Linear response theory for detectors consisting of discrete arrays

Nonstationary model of oblique x-ray incidence in amorphous selenium detectors: II. Transfer functions

PURPOSE

One limitation of experimental techniques for quantifying resolution and noise in detectors is that the measurement is made in a region-of-interest (ROI). With theoretical modeling, these properties can be measured at a point, allowing for quantification of spatial anisotropy. This paper calculates nonstationary transfer functions for amorphous selenium (a-Se) detectors in breast imaging. We use this model to demonstrate the performance advantage of a "next-generation" tomosynthesis (NGT) system, which is capable of x-ray source motion with more degrees of freedom than a clinical tomosynthesis system.

METHODS

Using Swank's formulation, the optical transfer function (OTF) and presampled noise power spectra (NPS) are determined based on the point spread function derived in Part 1. The modulation transfer function (MTF) is found from the normalized modulus of the OTF. To take into account the presence of digitization, the presampled NPS is convolved with a two-dimensional comb function, for which the period along each direction is the reciprocal of the detector element size. The detective quantum efficiency (DQE) is then determined from combined knowledge of the OTF and NPS.

RESULTS

First, the model is used to demonstrate the loss of image quality due to oblique x-ray incidence. The MTF is calculated along various polar angles, corresponding to different orientations of the input frequency. The MTF is independent of the incidence angle if the polar angle is perpendicular to the ray incidence direction. However, along other polar angles, oblique incidence results in MTF degradation at high frequencies. The MTF degradation is most substantial along the ray incidence direction. Unlike the MTF, the normalized NPS (NNPS) is independent of the incidence angle. To measure the relative signal-to-noise, the DQE is also calculated. Oblique incidence yields high-frequency DQE degradation, which is more pronounced than the MTF degradation. This arises because the DQE is proportionate with the square of the MTF. Ultimately, this model is used to evaluate how the image quality varies over the detector area. For various projection images, we calculate the variation in the incidence angle over this area. With the NGT system, the source can be positioned in such a way that this variation is minimized, and hence the DQE exhibits less anisotropy. To achieve this improvement in the image quality, the source needs to have a component of motion in the posteroanterior (PA) direction, which is perpendicular to the conventional direction of source motion in tomosynthesis.

CONCLUSIONS

In a-Se detectors, the DQE at high frequencies is degraded due to oblique incidence. The DQE degradation is more pronounced than the MTF degradation. This model is used to quantify the spatial variation in DQE over the detector area. The use of PA source motion is a strategy for minimizing this variation and thus improving the image quality.

Oblique reconstructions in tomosynthesis. I. Linear systems theory

PURPOSE

By convention, slices in a tomosynthesis reconstruction are created on planes parallel to the detector. It has not yet been demonstrated that slices can be generated along oblique directions through the same volume, analogous to multiplanar reconstructions in computed tomography (CT). The purpose of this work is to give a proof-of-principle justification for oblique reconstructions in tomosynthesis, which acquires projection images over a smaller angular range than CT.

METHODS

To investigate the visibility of individual frequencies in an oblique reconstruction, a theoretical framework is developed in which the reconstruction of a sinusoidal input is calculated. The test frequency is pitched at an angle in a 2D parallel-beam acquisition geometry. Reconstructions are evaluated along the pitch of the object. The modulation transfer function (MTF) is calculated from the relative signal at various test frequencies. The MTF determines whether modulation is within detectable limits in oblique reconstructions. In the previous linear systems (LS) model [B. Zhao and W. Zhao, "Three-dimensional linear system analysis for breast tomosynthesis," Med. Phys. 35(12), 5219-5232 (2008)], the MTF was calculated only in reconstructed slices parallel to the detector. This paper generalizes the MTF calculation to reconstructed slices at all possible pitches. Unlike the previous LS model, this paper also analyzes the effect of object thickness on the MTF. A second test object that is considered is a rod whose long axis is pitched similar to the sinusoidal input. The rod is used to assess whether the length of an object can be correctly estimated in oblique reconstructions.

RESULTS

To simulate the conventional display of the reconstruction, slices are first created along a 0° pitch. This direction is perpendicular to the rays of the central projection. The authors show that the input frequency of a pitched sinusoidal object cannot be determined using these slices. By changing the pitch of the slice to match the object, it is shown that the input frequency is properly resolved. To prove that modulation is preserved in pitched slices, the MTF is also calculated. Modulation is within detectable limits over a broad range of pitches if the object is thin, but is detectable over a narrower range of pitches if the object is thick. Turning next to the second test object, it is shown that the length of a pitched rod can be correctly determined in oblique reconstructions. Concordant with the behavior of the MTF, the length estimate is accurate over a broad range of pitches if the object is thin, but is correct over a narrower range of pitches if the object is thick.

CONCLUSIONS

This work justifies the feasibility of oblique reconstructions in tomosynthesis. It is demonstrated that pitched test objects are most easily visualized with oblique reconstructions instead of conventional reconstructions. In order to achieve high image quality over a broad range of pitches, the object must be thin. By analyzing the effect of reconstruction pitch and object thickness on image quality, this paper generalizes the previous LS model for tomosynthesis.

Observation of super-resolution in digital breast tomosynthesis

PURPOSE

Digital breast tomosynthesis (DBT) is a 3D x-ray imaging modality in which tomographic sections of the breast are generated from a limited range of tube angles. Because oblique x-ray incidence shifts the image of an object in subpixel detector element increments with each increasing projection angle, it is demonstrated that DBT is capable of super-resolution (i.e., subpixel resolution).

METHODS

By convention, DBT reconstructions are performed on planes parallel to the breast support at various depths of the breast volume. In order for resolution in each reconstructed slice to be comparable to the detector, the pixel size should match that of the detector elements; hence, the highest frequency that can be resolved in the plane of reconstruction is the alias frequency of the detector. This study considers reconstruction grids with much smaller pixelation to visualize higher frequencies. For analytical proof of super-resolution, a theoretical framework is developed in which the reconstruction of a high frequency sinusoidal input is calculated using both simple backprojection (SBP) and filtered backprojection. To study the frequency spectrum of the reconstruction, its Fourier transform is also determined. The experimental feasibility of super-resolution was investigated by acquiring images of a bar pattern phantom with frequencies higher than the detector alias frequency.

RESULTS

Using analytical modeling, it is shown that the central projection cannot resolve frequencies exceeding the detector alias frequency. The Fourier transform of the central projection is maximized at a lower frequency than the input as evidence of aliasing. By contrast, SBP reconstruction can resolve the input, and its Fourier transform is correctly maximized at the input frequency. Incorporating filters into the reconstruction smoothens pixelation artifacts in the spatial domain and reduces spectral leakage in the Fourier domain. It is also demonstrated that the existence of super-resolution is dependent on position in the reconstruction and on the directionality of the input frequency. Consistent with the analytical results, experimental reconstructions of bar patterns showed visibility of frequencies greater than the detector alias frequency. Super-resolution was present at positions predicted from analytical modeling.

CONCLUSIONS

This work demonstrates the existence of super-resolution in DBT. Super-resolution has the potential to impact the visualization of fine structural details in the breast, such as microcalcifications and other subtle signs of cancer.

To bin or not to bin? The effect of CT system limiting resolution on noise and detectability

We examine the noise advantages of having a computed tomography (CT) detector whose spatial resolution is significantly better (e.g. a factor of 2) than needed for a desired resolution in the reconstructed images. The effective resolution of detectors in x-ray CT is sometimes degraded by binning cells because the small cell size and fine sampling are not needed to achieve the desired resolution (e.g. with flat panel detectors). We studied the effect of the binning process on the noise in the reconstructed images and found that while the images in the absence of noise can be made identical for the native and the binned system, for the same system MTF in the presence of noise, the binned system always results in noisier reconstructed images. The effect of the increased noise in the reconstructed images on lesion detection is scale (frequency content) dependent with a larger difference between the high resolution and binned systems for imaging fine structure (small objects). We show simulated images reconstructed with both systems for representative objects and quantify the impact of the noise on the detection of the lesions based on mathematical observers. Through both subjective assessment of the reconstructed images and the quantification using mathematical observers, we show that for a CT system where the photon noise is dominant, higher resolution in the detectors leads to better noise performance in the reconstructed images at any resolution.

Ideal-observer detectability in photon-counting differential phase-contrast imaging using a linear-systems approach

PURPOSE

To provide a cascaded-systems framework based on the noise-power spectrum (NPS), modulation transfer function (MTF), and noise-equivalent number of quanta (NEQ) for quantitative evaluation of differential phase-contrast imaging (Talbot interferometry) in relation to conventional absorption contrast under equal-dose, equal-geometry, and, to some extent, equal-photon-economy constraints. The focus is a geometry for photon-counting mammography.

METHODS

Phase-contrast imaging is a promising technology that may emerge as an alternative or adjunct to conventional absorption contrast. In particular, phase contrast may increase the signal-difference-to-noise ratio compared to absorption contrast because the difference in phase shift between soft-tissue structures is often substantially larger than the absorption difference. We have developed a comprehensive cascaded-systems framework to investigate Talbot interferometry, which is a technique for differential phase-contrast imaging. Analytical expressions for the MTF and NPS were derived to calculate the NEQ and a task-specific ideal-observer detectability index under assumptions of linearity and shift invariance. Talbot interferometry was compared to absorption contrast at equal dose, and using either a plane wave or a spherical wave in a conceivable mammography geometry. The impact of source size and spectrum bandwidth was included in the framework, and the trade-off with photon economy was investigated in some detail. Wave-propagation simulations were used to verify the analytical expressions and to generate example images.

RESULTS

Talbot interferometry inherently detects the differential of the phase, which led to a maximum in NEQ at high spatial frequencies, whereas the absorption-contrast NEQ decreased monotonically with frequency. Further, phase contrast detects differences in density rather than atomic number, and the optimal imaging energy was found to be a factor of 1.7 higher than for absorption contrast. Talbot interferometry with a plane wave increased detectability for 0.1-mm tumor and glandular structures by a factor of 3-4 at equal dose, whereas absorption contrast was the preferred method for structures larger than ∼0.5 mm. Microcalcifications are small, but differ from soft tissue in atomic number more than density, which is favored by absorption contrast, and Talbot interferometry was barely beneficial at all within the resolution limit of the system. Further, Talbot interferometry favored detection of "sharp" as opposed to "smooth" structures, and discrimination tasks by about 50% compared to detection tasks. The technique was relatively insensitive to spectrum bandwidth, whereas the projected source size was more important. If equal photon economy was added as a restriction, phase-contrast efficiency was reduced so that the benefit for detection tasks almost vanished compared to absorption contrast, but discrimination tasks were still improved close to a factor of 2 at the resolution limit.

CONCLUSIONS

Cascaded-systems analysis enables comprehensive and intuitive evaluation of phase-contrast efficiency in relation to absorption contrast under requirements of equal dose, equal geometry, and equal photon economy. The benefit of Talbot interferometry was highly dependent on task, in particular detection versus discrimination tasks, and target size, shape, and material. Requiring equal photon economy weakened the benefit of Talbot interferometry in mammography.

