Maximum Entropy Image Reconstruction Research Articles

Resolving the obscuring torus in NGC 1068 with the power of infrared interferometry: revealing the inner funnel of dust

We present new interferometric data obtained with MIDI (MID infrared Interferometric instrument) for the Seyfert II galaxy NGC 1068, with an extensive coverage of sixteen uv points. These observations resolve the nuclear mid-infrared emission from NGC 1068 in unprecedented detail with a maximum resolution of 7 mas. For the first time, sufficient uv points have been obtained, allowing us to generate an image of the source using maximum entropy image reconstruction. The features of the image are similar to those obtained by modelling. We find that the mid-infrared emission can be represented by two components, each with a Gaussian brightness distribution. The first, identified as the inner funnel of the obscuring torus, is hot (800K), 1.35 parsec long, and 0.45 parsec thick in FWHM at a PA=-42 degrees (from north to east). It has an absorption profile different than standard interstellar dust and with evidence for clumpiness. The second component is 3 by 4 pc in FWHM with T=300K, and we identify it with the cooler body of the torus. The compact component is tilted by 45 degrees with respect to the radio jet and has similar size and orientation to the observed water maser distribution. We show how the dust distribution relates to other observables within a few parsecs of the core of the galaxy such as the nuclear masers, the radio jet, and the ionization cone. We compare our findings to a similar study of the Circinus galaxy and other relevant studies. Our findings shed new light on the relation between the different parsec-scale components in NGC 1068 and the obscuring torus.

Maximum entropy regularization of time-dependent geomagnetic field models

SUMMARY We incorporate a maximum entropy image reconstruction technique into the process of modelling the time-dependent geomagnetic field at the core‐mantle boundary (CMB). In order to deal with unconstrained small lengthscales in the process of inverting the data, some core field models are regularized using a priori quadratic norms in both space and time. This artificial damping leads to the underestimation of power at large wavenumbers, and to a loss of contrast in the reconstructed picture of the field at the CMB. The entropy norm, recently introduced to regularize magnetic field maps, provides models with better contrast, and involves a minimum of a priori information about the field structure. However, this technique was developed to build only snapshots of the magnetic field. Previously described in the spatial domain, we show here how to implement this technique in the spherical harmonic domain, and we extend it to the time-dependent problem where both spatial and temporal regularizations are required. We apply our method to model the field over the interval 1840‐1990 from a compilation of historical observations. Applying the maximum entropy method in space—for a fit to the data similar to that obtained with a quadratic regularization—effectively reorganizes the magnetic field lines in order to have a map with better contrast. This is associated with a less rapidly decaying spectrum at large wavenumbers. Applying the maximum entropy method in time permits us to model sharper temporal changes, associated with larger spatial gradients in the secular variation, without producing spurious fluctuations on short timescales. This method avoids the smearing back in time of field features that are not constrained by the data. Perspectives concerning future applications of the method are also discussed.

Doppler imaging and surface differential rotation of young open cluster stars - I. HD 307938 (R58) in IC 2602

In this paper we present Doppler images of a young active G dwarf (HD 307938) in the southern open cluster IC 2602. Spectroscopic data were obtained over a four-night period in 2000 January at the 3.9-m Anglo-Australian Telescope using the University College London Echelle Spectrograph. Simultaneous photometric observations (in the V and R bands) were obtained at the 1.0-m Australian National University telescope. By applying least-squares deconvolution (LSD) to the 2500+ photospheric lines in each echelle spectrum a single high signal-to-noise ratio LSD profile was produced for each phase of the spectroscopic observations. Maximum-entropy image reconstruction, incorporating both the LSD profiles and the photometric data, was used to produce maps of the surface features of the star, with the inclusion of the photometric data producing an increase (compared with the use of spectroscopic data alone) in the spot occupancy in both low- and mid-latitude regions of the star. The maps show that HD 307938 possesses a large, broken polar spot extending down to ∼60° latitude, as well as lower-latitude spots similar to other rapidly rotating G dwarfs. By incorporating a solar-like differential rotation law into the imaging process the surface differential rotation of HD 307938 was determined. This gave a surface shear of dΩ= 0.025 ± 0.015 rad d−1 (for an inclination angle of 60°). Thus the equator of HD 307938 laps the poles every ∼250 d and has a photospheric shear around half that of the Sun.

