Effect on null spaces of list-mode imaging systems due to increasing the number of attributes.

There are two types of uncertainty in image reconstructions from list-mode data: statistical and deterministic. One source of statistical uncertainty is the finite number of attributes of the detected particles, which are sampled from a probability distribution on the attribute space. A deterministic source of uncertainty is the effect that null functions of the imaging operator have on reconstructed pixel or voxel values. Quantifying the reduction in this deterministic source of uncertainty when more attributes are measured for each detected particle is the subject of this work. Specifically, upper bounds on an error metric are derived to quantify the error introduced in the reconstruction by the presence of null functions, and these upper bounds are shown to be reduced when the number of attributes is increased. These bounds are illustrated with an example of a two-dimensional single photon emission computed tomography (SPECT) system where the depth of interaction in the scintillation crystal is added to the attribute vector.

Performing tomographic reconstructions from a satellite looking toward Earth. Part 1: implementation and limitations.

For imaging instruments that are in space looking toward the Earth, there are a variety of nuisance signals that can get in the way of performing certain imaging tasks, such as reflections from clouds, reflections from the ground, and emissions from the OH-airglow layer. A method for separating these signals is to perform tomographic reconstructions from the collected data. A lingering struggle for this method is altitude-axis resolution and different methods for helping with it are discussed. An implementation of the maximum likelihood expectation maximization algorithm is given and analyzed.

Performing tomographic reconstructions from a satellite looking toward Earth. Part 2: analysis of image quality.

This paper is part 2 of two papers that explore performing tomographic reconstructions from a space platform. A simplified model of short-wave infrared emissions in the atmosphere is given. Simulations were performed that tested the effectiveness of reconstructions given signal amplitude, frequency, signal-to-noise ratio, number of iterations run, and others. Maximum likelihood expectation maximization is shown to be effective for reconstructing low signal cases.

Effect on null spaces of list-mode imaging systems due to increasing the size of attribute space.

An upper bound is derived for a figure of merit that quantifies the error in reconstructed pixel or voxel values induced by the presence of null functions for any list-mode system. It is shown that this upper bound decreases as the region in attribute space occupied by the allowable attribute vectors expands. This upper bound allows quantification of the reduction in this error when this type of expansion is implemented. Of course, reconstruction error is also caused by system noise in the data, which has to be treated statistically, but we will not be addressing that problem here. This method is not restricted to pixelized or voxelized reconstructions and can in fact be applied to any region of interest. The upper bound for pixelized reconstructions is demonstrated on a list-mode 2D Radon transform example. The expansion in the attribute space is implemented by doubling the number of views. The results show how the pixel size and number of views both affect the upper bound on reconstruction error from null functions. This reconstruction error can be averaged over all pixels to give a single number or can be plotted as a function on the pixel grid. Both approaches are demonstrated for the example system. In conclusion, this method can be applied to any list-mode system for which the system operator is known and could be used in the design of the systems and reconstruction algorithms.

Probability of error for detecting a change in a parameter and Bayesian Fisher information.

The van Trees inequality relates the ensemble mean squared error of an estimator to a Bayesian version of the Fisher information. The Ziv-Zakai inequality relates the ensemble mean squared error of an estimator to the minimum probability of error for the task of detecting a change in the parameter. In this work we complete this circle by deriving an inequality that relates this minimum probability of error to the Bayesian version of the Fisher information. We discuss this result for both scalar and vector parameters. In the process we discover that an important intermediary in the calculation is the total variation of the posterior probability distribution function for the parameter given the data. This total variation is of interest in its own right since it may be easier to compute than the other figures of merit discussed here.

Quantifying the loss of information from binning list-mode data.

List-mode data are increasingly being used in single photon emission computed tomography (SPECT) and positron emission tomography (PET) imaging, among other imaging modalities. However, there are still many imaging designs that effectively bin list-mode data before image reconstruction or other estimation tasks are performed. Intuitively, the binning operation should result in a loss of information. In this work, we show that this is true for Fisher information and provide a computational method for quantifying the information loss. In the end, we find that the information loss depends on three factors. The first factor is related to the smoothness of the mean data function for the list-mode data. The second factor is the actual object being imaged. Finally, the third factor is the binning scheme in relation to the other two factors.

Fisher information analysis of list-mode SPECT emission data for joint estimation of activity and attenuation distribution.

