Bernard A. Mair

Bernard A. Mair
University of Florida | UF

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67
Publications
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Introduction
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Publications

Publications (67)
Article
In this article, a new method is introduced for estimating the motion of the heart due to respiration in gated cardiac SPECT using a rigid-body model with rotation parametrized by a unit quaternion. The method is based on minimizing the sum of squared errors between the reference and the deformed frames resulting from the usual optical flow constra...
Article
Full-text available
We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local...
Conference Paper
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 normaliz...
Article
Full-text available
This paper describes a new method for estimating the 3D, non-rigid object motion in a time sequence of images. The method is a generalization of a standard optical flow algorithm that is incorporated into a successive quadratic approximation framework. The method was evaluated for gated cardiac emission tomography using images obtained from a mathe...
Article
Purpose: Displacement of the heart due to respiration introduces blur into myocardial perfusion images. Similar to ECG‐gating of the cardiac cycle, a respiratory monitoring device may be used to bin events according to respiratory phase. The images may then be compensated with an estimate of the frame‐to‐frame motion. We have found that the accurac...
Article
We develop a "fast" algorithm for obtaining regularized estimates of emission means in positron emission tomography (PET). The algorithm, which iteratively minimizes a penalized maximum likelihood (PML) objective function (i.e., negative log likelihood function plus a scaled penalty function), is based on a "pattern search" and a previously develop...
Conference Paper
Green's one-step-late (OSL) algorithm is gaining importance as an algorithm for the motion-compensated reconstruction of cardiac images. However, the original OSL algorithm does not converge, so various modifications have been made to mitigate this problem. In this paper we propose and test a modification of the OSL algorithm. The algorithm include...
Conference Paper
Full-text available
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...
Article
In this paper, we propose and test a new iterative algorithm to simultaneously estimate the nonrigid motion vector fields and the emission images for a complete cardiac cycle in gated cardiac emission tomography. We model the myocardium as an elastic material whose motion does not generate large amounts of strain. As a result, our method is based o...
Article
The purpose of this work was to develop and test a new motion estimation algorithm for gated cardiac emission tomography that is designed to be more accurate than conventional methods particularly for relatively large frame-to-frame displacements. The proposed method consists of a sequential application of the conventional Horn-Schunck algorithm to...
Article
We present penalized weighted least-squares (PWLS) and penalized maximum-likelihood (PML) methods for reconstructing transmission images from positron emission tomography transmission data. First, we view the problem of minimizing the weighted least-squares (WLS) and maximum likelihood objective functions as a sequence of nonnegative least-squares...
Conference Paper
This paper describes an evaluation of a simultaneous reconstruction and motion estimation algorithm for gated myocardial emission tomography using a multi-image, channelized Hotelling observer model. The test data for the evaluation were obtained using the NURBS-based cardiac torso (NCAT) 4D phantom with simulated myocardial defect. Results indicat...
Article
The purpose of this chapter is to present an introduction to thin-plate spline interpolation and indicate how it can be a useful tool in medical imaging applications. After a brief review of the strengths and weaknesses of polynomial and Fourier interpolation, the ideas fundamental to the success of cubic spline interpolation are discussed. These i...
Article
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 s...
Conference Paper
The primary goal of this work was to develop and evaluate a new method for simultaneous 3D motion estimation and image reconstruction for gated cardiac emission computed tomography (ECT). The method, termed the MR algorithm, employs a two-step, iterative procedure. The method was evaluated using both simulated and physical phantoms designed to mimi...
Article
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...
Conference Paper
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 myocardiu...
Article
In this paper, we introduce a new algorithm for estimating non-negative parameters from Poisson observations of a linear transformation of the parameters. The proposed objective function fits both a weighted least squares (WLS) and a minimum χ2 estimation framework, and results in a convex optimization problem. Unlike conventional WLS methods, the...
Article
The standard model for positron emission tomography is a First Kind Fredholm integral equation relating the emission means to the detection means in which the kernel is the probability that an annihilation at a point in image space is detected in a detector tube. This paper contains an overview of recent results on the precise mathematical represen...
Conference Paper
In this paper, we present novel maximum likelihood reconstruction algorithms for positron emission tomography (PET). The key idea behind the algorithms is that the set of maximum likelihood estimates is equivalent to the intersection of certain convex sets. Given this equivalence, we exploit results from set theoretic estimation and develop subgrad...
Conference Paper
The focus of this work has been to develop a processing method for gated cardiac ECT that simultaneously reconstructs the pixel intensities of the gated images and estimates the motion of the cardiac wall. The simultaneous reconstruction and motion estimation is achieved using conjugate gradient optimization with an objective function that is depen...