Beyond noise power in 3D computed tomography: the local NPS and off-diagonal elements of the Fourier domain covariance matrix

PURPOSE

To investigate the correlation and stationarity of noise in volumetric computed tomography (CT) using the local discrete noise-power spectrum (NPS) and off-diagonal elements of the covariance matrix of the discrete Fourier transform of noise-only images (denoted Σ(DFT)). Experimental conditions were varied to affect noise correlation and stationarity, the effects were quantified in terms of the NPS and Σ(DFT), and practical considerations in CT performance characterization were identified.

METHODS

Cone-beam CT (CBCT) images were acquired using a benchtop system comprising an x-ray tube and flat-panel detector for a range of acquisition techniques (e.g., dose and x-ray scatter) and three phantom configurations hypothesized to impart distinct effects on the NPS and Σ(DFT): (A) air, (B) a 20-cm-diameter water cylinder with a bowtie filter, and (C) the cylinder without a bowtie filter. The NPS and off-diagonal elements of the Σ(DFT) were analyzed as a function of position within the reconstructions.

RESULTS

The local NPS varied systematically throughout the axial plane in a manner consistent with changes in fluence transmitted to the detector and view sampling effects. Variability in fluence was manifest in the NPS magnitude-e.g., a factor of ~2 variation in NPS magnitude within the axial plane for case C (cylinder without bowtie), compared to nearly constant NPS magnitude for case B (bowtie filter matched to the cylinder). View sampling effects were most prominent in case A (air) where the variance increased at greater distance from the center of reconstruction and in case C (cylinder) where the NPS exhibited correlations in the radial direction. The effects of detector lag were observed as azimuthal correlation. The cylinder (without bowtie) had the strongest nonstationarity because of the larger variability in fluence transmitted to the detector. The diagonal elements of the Σ(DFT) were equivalent to the NPS estimated from the periodogram, and the average off-diagonal elements of the Σ(DFT) exhibited amplitude of ~1% of the NPS for the experimental conditions investigated. Furthermore, the off-diagonal elements demonstrated fairly long tails of nearly constant amplitude, with magnitude somewhat reduced for experimental conditions associated with greater stationarity (viz., lower Σ(DFT) tails for cases A and B in comparison to case C).

CONCLUSIONS

Volumetric CT exhibits nonstationarity in the NPS as hypothesized in relation to fluence uniformity and view sampling. Measurement of the NPS should seek to minimize such changes in noise correlations and include careful reporting of experimental conditions (e.g., phantom design and use of a bowtie filter) and spatial dependence (e.g., analysis at fixed radius within a phantom). Off-diagonal elements of the Σ(DFT) similarly depend on experimental conditions and can be readily computed from the same data as the NPS. This work begins to check assumptions in NPS analysis examine the extent to which NPS is an appropriate descriptor of noise correlations, and investigate the magnitude of off-diagonal elements of the Σ(DFT). While the magnitude of such off-diagonal elements appears to be low, their cumulative effect on space-variant detectability remains to be investigated-e.g., using task-specific figures of merit.

Optimization of phosphor-based detector design for oblique x-ray incidence in digital breast tomosynthesis

PURPOSE

In digital breast tomosynthesis (DBT), a volumetric reconstruction of the breast is generated from a limited range of x-ray projections. One trade-off of DBT is resolution loss in the projections due to non-normal (i.e., oblique) x-ray incidence. Although degradation in image quality due to oblique incidence has been studied using empirical data and Monte Carlo simulations, a theoretical treatment has been lacking. The purpose of this work is to extend Swank's calculations of the transfer functions of turbid granular phosphors to oblique incidence. The model is ultimately used as a tool for optimizing the design of DBT detectors.

METHODS

A quantum-limited system and 20 keV x-rays are considered. Under these assumptions, the modulation transfer function (MTF) and noise power spectra (NPS) are derived using the diffusion approximation to the Boltzmann equation to model optical scatter within the phosphor. This approach is applicable to a nonstructured scintillator such as gadolinium oxysulfide doped with terbium (Gd(2)O(2)S:Tb), which is commonly used in breast imaging and which can reasonably approximate other detector materials. The detective quantum efficiency (DQE) is then determined from the Nishikawa formulation, where it is written as the product of the x-ray quantum detection efficiency, the Swank factor, and the Lubberts fraction. Transfer functions are calculated for both front- and back-screen configurations, which differ by positioning the photocathode at the exit or entrance point of the x-ray beam, respectively.

RESULTS

In the front-screen configuration, MTF and DQE are found to have considerable angular dependence, while NPS is shown to vary minimally with projection angle. As expected, the high frequency MTF and DQE are degraded substantially at large angles. By contrast, all transfer functions for the back-screen configuration have the advantage of significantly less angular dependence. Using these models, we investigated the possibility for optimizing the design of DBT detectors. As an example optimization strategy, the phosphor thickness which maximizes the DQE at a fixed frequency is analyzed. This work demonstrates that the optimal phosphor thickness for the front-screen is angularly dependent, shifting to lower thickness at higher angles. Conversely, the back-screen is not optimized by a single thickness but instead attains reasonably high DQE values over a large range of thicknesses. Although the back-screen configuration is not suited for current detectors using a glass substrate, it may prove to be preferred in future detectors using newly proposed plastic thin-film transistor (TFT) substrates.

CONCLUSIONS

Using the diffusion approximation to the Boltzmann equation to model the spread of light in a scintillator, this paper develops an analytical model of MTF, NPS, and DQE for a phosphor irradiated obliquely. The model is set apart from other studies on oblique incidence in being derived from first principles. This work has applications in the optimization of DBT detector design.

