Investigation of Lesion Detection in MAP Reconstruction with Non-Gaussian Priors
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Statistical image reconstruction methods based on maximum a posteriori (MAP) principle have been developed for emission tomography. The prior distribution of the unknown image plays an important role in MAP reconstruction. The most commonly used prior is the Gaussian prior, whose logarithm has a quadratic form. Gaussian priors are relatively easy to analyze. It has been shown that the effect of a Gaussian prior can be approximated by linear-filtering a maximum likelihood (ML) reconstruction. As a result, sharp edges in reconstructed images are not preserved. To preserve sharp transitions, non-Gaussian priors have been proposed. In this paper, we study the effect of non-Gaussian priors on lesion detection and region of interest quantification in MAP reconstructions using computer simulation. We compare three representative priors - Gaussian prior, Huber prior, and Geman-McClure prior. The results show that for detection and quantification of small lesions, using non-Gaussian priors is not beneficial.Keywords:
Gaussian network model
Using the Fleishman approximation based on first 4 central moments of stochastic responses of Non - Gaussian structures, the responses without complete probability information are transformed into the process followed by standard Gaussian distribution in the paper. And then the formula estimating the mean up - crossing rate of the original responses is proposed, which can be modified by the initial condition , band width and clump size. Based on the traditional Poisson model, the probabilistic model for the first yield of Non - Gaussian structures is formulated. Numerical examples show that the model can overcome the problems leaded by the traditional Gaussian model.
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We formulate a hierarchical version of the Gaussian Process model. In particular, we assume there to be data on several units randomly drawn from the same population. For each unit, several responses are available that arise from a Gaussian Process model. The parameters characterizing the Gaussian Process model for the units are modeled to arise from normal or gamma distributions. Results for two simulations are given that compare the performance of the hierarchical and non-hierarchical models.
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Bayesian maximum a posteriori estimation (MAP) is a very popular way to recover unknown signals and images by using jointly observed data and priors formulated as a probability law. In a variational context, a MAP estimate minimizes an objective function where the priors are seen as a regularization or diffusion term. Independently of such interpretations, MAP estimates are implicit functions of the data and of the functions expressing the priors. This point of view enabled the author to exhibit analytical relations between prior functions and the features of the relevant estimates. These results entail important consequences and questions which are the subject of this paper. Namely, they reveal an essential gap between prior models and the way these are effectively involved in a MAP estimate. Hence the question about the rationale of MAP estimation. At the same time, they give precious indications about the hyperparameters and suggest how to construct estimators which indeed respect the priors.
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Maximum a-posteriori (MAP) estimation has the advantage of incorporating prior knowledge in the image reconstruction procedure which makes it a superior estimation technique compared to convolution back-projection (CBP), maximum likelihood (ML) etc. The inclusion of prior knowledge greatly improves the image quality. However excess smoothening occurs as the MAP-iterations are continued. In biomedical imaging sharp reconstruction is of potential use. To meet these requirements a new prior is proposed which is capable of enhancing the edges by recognizing the correlated neighbors while restoring homogeneity in the uniform regions of the reconstruction. The proposed prior serves as a post-processing technique in Bayesian domain, once an approximate smooth reconstruction is generated by MAP-algorithm. Simulated experiments show improved sharp reconstruction with the proposed post-processing technique.
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Statistical image reconstruction methods based on maximum a posteriori (MAP) principle have been developed for emission tomography. The prior distribution of the unknown image plays an important role in MAP reconstruction. The most commonly used prior are Gaussian priors, whose logarithm has a quadratic form. Gaussian priors are relatively easy to analyze. It has been shown that the effect of a Gaussian prior can be approximated by linear filtering a maximum likelihood (ML) reconstruction. As a result, sharp edges in reconstructed images are not preserved. To preserve sharp transitions, non-Gaussian priors have been proposed. However, their effect on clinical tasks is less obvious. In this paper, we compare MAP reconstruction with Gaussian and non-Gaussian priors for lesion detection and region of interest quantification using computer simulation. We evaluate three representative priors: Gaussian prior, Huber prior, and Geman-McClure prior. We simulate imaging a prostate tumor using positron emission tomography (PET). The detectability of a known tumor in either a fixed background or a random background is measured using a channelized Hotelling observer. The bias-variance tradeoff curves are calculated for quantification of the total tumor activity. The results show that for the detection and quantification tasks, the Gaussian prior is as effective as non-Gaussian priors.
