Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio

Invertibility of multi-energy X-ray transform

PURPOSE

The goal is to provide a sufficient condition for the invertibility of a multi-energy (ME) X-ray transform. The energy-dependent X-ray attenuation profiles can be represented by a set of coefficients using the Alvarez-Macovski (AM) method. An ME X-ray transform is a mapping from N AM coefficients to N noise-free energy-weighted measurements, where N ≥ 2 .

METHODS

We apply a general invertibility theorem to prove the equivalence of global and local invertibility for an ME X-ray transform. We explore the global invertibility through testing whether the Jacobian of the mapping J ( A ) has zero values over the support of the mapping. The Jacobian of an arbitrary ME X-ray transform is an integration over all spectral measurements. A sufficient condition for J ( A ) ≠ 0 for all A is that the integrand of J ( A ) is ≥ 0 (or ≤ 0 ) everywhere. Note that the trivial case of the integrand equals 0 everywhere is ignored. Using symmetry, we simplified the integrand of the Jacobian to three factors that are determined by the total attenuation, the basis functions, and the energy-weighting functions, respectively. The factor related to the total attenuation is always positive; hence, the invertibility of the X-ray transform can be determined by testing the signs of the other two factors. Furthermore, we use the Cramér-Rao lower bound (CRLB) to characterize the noise-induced estimation uncertainty and provide a maximum-likelihood (ML) estimator.

RESULTS

The factor related to the basis functions is always negative when the photoelectric/Compton/Rayleigh basis functions are used and K-edge materials are not considered. The sign of the energy-weighting factor depends on the system source spectra and the detector response functions. For four special types of X-ray detectors, the sign of this factor stays the same over the integration range. Therefore, when these four types of detectors are used for imaging non-K-edge materials, the ME X-ray transform is globally invertible. The same framework can be used to study an arbitrary ME X-ray imaging system, for example, when K-edge materials are present. Furthermore, the ML estimator we presented is an unbiased, efficient estimator and can be used for a wide range of scenes.

CONCLUSIONS

We have provided a framework to study the invertibility of an arbitrary ME X-ray transform and proved the global invertibility for four types of systems.

Evaluation of imaging systems using the posterior variance of emission counts

We investigate an approach to evaluation of emission-tomography (ET) imaging systems used for region-of-interest (ROI) estimation tasks. In the evaluation we employ the concept of "emission counts" (EC), which are the number of events per voxel emitted during a scan. We use the reduction in posterior variance of ROI EC, compared to the prior ROI EC variance, as the metric of primary interest, which we call the "posterior variance reduction index" (PVRI). Systems that achieve a higher PVRI are considered superior to systems with lower PVRI. The approach is independent of the reconstruction method and is applicable to all photon-limited data types including list-mode data. We analyzed this approach using a model of 2-D tomography, and compared our results to the classical theory of tomographic sampling. We found that performance evaluations using the PVRI index were consistent with the classical theory. System evaluation based on EC posterior variance is an intuitively appealing and physically meaningful method that is useful for evaluation of system performance in ROI quantitation tasks.

Scanning linear estimation: improvements over region of interest (ROI) methods

In tomographic medical imaging, a signal activity is typically estimated by summing voxels from a reconstructed image. We introduce an alternative estimation scheme that operates on the raw projection data and offers a substantial improvement, as measured by the ensemble mean-square error (EMSE), when compared to using voxel values from a maximum-likelihood expectation-maximization (MLEM) reconstruction. The scanning-linear (SL) estimator operates on the raw projection data and is derived as a special case of maximum-likelihood estimation with a series of approximations to make the calculation tractable. The approximated likelihood accounts for background randomness, measurement noise and variability in the parameters to be estimated. When signal size and location are known, the SL estimate of signal activity is unbiased, i.e. the average estimate equals the true value. By contrast, unpredictable bias arising from the null functions of the imaging system affect standard algorithms that operate on reconstructed data. The SL method is demonstrated for two different tasks: (1) simultaneously estimating a signal's size, location and activity; (2) for a fixed signal size and location, estimating activity. Noisy projection data are realistically simulated using measured calibration data from the multi-module multi-resolution small-animal SPECT imaging system. For both tasks, the same set of images is reconstructed using the MLEM algorithm (80 iterations), and the average and maximum values within the region of interest (ROI) are calculated for comparison. This comparison shows dramatic improvements in EMSE for the SL estimates. To show that the bias in ROI estimates affects not only absolute values but also relative differences, such as those used to monitor the response to therapy, the activity estimation task is repeated for three different signal sizes.

Accuracy of the gaussian point spread function model in 2D localization microscopy

The gaussian function is simple and easy to implement as Point Spread Function (PSF) model for fitting the position of fluorescent emitters in localization microscopy. Despite its attractiveness the appropriateness of the gaussian is questionable as it is not based on the laws of optics. Here we study the effect of emission dipole orientation in conjunction with optical aberrations on the localization accuracy of position estimators based on a gaussian model PSF. Simulated image spots, calculated with all effects of high numerical aperture, interfaces between media, polarization, dipole orientation and aberrations taken into account, were fitted with a gaussian PSF based Maximum Likelihood Estimator. For freely rotating dipole emitters it is found that the gaussian works fine. The same, theoretically optimum, localization accuracy is found as if the true PSF were a gaussian, even for aberrations within the usual tolerance limit of high-end optical imaging systems such as microscopes (Marechal's diffraction limit). For emitters with a fixed dipole orientation this is not the case. Localization errors are found that reach up to 40 nm for typical system parameters and aberration levels at the diffraction limit. These are systematic errors that are independent of the total photon count in the image. The gaussian function is therefore inappropriate, and more sophisticated PSF models are a practical necessity.

Quantitative imaging in breast tomosynthesis and CT: comparison of detection and estimation task performance

PURPOSE

This work investigates a framework for modeling volumetric breast imaging to compare detection and estimation task performance and optimize quantitative breast imaging.

METHODS

Volumetric reconstructions of a breast phantom, which incorporated electronic, quantum, and anatomical noise with embedded spherical lesions, were simulated over a range of acquisition angles varying from 4 degrees to 204 degrees with a constant total acquisition dose of 1.5 mGy. A maximum likelihood estimator was derived in terms of the noise power spectrum, which yielded figures of merit for quantitative imaging performance in terms of accuracy and precision. These metrics were computed for estimation of lesion area, volume, and location. Estimation task performance was optimized as a function of acquisition angle and compared to the performance of a more conventional lesion detection task.

RESULTS

Results revealed tradeoffs between electronic, quantum, and anatomical noise. The detection of a 4 mm sphere was optimal at an acquisition angle of 84 degrees, where reconstructed images using a smaller acquisition angle exhibited increased anatomical noise and reconstructed images using a larger acquisition angle exhibited increased quantum and electronic noise. For all estimation tasks, accuracy was found to be fairly constant as a function acquisition angle indicating adequate system calibration, whereas a more significant dependence on acquisition angle was observed for precision performance. Precision for the 2D area estimation task was optimal at approximately 104 degrees, while precision of the 3D volume estimation task was optimal at larger angles (approximately 124 degrees). Precision for the localization task showed orientation dependence where localization was significantly inferior in the depth direction. Overall, precision for localization was optimal at larger angles (i.e., > 125 degrees) compared to the size estimation tasks. Results suggested that for quantitative imaging tasks, the acquisition angle should be larger than currently used in conventional breast tomosynthesis for lesion detection.

CONCLUSIONS

Analysis of quantitative imaging performance using Fourier-based metrics highlights the difference between estimation and detection task in volumetric breast imaging and provides a meaningful framework for optimizing the performance of breast imaging systems for quantitative imaging applications.

Noise considerations of three-point water-fat separation imaging methods

Separation of water from fat tissues in magnetic resonance imaging is important for many applications because signals from fat tissues often interfere with diagnoses that are usually based on water signal characteristics. Water and fat can be separated with images acquired at different echo time shifts. The three-point method solves for the unknown off-resonance frequency together with the water and fat densities. Noise performance of the method, quantified by the effective number of signals averaged (NSA), is an important metric of the water and fat images. The authors use error propagation theory and Monte Carlo simulation to investigate two common reconstructive approaches: an analytic-solution based estimation and a least-squares estimation. Two water-fat chemical shift (CS) encoding strategies, the symmetric (-theta, 0, theta) and the shifted (0, theta, 2theta) schemes are studied and compared. Results show that NSAs of water and fat can be different and they are dependent on the ratio of intensities of the two species and each of the echo time shifts. The NSA is particularly poor for the symmetric (-theta, 0, theta) CS encoding when the water and fat signals are comparable. This anomaly with equal amounts of water and fat is analyzed in a more intuitive geometric illustration. Theoretical prediction of NSA matches well with simulation results at high signal-to-noise ratio (SNR), while deviation arises at low SNR, which suggests that Monte Carlo simulation may be more appropriate to accurately predict noise performance of the algorithm when SNR is low.