Task-based modeling and optimization of a cone-beam CT scanner for musculoskeletal imaging

PURPOSE

This work applies a cascaded systems model for cone-beam CT imaging performance to the design and optimization of a system for musculoskeletal extremity imaging. The model provides a quantitative guide to the selection of system geometry, source and detector components, acquisition techniques, and reconstruction parameters.

METHODS

The model is based on cascaded systems analysis of the 3D noise-power spectrum (NPS) and noise-equivalent quanta (NEQ) combined with factors of system geometry (magnification, focal spot size, and scatter-to-primary ratio) and anatomical background clutter. The model was extended to task-based analysis of detectability index (d') for tasks ranging in contrast and frequency content, and d' was computed as a function of system magnification, detector pixel size, focal spot size, kVp, dose, electronic noise, voxel size, and reconstruction filter to examine trade-offs and optima among such factors in multivariate analysis. The model was tested quantitatively versus the measured NPS and qualitatively in cadaver images as a function of kVp, dose, pixel size, and reconstruction filter under conditions corresponding to the proposed scanner.

RESULTS

The analysis quantified trade-offs among factors of spatial resolution, noise, and dose. System magnification (M) was a critical design parameter with strong effect on spatial resolution, dose, and x-ray scatter, and a fairly robust optimum was identified at M ∼ 1.3 for the imaging tasks considered. The results suggested kVp selection in the range of ∼65-90 kVp, the lower end (65 kVp) maximizing subject contrast and the upper end maximizing NEQ (90 kVp). The analysis quantified fairly intuitive results-e.g., ∼0.1-0.2 mm pixel size (and a sharp reconstruction filter) optimal for high-frequency tasks (bone detail) compared to ∼0.4 mm pixel size (and a smooth reconstruction filter) for low-frequency (soft-tissue) tasks. This result suggests a specific protocol for 1 × 1 (full-resolution) projection data acquisition followed by full-resolution reconstruction with a sharp filter for high-frequency tasks along with 2 × 2 binning reconstruction with a smooth filter for low-frequency tasks. The analysis guided selection of specific source and detector components implemented on the proposed scanner. The analysis also quantified the potential benefits and points of diminishing return in focal spot size, reduced electronic noise, finer detector pixels, and low-dose limits of detectability. Theoretical results agreed quantitatively with the measured NPS and qualitatively with evaluation of cadaver images by a musculoskeletal radiologist.

CONCLUSIONS

A fairly comprehensive model for 3D imaging performance in cone-beam CT combines factors of quantum noise, system geometry, anatomical background, and imaging task. The analysis provided a valuable, quantitative guide to design, optimization, and technique selection for a musculoskeletal extremities imaging system under development.

A comparative analysis of OTF, NPS, and DQE in energy integrating and photon counting digital x-ray detectors

PURPOSE

One of the benefits of photon counting (PC) detectors over energy integrating (EI) detectors is the absence of many additive noise sources, such as electronic noise and secondary quantum noise. The purpose of this work is to demonstrate that thresholding voltage gains to detect individual x rays actually generates an unexpected source of white noise in photon counters.

METHODS

To distinguish the two detector types, their point spread function (PSF) is interpreted differently. The PSF of the energy integrating detector is treated as a weighting function for counting x rays, while the PSF of the photon counting detector is interpreted as a probability. Although this model ignores some subtleties of real imaging systems, such as scatter and the energy-dependent amplification of secondary quanta in indirect-converting detectors, it is useful for demonstrating fundamental differences between the two detector types. From first principles, the optical transfer function (OTF) is calculated as the continuous Fourier transform of the PSF, the noise power spectra (NPS) is determined by the discrete space Fourier transform (DSFT) of the autocovariance of signal intensity, and the detective quantum efficiency (DQE) is found from combined knowledge of the OTF and NPS. To illustrate the calculation of the transfer functions, the PSF is modeled as the convolution of a Gaussian with the product of rect functions. The Gaussian reflects the blurring of the x-ray converter, while the rect functions model the sampling of the detector.

RESULTS

The transfer functions are first calculated assuming outside noise sources such as electronic noise and secondary quantum noise are negligible. It is demonstrated that while OTF is the same for two detector types possessing an equivalent PSF, a frequency-independent (i.e., "white") difference in their NPS exists such that NPS(PC) > or = NPS(EI) and hence DQE(PC) < or = DQE(EI). The necessary and sufficient condition for equality is that the PSF is a binary function given as zero or unity everywhere. In analyzing the model detector with Gaussian blurring, the difference in NPS and DQE between the two detector types is found to increase with the blurring of the x-ray converter. Ultimately, the expression for the additive white noise of the photon counter is compared against the expression for electronic noise and secondary quantum noise in an energy integrator. Thus, a method is provided to determine the average secondary quanta that the energy integrator must produce for each x ray to have superior DQE to a photon counter with the same PSF.

CONCLUSIONS

This article develops analytical models of OTF, NPS, and DQE for energy integrating and photon counting digital x-ray detectors. While many subtleties of real imaging systems have not been modeled, this work is illustrative in demonstrating an additive source of white noise in photon counting detectors which has not yet been described in the literature. One benefit of this analysis is a framework for determining the average secondary quanta that an energy integrating detector must produce for each x ray to have superior DQE to competing photon counting technology.

The Generalized NEQ and Detectability Index for Tomosynthesis and Cone-Beam CT: From Cascaded Systems Analysis to Human Observers

PURPOSE

In the early development of new imaging modalities - such as tomosynthesis and cone-beam CT (CBCT) - an accurate predictive model for imaging performance is particularly valuable in identifying the physical factors that govern image quality and guiding system optimization. In this work, a task-based cascaded systems model for detectability index is proposed that describes not only the signal and noise propagation in the 2D (projection) and 3D (reconstruction) imaging chain but also the influence of background anatomical noise. The extent to which generalized detectability index provides a valid metric for imaging performance was assessed through direct comparison to human observer experiments.