The potential to perform attenuation and scatter compensation (ASC) in single-photon emission computed tomography (SPECT) imaging without a separate transmission scan is highly significant. In this context, attenuation in SPECT is primarily due to Compton scattering, where the probability of Compton scatter is proportional to the attenuation coefficient of the tissue and the energy of the scattered photon and the scattering angle are related. Based on this premise, we investigated whether the SPECT scattered-photon data acquired in list-mode (LM) format and including the energy information can be used to estimate the attenuation map. For this purpose, we propose a Fisher-information-based method that yields the Cramer-Rao bound (CRB) for the task of jointly estimating the activity and attenuation distribution using only the SPECT emission data. In the process, a path-based formalism to process the LM SPECT emission data, including the scattered-photon data, is proposed. The Fisher information method was implemented on NVIDIA graphics processing units (GPU) for acceleration. The method was applied to analyze the information content of SPECT LM emission data, which contains up to first-order scattered events, in a simulated SPECT system with parameters modeling a clinical system using realistic computational studies with 2-D digital synthetic and anthropomorphic phantoms. The method was also applied to LM data containing up to second-order scatter for a synthetic phantom. Experiments with anthropomorphic phantoms simulated myocardial perfusion and dopamine transporter (DaT)-Scan SPECT studies. The results show that the CRB obtained for the attenuation and activity coefficients was typically much lower than the true value of these coefficients. An increase in the number of detected photons yielded lower CRB for both the attenuation and activity coefficients. Further, we observed that systems with better energy resolution yielded a lower CRB for the attenuation coefficient. Overall, the results provide evidence that LM SPECT emission data, including the scattered photons, contains information to jointly estimate the activity and attenuation coefficients.

Relation between Bayesian Fisher information and Shannon information for detecting a change in a parameter.

We derive a connection between the performance of statistical estimators and the performance of the ideal observer on related detection tasks. Specifically, we show how the task-specific Shannon information for the task of detecting a change in a parameter is related to the Fisher information and to the Bayesian Fisher information. We have previously shown that this Shannon information is related via an integral transform to the minimum probability of error on the same task. We then outline a circle of relations starting with this minimum probability of error and ensemble mean squared error for an estimator via the Ziv-Zakai inequality, then the ensemble mean squared error and the Bayesian Fisher information via the van Trees inequality, and finally the Bayesian Fisher information and the Shannon information for a detection task via the work presented here.

Optimal channels for channelized quadratic estimators.

We present a new method for computing optimized channels for estimation tasks that is feasible for high-dimensional image data. Maximum-likelihood (ML) parameter estimates are challenging to compute from high-dimensional likelihoods. The dimensionality reduction from M measurements to L channels is a critical advantage of channelized quadratic estimators (CQEs), since estimating likelihood moments from channelized data requires smaller sample sizes and inverting a smaller covariance matrix is easier. The channelized likelihood is then used to form ML estimates of the parameter(s). In this work we choose an imaging example in which the second-order statistics of the image data depend upon the parameter of interest: the correlation length. Correlation lengths are used to approximate background textures in many imaging applications, and in these cases an estimate of the correlation length is useful for pre-whitening. In a simulation study we compare the estimation performance, as measured by the root-mean-squared error (RMSE), of correlation length estimates from CQE and power spectral density (PSD) distribution fitting. To abide by the assumptions of the PSD method we simulate an ergodic, isotropic, stationary, and zero-mean random process. These assumptions are not part of the CQE formalism. The CQE method assumes a Gaussian channelized likelihood that can be a valid for non-Gaussian image data, since the channel outputs are formed from weighted sums of the image elements. We have shown that, for three or more channels, the RMSE of CQE estimates of correlation length is lower than conventional PSD estimates. We also show that computing CQE by using a standard nonlinear optimization method produces channels that yield RMSE within 2% of the analytic optimum. CQE estimates of anisotropic correlation length estimation are reported to demonstrate this technique on a two-parameter estimation problem.

Shannon information and receiver operating characteristic analysis for multiclass classification in imaging.

We show how Shannon information is mathematically related to receiver operating characteristic (ROC) analysis for multiclass classification problems in imaging. In particular, the minimum probability of error for the ideal observer, as a function of the prior probabilities for each class, determines the Shannon Information for the classification task, also considered as a function of the prior probabilities on the classes. In the process, we show how an ROC hypersurface that has been studied by other researchers is mathematically related to a Shannon information ROC (SIROC) hypersurface. In fact, the ROC hypersurface completely determines the SIROC hypersurface via a non-local integral transform on the ROC hypersurface. We also show that both hypersurfaces are convex and satisfy other geometrical relationships via the Legendre transform.

Shannon information for joint estimation/detection tasks and complex imaging systems.

