The Impact of Cross-Talk in a Flat Panel Detector on CT Image Quality

Spatial resolution and image noise are two aspects of image quality of an X-ray computed tomography (CT) system and are determined by the X-ray source, the detector, and mathematical operations for image reconstruction. In CT scanners with flat panel detectors (FPDs), there is cross-talk (signal leakage) between detector pixels. The contribution of the cross-talk to spatial resolution and noise in reconstructed images has not been adequately modeled. Previously, we estimated cross-talk from autocovariance measurements in air, and modeled the impact of cross-talk on spatial resolution. We have extended that work to calculate the impact of cross-talk on signal-to-noise ratio in radiographs and to reconstructed image noise. We modeled the spatial resolution and noise of a CT scanner that uses a flat-panel detector with 0.2-mm pixels and a gadolinium oxysulfide scintillator, and a 450 kVp, dual-focus X-ray tube. Our noise model agrees with measurements from experimental data and simulations to within 10%. We show that cross-talk in FPDs can reduce resolution by over 30%, reduce noise by approximately a factor of two, and introduce correlation in the noise, and therefore, cannot be disregarded when assessing CT image quality.


I. INTRODUCTION
F LAT panel detectors (FPDs) used in X-ray computed tomography (CT) and in digital radiography are area detectors with square pixels of side length on the order of 10-100 μm [1].X-rays incident on an FPD interact with a scintillator.The deposited X-ray energy generates light which percolates through the scintillator to amorphoussilicon photodiodes on the surface of an underlying glass sheet.Light absorbed by the photodiodes generates electron-hole pairs; that deposited charge is digitized to provide a digital image of the absorbed X-ray energy.A small fraction of the incident X-ray energy (2%-5% for energies below 1 MeV) is directly deposited in the photodiodes [2].
With every X-ray interaction in the scintillator, light is emitted in all directions and may interact with photodiodes belonging to neighboring pixels; we refer to this as optical cross-talk.There is also X-ray scatter within the scintillator and all components of the FPD that leads to X-ray deposition within neighboring pixels, that we call X-ray cross-talk [3].Other sources of cross-talk, such as circuit-based cross-talk, are considered negligible compared with optical and X-ray cross-talk [4].The total crosstalk in the FPD is the combination of all sources, and we do not separately examine the effect of each.Crosstalk blurs the signal, reducing contrast and perspicuity of small structures.The blur increases monotonically with scintillator thickness.Cross-talk reduces and adds correlation to image noise.Previously, we estimated the cross-talk from scans of air, and modeled and demonstrated its impact on spatial resolution [5].However, spatial resolution should be examined together with noise.Therefore, in this article, we extend that work to create and validate an image noise model that includes the impact of cross-talk.
Often, noise, resolution, and other aspects of image quality are measured using a known test-object, but are not modeled to explain the contributions of the X-ray system components or image reconstruction operations [6], [7], [8], [9], [10].The detector blur has been modeled without considering cross-talk [11], [12].Calculations of noise correlation in reconstructed images have been made without considering cross-talk [13], [14].Conversely, calculations of cross-talk have been made without considering its propagation on reconstructed image uncertainty [15].Our contribution is to bridge this gap; we developed an analytical model for the propagation of noise correlated by cross-talk in reconstructed images.This model draws upon well-known statistical propagation techniques [4], [16], [17] and is generalizable.The motivation for the work is threefold: 1) an explanation for the experimentally observed resolution and noise is useful in understanding the contribution of hardware and software components, to estimate the physical limits of scanner performance and determine areas for improvement; 2) models of performance can be used in scan planning; and 3) if cross-talk is quantified correctly, a compensation can be implemented.This capability is useful for quantitative analysis of images.
To validate the analytical model, we simulated and reconstructed projection data with known noise and a known cross-talk function.To understand the effect of cross-talk on noise in projection data, we carried out Monte Carlo (MC) simulations including the photon spectra incident on the FPD and the detector absorption efficiency without crosstalk.We compared the noise model against experimental data consisting of air scans obtained with several spectra and a water phantom scan.We estimate that cross-talk boosts the measured signal-to-noise ratio (SNR) in raw projection data by a factor of 2.2.Our analysis shows that while cross-talk is radially symmetric in our square-pixel FPD, the impact of the cross-talk on image noise is not isotropic because image reconstruction is not spherically symmetric.
Section II describes our scanner and reconstruction methods.The models for cross-talk and spatial resolution that we previously developed [5] are explained in Sections III and IV, respectively.We use the cross-talk model to develop a new noise model in Section V.The experiments and simulations are described in Section VI, followed by results in Section VII.The limitations of this work and conclusions are in Sections VIII and IX, respectively.

II. SCANNER DESCRIPTION
Our scanner consists of a fixed X-ray source and detector with a rotating stage on which an object to be imaged is mounted.The source-to-center distance is 1100 mm and the source-to-detector distance is 1400 mm.The source is a YXLON model Y.TU 450-D11 X-ray tube, with a 450 kVp maximum rating and two focal spots of nominal sizes 0.4 and 1 mm.The detector is a Varex 1621AN FPD with 0.2-mm pixel side, and a 0.2-mm thick gadolinum oxysulfide (Gd 2 O 2 S:Tb) scintillator [26].Our experiments were of two kinds, acquiring and reconstructing real scanner data, and simulating projections with known added noise and correlation.Projection data from the scanner are acquired in a step-and-shoot mode.The object moves between views but is stationary while the FPD is collecting data at a view position.Image reconstruction from raw projection data consists of dark current subtraction, the application of the Beer-Lambert law, a differentiable beam hardening correction (where required), and filtered backprojection (FBP).

III. CROSS-TALK MODEL
We estimate the modulation transfer function (MTF) of the cross-talk by calculating autocovariance in air, and using the Weiner-Khinchin theorem [18].In an ideal CT scanner, the random process of photon emission and detection is stationary (identical at every pixel) and ergodic (we can take samples over time instead of space).Therefore, the covariance between two pixels calculated using repeated projections is a sample of the autocovariance of the stationary random process at that interpixel distance.We take 300 projections of air and compute the covariance between pairs of pixels, i.e., correlation after subtracting the mean.We compute covariance within a neighborhood of 40 pixels and take the mean of the covariance over pixels.
Let the number of photons in a single projection at detector pixels i and j be represented by random variables y i and y j , respectively, and their expected values be denoted by ȳi and ȳj .The covariance is given by where the expectation is taken over projections.For a stationary process, the autocovariance is simply the average covariance for the detector pixels and ȳi = ȳj = ȳ for each detector.So, taking an expectation of the covariance over pixels In the above expression, C yy is a function of the displacement, , between pixels.The expression shows that autocovariance is given by subtracting a constant from autocorrelation.During gain correction (also called flatfielding), we subtract the mean in log-attenuation data.The Wiener-Khinchin theorem gives us the relationship between autocorrelation and noise power spectrum (NPS).We have modified it to subtract the zero-frequency ("DC") value ȳi ȳj = ȳ2 .In the equation below, S xx represents the NPS of a process whose autocovariance is C xx , and F represents the spatial Fourier transform Consider the cross-talk to be a spatial filter, that when applied to the random process of incident X-rays, gives us the random process of the scanner.If we represent the incident random process by x and the scanner random process by y, and the filter by H c , then where S xx and S yy , respectively, represent the NPS of the incident process and the scanner process.Now we assume that the X-ray flux at each detector pixel follows a distribution that is independent and identically distributed (IID).
The autocovariance of such a process is zero everywhere except at = 0, where it is the variance.This describes a delta function and therefore, its Fourier transform S xx = 1.
Since S xx is a constant, S yy = |H c | 2 .Using the assumption that the cross-talk is positive, symmetric, and real, the point spread function (PSF) of the cross-talk, We computed covariance from scans in air taken with a 100 kVp spectrum and 2-mm aluminum added filtration, as well as other spectra.The mean normalized covariance over ≈15 000 pixels at the center of the FPD, was computed as the sample autocovariance.In a practical scanner, since the pixels have different gains, we normalize by the variance and then assume that the SNR for the pixels is constant.Note that we have used ergodicity to compute the autocovariance, that is, we are able to avoid the effect of nonuniform gain of the pixels by taking the expectation in (1) over time, normalizing by gain, and then averaging the normalized result over pixels.Since each detector has a different gain, but receives the same incident flux density, we expect that the SNR is constant, and we can use the assumption of stationarity with the normalized data.This method cannot differentiate between X-ray cross-talk, which increases with X-ray energy [3], [19], and optical cross-talk, which is nearly energy-independent and dominates at the lower energies.

