Donald W. Wilson

Noise properties of the EM algorithm. I. Theory

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.

FastSPECT II: a second-generation high-resolution dynamic SPECT imager

FastSPECT II is a recently commissioned 16-camera small-animal SPECT imager built with modular scintillation cameras and list-mode data-acquisition electronics. The instrument is housed in a lead-shielded enclosure and has exchangeable aperture assemblies and adjustable camera positions for selection of magnification, pinhole size, and field of view. The calibration of individual cameras and measurement of an overall system imaging matrix (1 mm/sup 3/ voxels) are supported via a five-axis motion-control system. Details of the system integration and results of characterization and performance measurements are presented along with first tomographic images. The dynamic imaging capabilities of the instrument are explored and discussed.

SemiSPECT: A small-animal single-photon emission computed tomography (SPECT) imager based on eight cadmium zinc telluride (CZT) detector arrays

The first full single-photon emission computed tomography (SPECT) imager to exploit eight compact high-intrinsic-resolution cadmium zinc telluride (CZT) detectors, called SemiSPECT, has been completed. Each detector consists of a CZT crystal and a customized application-specific integrated circuit (ASIC). The CZT crystal is a 2.7 cm x 2.7 cm x -0.2 cm slab with a continuous top electrode and a bottom electrode patterned into a 64 x 64 pixel array by photolithography. The ASIC is attached to the bottom of the CZT crystal by indium-bump bonding. A bias voltage of -180 V is applied to the continuous electrode. The eight detectors are arranged in an octagonal lead-shielded ring. Each pinhole in the eight-pinhole aperture placed at the center of the ring is matched to each individual detector array. An object is imaged onto each detector through a pinhole, and each detector is operated independently with list-mode acquisition. The imaging subject can be rotated about a vertical axis to obtain additional angular projections. The performance of SemiSPECT was characterized using 99mTc. When a 0.5 mm diameter pinhole is used, the spatial resolution on each axis is about 1.4 mm as estimated by the Fourier crosstalk matrix, which provides an algorithm-independent average resolution over the field of view. The energy resolution achieved by summing neighboring pixel signals in a 3 x 3 window is about 10% full-width-at-half-maximum of the photopeak. The overall system sensitivity is about 0.5 x 10(-4) with the energy window of +/-10% from the photopeak. Line-phantom images are presented to visualize the spatial resolution provided by SemiSPECT, and images of bone, myocardium, and human tumor xenografts in mice demonstrate the feasibility of preclinical small-animal studies with SemiSPECT.

Compact CT/SPECT small-animal imaging system

We have developed a dual-modality CT/SPECT imaging system for small-animal imaging applications. The X-ray system comprises a commercially available micro-focus X-ray tube and a CCD-based X-ray camera. X-ray transmission measurements are performed based on cone-beam geometry. Individual projections are acquired by rotating the animal about a vertical axis in front of the CCD detector. A high-resolution CT image is obtained after reconstruction using an ordered subsets-expectation maximization (OS-EM) reconstruction algorithm. The SPECT system utilizes a compact semiconductor camera module previously developed in our group. The module is mounted perpendicular to the X-ray tube/CCD combination. It consists of a 64 /spl times/ 64 pixellated CdZnTe detector and a parallel-hole tungsten collimator. The field of view is 1 square inch. Planar projections for SPECT reconstruction are obtained by rotating the animal in front of the detector. Gamma-ray and X-ray images are presented of phantoms and mice. Procedures for merging the anatomical and functional images are discussed.

Tomographic small-animal imaging using a high-resolution semiconductor camera

We have developed a high-resolution, compact semiconductor camera for nuclear medicine applications. The modular unit has been used to obtain tomographic images of phantoms and mice. The system consists of a 64 x 64 CdZnTe detector array and a parallel-hole tungsten collimator mounted inside a 17 cm x 5.3 cm x 3.7 cm tungsten-aluminum housing. The detector is a 2.5 cm x 2.5 cm x 0.15 cm slab of CdZnTe connected to a 64 x 64 multiplexer readout via indium-bump bonding. The collimator is 7 mm thick, with a 0.38 mm pitch that matches the detector pixel pitch. We obtained a series of projections by rotating the object in front of the camera. The axis of rotation was vertical and about 1.5 cm away from the collimator face. Mouse holders were made out of acrylic plastic tubing to facilitate rotation and the administration of gas anesthetic. Acquisition times were varied from 60 sec to 90 sec per image for a total of 60 projections at an equal spacing of 6 degrees between projections. We present tomographic images of a line phantom and mouse bone scan and assess the properties of the system. The reconstructed images demonstrate spatial resolution on the order of 1-2 mm.