Shannon information is defined for imaging tasks where signal detection is combined with parameter estimation. The first task considered is when the parameters are associated with the signal and parameter estimates are only produced when the signal is present. The second task examined is when the parameters are associated with the object being imaged, and parameter estimates are produced whether the signal is present or not. In each case, the Shannon information expression has a simple additive form.

Characteristic functionals in imaging and image-quality assessment: tutorial.

Characteristic functionals are one of the main analytical tools used to quantify the statistical properties of random fields and generalized random fields. The viewpoint taken here is that a random field is the correct model for the ensemble of objects being imaged by a given imaging system. In modern digital imaging systems, random fields are not used to model the reconstructed images themselves since these are necessarily finite dimensional. After a brief introduction to the general theory of characteristic functionals, many examples relevant to imaging applications are presented. The propagation of characteristic functionals through both a binned and list-mode imaging system is also discussed. Methods for using characteristic functionals and image data to estimate population parameters and classify populations of objects are given. These methods are based on maximum likelihood and maximum a posteriori techniques in spaces generated by sampling the relevant characteristic functionals through the imaging operator. It is also shown how to calculate a Fisher information matrix in this space. These estimators and classifiers, and the Fisher information matrix, can then be used for image quality assessment of imaging systems.

Correcting partial volume effects in biexponential T2 estimation of small lesions.

PURPOSE: T2 mapping provides a quantitative approach for focal liver lesion characterization. For small lesions, a biexponential model should be used to account for partial volume effects (PVE). However, conventional biexponential fitting suffers from large uncertainty of the fitted parameters when noise is present. The purpose of this work is to develop a more robust method to correct for PVE affecting small lesions. METHODS: We developed a region of interest-based joint biexponential fitting (JBF) algorithm to estimate the T2 of lesions affected by PVE. JBF takes advantage of the lesion fraction variation among voxels within a region of interest. JBF is compared to conventional approaches using Cramér-Rao lower bound analysis, numerical simulations, phantom, and in vivo data. RESULTS: JBF provides more accurate and precise T2 estimates in the presence of PVE. Furthermore, JBF is less sensitive to region of interest drawing. Phantom and in vivo results show that JBF can be combined with a reconstruction method for highly undersampled data, enabling the characterization of small abdominal lesions from data acquired in a single breath hold. CONCLUSION: The JBF algorithm provides more accurate and stable T2 estimates for small structures than conventional techniques when PVE is present. It should be particularly useful for the characterization of small abdominal lesions.

Shannon information and ROC analysis in imaging.

Shannon information (SI) and the ideal-observer receiver operating characteristic (ROC) curve are two different methods for analyzing the performance of an imaging system for a binary classification task, such as the detection of a variable signal embedded within a random background. In this work we describe a new ROC curve, the Shannon information receiver operator curve (SIROC), that is derived from the SI expression for a binary classification task. We then show that the ideal-observer ROC curve and the SIROC have many properties in common, and are equivalent descriptions of the optimal performance of an observer on the task. This equivalence is described mathematically by an integral transform that maps the ideal-observer ROC curve onto the SIROC. This then leads to an integral transform relating the minimum probability of error, as a function of the odds against a signal, to the conditional entropy, as a function of the same variable. This last relation then gives us the complete mathematical equivalence between ideal-observer ROC analysis and SI analysis of the classification task for a given imaging system. We also find that there is a close relationship between the area under the ideal-observer ROC curve, which is often used as a figure of merit for imaging systems and the area under the SIROC. Finally, we show that the relationships between the two curves result in new inequalities relating SI to ROC quantities for the ideal observer.

Impact of the Fano Factor on Position and Energy Estimation in Scintillation Detectors.

The Fano factor for an integer-valued random variable is defined as the ratio of its variance to its mean. Light from various scintillation crystals have been reported to have Fano factors from sub-Poisson (Fano factor < 1) to super-Poisson (Fano factor > 1). For a given mean, a smaller Fano factor implies a smaller variance and thus less noise. We investigated if lower noise in the scintillation light will result in better spatial and energy resolutions. The impact of Fano factor on the estimation of position of interaction and energy deposited in simple gamma-camera geometries is estimated by two methods - calculating the Cramér-Rao bound and estimating the variance of a maximum likelihood estimator. The methods are consistent with each other and indicate that when estimating the position of interaction and energy deposited by a gamma-ray photon, the Fano factor of a scintillator does not affect the spatial resolution. A smaller Fano factor results in a better energy resolution.

Phantom study of tear film dynamics with optical coherence tomography and maximum-likelihood estimation.