IV. SPATIAL RESOLUTION MODEL
Spatial resolution in CT can be modeled as a product of the MTFs of the source and detector pixel apertures as projected onto the imaging plane, as well as the FBP kernel, and the interpolation function in backprojection.The model assumes step-and-shoot scanning.We use the specifications of source focal spot size (0.4 or 1 mm) and detector pixel pitch (0.2 mm) supplied by the vendors.Given the geometry in our system, these sizes project to about 0.086 and 0.213 mm, respectively, at the axis of rotation.We approximate each focal spot with a rectangular aperture.Similarly, we use a rectangular function to represent the detector pixel aperture, which projects to 0.157 mm at the center of rotation.Rectangular functions in the spatial domain correspond to sinc functions in the frequency domain.These functions are shown in Fig. 1.We use the Shepp-Logan reconstruction kernel [20].The backprojector uses linear interpolation, which can be represented by a triangular function in the spatial domain, and therefore is a squared sinc in the frequency domain.The first zeros of the detector aperture and the backprojector responses are at the sampling frequency, 1/0.157 = 6.36 lp/mm.While the frequency domain representation of the above apertures is well known, the significant finding of [5] deals with quantifying the blur due to cross-talk, labeled "ctk" in Fig. 1, which was determined as described in Section III.The crosstalk has the narrowest MTF of the component functions and therefore causes the greatest drop in resolution.

V. NOISE MODEL
Image noise is determined by the spectrum incident at the detector, cross-talk, the convolution kernel, and backprojection.Raw projection data is reported in arbitrary detector counts that represent the energy deposited in the detector.We develop an SNR model for raw projection data with MC simulations, since count variance alone is meaningless.Log-attenuation projection data (after the Beer-Lambert law is applied), and consequently reconstructed images, are normalized for air to have zero mean, so variance or standard deviation are the meaningful quantities for noise in those data (not SNR).We model noise in log-attenuation data and reconstructed data with analytical expressions.

A. DEFINITIONS 1) SIGNAL-TO-NOISE RATIO IN PROJECTION DATA
Let a reading at pixel i and projection view v be represented by a i,v .The pixel SNR for the pixel i is defined here as the ratio of the mean to standard deviation.The mean pixel SNR is defined over all pixels (abbreviated to A) where the signal, āi , is the mean computed over V views āi = 1 V v a i,v .Note that the "signal" in this definition is not the difference between pixel readings, so (5) is not a contrast-to-noise ratio.The standard deviation is computed over projections (repeated radiographs) We calculate the SNR for each pixel, and average the SNR over detector pixels.We do not calculate standard deviation across pixels within a projection because the random process of detection is not truly ergodic, i.e., the detector pixel gains are different, and the difference in gain would be falsely attributed to noise.Cross-talk introduces spatial correlations in data within a view but not across views.There is a separate afterglow with time (also called lag), but we have neglected this in our calculations because we have assumed that the decay is fast compared with our integration time for one view [15], [21].

2) IMAGE NOISE
Image noise is defined as the standard deviation of image pixels in an axial reconstructed slice.We measure this standard deviation in a region-of-interest at the center of the slice.