A prototype instrument for single pinhole small animal adaptive SPECT imaging

The authors have designed and constructed a small-animal adaptive SPECT imaging system as a prototype for quantifying the potential benefit of adaptive SPECT imaging over the traditional fixed geometry approach. The optical design of the system is based on filling the detector with the region of interest for each viewing angle, maximizing the sensitivity, and optimizing the resolution in the projection images. Additional feedback rules for determining the optimal geometry of the system can be easily added to the existing control software. Preliminary data have been taken of a phantom with a small, hot, offset lesion in a flat background in both adaptive and fixed geometry modes. Comparison of the predicted system behavior with the actual system behavior is presented, along with recommendations for system improvements.

The multi-module, multi-resolution system (M3R): A novel small-animal SPECT system

We have designed and built an inexpensive, high-resolution, tomographic imaging system, dubbed the multi-module, multi-resolution system, or M3R. Slots machined into the system shielding allow for the interchange of pinhole plates, enabling the system to operate over a wide range of magnifications and with virtually any desired pinhole configuration. The flexibility of the system allows system optimization for specific imaging tasks and also allows for modifications necessary due to improved detectors, electronics, and knowledge of system construction (e.g., system sensitivity optimization). We provide an overview of M3R, focusing primarily on system design and construction, aperture construction, and calibration methods. Reconstruction algorithms will be described and reconstructed images presented.

A Low-Cost Approach to High-Resolution, Single-Photon Imaging Using Columnar Scintillators and Image Intensifiers

Results are presented for a low-cost, ultra-high resolution gamma camera for small-animal SPECT and molecular imaging. The detector, known as Bazooka SPECT, employs a second-generation image intensifier which is directly coupled to a columnar CsI(Tl) scintillator. Operating in photon-counting mode, individual gamma-ray interactions are seen as clusters of signal, and significant improvement in spatial resolution is obtained by estimating the interaction position via Anger (centroid) estimation for individual clusters. Amplification of scintillator light prior to the optical path is advantageous compared to similar, low-light CCD gamma-cameras which solely apply gain in the CCD. With amplification from the image intensifier, the system is no longer limited by light loss from the optical path. This allows for a customizable optical system via lenses in a macro photography configuration and for the use of a low-cost, highspeed CCD. Experimental results show that Bazooka SPECT has an intrinsic resolution of approximately 150mum when the entire 25mm intensifier output screen is imaged onto the CCD. Better than 100mum resolution is attainable with less demagnification from the optical system. Initial indications are that this detector will serve as an attractive, inexpensive modular camera for high-resolution, multiple-pinhole, small-animal SPECT and molecular imaging.

Multi-pinhole collimator design for small-object imaging with SiliSPECT: a high-resolution SPECT

We have designed a multi-pinhole collimator for a dual-headed, stationary SPECT system that incorporates high-resolution silicon double-sided strip detectors. The compact camera design of our system enables imaging at source-collimator distances between 20 and 30 mm. Our analytical calculations show that using knife-edge pinholes with small-opening angles or cylindrically shaped pinholes in a focused, multi-pinhole configuration in combination with this camera geometry can generate narrow sensitivity profiles across the field of view that can be useful for imaging small objects at high sensitivity and resolution. The current prototype system uses two collimators each containing 127 cylindrically shaped pinholes that are focused toward a target volume. Our goal is imaging objects such as a mouse brain, which could find potential applications in molecular imaging.

Calibration of Scintillation Cameras and Pinhole SPECT Imaging Systems

Linear digital-imaging systems are most accurately described as mappings from an object, which is a function of continuous variables, to a discrete set of measurements. When the object is approximated by a discrete vector (e.g.,a linear combination of some expansion functions), the image-forming mechanism can be formulated as: