Maximum entropy image reconstruction in X-ray and diffraction tomography

On the Image Reconstruction of Capacitively Coupled Electrical Resistance Tomography (CCERT) with Entropy Priors

Regularization with priors is an effective approach to solve the ill-posed inverse problem of electrical tomography. Entropy priors have been proven to be promising in radiation tomography but have received less attention in the literature of electrical tomography. This work aims to investigate the image reconstruction of capacitively coupled electrical resistance tomography (CCERT) with entropy priors. Four types of entropy priors are introduced, including the image entropy, the projection entropy, the image-projection joint entropy, and the cross-entropy between the measurement projection and the forward projection. Correspondingly, objective functions with the four entropy priors are developed, where the first three are implemented under the maximum entropy strategy and the last one is implemented under the minimum cross-entropy strategy. Linear back-projection is adopted to obtain the initial image. The steepest descent method is utilized to optimize the objective function and obtain the final image. Experimental results show that the four entropy priors are effective in regularization of the ill-posed inverse problem of CCERT to obtain a reasonable solution. Compared with the initial image obtained by linear back projection, all the four entropy priors make sense in improving the image quality. Results also indicate that cross-entropy has the best performance among the four entropy priors in the image reconstruction of CCERT.

Adaptation and focusing of optode configurations for fluorescence optical tomography by experimental design methods

Fluorescence tomography excites a fluorophore inside a sample by light sources on the surface. From boundary measurements of the fluorescent light, the distribution of the fluorophore is reconstructed. The optode placement determines the quality of the reconstructions in terms of, e.g., resolution and contrast-to-noise ratio. We address the adaptation of the measurement setup. The redundancy of the measurements is chosen as a quality criterion for the optodes and is computed from the Jacobian of the mathematical formulation of light propagation. The algorithm finds a subset with minimum redundancy in the measurements from a feasible pool of optodes. This allows biasing the design in order to favor reconstruction results inside a given region. Two different variations of the algorithm, based on geometric and arithmetic averaging, are compared. Both deliver similar optode configurations. The arithmetic averaging is slightly more stable, whereas the geometric averaging approach shows a better conditioning of the sensitivity matrix and mathematically corresponds more closely with entropy optimization. Adapted illumination and detector patterns are presented for an initial set of 96 optodes placed on a cylinder with focusing on different regions. Examples for the attenuation of fluorophore signals from regions outside the focus are given.

Investigation of a 2D two-point maximum entropy regularization method for signal-to-noise ratio enhancement: application to CT polymer gel dosimetry

This study presents a new method of image signal-to-noise ratio (SNR) enhancement by utilizing a newly developed 2D two-point maximum entropy regularization method (TPMEM). When utilized as an image filter, it is shown that 2D TPMEM offers unsurpassed flexibility in its ability to balance the complementary requirements of image smoothness and fidelity. The technique is evaluated for use in the enhancement of x-ray computed tomography (CT) images of irradiated polymer gels used in radiation dosimetry. We utilize a range of statistical parameters (e.g. root-mean square error, correlation coefficient, error histograms, Fourier data) to characterize the performance of TPMEM applied to a series of synthetic images of varying initial SNR. These images are designed to mimic a range of dose intensity patterns that would occur in x-ray CT polymer gel radiation dosimetry. Analysis is extended to a CT image of a polymer gel dosimeter irradiated with a stereotactic radiation therapy dose distribution. Results indicate that TPMEM performs strikingly well on radiation dosimetry data, significantly enhancing the SNR of noise-corrupted images (SNR enhancement factors >15 are possible) while minimally distorting the original image detail (as shown by the error histograms and Fourier data). It is also noted that application of this new TPMEM filter is not restricted exclusively to x-ray CT polymer gel dosimetry image data but can in future be extended to a wide range of radiation dosimetry data.

Vector entropy imaging theory with application to computerized tomography

Medical imaging theory for x-ray CT and PET is based on image reconstruction from projections. In this paper a novel vector entropy imaging theory under the framework of multiple criteria decision making is presented. We also study the most frequently used image reconstruction methods, namely, least square, maximum entropy, and filtered back-projection methods under the framework, of the single performance criterion optimization. Finally, we introduce some of the results obtained by various reconstruction algorithms using computer-generated noisy projection data from the Hoffman phantom and real CT scanner data. Comparison of the reconstructed images indicates that the vector entropy method gives the best in error (difference between the original phantom data and reconstruction), smoothness (suppression of noise), grey value resolution and is free of ghost images.

Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite for the use of such a prior is the success of the optimization stage. The posterior energy corresponding to a PG MRF is generally multimodal and its minimization is particularly problematic. In this context, general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support. In this paper, global optimization is dealt with by extending the graduated nonconvexity (GNC) algorithm to ill-posed linear inverse problems. GNC has been pioneered by Blake and Zisserman in the field of image segmentation. The resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice. We show that the original GNC does not correctly apply to ill-posed problems. Our extension is based on a proper theoretical analysis, which provides further insight into the GNC. The performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography.

A Kalman filtering approach to stochastic global and region-of-interest tomography

We define two forms of stochastic tomography. In global tomography, the goal is to reconstruct an object from noisy observations of all of its projections. In region-of-interest (ROI) tomography, the goal is to reconstruct a small portion of an object (an ROI) from noisy observations of its projections densely sampled in and near the ROI and sparsely sampled away from the ROI. We solve both problems by expanding the object and its projections in a circular harmonic (Fourier) series in the angular variable so that the Radon transform becomes Abel transforms of integer orders applied to the harmonics. The algorithm has three major components. First, we fit state-space models to each order of Abel transform and thus represent the Radon transform operation as a parallel bank of systems, each of which computes the appropriate Abel transform of a circular harmonic. A variable transformation here allows either the global or ROI problem to be solved. Second, the object harmonics are modeled as a Brownian branch. This is a two-point boundary value system, which is Markovianized into a form suitable for the Kalman filter. Finally, a parallel bank of Kalman smoothing filters independently estimates each circular harmonic from the noisy projection data. Numerical examples illustrate the proposed procedure.

A class of robust entropic functionals for image restoration

This paper considers the concept of robust estimation in regularized image restoration. Robust functionals are employed for the representation of both the noise and the signal statistics. Such functionals allow the efficient suppression of a wide variety of noise processes and permit the reconstruction of sharper edges than their quadratic counterparts. A new class of robust entropic functionals is introduced, which operates only on the high-frequency content of the signal and reflects sharp deviations in the signal distribution. This class of functionals can also incorporate prior structural information regarding the original image, in a way similar to the maximum information principle. The convergence properties of robust iterative algorithms are studied for continuously and noncontinuously differentiable functionals. The definition of the robust approach is completed by introducing a method for the optimal selection of the regularization parameter. This method utilizes the structure of robust estimators that lack analytic specification. The properties of robust algorithms are demonstrated through restoration examples in different noise environments.

Bayesian approach with the maximum entropy principle in image reconstruction from microwave scattered field data

Microwave imaging is of great interest in medical applications owing to its high sensitivity with respect to dielectric properties. It allows detection of very small inhomogeneities. The image reconstruction employing the microwave inverse scattering consists of reconstructing the image of an object from the scattered field measured behind the object. This reconstruction runs up against the nonuniqueness of the solution of the inverse scattering problem. The authors propose to solve the ill-posed inverse problem by a statistical regularization method based on the Bayesian maximum a posteriori estimation where the principle of maximum entropy is used for assigning the a priori laws. The results obtained demonstrate the power and potential of this method in image reconstruction.

Observations on maximum entropy processing of MR images

A maximum entropy (MAXENT) criteria for MR image processing optimizations has previously shown poor performance, but this note observes that there are two entirely different kinds of "data transmission" applications which appear to have been intermixed. In the two cases, "image entropy" actually refers to different kinds of data variables. The previous literature formulations are for transfer of data in which pixel-locations are the transmitted variable, and these pixels may be neither uniform nor constant. The second application concerns the MRI data set for display. Its data variables are image pixel-values of magnetization intensity, and the data transfer mode has the sense of visual display. When MAXENT criteria are modified to address an array of pixel-value intensities, and use a pixel-value information entropy rather than pixel-locations entropy, then successful data processing results. Restoring display visualization from highly nonuniform surface coils for lumbar spine scans are demonstrated, as an example of MAXENT usefulness.

The application of the maximum entropy method to electron microscopic tomography

The maximum entropy method has been applied to single axis tilt electron microscopic tomography. Its application requires that the problem be correctly formulated and that the model for the noise in electron micrographs be developed. A suitable noise model was determined empirically. The maximum entropy method was applied to a reconstruction of a test object from projections to which noise had been added. These reconstructions were superior to those obtained by reciprocal space weighted back protection. The method was also robust towards the incorrect specification of the noise, the penalty being an increase in the time required for convergence rather than degradation of the quality of the reconstructed image. In the reconstruction of negatively stained chromatin fibres it was possible to obtain satisfactory images utilizing all the information in the projections, in contrast to conventional methods in which high resolution data are removed by the application of Fourier space filters.