B. MC MODEL OF RAW PROJECTION DATA SNR
MC simulations of X-ray spectra and the detector response function allowed us to model the noise in projection data.The MC simulations did not include cross-talk.
Our CT scanner was modeled using MC N-Particle Transport software, MCNP 6.2 [22].The source was modeled by directing a beam of monoenergetic electrons onto a tungsten anode according to the specifications of our system.Photon energy spectra and relative intensities at the detector were recorded using a series of 21 F5:p point detector tallies.The results of the simulations were dynamically scaled to analog-to-digital unit (ADU)/(0.04mm 2 pixel) by MCNP using an energy-dependent detector response function characteristic of this detector.Several spectra were simulated with 100-400 keV electron beams, along a single X-ray path.
In each simulation, one of two added filters were modeled, either 2 mm aluminum (Al), or 2 mm each of Al and copper (Cu) to simulate the scans discussed in Section VI.Two photon spectra are shown in Fig. 2.
Fig. 3 shows the energy deposition efficiency.The product of the incident X-ray spectrum and efficiency gives the number of photons absorbed by the scintillator.Assuming a Poisson distribution, the SNR is given by the square root of the number of photons.Accounting for the cross-talk, the variance decreases by a factor equal to the squared sum of cross-talk coefficients, so the SNR becomes where E and η, respectively, denote energy and efficiency.
The spectrum, i.e., the number of photons, is denoted by S(E) and abbreviated by S, and the SNR is MC (S).In the above equation, i denotes pixels in the support of h c .
We have a polychromatic beam, where the number of photons at each energy per view is Poisson distributed.The sum of Poisson-distributed random variables is also a Poisson-distributed random variable.The random processes of X-ray interactions with an object and with a scintillator are governed by binomial distributions, therefore, by the law of iterated expectations, the random variable representing the number of X-ray photons absorbed in a detector pixel per view is also Poisson distributed [23].The number of optical photons generated by each X-ray interaction is Poisson-distributed, which leads to an overall non-Poisson distribution [24].There are approximately a few thousand light photons, per X-ray photon [21].Further, there are partial X-ray energy depositions, and a nonbinomial conversion to electron-hole pairs-about 10 electron-hole pairs per keV deposited.Therefore, the noise process is no longer a Poisson process.However, the shot noise dominates and the Poisson approximation holds for the spectra we tested.

C. ANALYTICAL MODEL FOR RECONSTRUCTED IMAGE NOISE
As stated earlier, the reconstruction process includes dark current correction and gain (air) correction, taking the logarithm, applying beam hardening correction, and FBP.We neglect the dark current subtraction because dark noise is small compared to photon noise.We also neglect the effect of gain correction.For air scans, the gain data was the mean of all air data.For the water scans, the gain calibration scan was acquired with a 16 frame-average, so the noise is a factor of four lower than our object scan, and due to object attenuation, still lower than the noise in object data.

1) TAKING THE LOGARITHM
Based on the above assumptions, the energy deposited in a detector pixel in a view can be converted to a Poissondistributed random variable representing an equivalent number of photons.Let this number of photons be denoted N. Then The Beer-Lambert law yields log-attenuation data where N 0 denotes the number of photons in air, μ is the linear attenuation coefficient, and l is the path length.The noise in the log-attenuation is therefore the noise in N.For air scans, N 0 = N.From [17], for any differentiable function P, we have Let the variance of the log-attenuation signal be σ L .Then 2) BEAM HARDENING CORRECTION To compensate for beam hardening, we apply a polynomial correction.Since a polynomial is a differentiable function, we can again calculate the noise with (11) For air scans, σ L = σ C .

