Analysis of Image Noise in 3D Cone-Beam CT: Spatial and Fourier Domain Approaches under Conditions of Varying Stationarity

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1 Analysis of Image Noise in 3D Cone-Beam CT: Spatial and Fourier Domain Approaches under Conditions of Varying Stationarity Angel R. Pineda *a, Jeffrey H. Siewerdsen b,c, Daniel J. Tward b a Department of Mathematics, California State University, Fullerton CA, USA 9834 b Ontario Cancer Institute, Princess Margaret Hospital, Toronto ON, Canada M5G M9 c Department of Medical Biophysics, University of Toronto, Toronto ON, Canada M5G M9 ABSTRACT The statistical properties of medical images are central in characterizing the performance of imaging systems. The noise in cone-beam CT (CBCT) is often characterized using Fourier-based metrics, such as the 3D noise-power spectrum (NPS). Under a stationarity assumption, the NPS provides a complete representation of the covariance of the images, since the covariance matrix of the Fourier transform of the image is diagonal. In practice, such assumptions are obeyed to varying degrees. The objective of this work is to investigate the degree to which such assumptions apply in CBCT and to experimentally characterize the NPS and off-diagonal elements under a range of experimental conditions. A benchtop CBCT system was used to acquire 3D image reconstructions of various objects (air and a water cylinder) across a range of experimental conditions that could affect stationarity (bowtie filter and dose). We test the stationarity assumption under such varying experimental conditions using both spatial and frequency domain measures of stationarity. The results indicate that experimental conditions affect the degree of stationarity and that under some imaging conditions, local descriptions of the noise need to be developed to appropriately describe CBCT images. The off-diagonal elements of the DFT covariance matrix may not always be ignored. Keywords: noise-power spectrum, image quality, covariance matrix, stationarity, image noise, cone-beam CT, 3D imaging, flat-panel detector 1. INTRODUCTION Evaluating the performance of imaging systems should take into account three elements: the task, the statistics, and the observer [1-4]. The task refers to the intended use of the images, such as the detection of a lesion or the estimation of the size of a vessel. The statistics characterize the sources of variation in the images. The observer deals with how to extract information from the data and may be a human or a computer. Each of these components is critical in evaluating the usefulness of a given imaging system. Noise in computed tomography (CT) is often characterized using measurements of the noise-power spectrum (NPS) by implicitly assuming that the noise is stationary or at least wide-sense stationary [4-7]. Since stationarity implies that the correlations of the image are the same for all the locations in the 3D volume, both the experimental procedure and the analysis of the noise are greatly simplified, because the NPS gives a complete description of the covariance of the noise i.e., the covariance matrix is diagonal, and the diagonal is the NPS. The stationarity assumption arises naturally from an ideal and continuous description of CT but may not be rigorously justified for real experimental data. In this paper, we experimentally investigate the validity of the stationarity assumption in characterizing the statistics of images produced in cone-beam computed tomography (CBCT). The purpose is to understand to what degree the NPS fully characterizes the magnitude and correlation of fluctuations in the image data as we change the experimental conditions. There has been an active discussion in the research community of the non-stationarity nature of medical images [8-9]. For this study, we test the extent to which we can control the stationarity in the images by changing the experimental imaging conditions. We consider both spatial and frequency domain characterizations of the noise to better understand the degree to which the stationarity assumption holds in real CBCT images. The degree of applicability of the stationarity assumption is an initial step in understanding the validity of summarizing image quality and object detectability using the NPS and related Fourier-based metrics, such as the noise-equivalent quanta (NEQ). * apineda@fullerton.edu; phone Medical Imaging 8: Physics of Medical Imaging, edited by Jiang Hsieh, Ehsan Samei, Proc. of SPIE Vol. 6913, 69131Q, (8) /8/$18 doi: / Proc. of SPIE Vol Q-1 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

2 . THEORY The evaluation of noise in CBCT images based on the NPS assumes that the images are stationary. While a finite and discrete image cannot be rigorously stationary, this approximation can be useful to different degrees depending on how well the NPS characterizes the covariance of the images in the Fourier domain. For a wide-sense stationary continuous random process, the Fourier transform diagonalizes the autocovariance function [4]. The discrete version of this property is that the image noise in the DFT space would have a diagonal covariance matrix with the NPS along the diagonal. There are different methods to estimate digital image noise-power spectrum. The technique used below is the directdigital method, which considers the ensemble average of the Fourier transform of zero-mean ( noise-only ) data which is equivalent to averaging periodograms: dxdydz NPS( fn) = DFT ( I) NPS() f = FLLL {( a x )} x y z n subvolumes where DFT(I) is the Discrete Fourier Transform of zero-mean (de-trended) images (I), f n is a 3-D spatial frequency, dx, dy, and dz are the dimensions of the reconstructed voxels, and L x, L y, and L z are the dimensions of the subvolumes used to estimate the NPS. (1) The elements of the full covariance DFT covariance matrix can be computed similarly as follows: dxdydz K = DFT I DFT I * ( ())( () ) nm n m LLL x y z subvolumes () Note that the diagonal of this matrix (for which n = m) is simply the NPS, illustrating the interpretation of the NPS as the variance of the Fourier components of the noise. In a stationary ensemble of images, the off-diagonal elements of this matrix are zero. In experiments described below, we test how well the DFT diagonalizes the covariance matrix of the images by computing the off-diagonal elements and comparing them to the NPS. Task-based evaluation of imaging systems using the Hotelling observer either in the frequency or spatial domain requires a description of the covariance of the data. The covariance matrix enters the computation of the detectability of a signal through the inverse of the covariance matrix. In the DFT space and if the DFT covariance matrix is diagonal then this would be simply related to 1/NPS. In the general case where the DFT covariance is not diagonal, the offdiagonal elements enter the detectability estimate in a complicated manner depending on the object being imaged. Future work in this project is to map the effect of the off-diagonal elements in computations of detectability. 3. EXPERIMENTAL METHODS 3.1 Experimental Setup The imaging bench of Fig. 1 was used in all experiments below. The bench incorporates an x-ray tube, a flat-panel detector (FPD), and motion control system (translation and rotation axes) for CBCT imaging under precise, reproducible conditions. The imaging components, system geometry, acquisition parameters, and reconstruction parameters are summarized in Table I. The FPD incorporated a scintillator (5 mg/cm CsI:Tl x-ray converter) and a (41 41 cm ) active matrix of a-si:h photodiodes and thin-film transistors at 4 µm pixel pitch (8% fill factor). The geometry approximated that common in CBCT-guided radiation therapy viz., a geometric magnification of ~1.54, giving a ~(5.6 x 5.6 x 5.6) cm 3 volumetric field of view. For each CBCT scan, 3 projections were acquired from a circular orbit over 36 o. The Feldkamp algorithm for 3D filtered backprojection was used to reconstruct volumetric images at (.5 x.5 x.5) mm 3 voxel size, equal to the detector pixel pitch divided by the magnification. The reconstruction filter consisted of a ramp and apodization window (equivalent to a Hann filter). Proc. of SPIE Vol Q- Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

3 Figure 1. Experimental setup for cone-beam CT imaging. The x-ray tube, object rotation, and detector are synchronized within a precisely controlled geometry, with images reconstructed using a modified Feldkamp algorithm. A variety of objects (illustrated in blue) were considered, including air and various water cylinders. Images were acquired with and without a bowtie filter and as a function of dose to examine factors that affect the degree of noise stationarity. Experimental Methods Imaging Components Image Acquisition Parameters X-Ray Tube W target /.4 mm FS kvp 1 kvp Generator 8 kw const. Added Filtration Variable (below) potential Flat-Panel Detector RID-164 mas / Projection Variable (below) X-ray Converter 5 mg/cm CsI:Tl Frame Rate 1 fps Pixel Pitch.4 mm Number of Projections 3 Format 14 x 14 Orbital Extent (Angular Increment) 36 o (1.15 o ) Projection FOV 41 x 41 cm Antiscatter Grid None Bowtie Filter Optional Imaging Geometry Reconstruction Parameters Source-to-Axis Distance (SAD) 93.5 cm Filter Hann Source-to-Detector Distance 144. cm Voxel Size.5 mm isotropic (SDD) Geometric Magnification 1.54 Volume Image Format (14 x 14) axial < 14 longitudinal Volumetric FOV up to 5.6 cm Imaging Phantom Configurations Phantom Added Filtration Radiation Dose Level (mas / Bowtie Filter projection) Air 5.1 mm Cu (.4, 1.,., and 4.) mas / projection No cm Diameter.1 mm Cu (.4,.8, 1.6, and 3.) mas / projection No Water Cylinder cm Diameter Water Cylinder.1 mm Cu (.4,.8, 1.6, and 3.) mas / projection Yes Table I. Summary of experimental parameters and imaging phantom configurations. A variety of experimental conditions were tested to investigate the influence on the NPS and DFT covariance matrix. The scope of experimental parameters that could potentially affect the degree of stationarity in CBCT is vast, including: geometry, scatter conditions (grid, air gap, and longitudinal FOV), detector type, pixel binning, reconstruction filters, voxel size, radiation dose level, use of a bowtie filter, and phantom configuration. In the present study, we investigated the last 3, giving the phantom configurations described in Table I: Air (no bowtie); Water Cylinder (with bowtie); and Water Cylinder (without bowtie), each imaged at 4 dose levels roughly spanning the sensitive range of the detector (bare-beam signal maintained below sensor saturation). The 1 resulting data sets formed the basis for experimental analysis of 3D NPS and DFT covariance matrix, described below. Proc. of SPIE Vol Q-3 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

4 The bowtie filter was an Elekta bowtie filter cassette (F , Elekta Oncology Systems, Atlanta GA), taken from an Elekta Synergy TM system for CBCT-guided radiation therapy and adapted to the imaging bench. The bowtie was precisely centered at the exit face of the collimator and was in place during flood-field calibration. Ideally, the bowtie would provide full lateral modulation of the x-ray beam projected through the water cylinder, such that the resulting mean detector signal (in pre-flood-field-corrected projections) was constant (1:1 modulation between the center and edge of the image). Without the bowtie filter, the signal modulation from the center to the edge of the image was 1:1. With the bowtie, the modulation improved to 5:1, and while this was less than ideal, it was believed sufficient to give a measurable effect on noise and stationarity. Future work will incorporate a stronger (thicker / higher density) bowtie. (b) Water + Bowtie Figure. Example axial images of the three phantom configurations, giving qualitative illustration of possible variation in noise stationarity. The hypothesis underlying these experiments was that various experimental configurations would result in varying degrees of noise stationarity, measurable in terms of: 1.) locally varying NPS; and.) relative magnitude of off-diagonal elements of the DFT covariance matrix. Specifically, we hypothesize that Air images would exhibit the highest degree of stationarity overall and that images of the Water Cylinder with a bowtie filter would exhibit higher stationarity than without a bowtie filter. The basis for this hypothesis is the degree of lateral uniformity in detector SNR highest for Air and worst for Water without a bowtie. Example axial image reconstructions for the three phantom configurations are shown in Fig., in qualitative agreement with these assertions. The stationarity of first-order statistics (i.e., the mean) appears highest for the Air phantom and worst for the Water (No Bowtie) case (highest level of x-ray scatter shading artifact). The stationarity of the second-order statistics (i.e., the correlations) deserves closer inspection. For the Air scan, there appears to be a rich variation of spatial-frequency dependence in the noise from the center (ring correlation) to intermediate radii (uniform mid-pass noise) to the periphery (rings and view aliasing). Similarly for the Water phantom configurations: one would expect radial dependence of the variance according to the transmittance of the cylinder (lower quantum noise at the periphery); however, the images suggest a tradeoff in noise texture among ring correlation, midpass quantum noise, and view aliasing. 3. Measurement of the 3D Noise-Power Spectrum and DFT Covariance Matrix The resulting 1 volume reconstructions (3 phantoms x 4 exposure levels) provided the basis for analysis of the NPS, local NPS, and DFT covariance matrix. Each volume was divided into non-overlapping sub-volumes ROIs of (64 x 64 x 64) voxels, 6 of which were taken from completely within the cylindrical volume. Prior to spectral estimation, each ROI was detrended by subtraction of a 3D planar fit, giving zero-mean data free from first-order signal non-uniformity (e.g., x-ray scatter shading artifact). The 3D NPS was estimated from the 3D FFT of the zero-mean data using equation (1). Any element of the full covariance matrix can be estimated using equation (). Because CT reconstruction applies a high-pass filter in the in-plane (x,y) direction but not in the longitudinal (z) direction, we focus on the frequencies in the x direction. We compute the NPS by averaging the periodograms (square of the magnitude of the normalized DFT) over the 6 zero-mean image blocks. The covariance matrix in the DFT domain has 64 3 x 64 3 complex elements. There are methods for dealing with such large covariance matrices [1] but for the purposes of this work we limit ourselves to estimating diagonals. We explore experimentally the components of the DFT Proc. of SPIE Vol Q-4 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

5 covariance matrix that are typically ignored in NPS analysis. For example, the 1 st off-diagonal in the x-direction describes the correlations with the nearest frequency in the x-direction which likely represents the greatest component of off-diagonal (non-stationary) noise. 4. RESULTS 4.1 The 3D Noise-Power Spectrum Figure 3 shows slices of the 3D NPS for the Air scan reconstructions. Note the asymmetric characteristic of the NPS between axial and sagittal planes, as previously reported and in agreement with theoretical expectations. The axial domain exhibits a mid-pass spectrum that is characteristic of filtered backprojection (ramp + apodization filters), whereas the NPS in the longitudinal direction is band-limited according to the low-pass characteristics of the detector (indirect detection FPD), interpolation filter, etc. Such analysis is the basis for assessment of noise-equivalent quanta (NEQ) and is becoming an increasingly important characterization of the noise in CT systems evaluation, quality assurance, and optimization. NPSInAXLPInsforAir Spatial Frequency, f (mmd) xlo' (a) (b) NPS In S.glttai Pluis for Air Spatial Frequency, f1 (mmd) I io' Figure 3. Orthogonal slices of the 3D NPS estimated from Air scan reconstructions: (a) axial NPS and (b) sagittal NPS. The asymmetry in the noise between axial and saggital planes is a characterization of asymmetric filtration in 3D filtered backprojection. Data shown here are from reconstructions of Air at mas per projection. The extent to which the 3D NPS is invariant with location throughout the reconstruction FOV was investigated in terms of the local NPS estimated from ROIs at a specific radius (assuming NPS constant at a given radius). The resulting local NPS were examined as a function of distance from the reconstruction center as illustrated in Figs. 4 and 5. A number of features come to light from such analysis. For the Water phantom, the reduction in NPS at larger radius is attributable to increased fluence (reduced attenuation) nearer the edges of the detector (i.e., reduced fluence higher attenuation at the center of the detector in the shadow of the thickest part of the phantom, resulting in higher NPS). A competing effect is view sampling, which becomes more significant at larger radius and tends to increase the NPS. Second, the extent to which the NPS varies with position for a given phantom configuration is evident in