An IDL Based Image Deconvolution Software Package

Transcription

1 An IDL Based Image Deconvolution Software Package F. Városi and W. B. Landsman Hughes STX Co., Code 685, NASA/GSFC, Greenbelt, MD Abstract. Using the Interactive Data Language (IDL), we have implemented some of the basic iterative algorithms for deconvolution of blurred images in a unified environment called DeConv Tool. Most algorithms have as their goal the optimization of some goodness of fit statistic, possibly combined with a smoothing criterion. All the statistics for all the implemented algorithms are computed by DeConv Tool for the purpose of determining the performance and convergence of any particular algorithm and comparing the different methods. DeConv Tool allows interactive monitoring of the statistics and the deconvolved image during computation. Results are stored in a structure convenient for making comparisons between methods and reviewing the deconvolution process. The DeConv Tool package can be acquired by anonymous FTP from the IDL Astronomy User s Library (Landsman 1993). 1. Introduction The standard model of image formation involves first knowing or estimating the point spread function (PSF), which is the response of the instrument to a point source of photons (e.g., a star). Effects such as atmospheric seeing and optical abberations are then presumably described by the PSF, and for convenience, assume that the PSF is spatially invariant over the imaging detector. Observed images are the result of convolution of the photon signal I with H, the PSF : D = I H + N, (1) where D is the observed image (the data), and N is the noise added during data aquisition, which we assume to follow either a Gaussian or Poisson distribution. Convolution is defined as (I H)(i, j) = k,l I(k, l)h(i k, j l), (2) where (i, j) are image pixel indices and H is assumed to be centered at (0, 0). Similarly, the correlation operator, used below, is defined as (I H)(i, j) = k,l I(k + i, l + j)h(k, l), (3) where now (i, j) are referred to as the lag indices. The objective of deconvolution is to reconstruct the image I given the data D, the PSF = H, and 1

2 knowledge of the noise N. The eq.(??) is part of what is referred to as the imaging model M. In reality the deconvolution problem has more unknown than known quantities because of noise and instrument limitations. For this reason a unique solution does not exist, so it is more realistic to consider the probabilities of solutions. The statistical formulation of the deconvolution problem involves the use of Bayes theorem to derive (see, e.g., Weir 1991, Piña and Puetter 1992) p(i D, M) = p(d I, M)p(I M) p(d M) (4) where p(i D, M) = probability of the restored image given the data and model, p(d I, M) = probability of the data for a given restored image and model. Assuming the model is fixed, the term p(d M) is independent of I so it is constant, and p(i M) is the prior probability of the image given the model. A solution to the deconvolution problem is obtained by maximizing the probability of the restored image, and this can be accomplished by maximizing p(d I, M), the goodness of fit, leading to maximum likelihood methods. Also, the product p(d I, M)p(I M) could be maximized, with p(i M) = e S and S entropy of I, then leading to maximum entropy methods. Hence the Bayesian formulation leads to a variety of algorithms for deconvolution, each having its own statistic to be optimized in order to reconstruct the image. 2. Algorithms Implemented The deconvolution algorithms implemented in DeConv Tool are: the Maximum Likelihood method for Poisson or Gaussian noise (MLP and MLG resp.), and the Maximum Residual Likelihood (MRL) method for Gaussian noise. An implementation of the Maximum Entropy (MEM) method is in progress, and other algorithms which are iterative in nature can be integrated into DeConv Tool. The convolution and correlation operations are computed via the Fast Fourier Transform (FFT) by default, and this assumes periodic boundary conditions. Convolutions can be performed directly if desired, but in any case the edge regions of observed image should not contain significant data. The Maximum Likelihood algorithms are derived by noting that when p(d I, M) is at a maximum I [ln p(d I, M)] = I j [ln p(d I, M)] = 0. (5) Applying this condition leads to iterative algorithms which converge toward a fixed point of the iteration (see, e.g., Meinel 1986, Shepp and Vardi 1982). For the case of Poisson noise in the imaging model n D (I H) k p(d I, M) = k D k! k=1 exp[ (I H) k ] (6) where the product is over all pixel indices k. Applying the condition (??) and 2

3 manipulating the result gives the following iterative algorithm: [ ( ) ] D I new = I old H, (7) I old H which is the Richardson-Lucy algorithm (Richardson 1972, Lucy 1974). The relevant statistic for the above MLP algorithm is the log-likelihood ln p(d I, M), and note that the restored image is always positive (if D 0). The case of Gaussian noise in the imaging model gives ( n p(d I, M) = exp [D k (I H) k ] 2 ) 2σN 2 (8) k=1 where the product is over all pixel indices k and σn 2 is the constant variance of the noise. Applying the condition (??) results in the iterative algorithm I new = I old + [D (I old H)] H, (9) called MLG, which is similar in concept to MLP, but can result in negative restored image values, and so must be forced positive. Defining the residual image as R = D (I H), (10) the relevant MLG statistic is the ratio of standard deviations, σ R /σ N, which should approach unity ( note that nσr 2 = Variance[ ln p(d I, M) ] ). Gaussian noise is the limit of Poisson noise at high intensities, as in the case of high background imaging. We have found that the MLP algorithm, eq.(??), can also achieve good image restoration results for the case of Gaussian noise. The residual image R should approach the noise N not only in mean and variance but in all properties. If the additive noise is Gaussian white noise, then there should be no correlation between pixels. Recalling the definition of autocorrelation, A N = N N, the white property of the noise gives A N (i, j) = { n σ 2 N if i = j = 0 0 otherwise, where denotes expectation and n equals the total number of pixels in the image. Since there is no spatial information in the white noise there should be no spatial information left in the residual image after deconvolution. This leads to a new figure of merit for the goodness of fit, the Maximum Residual Likelihood (MRL) statistic, which explicitly incorporates spatial information into the deconvolution. The MRL statistic, developed by Piña and Puetter (1992), is defined as ( ) AR A N 2 (11) χ 2 (A R ) = i,j σ(a N ) where the sum is over nonredundant lags (i, j), since A R (i, j) = A R ( i, j). The range of practical lags is determined by the FWHM of the PSF. This MRL 3