A priori information and optimisation in polarimetry

Polarimetric measurements are designed to obtain information pertaining to the system under study, however noise in the system limits the precision and hence information obtainable. Exploitation of a priori knowledge of the system allows for an improvement in the precision of experimental data. In this vein we present a framework for system design and optimisation based upon the Fisher information matrix, which allows easy incorporation of such a priori information. As such the proposed figure of merit is more complete than the commonly used condition number. Conditions of equivalence are considered, however a number of examples highlight the failings of the condition number under more general scenarios. Bounds on the achievable informational gains via multiple polarimeter arms are also given. Finally we present analytic results concerning error distribution in a Mueller matrix polar decomposition, allowing for a more accurate noise analysis in polarimetric experiments.

Estimating random signal parameters from noisy images with nuisance parameters: linear and scanning-linear methods

In a pure estimation task, an object of interest is known to be present, and we wish to determine numerical values for parameters that describe the object. This paper compares the theoretical framework, implementation method, and performance of two estimation procedures. We examined the performance of these estimators for tasks such as estimating signal location, signal volume, signal amplitude, or any combination of these parameters. The signal is embedded in a random background to simulate the effect of nuisance parameters. First, we explore the classical Wiener estimator, which operates linearly on the data and minimizes the ensemble mean-squared error. The results of our performance tests indicate that the Wiener estimator can estimate amplitude and shape once a signal has been located, but is fundamentally unable to locate a signal regardless of the quality of the image. Given these new results on the fundamental limitations of Wiener estimation, we extend our methods to include more complex data processing. We introduce and evaluate a scanning-linear estimator that performs impressively for location estimation. The scanning action of the estimator refers to seeking a solution that maximizes a linear metric, thereby requiring a global-extremum search. The linear metric to be optimized can be derived as a special case of maximum a posteriori (MAP) estimation when the likelihood is Gaussian and a slowly varying covariance approximation is made.

Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images

When building an imaging system for detection tasks in medical imaging, we need to evaluate how well the system performs before we can optimize it. One way to do the evaluation is to calculate the performance of the Bayesian ideal observer. The ideal-observer performance is often computationally expensive, and it is very useful to have an approximation to it. We use a parameterized probability density function to represent the corresponding densities of data under the signal-absent and the signal-present hypotheses. We develop approximations to the ideal-observer detectability as a function of signal parameters involving the Fisher information matrix, which is normally used in parameter estimation problems. The accuracy of the approximation is illustrated in analytical examples and lumpy-background simulations. We are able to predict the slope of the detectability as a function of the signal parameter. This capability suggests that the Fisher information matrix itself evaluated at the null parameter value can be used as the figure of merit in imaging system evaluation. We are also able to provide a theoretical foundation for the connection between detection tasks and estimation tasks.

Tracking transplanted cells using dual-radionuclide SPECT

The purpose of this study was to characterize the performance of single photon emission computed tomography (SPECT) in tasks associated with tracking transplanted cells. Previous studies identified matters of hardware design, whereas we focus on biological variables impacting system performance, such as cell colony growth and non-specific radiolabelling. Using experimental data, a digital phantom was developed of in vitro 111In-radiolabelled stem cells, transfected with a reporter gene, transplanted into canine infarcted myocardium and interrogated using a peripherally injected 131I-radiolabelled reporter probe. Single- and dual-head SPECT acquisition was simulated. Performance was characterized using an estimation task, where the precision of parameter estimates (111In and 131I radiolabel quantity, cell colony size and location, and background) was tracked as the phantom evolved to simulate 111In-label efflux, cell colony growth and improved reporter probe specificity. In vitro pre-labelling of transplanted cells improved precision of parameter estimates via a priori size and location information. Precision of radiolabel quantity estimates improved with cell colony growth, despite 111In radiolabel dilution; size and location parameters were influenced little. Precision of radiolabel quantity estimates improved with reduced reporter probe non-specific uptake. The performance of SPECT in cell tracking is influenced strongly by biological variables. These should be considered when planning experiments or developing SPECT technology for cell tracking.