B. A. Mair

Statistical inverse estimation in Hilbert scales

The recovery of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable. For greater flexibility the general problem is embedded in an abstract Hilbert scale. In the applications Sobolev scales are used. For the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product, where the operator is bounded but not necessarily compact. A lower bound to certain minimax rates is included, and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate. Examples include errors-in-variables (deconvolution) and indirect nonparametric regression. Special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation.

Tikhonov regularization for finitely and infinitely smoothing operators

The main goal of this paper is to obtain a unified theory of Tikhonov regularization, incorporating explicit asymptotic rates of convergence based on a priori assumptions, which cover both the fini...

Smoothing and Edge Detection by Time-Varying Coupled Nonlinear Diffusion Equations

In this paper, we develop new methods for de-noising and edge detection in images by the solution of nonlinear diffusion partial differential equations. Many previous methods in this area obtain a de-noising u of the noisy image I as the solution of an equation of the form ?tu=L(g(|?v|), ?u, u?I), where g controls the speed of the diffusion and defines the edge map. The usual choice for g(s) is (1+ks2)?1 and the function v is always some smoothing of u. Previous choices include v=u, v=G?* u, and v=G?*I. Numerical results indicate that the choice of v plays a very important role in the quality of the images obtained. Notice that all these choices involve an isotropic smoothing of u, which sometimes fails to preserve important corners and junctions, and this may also fail to resolve small features which are closely grouped together. This paper obtains v as the solution of a nonlinear diffusion equation which depends on u. The equation can be obtained as the energy descent equation for the total variation of v penalized by the mean squared error between u and v. The parameters in this energy descent equation are regarded as functions of time rather than constants, to allow for a reduction in the amount of smoothing as time progresses. Numerical tests indicate that our new method is faster and able to resolve small details and junctions better than standard methods.

Three-dimensional motion estimation with image reconstruction for gated cardiac ECT

The primary goal of this work was to develop and evaluate a new method for simultaneous three-dimensional motion estimation and image reconstruction for gated cardiac emission computed tomography (ECT). The method employs a two-step iterative procedure for obtaining the motion and reconstructed image estimates. The method was evaluated using both simulated and physical phantoms designed to mimic myocardial perfusion imaging in ECT. In both the simulated and physical phantom studies, the images reconstructed by the simultaneous method showed improved noise characteristics relative to a standard iterative algorithm. The percent error of the motion estimate ranged from 16% to 59% for the simulated phantom study, and 45% to 65% for the physical phantom study, depending on the position within the myocardium.

A statistical stopping rule for MLEM reconstructions in PET

In this paper we propose and test a new method for terminating the maximum likelihood expectation maximization algorithm for reconstructing positron emission tomography images. It produces both a unique stopping iteration and a set of feasible iterates. The method is based on a stochastic multi-scale analysis which involves partial sums of normalized differences between the projected images and the detector data for each row of the sinogram. Previous methods involved only the single total sum of these differences for all detectors and were unable to produce feasible stopping iterations in the case of modeling errors in the system point spread function. The proposed method (SMAP) is compared with the previous statistical stopping criteria (LV) proposed by Veklerov and Llacer using ensembles of simulated data obtained from a Hoffman brain phantom and a thorax phantom. In these tests, the proposed method produced stopping iterations which were robust relative to modeling errors in the system matrix and improved the signal-to-noise ratio and contrast recovery coefficient of hot regions over the previous LV method.

Simultaneous motion estimation and image reconstruction from gated data

In this paper we investigate the problem of simultaneously estimating the non-rigid motion vector field and the emission images in dynamic medical imaging procedures, such as gated cardiac ECT. We consider the case of two datasets, which, for instance, may be applied to the problem of determining the motion and emission intensities of the myocardium between end-diastole and end-systole gated cardiac data. Our method is based on minimizing an objective function consisting of a maximum likelihood image reconstruction term; the strain energy of an elastic material; and an image matching term that ensures a measure of agreement between the reference and deformed gated images. Minimization is achieved by a two-step iterative algorithm which alternately updates the motion vector field and the gated images. Simulations using exact and noisy datasets for a simple 2D phantom demonstrate the feasibility of the proposed method.

A multi-scale stopping criterion for MLEM reconstructions in PET

In this paper we propose and test a new method for terminating the maximum likelihood expectation maximization algorithm for reconstructing positron emission tomography images. The method is based on a stochastic multiresolution analysis which involves all partial sums (scales) of normalized differences between the projected images and the detector data for each row of the sinogram. Previous methods involved only the single total sum of these differences for all detectors. Our method is theoretically founded on recent results from probability theory on the almost sure behaviour of the maximum of the partial sum process for Poisson data. Preliminary tests indicate that this method produces predictions for the optimal stopping iterations which are robust relative to modeling errors in the system matrix and has a signal-to-noise ratio which is 80% of the maximal SNR available from the MLEM iterates.

Hidden Markov model based attenuation correction for positron emission tomography

In this paper, we present a new algorithm for segmenting short-duration transmission images in positron emission tomography (PET). Additionally, we show how the information provided by the segmentation algorithm can be used to obtain accurate attenuation correction factors. The key idea behind the segmentation algorithm is that transmission images can be viewed as hidden Markov models (HMMs). Using this viewpoint and a training procedure, it is possible to incorporate both a priori anatomical information and the statistical properties of the estimator used to reconstruct the transmission images. The main advantages of the proposed segmentation algorithm, referred to as the HMM segmentation algorithm, are that it is robust and directly addresses the inhomogeneity of the lung region. Once an attenuation image is segmented; the pixel values in the various regions are replaced by more accurate attenuation coefficient values. Then, the resulting image is smoothed with a Gaussian filter and reprojected to obtain the desired attenuation correction factors. Using data from a thorax phantom and a patient, we demonstrate the effectiveness of the HMM-based attenuation correction method.

A weighted least-squares method for PET

In this paper, the authors present a reconstruction algorithm for positron emission tomography that minimizes a weighted least-squares (WLS) objective function. The weights are based on the covariance matrix of the model error and depend on the unknown parameters. The algorithm guarantees nonnegative estimates, and in simulation studies it converged faster and had significantly better resolution and contrast than the ML-EM algorithm. Although simulations suggest that the proposed algorithm is globally convergent, a proof of convergence has not been found yet. Nevertheless, the authors are able to show that it produces estimates that decrease the objective function monotonically with increasing iterations.

POSITRON EMISSION TOMOGRAPHY, BOREL MEASURES AND WEAK CONVERGENCE

In this paper, we develop a refined version of the usual Poisson model for positron emission tomography (PET), in which the data space is finite dimensional, but the unknown emission intensity is represented by a Borel measure on the region of interest. We demonstrate that maximum likelihood (ML) estimators exist in the space of Borel measures and analyse an extension of the finite dimensional EM algorithm for reconstructing the emission intensity. We present evidence that convergence of this functional iteration should be considered in the weak topology and obtain partial convergence results, which contain all the known convergence results to date as special cases. General conditions are obtained under which an ML estimator can be represented by a bounded function. In particular, we show that the regularity of ML estimators depends heavily on properties of the probabilities governing the PET mathematical model. We also show that, in some cases, no ML estimator can be represented by a bounded function. Although this paper is motivated by PET, the results apply to general inverse problems in which the unknown measure, and the kernel representing the blurring operator are all positive.