Maximum-entropy image reconstruction using wavelets

The maximum-entropy method (MEM) is often used for enhancing astronomical images and, in particular, has recently been applied to cosmic microwave background (CMB) observations. Wavelet functions are also now used widely in astronomy, since they allow the sparse and efficient representation of a signal at different scales, and the application of wavelets to the denoising of CMB maps has been investigated. In this paper, we give a systematic discussion of how to combine these two approaches by the use of the MEM in wavelet bases for the denoising and deconvolution of general images and, in particular, CMB maps. We find that the MEM in the à trous wavelet basis has lower reconstruction residuals than conventional pixel-basis MEM in the case when the signal-to-noise ratio is low and the point spread function is narrow. Furthermore, the Bayesian evidence for the wavelet MEM reconstructions is generally higher for a wide range of images. From a Bayesian point of view, the wavelet basis thus provides a better model of the image.

Maximum Entropy Reconstruction Methods in Electron Paramagnetic Resonance Imaging

Electron Paramagnetic Resonance (EPR) is a spectroscopic technique that detects and characterizes molecules with unpaired electrons (i.e., free radicals). Unlike the closely related nuclear magnetic resonance (NMR) spectroscopy, EPR is still under development as an imaging modality. Athough a number of physical factors have hindered its development, EPR's potential is quite promising in a number of important application areas, including in vivo oximetry. EPR images are generally reconstructed using a tomographic imaging technique, of which filtered backprojection (FBP) is the most commonly used. We apply two iterative methods for maximum-entropy image reconstruction in EPR. The first is the multiplicative algebraic reconstruction technique (MART), a well-known row-action method. We propose a second method, known as LSEnt (least-squares entropy), that maximizes entropy and performs regularization by maintaining a desired distance from the measurements. LSEnt is in part motivated by the barrier method of interior-point programming. We present studies in which images of two physical phantoms, reconstructed using FBP, MART, and LSEnt, are compared. The images reconstructed using MART and LSEnt have lower variance, better contrast recovery, subjectively better resolution, and reduced streaking artifact than those reconstructed using FBP. These results suggest that maximum-entropy reconstruction methods (particularly the more flexible LSEnt) may be critical in overcoming some of the physical challenges of EPR imaging.

Surface differential rotation and prominences of the Lupus post T Tauri star RX J1508.6-4423

We present in this paper a spectroscopic monitoring of the Lupus post T Tauri star RX J1508.6-4423 carried out at two closely separated epochs (1998 May 06 and 10) with the UCL Echelle Spectrograph on the 3.9-m Anglo-Australian Telescope. Applying least-squares convolution and maximum entropy image reconstruction techniques to our sets of spectra, we demonstrate that this star features on its surface a large cool polar cap with several appendages extending to lower latitudes, as well as one spot close to the equator. The images reconstructed at both epochs are in good overall agreement, except for a photospheric shear that we interpret in terms of latitudinal differential rotation. Given the spot distribution at the epoch of our observations, differential rotation could only be investigated between latitudes 15° and 60°. We find in particular that the observed differential rotation is compatible with a solar-like law (i.e., with rotation rate decreasing towards high latitudes proportionally to sin2l, where l denotes the latitude) in this particular latitude range. Assuming that such a law can be extrapolated to all latitudes, we find that the equator of RX J1508.6-4423 does one more rotational cycle than the pole every 50±10 d, implying a photospheric shear 2 to 3 times stronger than that of the Sun. We also discover that the Hα emission profile of RX J1508.6-4423 is most of the time double-peaked and strongly modulated with the rotation period of the star. We interpret this rotationally modulated emission as being caused by a dense and complex prominence system, the circumstellar distribution of which is obtained through maximum entropy Doppler tomography. These maps show in particular that prominences form a complete and inhomogeneous ring around the star, precisely at the corotation radius. We use the total Hα and Hβ emission flux to estimate that the mass of the whole prominence system is about 1020 g. From our observation that the whole cloud system surrounding the star is regenerated in less than 4 d, we conclude that the braking time-scale of RX J1508.6-4423 is shorter than 1 Gyr, and that prominence expulsion is thus likely to contribute significantly to the rotational spindown of young low-mass stars.