3) NOISE PROPAGATED BY FILTERED BACKPROJECTION IN SINGLE SLICE
We first derive the image noise (standard deviation in a central region) considering 1-D projections and 2-D reconstruction, i.e., a single slice.Then, we build a model for 2-D projections and fan beam reconstruction, which also pertains to the central slice of a cone beam system.Finally, we introduce the effect of interpolation along the z-axis on image noise in the 3-D volume.1) 1-D projections and 2-D reconstruction: Consider the reconstruction of V views, where the IID noise in each is given by σ L and the detector pixel sample spacing is τ .Following [13] and [16], image noise in an axial slice produced by 2-D FBP is given by where K is a scalar depending on the filter and interpolation function in backprojection.Since the filtering and interpolation operate on the same dimension of data, we multiply the MTFs of the filter and the interpolation in backprojection, similar to [13].
Since detector cross-talk also correlates the signal along the same dimension, the MTF of cross-talk can be included in the calculation of K as follows: where K c is the frequency space representation of the crosstalk, K f is that of the reconstruction filter, and K b is that of the backprojection.The limits of integration are two sampling periods, which allows us to capture aliasing.Using Parseval's theorem [25], we can make the same calculations in the spatial domain rather than the frequency domain by convolving the corresponding spatial domain functions Backprojection itself (not including interpolation) is the sum of random variables, as reflected by √ V in the denominator of (14).Since summation along views is not in the same dimension as the interpolation among pixels during backprojection, interpolation is separately accounted for in K.For the rest of this section, we replace the integration with summation.For better readability, we will omit arguments and subscripts where they are unambiguous.2) 2-D projections and 2-D reconstruction: Our detector is a 2-D flat panel, and the data is therefore correlated by the 2-D cross-talk before we apply 1-D filtering along the rows.Each row contributes signal from an independent incident flux, and a reconstruction filter and interpolation are applied to the signal.
We first calculate K with 2-D crosstalk and 2-D FBP.This is appropriate where there is no row-interpolation, i.e., at the central slice if it exists.This gives the maximum possible standard deviation in images.For a given pixel, pixels in neighboring rows with the same assumed projection data noise σ L contribute signal according to the estimated cross-talk function h c .We can therefore consider their contributions in the spatial domain as follows: In the frequency domain, this is equivalent to where H c j (w) = F(h(i, j)) is the 1-D Fourier transform of row j of the cross-talk PSF.Fig. 4 shows the Fourier transform of the cross-talk coefficients in the center row.
3) Cone beam backprojection: With cone beam backprojection, there is a further interpolation along the axis of rotation ("z-axis").If the central slice is centered at a projection row, there is no z-axis interpolation at the central slice; the zaxis interpolation increases with distance from the central plane.The maximum interpolation occurs when we must interpolate a ray that is in-between two pixels.If the data in neighboring rows were IID (uncorrelated), we could model the average effect of interpolation at slices that contain such rays as where H b is the same linear interpolation kernel we used in (15).Therefore, in a slice with maximum interpolation and uncorrelated data, the noise is 0.82 σ I,z 0 .Since we have 2-D cross-talk, the data in adjacent rows is already correlated before backprojection.We calculate a fraction r z that when multiplied by the noise in the central slice, gives the noise in a slice with maximum interpolation where K b is the squared sinc that represents linear interpolation, and H c z is the Fourier transform of the central row of the cross-talk PSF.

VI. EXPERIMENTS A. SCANNER DATA
We collected scanner data with three objectives: 1) to characterize cross-talk; 2) to measure spatial resolution; and 3) to measure noise.All images were reconstructed with FBP using either the Shepp-Logan filter or the Ramp filter, using the livermore tomography tools (LTTs) software [27].
For each scan, 300 projections were taken with various kVs and two filters: either 2 mm of Al or a combination of 2 mm Al and 2 mm Cu.We used either a 20-mm collimator slit or no collimation.In all these scans, the tube current was varied to maintain 35 000-45 000 detector counts on the FPD.The integration time was 0.267 s for the scans made with the Al filter, and 1 s for scans made with the Al-Cu filter.Of these scans, 100 and 400 kV scans were used in the calculation of variance and consequently, the cross-talk filter as described in Section III.All air scans were used to calculate the SNR in air raw projection data, log-attenuation data, and image noise in reconstructed slices of air.We scanned a water phantom of diameter 83 mm with the 100 kV and 2mm Al filter.
For experimental measurement of spatial resolution, we scanned a 0.05-mm diameter tungsten wire stretched in a plastic cylinder that we designed and fabricated at our laboratory.The wire serves as an impulse function.The image is therefore an impulse response, and its Fourier transform yields the MTF.The experiment was performed with various scan tube voltage and filter combinations, with the cylinder empty or filled with water.Images were reconstructed with 0.01-mm voxel spacing so that the measured MTF was not reconstruction limited.

B. SIMULATED PROJECTION DATA
In order to validate the analytical model of noise ( 14)-( 20

A. CROSS-TALK MODEL
A plot of the autocovariance along one dimension is shown in Fig. 5.The autocovariance function is approximately radially symmetric.The autocovariance measured from the two scan techniques was similar.In the calculations of resolution and noise, we used the model extracted from the 100-kV data with the Al filter.The frequency-space representation of the cross-talk model, [H in (4)], is shown by the plot labeled "ctk" in Fig. 1.

