We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.

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Positron-emission tomography

This paper examines the spatial resolution properties of penalized-likelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce space-variant local impulse response functions, even for space-invariant tomographic systems. Paradoxically, for emission image reconstruction, the local resolution is generally poorest in high-count regions. We show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections. This analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution. The modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized-likelihood methods.

/pdf/spatial-resolution-properties-of-penalized-likelihood-image-3180id06ty.pdf

Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs

Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a "plug-in" approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems.

/pdf/mean-and-variance-of-implicitly-defined-biased-estimators-f12i9uzrc3.pdf

Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography

Derives approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three-dimensional (3-D) positron emission tomography (PET) data using maximum a posteriori (MAP) estimation. These expressions explicitly account for the spatially variant detector response and sensitivity of a 3-D tomograph. The resulting spatially variant impulse response and covariance are computed using 3-D Fourier transforms. A truncated Gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in MAP reconstruction. Using Monte Carlo simulations and phantom data from the microPET small animal scanner, the authors show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of MAP reconstruction for priors with quadratic energy functions. They also describe how these analytical results can be used to achieve near-uniform contrast recovery throughout the reconstructed volume.

https://neuroimage.usc.edu/paperspdf/QiLeahyTMI99.pdf

Resolution and noise properties of MAP reconstruction for fully 3-D PET

The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. It is a non-linear iterative algorithm that attempts to find the ML estimate of the object that produced a data set. The convergence of the algorithm and other deterministic properties are well established, but relatively little is known about how noise in the data influences noise in the final reconstructed image. In this paper we present a detailed treatment of these statistical properties. The specific application we have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. We show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in the second paper in this series we demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.

Noise properties of the EM algorithm: I. Theory.

For pt.I see ibid., vol.39, no.5, p.833-46 (1994). In pt.I the authors derived a theoretical formulation for estimating the statistical properties of images reconstructed using the iterative maximum-likelihood expectation-maximization (ML-EM) algorithm. To gain insight into this complex problem, two levels of approximation were considered in the theory. These techniques revealed the dependence of the variance and covariance of the reconstructed image noise on the source distribution, imaging system transfer function, and iteration number. Here, a Monte Carlo approach was taken to study the noise properties of the ML-EM algorithm and to test the predictions of the theory. The study also served to evaluate the approximations used in the theory. Simulated data from phantoms were used in the Monte Carlo experiments. The ML-EM statistical properties were calculated from sample averages of a large number of images with different noise realizations. The agreement between the more exact form of the theoretical formulation and the Monte Carlo formulation was better than 10% in most cases examined, and for many situations the agreement was within the expected error of the Monte Carlo experiments. Results from the studies provide valuable information about the noise characteristics of ML-EM reconstructed images. Furthermore, the studies demonstrate the power of the theoretical and Monte Carlo approaches for investigating noise properties of statistical reconstruction algorithms.

http://iopscience.iop.org/article/10.1088/0031-9155/39/5/005/pdf

Noise properties of the EM algorithm: II. Monte Carlo simulations.

Noise properties of emission tomographic reconstructed images are compared using maximum-likelihood-expectation-maximization (ML-EM) and filtered-backprojection (FBP) algorithms. Noise comparisons are made in terms of the covariance matrix which gives information on the noise magnitude and noise correlations. Noise properties are studied as a function of iteration for ML-EM and as a function of noise apodization filter for FBP. It is shown that FBP reconstruction spreads noise variance from image regions containing high count densities into regions of low count densities. It is demonstrated that at lower FBP filter cutoff frequencies the noise is correlated over relatively long distances and the correlation function has deep negative sidelobes. For ML-EM reconstruction it is shown that little noise variance is spread from high count density regions into low count regions and that at lower iteration number the noise is correlated over shorter distances than for FBP. For ML-EM, the correlation function has no negative sidelobes at low iteration numbers. It is concluded from these observations that ML-EM reconstruction offers properties that may exceed FBP in terms of detectability for certain emission tomographic imaging situations. >

Noise properties of filtered-backprojection and ML-EM reconstructed emission tomographic images

A theoretical formulation for characterization of SPECT (single photon emission computed tomography) image noise for a filtered-backprojection reconstructed image was derived. The formulation takes into account all segments of the reconstruction process including the effects of discrete filtering and backprojection as well as nonstationarity in both the projection data and the reconstructed image. It allows calculations of the local noise properties of a reconstructed image in the spatial and frequency domains. This facilitates studies of nonstationary image noise. The formulation was validated by comparing the local autocovariance matrix and noise power spectrum calculated using the theoretical formulation with these quantities calculated from an ensemble of 12000 reconstructed images. The assumption of stationarity for SPECT reconstructed images is examined. The results show that SPECT image noise is nonstationary over large uniform regions, except near the center of the uniform region and over small regions of interest near image edges. >

http://ieeexplore.ieee.org/iel2/623/6551/00259213.pdf

Non-stationary noise characteristics for SPECT images

We investigated the spatial resolution and quantitative properties of SPECT images reconstructed using filteredbackprojection (FB) and maximum likelihood-expectation maximization (ML-EM) algorithms. We studied ML-EM as a function of iteration number and projectorhackprojector (php) model, and FE3 as a function of filter. To quantify the image resolution for a shift-variant process we defined a local point response function (PRF). To justify the use of the local PRF to quantify resolution for nonlinear ML-EM we tested the regime over which the algorithm behaved linearly with respect to input perturbations. Using Fourier domain analogs of the local PRF we demonstrated that ML-EM with an accurate php can lead to images of better resolution than FB with a Butterworth or Metz filter or ML-EM with a php that does not fully model the imaging process. We also showed that the resolution improves with increasing MLEM iteration number, and the extent of the improvement depends on the accuracy of the php model. We conclude that the local PRF can be used to characterize spatial resolution in SPECT images, and that ML-EM offers the possibility of superior resolution, particularly if the algorithm contains a good model of the imaging process.

D.W. Wilson

Papers

Noise properties of the EM algorithm: II. Monte Carlo simulations.

Noise properties of filtered-backprojection and ML-EM reconstructed emission tomographic images

Non-stationary noise characteristics for SPECT images

Spacial resolution properties of FB and ML-EM reconstruction methods