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Fluorescence molecular tomography (FMT) is an attractive imaging tool for quantitatively and three-dimensionally resolving fluorophore distributions in small animals, but it suffers from low spatial resolution due to its inherent ill-posed nature. Structural priors obtained from a secondary modality system such as x-ray computed tomography or magnetic resonance imaging can help to improve FMT reconstruction results. However, challenge remains in how to fully take advantage of the structural priors while effectively avoid undesirable influence caused by an immoderate usage. In this paper, we propose a new method to resolve the FMT inverse problem based on maximum a posteriori (MAP) estimation with structural priors (MAP-SP) in a Bayesian framework. Instead of imposing the structural priors directly on the reconstruction results, the MAP-SP method utilizes them to constrain the unknown hyperparameters of the prior information model which is essential for the Bayesian framework. Then, a low dimensional inverse problem and an alternating optimization scheme are used to automatically calculate the unknown hyperparameters, which make the FMT reconstruction process self-adaptive. Simulation and phantom results show that the proposed MAP-SP method can effectively make use of the structural priors and leads to improvements in reconstruction quality as compared with traditional regularization methods.
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In the reconstruction of positron emission tomography images, each slice of the image volume is individually reconstructed from a sinogram, in which the statistics of the data elements are Poisson and the image data is hidden by the mechanism of projection. We propose a method of image reconstruction which incorporates the given data set and also reflects the a prior knowledge that the image consists of smooth noiseless regions that are separated by sharp edges. This method uses both maximum likelihood and maximum a posteriori techniques in a manner that is similar to techniques used by others, but our method incorporates a bounded prior term and adaptive annealing techniques. These advancements prevent excessive smoothing and address the difficulties presented by parameter selection and image convergence.
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Statistical image reconstruction methods based on maximum a posteriori (MAP) principle have been developed for emission tomography. The prior distribution of the unknown image plays an important role in MAP reconstruction. The most commonly used prior is the Gaussian prior, whose logarithm has a quadratic form. Gaussian priors are relatively easy to analyze. It has been shown that the effect of a Gaussian prior can be approximated by linear-filtering a maximum likelihood (ML) reconstruction. As a result, sharp edges in reconstructed images are not preserved. To preserve sharp transitions, non-Gaussian priors have been proposed. In this paper, we study the effect of non-Gaussian priors on lesion detection and region of interest quantification in MAP reconstructions using computer simulation. We compare three representative priors - Gaussian prior, Huber prior, and Geman-McClure prior. The results show that for detection and quantification of small lesions, using non-Gaussian priors is not beneficial.
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An attempt is made to determine how Gibbs priors can be designed to optimize the reconstruction of objects of specific sizes and contrasts using a MAP-EM (maximum a posteriori, expectation maximization) algorithm. Two-dimensional parallel projection datasets were realistically simulated for phantoms with various object sizes and contrasts. The resulting datasets were reconstructed using a MAP-EM algorithm with a Gibbs prior whose potential function is determined by a set of parameters. Analysis of the contrast and root-mean-squared-errors (RMSEs) of reconstructed objects revealed a tradeoff between noise reduction and contrast for the MAP-EM approach. It is found that the Gibbs priors can be designed to reduce noise and maintain edge sharpness, as compared to ML-EM (maximum-likelihood, EM), only for certain high-contrast objects, but that such priors may smooth over low-contrast objects. Methods for designing priors to optimize the reconstruction of high- or low-contrast objects are demonstrated. It is concluded that MAP-EM significantly reduces noise at the price of some object contrast and that Gibbs priors should be chosen carefully to avoid smoothing out important small and/or low-contrast objects.< >
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