B. SPATIAL RESOLUTION
We calculated spatial resolution in radial and tangential directions.The radial direction is along the ray from the center of rotation to the wire.The tangential direction is perpendicular to the radial direction.A plot of the spatial  resolution model is shown along with the experimentally determined MTF of the scanner in Fig. 6.The scan for this measurement was made with 100 kV tube voltage, a 2-mm Al filter, and an empty wire cylinder.The reconstruction field of view (FOV) was 5 mm, centered at the wire.The MTFs in radial and tangential directions are nearly identical and match the model MTF within 5% at the 50% and 10% dropoff points.From Fig. 1, the main finding is that cross-talk dominates the blurring.By multiplying components other than cross-talk, we can see that the cross-talk lowers the total MTF by over 30% at the 50% point.
MTFs measured from images acquired with different techniques are shown in Fig. 7.At higher energies, particularly, when the jar was filled with water, the radial MTF measurement was affected by ring artifacts from imperfect calibration.For these images, the radial measurements were made with a reduced FOV of 2.5 mm.PSFs are shown in [5].

C. NOISE 1) SNR IN PROJECTIONS
Fig. 8 shows the SNR per mA-s in air for various spectra.The discrepancy in modeled SNR [ MC , (7)] and real data SNR [ meas , (5)] is within 3% for the 2-mm Al added filter shown.The discrepancy was greater, i.e., as high as 13% for the Al-Cu filter (plot not shown).

2) IMAGE NOISE
The values of K for the Shepp-Logan and Ramp filters, with and without cross-talk are given in Table 1.We applied these values of K in ( 14) to obtain predicted noise, that we could compare with reconstructions from LTT-simulated data and real scanner data.When computing image noise based on the empirical projection noise, we will account for cross-talk with K, so we must divide the empirical SNR by h 2 c before using (14).Table 2 shows the noise measured in simulated images, and the noise predicted for the given simulation parameters.The value of K is given by (15) for the 1-D correlation kernel and by (18) for the 2-D correlation kernel.The model and measured values match to within 0.5%.
Table 3 shows the noise measured in air scans and the noise predicted by the model for a few different kV.The first row of data contains noise measured from scanner images.For the prediction, we use the K from (18).The second row contains σ L directly measured from the projection data, and the error is within 0.5%.The third row calculates σ L using the coefficients of the empirically fitted SNR.We used the fit instead of MCNP-derived SNR because it is a better approximation to the measured SNR.For this comparison, the SNR is scaled by the square root of the mA-s, and divided by h 2 c and then (13) applied to get the estimated σ L .Note that the SNR fit was obtained from the same data in which we measure image noise.However, this analysis validates two key issues: 1) that the Poisson approximation holds for detected data and 2) that the image noise calculations of Section IV are a good approximation.
We validated (20) against three simulations to assess its generalizability.In each simulation, a hypothetical cross-talk PSF was convolved with noisy cone beam projection data, an image volume was reconstructed and noise was measured in image slices.The r z,corr was evaluated for each PSF.The three PSFs are the cross-talk estimated from the scanner, a 1-D cross-talk function in the axial direction (0.25 0.5 0.25), and a mean value in a 5×5 pixel region.For each of these PSFs, we convolved it with noisy (IID) projections, reconstructed an image volume and measured the noise in each slice.We computed the ratio of noise in slices far from the central plane to that in the central plane.Table 4 shows the values of r z,corr for the three functions.In all three cases, the error between r z,corr and the measured ratio of noise was less than 1.5%.Fig. 9 shows the noise in each cone-beam reconstructed slice from a simulation using the cross-talk kernel estimated from the scanner.
Noise relative to the maximum noise in cone beam images is shown in Fig. 10.Image noise from four real scans with different spectra and without collimation are plotted.The peak represents the central plane, and the noise decreases with the distance from the central plane.The left and right sides are asymmetric due to a heel effect in our scanner where the flux density varies along the axis of rotation [28].The flux density difference increases with  tube voltage.These plots illustrate that the noise due to interpolation along the z-axis approximately matches our model.The results above validated our models in air.For further verification of the noise model, we predict image noise in the center of a 84-mm diameter water phantom.The error between prediction using the measured noise in logattenuation projections, and the noise in reconstructed images was about 3%.The error of prediction was 13% using the MCNP SNR followed by Beer's law.A contributing factor to the greater discrepancy in the latter prediction is object scatter.