METHODS

Detectability index (d') was generalized to include anatomical background noise in the same manner as the generalized noise-equivalent quanta (NEQ) proposed by Barrett et al. (Proc. SPIE Med. Imaging, Vol. 1090, 1989). Anatomical background noise was measured from a custom phantom designed to present power-law spectral density comparable to various anatomical sites (e.g., breast and lung). Theoretical calculations of d' as a function of the source-detector orbital extent (θtot) was obtained from a 3D cascaded systems analysis model for tomosynthesis and cone-beam CT (CBCT). Four model observers were considered in the calculation of d': prewhitening (PW), non-prewhitening (NPW), prewhitening with eye filter and internal noise (PWE), and non-prewhitening with eye filter and internal noise (NPWE). Human observer performance was measured from 9AFC tests for a variety of idealized imaging tasks presented within a clutter phantom. Theoretical results (d') were converted to area under the ROC curve (Az ) and compared directly to human observer performance as a function of imaging task and orbital extent.

RESULTS

Theoretical results demonstrated reasonable correspondence with human observer response for all tasks across the continuum in θtot ranging from low-angle tomosynthesis (θtot ~10°) to CBCT (θtot ~180°). Both theoretical and experimental Az were found to increase with acquisition angle, consistent with increased rejection of out-of-plane clutter for larger tomosynthesis angle. Of the four theoretical model observers considered, the prewhitening models tended to overestimate real observer performance, while the non-prewhitening models demonstrated reasonable agreement.

CONCLUSIONS

Generalized detectability index was shown to provide a meaningful metric for imaging performance, helping to bridge the gap between real observer performance and prevalent Fourier-based metrics based in first principles of spatial-frequency-dependent NEQ and imaging task.

Noise aliasing and the 3D NEQ of flat-panel cone-beam CT: effect of 2D/3D apertures and sampling

The ability to tune an imaging system to be optimal for a specific task is an essential component of image quality. This article discusses the ability to tune the noise-equivalent quanta (NEQ) of cone-beam computed tomography (CBCT) by managing noise aliasing through binning of data at different points in the reconstruction cascade. The noise power spectrum, modulation transfer function, and NEQ for CBCT are calculated using cascaded systems analysis. Binning is treated as a modular process, insertable between any two stages (in both the 2D projection domain and in the 3D reconstruction domain), consisting of the application of an aperture, followed by the resampling of data (which introduces noise aliasing). Several conditions were examined to demonstrate the validity of the model and to describe the effect on the image quality of some common reconstruction and visualization techniques. It was found that when downsampling data for increased reconstruction speed, binning in 2D results in a superior low-frequency NEQ, while binning in 3D results in a superior high-frequency NEQ. Furthermore, visualization procedures such as slice averaging were found not to degrade the NEQ provided the sampling interval is unchanged. Finally methods for reducing noise aliasing by oversampling are examined, and a method to eliminate noise aliasing without increasing reconstruction time is proposed. These results demonstrate the ease with which the NEQ of CBCT can be modified and thus optimized for specific tasks and show how such analysis can be used to improve image quality.

Cascaded systems analysis of the 3D noise transfer characteristics of flat-panel cone-beam CT

The physical factors that govern 2D and 3D imaging performance may be understood from quantitative analysis of the spatial-frequency-dependent signal and noise transfer characteristics [e.g., modulation transfer function (MTF), noise-power spectrum (NPS), detective quantum efficiency (DQE), and noise-equivalent quanta (NEQ)] along with a task-based assessment of performance (e.g., detectability index). This paper advances a theoretical framework based on cascaded systems analysis for calculation of such metrics in cone-beam CT (CBCT). The model considers the 2D projection NPS propagated through a series of reconstruction stages to yield the 3D NPS and allows quantitative investigation of tradeoffs in image quality associated with acquisition and reconstruction techniques. While the mathematical process of 3D image reconstruction is deterministic, it is shown that the process is irreversible, the associated reconstruction parameters significantly affect the 3D DQE and NEQ, and system optimization should consider the full 3D imaging chain. Factors considered in the cascade include: system geometry; number of projection views; logarithmic scaling; ramp, apodization, and interpolation filters; 3D back-projection; and 3D sampling (noise aliasing). The model is validated in comparison to experiment across a broad range of dose, reconstruction filters, and voxel sizes, and the effects of 3D noise correlation on detectability are explored. The work presents a model for the 3D NPS, DQE, and NEQ of CBCT that reduces to conventional descriptions of axial CT as a special case and provides a fairly general framework that can be applied to the design and optimization of CBCT systems for various applications.