Doppler Imaging of Two Zero‐Age Main‐Sequence Stars in the Pleiades: HII 686 and HII 3163

Maximum entropy image reconstruction (Doppler imaging) is applied to HII 686 (K4 V, v sin i = 64 km s-1) and HII 3163 (K0 V, v sin i = 70 km s-1), two zero-age main-sequence stars of the Pleiades cluster. Their surface temperature distributions are determined in order to study the nature of magnetic activity in ultrafast rotators of known evolutionary status. Specific intensity line profiles used in the Doppler imaging (DI) algorithm are synthesized. The sensitivity of the reconstructions to the choice of input stellar and atomic data is investigated. The quality of the fit to the average disk-integrated line profile is found to be more important to DI analysis than the exact parameters used to generate the fit when the shape of the disk-integrated line profile is dominated by rotational broadening. Fifteen absorption lines are used to generate 15 individual reconstructions of each Pleiades target. Averaging reduces noise in the reconstructions. However, simulations reveal that spurious features tend to cluster systematically at subobserver longitudes and can therefore persist in the images even after averaging several reconstruction together. Such phase-ghosting becomes more severe as the signal-to-noise ratio decreases and does not appear to depend on the spot morphology. HII 686 shows one predominant feature at 77° latitude, which is 860 K cooler than the surrounding photosphere. HII 3163 shows a predominant feature centered at approximately 70° latitude, which has two temperature minima (780 K cooler than the photosphere) separated by approximately 130° in longitude. However, photometric observations suggest that this double-lobed morphology is a consequence of poor phase sampling. One temperature minimum located midway in longitude better reproduces the observed light curve. We are unable to make any reliable detections of low-latitude features on the surface of either star. The results are discussed in light of recent theoretical observations and existing image reconstructions of similar objects. No correlations are found among reconstructions that could be attributed to line characteristics: equivalent width, excitation potential, or depth of formation. This is also true for the Li I and K I resonance lines. Despite their temperature sensitivity, they yield results that are consistent with those determined from lines that are not temperature sensitive. For each object, the chromospheric residual emission (Hα, Ca II 8498 A) is measured as a function of rotational phase. Maximum emission does not consistently occur at the phase of spot transit.

Magnetic cycles of HR 1099 and LQ Hydrae

We present in this paper a 6-yr time series of magnetic (and brightness) surface images of the K1 subgiant of the RS CVn system HR 1099 (=V711 Tauri) and of the young K0 dwarf LQ Hydrae, reconstructed (with the help of a dedicated maximum entropy image reconstruction software) from Zeeman--Doppler imaging observations collected at the Anglo-Australian Telescope. All the stellar magnetic images that we reconstruct host at least one high-contrast feature in which the field is predominantly azimuthal, thus confirming that such surface structures (already detected in previous similar studies) are indeed real. We take this as strong evidence that dynamo processes of late-type rapid rotators operate throughout the whole stellar convective envelope, rather than being confined to an interface layer between the convective and radiative zones as in the Sun. The latitudinal polarity pattern of azimuthal and radial fields that we observe at the surface of both stars suggests that these magnetic regions respectively reveal the toroidal and poloidal components of the large-scale dynamo field. The spatial structure of these two magnetic field components becomes increasingly more complex (with higher axisymmetric spherical harmonic degrees) for larger rotation rates and deeper convective zones. The strength of the toroidal and poloidal components is typically a few hundred G, more than two orders of magnitude stronger than in the Sun. Long-term evolution of the toroidal and poloidal components of the large-scale field is clearly detected in our time series. We also report the detection of small fluctuations in the orbital period of HR 1099, with a peak-to-peak amplitude of about 36±1 s (i.e. 0.015 per cent) and a period of about 18±2 yr (assuming sinusoidal variations around the nominal value). The most plausible way of explaining such fluctuations is that the quadrupole moment of the K1 subgiant is varying with time, and that this modulation is driven by the magnetic activity cycle of the primary star itself (through a periodic exchange between kinetic and magnetic energy within the convective zone). It provides in particular an independent confirmation that a dynamo operates within the whole convective zone, and suggests that the average azimuthal field in the convective envelope is of the order of 6 kG.

