Towards continuous-to-continuous 3D imaging in the real world.

Imaging systems are often modeled as continuous-to-discrete mappings that map the object (i.e. a function of continuous variables such as space, time, energy, wavelength, etc) to a finite set of measurements. When it comes to reconstruction, some discretized version of the object is almost always assumed, leading to a discrete-to-discrete representation of the imaging system. In this paper, we discuss a method for single-photon emission computed tomography (SPECT) imaging that avoids discrete representations of the object or the imaging system, thus allowing reconstruction on an arbitrarily fine set of points.

Design and performance of a small-animal imaging system using synthetic collimation.

This work outlines the design and construction of a single-photon emission computed tomography imaging system based on the concept of synthetic collimation. A focused multi-pinhole collimator is constructed using rapid-prototyping and casting techniques. The collimator projects the centre of the field of view (FOV) through 46 pinholes when the detector is adjacent to the collimator, with the number reducing towards the edge of the FOV. The detector is then moved further from the collimator to increase the magnification of the system. The object distance remains constant, and each new detector distance is a new system configuration. The level of overlap of the pinhole projections increases as the system magnification increases, but the number of projections subtended by the detector is reduced. There is no rotation in the system; a single tomographic angle is used in each system configuration. Image reconstruction is performed using maximum-likelihood expectation-maximization and an experimentally measured system matrix. The system matrix is measured for each configuration on a coarse grid, using a point source. The pinholes are individually identified and tracked, and a Gaussian fit is made to each projection. The coefficients of these fits are used to interpolate the system matrix. The system is validated experimentally with a hot-rod phantom. The Fourier crosstalk matrix is also measured to provide an estimate of the average spatial resolution along each axis over the entire FOV. The 3D synthetic-collimator image is formed by estimating the activity distribution within the FOV and summing the activities in the voxels along the axis perpendicular to the collimator face.

ADAPTIVE SMALL-ANIMAL SPECT/CT.

We are exploring the concept of adaptive multimodality imaging, a form of non-linear optimization where the imaging configuration is automatically adjusted in response to the object. Preliminary studies suggest that substantial improvement in objective, task-based measures of image quality can result. We describe here our work to add motorized adjustment capabilities and a matching CT to our existing FastSPECT II system to form an adaptive small-animal SPECT/CT.

Multi-energy, single-isotope imaging using stacked detectors.

We investigated a scheme for concurrently detecting low- and high-energy emissions from (123)I with a stacked silicon double-sided strip detector (DSSD) and modular scintillation camera (Modcam) from the FastSPECT II design. We sequentially acquired both low- and high-energy emission images of an (123)I object with a prototype DSSD and a Modcam. A sandwich aperture increases spatial resolution in the low-magnification DSSD image via a smaller pinhole diameter and allows a higher magnification image on the Modcam. Molybdenum, the insert material, efficiently stops 20-30 keV photons due to its ∼20 keV K-edge. Theoretically, less than 10% of 159 keV photons interact in 0.035 cm thick sheet of molybdenum, while this thickness stops virtually all ∼30 keV photons. Thus, photons from both energy regions will be incident upon their respective detectors with little cross talk. With a multi-pinhole collimator, we can decode multiplexed images on the Modcam by making use of the lower-magnification DSSD image. This approach can provide an increase in system sensitivity compared to single-detector configurations. Using MCNP5 we examined the potential benefits and drawbacks of stacked detectors and the sandwich aperture for small-animal pinhole SPECT via the synthetic-collimator method. Simulation results encourage us to construct the novel aperture and use it with our new DSSDs designed for mounting in a transmission configuration.

Fast maximum-likelihood estimation methods for scintillation cameras and other optical sensors.

Maximum-likelihood estimation methods offer many advantages for processing experimental data to extract information, especially when combined with carefully measured calibration data. There are many tasks relevant to x-ray and gamma-ray detection that can be addressed with a new, fast ML-search algorithm that can be implemented in hardware or software. Example applications include gamma-ray event position, energy, and timing estimation, as well as general applications in optical testing and wave-front sensing.

Maximum-likelihood Estimation of 3D Event Position in Monolithic Scintillation Crystals: Experimental Results.