Cascaded systems analysis of noise reduction algorithms in dual-energy imaging

An important aspect of dual-energy (DE) x-ray image decomposition is the incorporation of noise reduction techniques to mitigate the amplification of quantum noise. This article extends cascaded systems analysis of imaging performance to DE imaging systems incorporating linear noise reduction algorithms. A general analytical formulation of linear DE decomposition is derived, with weighted log subtraction and several previously reported noise reduction algorithms emerging as special cases. The DE image noise-power spectrum (NPS) and modulation transfer function (MTF) demonstrate that noise reduction algorithms impart significant, nontrivial effects on the spatial-frequency-dependent transfer characteristics which do not cancel out of the noise-equivalent quanta (NEQ). Theoretical predictions were validated in comparison to the measured NPS and MTF. The resulting NEQ was integrated with spatial-frequency-dependent task functions to yield the detectability index, d', for evaluation of DE imaging performance using different decomposition algorithms. For a 3 mm lung nodule detection task, the detectability index varied from d' < 1 (i.e., nodule barely visible) in the absence of noise reduction to d' > 2.5 (i.e., nodule clearly visible) for "anti-correlated noise reduction" (ACNR) or "simple-smoothing of the high-energy image" (SSH) algorithms applied to soft-tissue or bone-only decompositions, respectively. Optimal dose allocation (A*, the fraction of total dose delivered in the low-energy projection) was also found to depend on the choice of noise reduction technique. At fixed total dose, multi-function optimization suggested a significant increase in optimal dose allocation from A* = 0.32 for conventional log subtraction to A* = 0.79 for ACNR and SSH in soft-tissue and bone-only decompositions, respectively. Cascaded systems analysis extended to the general formulation of DE image decomposition provided an objective means of investigating DE imaging performance across a broad range of acquisition and decomposition algorithms in a manner that accounts for the spatial-frequency-dependent imaging task.

Evaluating the impact of X-ray spectral shape on image quality in flat-panel CT breast imaging

In recent years, there has been an increasing interest in exploring the feasibility of dedicated computed tomography (CT) breast imaging using a flat-panel digital detector in a truncated cone-beam imaging geometry. Preliminary results are promising and it appears as if three-dimensional tomographic imaging of the breast has great potential for reducing the masking effect of superimposed parenchymal structure typically observed with conventional mammography. In this study, a mathematical framework used for determining optimal design and acquisition parameters for such a CT breast imaging system is described. The ideal observer signal-to-noise ratio (SNR) is used as a figure of merit, under the assumptions that the imaging system is linear and shift invariant. Computation of the ideal observer SNR used a parallel-cascade model to predict signal and noise propagation through the detector, as well as a realistic model of the lesion detection task in breast imaging. For all evaluations, the total mean glandular dose for a CT breast imaging study was constrained to be approximately equivalent to that of a two-view conventional mammography study. The framework presented was used to explore the effect of x-ray spectral shape across an extensive range of kVp settings, filter material types, and filter thicknesses. The results give an indication of how spectral shape can affect image quality in flat-panel CT breast imaging.

Optimization of dual-energy imaging systems using generalized NEQ and imaging task

Dual-energy (DE) imaging is a promising advanced application of flat-panel detectors (FPDs) with a potential host of applications ranging from thoracic and cardiac imaging to interventional procedures. The performance of FPD-based DE imaging systems is investigated in this work by incorporating the noise-power spectrum associated with overlying anatomical structures ("anatomical noise" modeled according to a 1/f characteristic) into descriptions of noise-equivalent quanta (NEQ) to yield the generalized NEQ (GNEQ). Signal and noise propagation in the DE imaging chain is modeled by cascaded systems analysis. A Fourier-based description of the imaging task is integrated with the GNEQ to yield a detectability index used as an objective function for optimizing DE image reconstruction, allocation of dose between low- and high-energy images, and selection of low- and high-kVp. Optimal reconstruction and acquisition parameters were found to depend on dose; for example, optimal kVp varied from [60/150] kVp at typical radiographic dose levels (approximately 0.5 mGy entrance surface dose, ESD) but increased to [90/150] kVp at high dose (ESD approximately 5.0 mGy). At very low dose (ESD approximately 0.05 mGy), detectability index indicates an optimal low-energy technique of 60 kVp but was largely insensitive to the choice of high-kVp in the range 120-150 kVp. Similarly, optimal dose allocation, defined as the ratio of low-energy ESD and the total ESD, varied from 0.2 to 0.4 over the range ESD=(0.05-5.0) mGy. Furthermore, two applications of the theoretical framework were explored: (i) the increase in detectability for DE imaging compared to conventional radiography; and (ii) the performance of single-shot vs double-shot DE imaging, wherein the latter is found to have a DQE approximately twice that of the former. Experimental and theoretical analysis of GNEQ and task-based detectability index provides a fundamental understanding of the factors governing DE imaging performance and offers a framework for system design and optimization.

Amorphous selenium flat panel detectors for digital mammography: Validation of a NPWE model observer with CDMAM observer performance experiments

Model observers have been developed which incorporate a specific imaging task, system performance, and human observer characteristics and can potentially overcome some of the limitations in using detective quantum efficiency for optimization and comparison of detectors. In this paper, a modified nonprewhitening matched filter (NPWE) model observer was developed and validated to predict object detectability for an amorphous selenium (a-Se) direct flat-panel imager (FPI) where aliasing is severe. A preclinical a-Se digital mammography FPI with 85 microm pixel size was used in this investigation. Its physical imaging properties including modulation transfer function (MTF), noise power spectrum, and DQE were fully characterized. An observer performance study was conducted by imaging the CDMAM 3.4 contrast-detail phantom designed specifically for digital mammography and presenting these images to a panel of seven observers. X-ray attenuation and scatter due to the phantom were determined experimentally for use in development of the model observer. The observer study results were analyzed via threshold averaging and signal detection theory (SDT) based techniques to produce contrast-detail curves where threshold contrast is plotted as a function of disk diameter. Validity of the model was established using SDT analysis of the experimental data. The effect of aliasing on the detectability of small diameter disks was determined using the NPWE model observer. The signal spectrum was calculated using the presampling MTF of the detector with and without including the aliased terms. Our results indicate that the NPWE model based on Fourier domain parameters provides reasonable prediction of object detectability for the signal-known-exactly task in uniform image noise for a-Se direct FPI.