Maximum entropy image reconstruction from sparsely sampled coherent field data

There are many practical problems in which it is required to detect and characterize hidden structures or remote objects by virtue of the scattered acoustic or electromagnetic fields they generate. It remains an open question, however, as to which reconstruction algorithms offer the most informative images for a given set of field measurements. Commonly used time-domain beamforming techniques, and their equivalent frequency-domain implementations, are conceptually simple and stable in the presence of noise, however, large proportions of missing measurements can quickly degrade the image quality. We apply a new algorithm based on the maximum entropy method (MEM) to the reconstruction of images from sparsely sampled coherent field data. The general principles and limitations of the new method are discussed in the framework of regularization theory, and the results of monostatic imaging experiments confirm that superior resolution and artifact suppression are obtained relative to a commonly used linear inverse filtering approach.

Information, Language, and Pixon‐Based Image Reconstructiona

From an information theoretic point of view, the in verse problem and the problem of data compression a re intimately related. Optimal compression seeks the most concis e representation of a data set, while Bayesian prob ability theory favors image reconstruction algorithms which minimally model the information present in the data. This shoul d not be surprising. It is in keeping with a scientists intuitive need to s atisfy the precepts of Occam’s Razor, i.e. not to over interpret one’s data. Information scientists might describe this process as quantifying the Algorithmic Information Content (AIC) of the image, and then using this “coordinate system” for optimal ima ge reconstruction. The present paper describes pixon-based image recon struction, a technique based upon AIC minimal image models. Because AIC is language dependent (description length and language complexity are inversely related) w e have based the practical implementation of our method on concise ( descriptive) languages for generic images, e.g. mul tiresolution basis functions. The present paper describes both the theory of pixon-based reconstruction and presents practic al examples demonstrating that pixon-based reconstruction produces result s consistently superior (often by large factors) to those of other methods, including the best examples of Maximum Likelihood and Maximum Entropy image reconstruction.

Extended dust shells around carbon stars resolved by “HIRAS”

We have examined forty-two carbon stars which show excess emission at 60 and/or 100 μm by applying maximum-entropy image reconstruction techniques to the IRAS 60 μm survey data. Thirteen stars are found to be extended in the reconstructed images. Four of them show a detached ring centered on the stellar position. In particular, U Ant may have a double detached dust shell. The implications of our results are discussed concerning the variation of mass loss on the AGB evolution.

Albedo maps of Pluto and Charon: Initial mutual event results

We present single-scattering albedo maps of the surfaces of Pluto and Charon based primarily on mutual event observations. The dataset contains 3374 photometric observations that cover 15 different satellite transit events, 14 satellite eclipse events, and other out-of-eclipse photometry spanning 1954 to 1986. The maps consist of a 59 × 29 grid of tiles for each body. We applied the technique of maximum entropy image reconstruction to invert the lightcurves, thus revealing surface maps of single-scattering albedo. The surface of Pluto is seen to have albedo features similar to our previous spot model maps (Buie and Tholen 1989). In particular, a south polar cap is evident in the map of Pluto. The north polar region is brighter than the equatorial regions but is not as bright as the south pole. Single-scattering albedos range from 0.98 in the south polar cap to a low near 0.2 at longitudes corresponding to the lightcurve minimum. The map of Charon is somewhat darker with single-scattering albedos as low as 0.03.

Multicriterion maximum entropy image reconstruction from projections

A solution algorithm for the image reconstruction problem with three criteria, maximum entropy, minimum nonuniformity and peakedness, and least square error between the original projection data and projection due to reconstruction is presented. Theoretical results of precedence properties which are respected by all noninferior solutions are first derived. These precedence properties are then incorporated into a multiple-criteria optimization framework to improve the computational efficiency. Comparisons of the new algorithm to the MART and MENT algorithms are carried out using computer-generated noise-free and Gaussian noisy projections. Results of the computational experiment and the efficiency of the multiobjective entropy optimization algorithm (MEOA) are reported.