Article
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 def...
Article
The popular Radon transform approximation used in the modeling and reconstruction of positron emission tomography (PET) images fails to account for the non-trivial size of PET detectors. Currently, all reconstruction algorithms which account for detector width are based on the iterative EMML (expectation maximization maximum likelihood) method whic...
Conference Paper
In this paper, we present weighted least-squares (WLS) and maximum likelihood (ML) algorithms for reconstructing transmission images from positron emission tomography (PET) transmission data. The key idea behind the algorithms is that the problem of minimizing the WLS and ML objective functions can be viewed as a sequence of least-squares minimizat...
Article
This paper describes a method for reconstructing two dimensional profiles from positron emission tomography (PET) emission data. Unlike the usual methods which assume the detectors are small, the method is based on an accurate system response model which is valid for PET detectors of arbitrary size. The emission profile is represented as an orthogo...
Article
In this paper we derive some properties of inverse estimators in the Wicksell unfolding problem, based on a regularized version of the inverse of the preconditioned operator. We focus on the integrated mean square error, data-driven selection of the regularization parameter, and asymptotic efficiency of linear functionals in the Hájek–LeCam sense.
Article
For decades, the Radon transform has been used as an approximate model for two-dimensional (2D) positron emission tomography (PET). Since this model assumes that detector tubes are represented by lines (hence have no area), PET reconstruction algorithms need to be modified to account for the nonzero width of detectors. To date, these modifications...
Conference Paper
Presents an attenuation correction method where attenuation images are first segmented and then attenuation density values are assigned to the various regions. A key idea behind the segmentation component of the method is that the attenuation image is viewed as a realization of a system that obeys a hidden Markov model. Some advantages of the segme...
Article
In this paper, we develop a refined version of the mathematical model introduced by Shepp and Vardi for positron emission tomography. This model replaces the usual finite-dimensional linear system by a non-standard integral equation in which the data-space is finite-dimensional, but the unknown emission intensities are represented by a mathematical...
Article
The authors investigate the problem of recovering the initial states of distributed parameter systems, governed by linear parabolic partial differential equations, from finite approximate data. It is shown that an approximation of the true initial state can be obtained from the solution of an appropriate finite-dimensional algebraic linear system a...
Article
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...
Conference Paper
In this paper, two iterative methods for correcting attenuation errors in positron emission tomography (PET) are proposed. The first method is a maximum likelihood (ML) method that simultaneously estimates the emission intensity and attenuation density. The second one is a least squares (LS) reprojection method, where the survival probabilities are...
Conference Paper
In this paper, the authors introduce a new iterative algorithm for reconstructing positron emission tomography (PET) images. This algorithm seeks to minimize an objective function of weighted least squares (WLS) type. However, unlike conventional WLS methods, the weights do not need to be estimated from the data, but are incorporated in the objecti...
Article
We present unpenalized and penalized weighted least-squares (WLS) reconstruction methods for positron emission tomography (PET), where the weights are based on the covariance of a model error and depend on the unknown parameters. The penalty function for the latter method is chosen so that certain a priori information is incorporated. The algorithm...
Article
A method is presented for deblurring an image blurred by the discrete Gaussian. The method, based on classical theorems of Jacobi and Ramanujan, not only provides exact formulas for the deblurring, but also condition numbers and error bounds estimating the agreement between the original and reconstructed image. The use of the Jacobi Triple Product...
Conference Paper
The authors present a wavelet based modification of the ML-EM algorithm for reconstructing positron emission tomography images. By using the filter bank implementation of the wavelet transform, this algorithm has the flexibility to incorporate a priori information, while maintaining the same computational complexity as the standard ML-EM algorithm....
Conference Paper
The authors present minimum distance methods for reconstructing transmission tomography images. Specifically, the methods the authors propose estimate the attenuation density by minimizing least-squares (weighted and unweighted) and Kullback-Leibler distances. The algorithms are compared using a simulated phantom and Poisson data
Article
Inverse estimation is concerned with estimation of a signal, where the signal is indirectly observed in the presence of random noise. By indirect we mean, more precisely, that we observe a transform of the signal. The transforms that we consider here are quite general and in particular not restricted to compact operators. This general abstract setu...
Article
The recover of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is available. 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...
Conference Paper
In 1995 Z. Reti presented a method for deblurring images blurred by the discrete Gaussian. The method is based on theorems borrowed from analytic number theory developed by Gauss, G. Jacobi (1829), and Ramanujan. One advantage of this method over similar ones developed for the continuous domain is that it provides exact formulas for the deblurring...