VIII. LIMITATIONS AND FUTURE WORK
Our spatial resolution and noise model was developed for our step-and-shoot acquisition and FBP reconstruction.Continuous scanning and operations, such as data resampling, would require additional modeling.The noise model was validated in the 100-400-kV energy range, but has not been tested against higher energy scans.We also plan to evaluate how well our detector model approximates measured MTF derived from edge-spread data.

IX. CONCLUSION
We have approximated the cross-talk as a digital filter and used it in models of spatial resolution and noise.The models allowed us to understand the impact of cross-talk on resolution and noise.The cross-talk was the largest component of resolution loss.We built a complete noise model based on MC simulations and analytical modeling.Despite several simplifying assumptions, our analytical model matched our measured noise to within 10%, for various combinations of scanning voltages and hardware filters.Our methods are easily reproducible and do not require sophisticated testobjects, difficult alignment procedures, or image processing as is common with other measurement methods.The methods are applicable to other CT scanners and detectors with the same assumptions.

FIGURE 1 .
FIGURE 1. Components of the MTF model and their product, i.e., the total MTF.The sampling frequency is 6.37 lp/mm.The sinc for the source aperture ("src") has a zero at 11.6 lp/mm.The detector aperture ("det"), and the backprojection ("back") has a zero at the sampling frequency.The filter ("filt") is the Shepp-Logan filter.The cross-talk ("ctk") is modeled by a digital filter.

FIGURE 2 .
FIGURE 2. Simulation of photon spectra with 100 kV and Al filter, and with 400 kV and Al-Cu filter.

FIGURE 3 .
FIGURE 3. Simulation of the relative energy deposition in a detector pixel.

FIGURE 4 .
FIGURE 4. Frequency responses for the filter, interpolation, and backprojection.Only one sampling period is shown for clarity.However, we count the effect of aliasing by considering a second lobe.Two different filters are shown, the Shepp-Logan filter (KSL) and the ramp filter (KRamp).The product function shown in this plot, Ktotal, uses the Shepp-Logan filter.
), we simulated log-attenuation projection data using LTT.IID Poisson noise with a known variance was added to each projection ray with a random number generator.Correlation was added by convolving the IID data either with a 1-D kernel or with a 2-D kernel.Images were reconstructed either with 2-D backprojection (1-D interpolation) or 3-D backprojection (2-D interpolation).The filters were either the Shepp-Logan filter or the Ramp filter.These simulations allowed us to validate the values of K for uncorrelated noise, 1-D correlation, or 2-D correlation.The 2-D and 3-D backprojection allowed us to validate the calculation of variable noise with cone-beam backprojection.When measuring noise from 2-D backprojection, we took the mean of several rows of images to give us better estimates of the maximum noise.Validating the various components with 1-D and 2-D simulations gives us a better understanding of noise propagation.In contrast to the MC simulations to build the spectrum model, these simulations were not used to build the image noise model.

FIGURE 6 .
FIGURE 6. MTFs of the wire scanned with a 0.4 mm focal spot.

FIGURE 7 .
FIGURE 7. MTFs of the wire scanned with a 1 mm focal spot, various scan techniques, with and without filling the cylinder.

FIGURE 8 .
FIGURE 8. SNR with 2 mm Al filter and various tube voltages.TABLE 1. Values of K for uncorrelated and correlated data, and for the Shepp-Logan and Ramp filters for image reconstruction.

TABLE 2 .TABLE 3 .
Image noise measured from reconstructions of LTT-simulated noisy projections.The unit is 10 -3 cm -1 .Image noise in air scans, in units of 10 -4 cm -1 .Comparing the first and third rows, the maximum error is between the predicted and measured values is 4%.TABLE 4. Values of rz,corr for different possible cross-talk kernels.The first column shows the result for the cross-talk kernel estimated from the scanner.

FIGURE 9 .
FIGURE 9. Noise in cone beam reconstructed slices from simulated data.The x-axis shows the distance from the central plane.

FIGURE 10 .
FIGURE 10.Image noise relative to the central plane.The plot is asymmetric due to the heel effect.