Multiobjective optimization approach to image reconstruction from projections

Abstract In this paper we describe how a multiobjective programming can be applied to the image reconstruction from projections under three conflict criteria. We propose a lexicographic minimal goal programming model and algorithm for multicriterion image reconstruction from projections. Comparisons of the multiobjective method to the single objective maximum entropy image reconstruction algorithm (MART) are carried out using computer generated noise-free and noisy projections. Improvement in image reconstruction with multiple criteria is demonstrated.

Maximum entropy image reconstruction

The authors justify the maximum entropy image reconstruction (MEIR) formulation proposed by Zhuang et al. (1987), solve an open problem of whether or not there exists a solution to the MEIR problem, prove the fast convergence of the MEIR algorithm proposed by Zhuang et al., and show that the system of differential equations which is the basis of the MEIR algorithm is of the Lyapunov type.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Maximum entropy methods in coherent radar imaging

AbstractThere are some compelling reasons for viewing the problem of image reconstruction from noisy or incomplete data as one of statistical estimation, i.e., of choosing, from the infinity of images consistent with the data, that image which, in some statistical sense, is most plausible. Among these reasons are the soundness of the philosophical underpinning of the resulting image reconstruction process, a greater realization of the image resolution which is inherent in the data, and freedom from many of the artifacts encountered in commonly used ad hoc reconstruction schemes. One successful technique employing a principle of statistical inference is the maximum entropy technique, in which the data‐consistent image with maximum configurational entropy is chosen. It is a computationally intensive approach involving a conjugate gradient search over a convex function of a vector in a space of dimensionality equal to the number of image pixels. This technique has been employed with success in situations where the data samples are modeled as linearly related to a real non‐negative object. We investigate application of maximum entropy image reconstruction to the problem of high‐resolution radar diagnostic imaging. The problem differs from others in which maximum entropy has been applied in that the object to be imaged is complex. Although the desired image is of the magnitude of the complex object and is thus real and non‐negative, there is no linear relationship beween object magnitude and data. Rather, the data are linearly related to the complex object. Several earlier proposed methods for applying the maximum entropy principle to this problem are identified and analyzed. A method that more closely approximates true Bayesian estimation is proposed.

Maximum entropy image reconstruction of DNA sequencing gel autoradiographs

DNA sequencing gel autoradiographs become increasingly difficult to read as one moves up the gel, due to crowding and overlapping of the bands. Maximum entropy image reconstructions of the autoradiograph improve the ease with which crowded sequence data can be read, and extend the region of the gel in which reading is possible. Superior reconstructions are obtained by using nonuniform models based on the initial reconstructions.

Multicriterion image reconstruction and implementation

In this paper, a multiobjective optimization method of the maximum entropy image reconstruction from projection is described. We apply a new iterative algorithm to solve this problem. Computer simulation results are given.

An algorithm for maximum entropy image reconstruction from noisy data

A Maximum Aposteriori Probability (MAP-) approach is considered for reconstructing an image from noisy projections. This approach needs, according to Bayes rule, two specified distributions. The entropy concept is used to specify the à priori distribution of the unknown image vector. The second distribution (which reflects measurement errors) is taken as an uncorrelated normal distribution. The MAP approach then leads to a discrete, strictly convex, nonquadratic optimization problem. The model contains a real parameter which controls the degree of smoothing imposed on the solution. We propose the use of cross-validation for picking the value of this parameter. We derive an algorithm to solve the optimization problem based on Bregman's convex programming scheme. First a simple generalization of Bregman's method is made which leads to increased efficiency for general linear constraints. Based on this generalization we write down the corresponding Bregman method for our reconstruction problem. The method requires that in each iteration a single nonlinear equation is solved. We propose to use one step of Newton's method to approximate the root. Finally a convergence proof is given for the new algorithm.

Maximum entropy image reconstruction in X-ray and diffraction tomography

The authors propose a Bayesian approach with maximum-entropy (ME) priors to reconstruct an object from either the Fourier domain data (the Fourier transform of diffracted field measurements) in the case of diffraction tomography, or directly from the original projection data in the case of X-ray tomography. The objective function obtained is composed of a quadratic term resulting from chi(2) statistics and an entropy term that is minimized using variational techniques and a conjugate-gradient iterative method. The computational cost and practical implementation of the algorithm are discussed. Some simulated results in X-ray and diffraction tomography are given to compare this method to the classical ones.