We present a simple 3D event position-estimation method using raw list-mode acquisition and maximum-likelihood estimation in a modular gamma camera with a thick (25mm) monolithic scintillation crystal. This method involves measuring 2D calibration scans with a well-collimated 511 keV source and fitting each point to a simple depth-dependent light distribution model. Preliminary results show that angled collimated beams appear properly reconstructed.

Real-time Data Acquisition and Maximum-Likelihood Estimation for Gamma Cameras.

We have developed modular gamma-ray cameras for biomedical imaging that acquire data with a raw list-mode acquisition architecture. All observations associated with a gamma-ray event, such as photomultiplier (PMT) signals and time, are assembled into an event packet and added to an ordered list of event entries that comprise the acquired data. In this work we present the design of the data-acquisition system, and discuss algorithms for a specialized computing engine to reside in the data path between the front and back ends of each camera and carry out maximum-likelihood position and energy estimations in real time while data was being acquired..

Ideal-observer performance under signal and background uncertainty.

We use the performance of the Bayesian ideal observer as a figure of merit for hardware optimization because this observer makes optimal use of signal-detection information. Due to the high dimensionality of certain integrals that need to be evaluated, it is difficult to compute the ideal observer test statistic, the likelihood ratio, when background variability is taken into account. Methods have been developed in our laboratory for performing this computation for fixed signals in random backgrounds. In this work, we extend these computational methods to compute the likelihood ratio in the case where both the backgrounds and the signals are random with known statistical properties. We are able to write the likelihood ratio as an integral over possible backgrounds and signals, and we have developed Markov-chain Monte Carlo (MCMC) techniques to estimate these high-dimensional integrals. We can use these results to quantify the degradation of the ideal-observer performance when signal uncertainties are present in addition to the randomness of the backgrounds. For background uncertainty, we use lumpy backgrounds. We present the performance of the ideal observer under various signal-uncertainty paradigms with different parameters of simulated parallel-hole collimator imaging systems. We are interested in any change in the rankings between different imaging systems under signal and background uncertainty compared to the background-uncertainty case. We also compare psychophysical studies to the performance of the ideal observer.

The Effects of Incorrect Modeling on Noise and Resolution Properties of ML-EM Images.

The effects of incorrect compensation for collimator blur in single-photon emission computed tomography (SPECT) images are studied in terms of the noise and resolution properties of the reconstructed images. Qualitative analysis of images of the Hoffman brain phantom reconstructed using nonlinear maximum-likelihood-expectation maximization (ML-EM) show the behavior of longer noise correlations for high-pass filtered images. These qualitative observations are confirmed with more quantitative noise measures. The differences also appear in images reconstructed using linear Landweber iteration. However, the signal-to-noise ratio, in terms of the noise-equivalent quanta, remains largely unchanged. We conclude that the compensation model affects SPECT image properties, though the effect on human task performance remains to be studied.

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.

Chemical-shift imaging utilizing the positional shifts along the readout gradient direction.

In this work, we describe a method that uses the linear phase acquired during the readout period due to chemical shift to generate individual magnetic resonance (MR) images of chemically shifted species. The method utilizes sets of Fourier (or k-space) data acquired with different directions of the readout gradient and a postprocessing algorithm to generate chemical shift images. The methodology is developed for both Cartesian data acquisition and for radial data acquisition. The method is presented here for two chemically shifted species but it can be extended to more species. In this work, we present the theory, show the results in phantoms and in human images, and discuss the artifacts and signal-to-noise ratio of the images obtained with the technique.

Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability.