A comparison between objective and subjective image quality measurements for a full field digital mammography system

This paper presents pre-sampling modulation transfer function (MTF), normalized noise power spectrum (NNPS) and detective quantum efficiency (DQE) results for an amorphous selenium (a-Se) full field digital mammography system. MTF was calculated from the image of an angled 0.5 mm thick Cu edge, acquired without additional beam filtration. NNPS data were acquired at detector air-kerma levels ranging from 9.1 microGy to 331 microGy, using a standard mammography x-ray spectrum of 28 kV, Mo/Mo target/filter combination and 4 cm of PMMA additional filtration. Prior to NNPS estimation, the image statistics were assessed using a variance image. This method was able to easily identify a detector artefact and should prove useful in routine quality assurance (QA) measurements. Detector DQE, calculated from the NNPS and MTF data, dropped to 0.3 for low detector air-kerma settings but reached an approximately constant value of 0.6 above 50 microGy at the detector. Subjective image quality data were also obtained at these detector air-kerma settings using the CDMAM contrast-detail (c-d) test object. The c-d data reflected the trend seen in DQE, with threshold contrast increasing at low detector air-kerma values. The c-d data were then compared against predictions made using two established models, the Rose model and a standard signal detection theory model. Using DQE(0), the Rose model gave results within approximately 15% on average for all the detector air-kerma values studied and for detail diameters down to 0.2 mm. Similar agreement was also found between the measured c-d data and the signal detection theory results, which were calculated using an ideal human visual response function and a system magnification of unity. The use of full spatial frequency DQE improved the agreement between the calculated and observer results for detail sizes below 0.13 mm.

Segmented crystalline scintillators: An initial investigation of high quantum efficiency detectors for megavoltage x-ray imaging

Electronic portal imaging devices (EPIDs) based on indirect detection, active matrix flat panel imagers (AMFPIs) have become the technology of choice for geometric verification of patient localization and dose delivery in external beam radiotherapy. However, current AMFPI EPIDs, which are based on powdered-phosphor screens, make use of only approximately 2% of the incident radiation, thus severely limiting their imaging performance as quantified by the detective quantum efficiency (DQE) (approximately 1%, compared to approximately 75% for kilovoltage AMFPIs). With the rapidly increasing adoption of image-guided techniques in virtually every aspect of radiotherapy, there exist strong incentives to develop high-DQE megavoltage x-ray imagers, capable of providing soft-tissue contrast at very low doses in megavoltage tomographic and, potentially, projection imaging. In this work we present a systematic theoretical and preliminary empirical evaluation of a promising, high-quantum-efficiency, megavoltage x-ray detector design based on a two-dimensional matrix of thick, optically isolated, crystalline scintillator elements. The detector is coupled with an indirect detection-based active matrix array, with the center-to-center spacing of the crystalline elements chosen to match the pitch of the underlying array pixels. Such a design enables the utilization of a significantly larger fraction of the incident radiation (up to 80% for a 6 MV beam), through increases in the thickness of the crystalline elements, without loss of spatial resolution due to the spread of optical photons. Radiation damage studies were performed on test samples of two candidate scintillator materials, CsI(Tl) and BGO, under conditions relevant to radiotherapy imaging. A detailed Monte Carlo-based study was performed in order to examine the signal, spatial spreading, and noise properties of the absorbed energy for several segmented detector configurations. Parameters studied included scintillator material, septal wall material, detector thickness, and the thickness of the septal walls. The results of the Monte Carlo simulations were used to estimate the upper limits of the modulation transfer function, noise power spectrum and the DQE for a select number of configurations. An exploratory, small-area prototype segmented detector was fabricated by infusing crystalline CsI(Tl) in a 2 mm thick tungsten matrix, and the signal response was measured under radiotherapy imaging conditions. Results from the radiation damage studies showed that both CsI(Tl) and BGO exhibited less than approximately 15% reduction in light output after 2500 cGy equivalent dose. The prototype CsI(Tl) segmented detector exhibited high uniformity, but a lower-than-expected magnitude of signal response. Finally, results from Monte Carlo studies strongly indicate that high scintillator-fill-factor configurations, incorporating high-density scintillator and septal wall materials, could achieve up to 50 times higher DQE compared to current AMFPI EPIDs.

Extension of x-ray imaging linear systems analysis to detectors with energy discrimination capability

A figure of merit, the broad-spectrum generalized detective quantum efficiency, which describes the performance of digital detectors designed for broad-spectrum x-ray imaging is derived from linear response theory. This measure of the imaging efficacy of an x-ray sensor is obtained when detector contrast modulation in the domain of x-ray energy is introduced in the Fourier-based analysis of digital systems. A method is proposed to scale existing figures of merit according to the energy-dependent response of the detector and the spectral shape of the x-ray beam. The new figure of merit obtained with this method provides an extended description of system performance when comparing energy-integrating, single-photon counting, and future energy-sensitive x-ray imaging sensors. The applicability of this linear system analysis is restricted to the tasks of low-contrast object detection in radiography. The method for scaling the figure of merit to take into consideration broad-spectrum conditions is applied to mammography for future energy-dependent detectors. An approximation valid in the typical mammographic x-ray energy range is used to calculate the broad-spectrum generalized detective quantum efficiency at zero spatial frequency, for several mammographic x-ray spectra. X-ray energy weighting in mammography is investigated in the context of simulated tumors and microcalcifications detection by comparing this figure of merit, calculated for different detector technologies, under ideal imaging conditions, at zero spatial frequency.