4 statistic follows the standard χ 2 distribution with f degrees of freedom, where f is number of terms in the sum of eq.(??). Another likelihood statistic is then defined by setting p(d I, M) = p( χ 2 (A R ) ), which is maximized when χ 2 (A R ) = f 2. Then the autocorrelation of the residuals is consistent with the white noise model and the residuals have minimal spatial structure. However, the assumption of white noise is sometimes unrealistic since detectors can exhibit noise which is slightly correlated. In such a case the correlation usually occurs over adjacent pixels, so the MRL statistic can still be applied by leaving out the single pixel lags from the sum in eq.(??). The goal of maximizing p( χ 2 (A R ) ), or equivalently, minimizing the difference between χ 2 (A R ) and its most probable value f 2, is accomplished by employing the conjugate-gradient method for n-dimensional function minimization. The resulting iterative MRL algorithm is computation intensive, since, at each iteration it requires computing I [χ 2 (A R )]. However, the MRL statistic proves to be a useful measure for the goodness of fit when using other deconvolution algorithms. We find that as iteration of a deconvolution algorithm proceeds, the point at which the MRL statistic reaches a minimum, or when it stops decreasing at a steady rate, indicates a reasonable cutoff point for the deconvolution iterations. 3. Software Features and Options DeConv Tool provides options for the setup of the observed image and PSF, and the user input is via mouse-menu choices and/or direct keyboard entry. A more advanced widget interface for setup and control will be implemented soon. Starting with an image containing the PSF, any subregion of the image can be selected and extracted, and the result is automatically centered and normalized for use in convolutions. Optionally, the observed PSF can be modeled analytically, where model parameters are estimated by fitting x-y profiles. If the PSF is not supplied, an analytic PSF will be used by inquiring the user for model profile parameters. The PSF model is either a Gaussian functional form or a modified Lorentzian form (Diego 1985), defined as H = α = β = C 1 + α P (1+β) (12) ( ) ( ) x x y y0 + r x r y ρ x (x x 0 ) 2 + ρ y (y y 0 ) 2, where (x 0, y 0 ) is the center pixel, (r x, r y ) specifies the radius at half-max of the PSF x-y profiles, and parameters P, ρ x, ρ y control the steepness of the profiles. The standard deviation, σ N, of Gaussian noise in the observed image is determined automatically by examining histograms of the local image variances around each pixel and finding the most probable variance. Background (mean sky) level is then determined by fitting a Gaussian having the determined σ N to the histogram of image data values. The user can adjust the sky level, and specify an alternate value for σ N if desired. Any subregion of the observed 4

5 image can be extracted and/or a larger region can be defined for the subsequent computations, to make, for example, the size a power of two for optimal FFT convolutions. Empty pixels can then be substituted with appropriate Gaussian noise or a constant value. Progress of the deconvolution is monitored automatically at every b m iterations (m = 1, 2, 3,...), where the base b can be specified to give more or less frequent monitoring, and default base b = 2 (setting b = 1 causes every iteration to be monitored). In addition, the user can press a key at any time to get the current status, or press P to pause with options to save results, adjust colors, or create a zoom window. During the monitoring events DeConv Tool displays: the current deconvolution result (I) alongside the data (D), the residual image (R = D I H), and the residual autocorrelation (A R = R R). Also plotted is the histogram of the residual image, with comparison to a Gaussian curve of same variance. For the case of Poisson noise distribution, since the variance equals the mean, the residuals are first divided by I H so that all variances are normalized. Each monitoring event also prints a line of statistics giving: the mean and standard deviation of residuals (Gaussian noise case), the log-likelihood and RMS difference (Poisson noise case), and the ratio of the residual autocorrelation MRL statistic to the most probable value: χ 2 (A R )/χ 2 (A N ). These statistics can be monitored to determine convergence and stopping iterations. Also printed are the total flux conservation ratio I/ D, the current maximum of the restored image, and FWHM of x-y profiles. All deconvolution statistics are stored in IDL structures and are automatically saved every b m iterations (m = 1, 2, 3,...). The observed image, PSF, and deconvolved images can also be saved, if requested. By save we mean using the IDL save procedure which writes to a file, using External Data Representation (XDR), the designated variables with IDL descriptors so that they can later be easily and completely restored into memory. Thus at a later time, the computations can be restored and replayed using the procedure DeConv Review, giving the same output as DeConv Tool, and further, the statistics for all iterations are then concatenated to form a structured array for convenient examination and graphical display. In summary, DeConv Tool is a modular IDL software package which provides a convenient and powerful environment for applying deconvolution algorithms. The modules (procedures and functions) comprising it can also be utilized separately for special image analysis tasks, and of course, any stage of the computations can be manipulated and examined using IDL. References Diego, F. 1985, PASP, 97, 1209 Landsman, W. B. 1993, this volume Lucy, L. B. 1974, AJ, 79, 745 Meinel, E. S. 1986, J. Opt. Soc. Am. A, 3, 787 Piña, R. K. and Puetter, R. C. 1992, PASP, 104, 1096 Richardson, B. H. 1972, J. Opt. Soc. Am., 62, 55 Shepp, L. A. and Vardi, Y. 1982, IEEE Trans. Med. Imag., MI-1, 113 Weir, N. 1991, in Proceedings of the 3rd ESO/ST-ECF Data Analysis Workshop 5