We consider detection of a nodule signal profile in noisy images meant to roughly simulate the statistical properties of tomographic image reconstructions in nuclear medicine. The images have two sources of variability arising from quantum noise from the imaging process and anatomical variability in the ensemble of objects being imaged. Both of these sources of variability are simulated by a stationary Gaussian random process. Sample images from this process are generated by filtering white-noise images. Human-observer performance in several signal-known-exactly detection tasks is evaluated through psychophysical studies by using the two-alternative forced-choice method. The tasks considered investigate parameters of the images that influence both the signal profile and pixel-to-pixel correlations in the images. The effect of low-pass filtering is investigated as an approximation to regularization implemented by image-reconstruction algorithms. The relative magnitudes of the quantum and the anatomical variability are investigated as an approximation to the effects of exposure time. Finally, we study the effect of the anatomical correlations in the form of an anatomical slope as an approximation to the effects of different tissue types. Human-observer performance is compared with the performance of a number of model observers computed directly from the ensemble statistics of the images used in the experiments for the purpose of finding predictive models. The model observers investigated include a number of nonprewhitening observers, the Hotelling observer (which is equivalent to the ideal observer for these studies), and six implementations of channelized-Hotelling observers. The human observers demonstrate large effects across the experimental parameters investigated. In the regularization study, performance exhibits a mild peak at intermediate levels of regularization before degrading at higher levels. The exposure-time study shows that human observers are able to detect ever more subtle lesions at increased exposure times. The anatomical slope study shows that human-observer performance degrades as anatomical variability extends into higher spatial frequencies. Of the observers tested, the channelized-Hotelling observers best capture the features of the human data.

High-pass filters give histograms with positive kurtosis.

If we filter an ensemble of natural images with a high-pass filter, the output is a random variable. The probability density for this variable will, under fairly general assumptions, be a Gaussian scale mixture. The kurtosis for this type of distribution is always positive.

A new design for a SPECT small-animal imager.

We demonstrate, using computer models, the feasibility of a new SPECT system for imaging small animals such as mice. This system consists of four modular scintillation cameras, four multiple-pinhole apertures, electronics, and tomographic reconstruction software. All of these constituents have been designed in our laboratory. The cameras are 120mm×120mm with a resolution of approximately 2mm, the apertures can have either single or multiple pinholes, and reconstruction is performed using the OS-EM algorithm. One major advantage of this system is the design flexibility it offers, as the cameras are easy to move and the aperture s are simple to modify. We explored a number of possible configurations. One promising configuration had the four camera faces forming four sides of a cube with multiple-pinhole apertures employed to focus the incoming high-energy photons. This system is rotated three times, so that data are collected from a total of sixteen camera angles. It is shown that this hybrid system has some superior properties to single-aperture-type systems. We conclude that this proposed system offers advantages over current imaging systems in terms of flexibility, simplicity, and performance.

Reconstruction of two- and three-dimensional images from synthetic-collimator data.

A novel SPECT collimation method, termed the synthetic collimator, is proposed. The synthetic collimator employs a multiple-pinhole aperture and a high-resolution detector. The problem of multiplexing, normally associated with multiple pinholes, is reduced by obtaining projections at a number of pinhole-detector distances. Projections with little multiplexing are collected at small pinhole-detector distances and high-resolution projections are collected at greater pinhole-detector distances. These projections are then reconstructed using the ML-EM algorithm. It is demonstrated through computer simulations that the synthetic collimator has superior resolution properties to a high-resolution parallel-beam (HRPB) collimator and a specially built ultra-high-resolution parallel-beam (UHRPB) collimator designed for our 0.38-mm pixel CdZnTe detectors. It is also shown that reconstructing images in three dimensions is superior to reconstructing them in two dimensions. The advantages of a high-resolution synthetic collimator over the parallel-hole collimators are apparently reduced in the presence of statistical noise. However, a high-sensitivity synthetic collimator was designed which again shows superior properties to the parallel-hole collimators. Finally, it is demonstrated that, for the cases studied, high-resolution detectors are necessary for the proper functionality of the synthetic collimator.

Reconstruction of MR images from data acquired on a general nonregular grid by pseudoinverse calculation.

A minimum-norm least-squares image-reconstruction method for the reconstruction of magnetic resonance images from non-Cartesian sampled data is proposed. The method is based on a general formalism for continuous-to-discrete mapping and pseudoinverse calculation. It does not involve any regridding or interpolation of the data and therefore the methodology differs fundamentally from existing regridding-based methods. Moreover, the method uses a continuous representation of objects in the image domain instead of a discretized representation. Simulations and experiments show the possibilities of the method in both radial and spiral imaging. Simulations revealed that minimum-norm least-squares image reconstruction can result in a drastic decrease of artifacts compared with regridding-based reconstruction. Besides, both in vivo and phantom experiments showed that minimum-norm least-squares image reconstruction leads to contrast improvement and increased signal-to-noise ratio compared with image reconstruction based on regridding. As an appendix, an analytical calculation of the raw data corresponding to the well-known Shepp and Logan software head phantom is presented.

Spatial Pileup Considerations for Pixellated Gamma -ray Detectors.

High-spatial-resolution solid-state detectors being developed for gamma-ray applications benefit from having pixel dimensions substantially smaller than detector slab thickness. This leads to an enhanced possibility of charge partially spreading to neighboring pixels as a result of diffusion (and secondary photon emission) transverse to the drift direction. An undesirable consequence is the effective magnification of the event "size" and the spatial overlap issues which result when two photons are absorbed in close proximity within the integration time of the detector/readout system. In this work, we develop the general statistics of spatial pileup in imaging systems and apply the results to detectors we are developing based on pixellated cadmium zinc telluride (CdZnTe) and a multiplexing application-specific integrated circuit (ASIC) readout. We consider the limitations imposed on total count rate capacity and explore in detail the consequences for the LISTMODE data-acquisition strategy. Algorithms are proposed for identifying and, where possible, resolving overlapping events by maximum-likelihood estimation. The efficacy and noise tolerance of these algorithms will be tested with a combination of simulated and experimental data in future work.

Approximations to ideal-observer performance on signal-detection tasks.

The ideal-observer performance, as measured by the area under the receiver's operating characteristic curve, is computed for six examples of signal-detection tasks. Exact values for this quantity, as well as approximations based on the signal-to-noise ratio of the log likelihood and the likelihood-generating function, are found. The noise models considered are normal, exponential, Poisson, and two-sided exponential. The signal may affect the mean or the variance in each case. It is found that the approximation from the likelihood-generating function tracks well with the exact area, whereas the log-likelihood signal-to-noise approximation can fail badly. The signal-to-noise ratio of the likelihood ratio itself is also computed for each example to demonstrate that it is not a good measure of ideal-observer performance.

List-mode likelihood: EM algorithm and image quality estimation demonstrated on 2-D PET.

Using a theory of list-mode maximum-likelihood (ML) source reconstruction presented recently by Barrett et al., this paper formulates a corresponding expectation-maximization (EM) algorithm, as well as a method for estimating noise properties at the ML estimate. List-mode ML is of interest in cases where the dimensionality of the measurement space impedes a binning of the measurement data. It can be advantageous in cases where a better forward model can be obtained by including more measurement coordinates provided by a given detector. Different figures of merit for the detector performance can be computed from the Fisher information matrix (FIM). This paper uses the observed FIM, which requires a single data set, thus, avoiding costly ensemble statistics. The proposed techniques are demonstrated for an idealized two-dimensional (2-D) positron emission tomography (PET) [2-D PET] detector. We compute from simulation data the improved image quality obtained by including the time of flight of the coincident quanta.

Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions.

We continue the theme of previous papers [J. Opt. Soc. Am. A 7, 1266 (1990); 12, 834 (1995)] on objective (task-based) assessment of image quality. We concentrate on signal-detection tasks and figures of merit related to the ROC (receiver operating characteristic) curve. Many different expressions for the area under an ROC curve (AUC) are derived for an arbitrary discriminant function, with different assumptions on what information about the discriminant function is available. In particular, it is shown that AUC can be expressed by a principal-value integral that involves the characteristic functions of the discriminant. Then the discussion is specialized to the ideal observer, defined as one who uses the likelihood ratio (or some monotonic transformation of it, such as its logarithm) as the discriminant function. The properties of the ideal observer are examined from first principles. Several strong constraints on the moments of the likelihood ratio or the log likelihood are derived, and it is shown that the probability density functions for these test statistics are intimately related. In particular, some surprising results are presented for the case in which the log likelihood is normally distributed under one hypothesis. To unify these considerations, a new quantity called the likelihood-generating function is defined. It is shown that all moments of both the likelihood and the log likelihood under both hypotheses can be derived from this one function. Moreover, the AUC can be expressed, to an excellent approximation, in terms of the likelihood-generating function evaluated at the origin. This expression is the leading term in an asymptotic expansion of the AUC; it is exact whenever the likelihood-generating function behaves linearly near the origin. It is also shown that the likelihood-generating function at the origin sets a lower bound on the AUC in all cases.