Efficiency of the Human Observer Compared to an Ideal Observer Based on a Generalized NEQ Which Incorporates Scatter and Geometric Unsharpness: Evaluation with a 2AFC Experiment

Under certain assumptions the detectability of the ideal observer can be defined as the integral of the system Noise Equivalent Quanta multiplied by the squared object spatial frequency distribution. Using the detector Noise-Equivalent-Quanta (NEQ(D)) for the calculation of detectability inadequately describes the performance of an x-ray imaging system because it does not take into account the effects of patient scatter and geometric unsharpness. As a result, the ideal detectability index is overestimated, and hence the efficiency of the human observer in detecting objects is underestimated. We define a Generalized-NEQ (GNEQ) for an x-ray system referenced at the object plane that incorporates the scatter fraction, the spatial distributions of scatter and focal spot, the detector MTF(D), and the detector Normalized-Noise-Power-Spectrum (NNPS(D)). This GNEQ was used in the definition of the ideal detectability for the evaluation of the human observer efficiency during a two Alternative Forced Choice (2-AFC) experiment, and was compared with the case where only the NEQ(D) was used in the detectability calculations. The 2-AFC experiment involved the detection of images of polyethylene tubes (diameters between 100-300 μm) filled with iodine contrast (concentrations between 0-120 mg/cm(3)) placed onto a uniform head equivalent phantom placed near the surface of a microangiographic detector (43 μm pixel size). The resulting efficiency of the human observer without regarding the effects of scatter and geometric unsharpness was 30%. When these effects were considered the efficiency was increased to 70%. The ideal observer with the GNEQ can be a simple optimization method of a complete imaging system.

Lessons from my dinners with the giants of modern image science

The author traces some critical moments in the history of Image Science in the last half century from first-hand or once-removed experience. The Image Science used in the field of medical imaging today had its origins in the analysis of photon detection developed for modern television, conventional photography, and the human visual system. Almost all "model observers" used in image assessment today converge to the model originally used by Albert Rose in his analysis of those classic photo-detectors. A more general statistical analysis of the various "defects" of conventional and unconventional photon-imaging technologies was provided by Shaw. A number of investigators in medical imaging elaborated the work of these pioneers into a synthesis with the general theory of signal detectability and extended this work to the various forms of CT, energy-spectral-dependent imaging, and the further complication of anatomical-background-noise limited imaging. The author calls for further extensions of this work to the problem of under-sampled and thus artefact-limited imaging that will be important issues for high-speed CT and MRI.

Figures of merit for detectors in digital radiography. I. Flat background and deterministic blurring

Digital radiography systems can be thought of as continuous linear shift-invariant systems followed by sampling. This view, along with the large number of pixels used for flat-panel systems, has motivated much of the work which attempts to extend figures of merit developed for analog systems, in particular noise equivalent quanta (NEQ) and detective quantum efficiency (DQE). A more general approach looks at the system as a continuous-to-discrete mapping and evaluates the signal-to-noise ratio (SNR) completely from the discrete data. In this paper, we study the effect of presampling blur on these figures of merit for a simple model that assumes that the background fluence is constant and that the blurring of the signal is deterministic. We find that for small signals, even in this idealized model, commonly used DQE/NEQ formulations do not accurately track the behavior of the fully digital SNR. Using these NEQ-based figures of merit would lead to different design decisions than using the ideal SNR. This study is meant to bring attention to the assumptions implicitly made when using Fourier methods.

Figures of merit for detectors in digital radiography. II. Finite number of secondaries and structured backgrounds

The current paradigm for evaluating detectors in digital radiography relies on Fourier methods. Fourier methods rely on a shift-invariant and statistically stationary description of the imaging system. The theoretical justification for the use of Fourier methods is based on a uniform background fluence and an infinite detector. In practice, the background fluence is not uniform and detector size is finite. We study the effect of stochastic blurring and structured backgrounds on the correlation between Fourier-based figures of merit and Hotelling detectability. A stochastic model of the blurring leads to behavior similar to what is observed by adding electronic noise to the deterministic blurring model. Background structure does away with the shift invariance. Anatomical variation makes the covariance matrix of the data less amenable to Fourier methods by introducing long-range correlations. It is desirable to have figures of merit that can account for all the sources of variation, some of which are not stationary. For such cases, we show that the commonly used figures of merit based on the discrete Fourier transform can provide an inaccurate estimate of Hotelling detectability.

Determination of the two-dimensional detective quantum efficiency of a computed radiography system

Based on a recently described method for determining the two-dimensional presampling modulation transfer function (MTF), the aperture mask method, a method for determining the two-dimensional detective quantum efficiency (DQE) of a digital radiographic system was developed. The method was applied to a new computed radiography (CR) system and comparisons with one-dimensional determinations of the presampling MTF and the DQE were performed. The aperture mask method was shown to agree with the conventional tilted slit method for determining the presampling MTF along the axes. For the particular CR system studied, the mean of one-dimensional determinations of the DQE in orthogonal directions led to a representative measure of the average DQE behavior of the system up to the Nyquist frequency along the axes, but a deviation was observed above this frequency. In conclusion, the method developed for determining the two-dimensional DQE can be used to determine the imaging properties of a digital radiographic detector system over almost the entire frequency domain, the exception being the lowest frequencies (< or = 0.1 mm(-1)) at which the validity and the reliability of the method are low.

Conditioning data for calculation of the modulation transfer function

A method for conditioning data used in the measurement of the modulation transfer function (MTF) is discussed. This method is based upon imposing the constraint that the edge spread function (ESF) is monotonic. The advantages of this technique, when applicable, are demonstrated with simulated examples for which the true MTF is known. The application of this technique in the measurement of the MTF of a digital detector in clinical use is also demonstrated.

Aliasing effects in digital images of line-pair phantoms

Line-pair phantoms are commonly used for evaluating screen-film systems. When imaged digitally, aliasing effects give rise to additional periodic patterns. This paper examines one such effect that medical physicists are likely to encounter, and which can be used as an indicator of super-resolution.