account we can achieve much lower relative errors in the reconstruction of the fractional abundances The following values are used for the parameters in the alternating minimization scheme: β = 2, µ =, µ 2 = 5 4 The convergence of the alternating direction method is theoretically guaranteed as long as the penalty parameter β is positive, see eg [4] We note that the conjugate gradient method required, iterations to solve each X-subproblem (for multiple PSFs) at the same accuracy level as the single PSF method By using the preconditioner (presented in Section 3) we were able to reduce the number of iterations required for convergence to 2 In Figure 55 we show the relative residual norms for the first 95 iterations of preconditioned conjugate gradient (PCG) and conjugate gradient (CG) without preconditioning It is clear from this figure that the relative residual norms for PCG decrease very quickly until they reach the default tolerance level of 6 whereas for CG the relative residual norms decrease very slowly Figures 56 and 57 show the reconstructed columns of X for both the single PSF and multiple PSF methods The reconstructed spectral signatures for the various materials are shown in Figure 58 This figure shows clearly that the reconstructed spectra using multiple PSFs are much better approximations to the true spectra than when using a single PSF As with the results shown in Figure 54, the single PSF is obtained averaging all PSFs Figure 59 is essentially the same as Figure 58, except we also include the spectral signatures of the observed (blurred) data Fig 5 Synthetic map representation of the hyperspectral satellite image The false colors are used to denote different materials, which are defined in Table 5

12 2 S BERISHA, J NAGY AND R PLEMMONS Table 5 Materials, corresponding colors (see Fig 5), fractional abundances of constituent endmembers, and fractional abundances of the materials used for the Hubble satellite simulation Percent Percent Material Color Constituent Endmembers Fractional Abundance light gray Em () 2 green Em 2 (7), Em 9 (3) 8 3 red Em 3 () 4 4 dark gray Em 4 (6), Em (4) 9 5 brown Em 5 () 7 6 gold Em 6 (4), Em (3), 32 Em 2 (3) 7 blue Em 7 () 3 8 white Em 8 () 6

13 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 3 Fig 52 First column: the true columns of G observed at different wavelengths (from top to bottom: 4nm, 7nm, 843nm, 25nm) Second column: the circular Moffat PSFs used to blur the columns of G at different wavelengths Third column: the corresponding blurred and noisy columns of G blurred with different Moffat PSFs corresponding to different wavelengths

14 4 S BERISHA, J NAGY AND R PLEMMONS Fig 53 Spectral signatures of eight materials assigned to the simulated Hubble Telescope model 9 Relative Errors 7 5 Multiple PSFs One PSF Iterations Fig 54 Relative errors for the computed fractional abundances using a single PSF and multiple PSFs

15 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 5 Relative residual norms PCG CG Iterations Fig 55 Relative residual norms for the first 95 iterations of PCG and CG Fig 56 Fractional abundances for materials to 4 (the first 4 materials in Table 5) First column: true fractional abundances; second column: the estimated fractional abundances using the single PSF approach; third column: the estimated fractional abundances using the multiple PSF approach

16 6 S BERISHA, J NAGY AND R PLEMMONS Fig 57 Fractional abundances for materials 5 to 8 (the last 4 materials in Table 5) First column: true fractional abundances; second column: the estimated fractional abundances using the single PSF approach; third column: the estimated fractional abundances using the multiple PSF approach

17 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF Fig 58 The true material spectral signatures (blue and +), the computed spectral signatures using the multiple PSF numerical approach (red and o), and the computed spectral signatures using the single PSF approach (magenta : and ) Note that the y-axis has not been scaled in order to show more clearly the differences between spectral signatures in the three cases

18 8 S BERISHA, J NAGY AND R PLEMMONS Fig 59 The true material spectral signatures (blue and +), the computed spectral signatures using the multiple PSF numerical approach (red and o), the computed spectral signatures using the single PSF approach (magenta : and ), and the original observed blurred and noisy material spectral signatures (black and ) Note that the y-axis has not been scaled in order to show more clearly the differences between spectral signatures in the four cases

19 HYPERSPECTRAL IMAGING WITH MULTIPLE PSF 9 6 Conclusions We have presented a numerical approach, based on the ADMM method, for deblurring and sparse unmixing of ground-based hyperspectral images of objects taken through the atmosphere at multiple wavelengths with narrow spectral channels Because the PSFs, which define the blurring operations, depend on the imaging system as well as the seeing conditions and is wavelength dependent, the reconstruction process is computationally intensive In particular, we found it important to use a preconditioned conjugate gradient method to solve a large-scale linear system needed at each ADMM iteration We showed how to efficiently construct an optimal Kronecker product-based preconditioner, and provided numerical experiments to illustrate the effectiveness of our approach In particular, we illustrated that much better accuracy can be obtained by using the multiple, wavelength dependent PSFs, and we showed that our preconditioner is quite effective in significantly reducing the number of conjugate gradient iterations REFERENCES [] M T Eismann, Hyperspectral Remote Sensing, SPIE Press, 22 [2] G H Golub and C F Van Loan, Matrix Computations, 4th Ed, Johns Hopkins University Press, Baltimore, 23 [3] M-D Iordache, J M Bioucas-Dias, and A Plaza, Total variation spatial regularization for sparse hyperspectral unmixing, Geoscience and Remote Sensing, IEEE Transactions on, 5 (22), pp [4] S M Jefferies and M Hart, Deconvolution from wavefront sensing using the frozen flow hypothesis, Optics express, 9 (2), pp [5] A J Lambert and G Nichols, Wavelength diversity in restoration from atmospheric turbulence effected surveillance imagery, in Frontiers in Optics 29, Laser Science XXV, Fall 29 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) [6] C Li, T Sun, K F Kelly, and Y Zhang, A compressive sensing and unmixing scheme for hyperspectral data processing, Image Processing, IEEE Transactions on, 2 (22), pp 2 2 [7] F Li, M K Ng, and R J Plemmons, Coupled segmentation and denoising/deblurring models for hyperspectral material identification, Numerical Linear Algebra with Applications, 9 (22), pp [8] M C Roggemann and B M Welsh, Imaging Through Turbulence, CRC press, 996 [9] D Serre, E Villeneuve, H Carfantan, L Jolissaint, V Mazet, S Bourguignon, and A Jarno, Modeling the spatial PSF at the VLT focal plane for MUSE WFM data analysis purpose, in SPIE Astronomical Telescopes and Instrumentation: Observational Frontiers of Astronomy for the New Decade, International Society for Optics and Photonics, 2, pp [] F Soulez, E Thiebaut, and L Denis, Restoration of hyperspectral astronomical data with spectrally varying blur, European Astronomical Society Publications Series, 59 (23), pp [] C F Van Loan and N P Pitsianis, Approximation with Kronecker products, in Linear Algebra for Large Scale and Real Time Applications, M S Moonen and G H Golub, eds, Kluwer Publications, 993, pp [2] E Villeneuve, H Carfantan, and D Serre, PSF estimation of hyperspectral data acquisition system for ground-based astrophysical observations, in Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2 3rd Workshop on, IEEE, 2, pp 4 [3] Q Zhang, H Wang, R Plemmons, and V Pauca, Tensor methods for hyperspectral data analysis: a space object material identification study, Journal of the Optical Society of America A, 25 (28), pp 3 32 [4] X-L Zhao, F Wang, T-Z Huang, M K Ng, and R J Plemmons, Deblurring and sparse unmixing for hyperspectral images, Geoscience and Remote Sensing, IEEE Transactions on, 5 (23), pp