Article
Positron emission tomography (PET) is a relatively new area of medical imaging, which has been in clinical use for about forty years. Due to its wide applicability in medical and psychological diagnostic procedures, researchers are interested in obtaining accurate quantitative information as to the metabolic activity rate of various parts of the hu...
Conference Paper
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 converge...
Conference Paper
The authors introduce a refined version of the mathematical model introduced by Shepp and Vardi (1982) for positron emission tomography. This model replaces the finite-dimensional Shepp-Vardi linear system by a nonstandard integral equation in which the data-space is finite-dimensional, but the unknown emission intensities are represented by a math...
Article
This paper analyzes a cross validation method of obtaining a stable, data- based, approximate solution to a general class of first kind Fredholm integral equations of the form $$g(t) = \int_0^1 h (t,\tau )f(\tau )d\tau $$ (1.1) from finite, discrete, inaccurate data $${Y_j} = g({T_j}) + {\varepsilon _j},j = 1,2,...,n$$ (1.2) where, as usual, the \(...
Article
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 finitely and infinitely smoothing forward operators, and to extend a classic result of Natterer to this more general framework. More specifically, it is show...
Article
Results are reported from an analytical investigation of the problem of reconstructing a boundary control, for a given time interval and within a specified class, on the basis of incomplete data from observation of a parabolic distributed-parameter system for a limited time and from a single point in the system spatial domain. The formulation and s...
Article
In this paper we consider the analytic and numerical analysis of a class of inverse problems arising from input-output systems governed by partial differential equations of parabolic type where the inputs and outputs are given as point actuators and sensors. In particular, it is assumed that an unknown input produces an (approximately) known or des...
Article
Let X be a space of homogeneous type and W a subset of X × (0, ∞). Then, under minimal conditions on W, we obtain a relationship between two modes of convergence at the boundary X for functions defined on W. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediat...
Article
The problem of determining an unknown boundary control of a parabolic distributed parameter system, evolving over finite or infinite time, from incomplete, approximate interior temperature measurements is investigated. One advantage of this inversion process is the availability of a priori error bounds based on the measurement errors and frequency...
Article
The determination of the surface temperature and heat flux of a body by means of interior temperature measurements is very important in many areas of science and industry (cf. [1], [8], [9]), and is usually referred to as inverse heat conduction problems (IHCP). Due to their wide applicability, much emphasis has been placed on the numerical solutio...
Article
The problem of determining the temperature at one end of a rod by means of interior temperature measurements has considerable practical importance and has been well studied, however, few exact solutions have been found. Most of the applicable results therefore, have been obtained from the (numerical) analysis of discretized systems. In this paper w...
Article
The problem of determining the surface temperature of a thin rod and thin rectangular plate from temperature readings at interior points is investigated. It is shown that this problem can be solved by an infinite differentiation operator acting on the known temperature readings. Examples are also obtained which show that the surface temperature can...
Article
Full-text available
In this paper, a general Fatou theorem is obtained for functions which are integrals of kernels against measures on Rn. These include solutions of Laplace’s equation on an upper half-space, parabolic equations on an infinite slab and the heat equation on a right half-space. Lebesgue almost everywhere boundary limits are obtained within regions whic...
Article
Full-text available
In this paper, the abstract Fatou-Naim-Doob theorem is used to investigate the boundary behavior of positive solutions of the heat equation on the semi-infinite slab X = Rn_1 x R+ x (0, X). The concept of semifine limit is introduced, and relationships are obtained between fine, semifine, parabolic, one-sided parabolic and two-sided parabolic limit...
Article
Full-text available
This paper investigates the boundary behaviour of positive solutions of the equation Lu = 0, where L is a uniformly parabolic second-order differential operator in divergence form having Holder-continuous coefficients on X = Rn X (0, T), where 0 < T < oo. In particular, the notion of semithinness for the potential theory on X associated with L is i...
Article
Full-text available
In this paper we present a fully data-driven selection algorithm for the stopping criterion for MLEM reconstructions in PET. The method can be generalized to various other reconstruction algorithms, and is based on a statistical analysis of the residuals between projected model and data. To this end we test whether the residuals are consistent with...
Article
In this thesis, an integral representation theorem is obtained for non-negative solutions of the heat equation on X = (//R)('n-1) x (0,(INFIN)) x (0,T) and their boundary behaviour is investigated by using the abstract Fatou-Naim-Doob theorem. The boundary behaviour of positive solutions of the equation Lu = 0 on Y = (//R)('n) x (0,T), where L is a...

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