In this Letter, we implement a maximum-likelihood estimator to interpret optical coherence tomography (OCT) data for the first time, based on Fourier-domain OCT and a two-interface tear film model. We use the root mean square error as a figure of merit to quantify the system performance of estimating the tear film thickness. With the methodology of task-based assessment, we study the trade-off between system imaging speed (temporal resolution of the dynamics) and the precision of the estimation. Finally, the estimator is validated with a digital tear-film dynamics phantom.

Simulating Silicon Photomultiplier Response to Scintillation Light.

The response of a Silicon Photomultiplier (SiPM) to optical signals is affected by many factors including photon-detection efficiency, recovery time, gain, optical crosstalk, afterpulsing, dark count, and detector dead time. Many of these parameters vary with overvoltage and temperature. When used to detect scintillation light, there is a complicated non-linear relationship between the incident light and the response of the SiPM. In this paper, we propose a combined discrete-time discrete-event Monte Carlo (MC) model to simulate SiPM response to scintillation light pulses. Our MC model accounts for all relevant aspects of the SiPM response, some of which were not accounted for in the previous models. We also derive and validate analytic expressions for the single-photoelectron response of the SiPM and the voltage drop across the quenching resistance in the SiPM microcell. These analytic expressions consider the effect of all the circuit elements in the SiPM and accurately simulate the time-variation in overvoltage across the microcells of the SiPM. Consequently, our MC model is able to incorporate the variation of the different SiPM parameters with varying overvoltage. The MC model is compared with measurements on SiPM-based scintillation detectors and with some cases for which the response is known a priori. The model is also used to study the variation in SiPM behavior with SiPM-circuit parameter variations and to predict the response of a SiPM-based detector to various scintillators.

T2 mapping from highly undersampled data by reconstruction of principal component coefficient maps using compressed sensing.

Recently, there has been an increased interest in quantitative MR parameters to improve diagnosis and treatment. Parameter mapping requires multiple images acquired with different timings usually resulting in long acquisition times. While acquisition time can be reduced by acquiring undersampled data, obtaining accurate estimates of parameters from undersampled data is a challenging problem, in particular for structures with high spatial frequency content. In this work, principal component analysis is combined with a model-based algorithm to reconstruct maps of selected principal component coefficients from highly undersampled radial MRI data. This novel approach linearizes the cost function of the optimization problem yielding a more accurate and reliable estimation of MR parameter maps. The proposed algorithm--reconstruction of principal component coefficient maps using compressed sensing--is demonstrated in phantoms and in vivo and compared with two other algorithms previously developed for undersampled data.

Asymptotic ideal observers and surrogate figures of merit for signal detection with list-mode data.

The asymptotic form for the likelihood ratio is derived for list-mode data generated by an imaging system viewing a possible signal in a randomly generated background. This calculation provides an approximation to the likelihood ratio that is valid in the limit of large number of list entries, i.e., a large number of photons. These results are then used to derive surrogate figures of merit, quantities that are correlated with ideal-observer performance on detection tasks, as measured by the area under the receiver operating characteristic curve, but are easier to compute. A key component of these derivations is the determination of asymptotic forms for the Fisher information for the signal amplitude in the limit of a large number of counts or a long exposure time. This quantity is useful in its own right as a figure of merit (FOM) for the task of estimating the signal amplitude. The use of the Fisher information in detection tasks is based on the fact that it provides an approximation for ideal-observer detectability when the signal is weak. For both the fixed-count and fixed-time cases, four surrogate figures of merit are derived. Two are based on maximum likelihood reconstructions; one uses the characteristic functional of the random backgrounds. The fourth surrogate FOM is identical in the two cases and involves an integral over attribute space for each of a randomly generated sequence of backgrounds.

Simulating photon-transport in uniform media using the radiative transport equation: a study using the Neumann-series approach.

We present the implementation, validation, and performance of a Neumann-series approach for simulating light propagation at optical wavelengths in uniform media using the radiative transport equation (RTE). The RTE is solved for an anisotropic-scattering medium in a spherical harmonic basis for a diffuse-optical-imaging setup. The main objectives of this paper are threefold: to present the theory behind the Neumann-series form for the RTE, to design and develop the mathematical methods and the software to implement the Neumann series for a diffuse-optical-imaging setup, and, finally, to perform an exhaustive study of the accuracy, practical limitations, and computational efficiency of the Neumann-series method. Through our results, we demonstrate that the Neumann-series approach can be used to model light propagation in uniform media with small geometries at optical wavelengths.