the results. Each phantom exhibits significant variation of NPS with radius, but for the Air data, the NPS varies less (i.e., more stationary) than for the water data consistent with hypothesis. Finally, the axial NPS is seen to exhibit a marked change not only in magnitude, but also in spatial-frequency content, changing from a radially symmetric donut at smaller radii to a more lobular characteristic at greater radii. Moreover, the change in such frequency characteristics is slightly different among the three phantom configurations a somewhat more pronounced, less radially symmetric shape observed for the water phantom data. Proc. of SPIE Vol Q-5 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

6 (a) Air (b) Water (with Bowtie Filter) (c) Water (No Bowtie)..5 (x1o)i R-.5 (x15) 4cm R 4 cm.. U- a). z (a)air i :: d.5 1:i.5 :Lo 6.5. Spatial Frequency (mm-') Spatial Frequency (mm-') Figure 4. The local 3D NPS measured for various phantom configurations. (a-c) Axial NPS estimated as a function of distance from reconstruction center (R ~4-8 cm). The Air phantom shows a near independence in NPS with varying radius, while the water phantoms show a distinct reduction in NPS with radius. (b) Water (with Bowtie Filter) (C) Water (No Bowtie) Figure 5. The local 3D NPS measured for various phantom configurations as a function of distance from reconstruction center. (a-c) Axial NPS are illustrated as a function of distance from center, demonstrating the increasingly asymmetric, lobular structure of the NPS at extended radii a clear demonstration of non-stationarity. 4. Off-Diagonal Elements of the DFT Covariance Matrix Figure 6 shows the 1 st off-diagonal element of the DFT covariance matrix, which quantifies the correlation of two neighboring frequencies in the x-direction. Such likely represents the greatest component of off-diagonal components in the DFT covariance matrix. Proc. of SPIE Vol Q-6 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

7 First Of.dI.gonL In x In AXL Pluis for Air xl o' First Of.dI.gonL In x In S.glttai Pluis for Air xl o' Spatial Frequency, f (mmd) (a) (b) Spatial Frequency, f1 (mmd) Figure 6. First off diagonal of the DFT covariance matrix in the x-frequency direction, i.e. the correlation of the given frequency with the next closest frequency. These components are commonly ignored in typical NPS analysis. By comparing the axial NPS [Fig. 3(a)] with the axial representation of the 1 st off-diagonal element [Fig. 6(a)], we see that the off-diagonal elements are small but not negligible with respect to the NPS. In this case, the two exhibit similar shape in the frequency domain and differ in magnitude by approximately a factor of 1. Subsequent analysis (not shown here) demonstrates that the nd, 3 rd, etc. off-diagonal elements are reduced still further, but persist in a manner that may not be entirely negligible in system noise characterization and evaluation of detectability. i O Radial Average of NPS and ODE in Axial Plane (Air) 6 NPS ' FirstODEinx I Frequency mm Figure 7. Axial NPS in comparison to off-diagonal elements (ODE) of the DFT covariance matrix. The plots represent radial averages of the axial plots in Figs. 3(a) and 6(a). Data are for Air scan reconstructions at mas per projection. 4.3 Simple Measures of Stationarity We are interested in quantifying the degree of stationarity of the images. A simple spatial domain measure of stationarity is the variability of the standard deviation along the radius in the axial plane of the volume. For a stationary random process, the standard deviation would be constant. Figure 8 demonstrates how the standard deviation of the noise changes with radius for the three phantom configurations described in Table I. The Air scan reconstructions are seen to exhibit the greatest degree of stationarity, with image noise increasing with radius by a few percent. The increase in noise at larger radius is attributed to view sampling (finite number of projections). The Water+Bowtie and Water (no bowtie) configurations are considerably less stationary, each demonstrating a reduction in noise at larger radius of Proc. of SPIE Vol Q-7 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

8 several percent. The reduction is attributed to higher detector signal (less attenuation and, therefore, higher incident fluence) at greater distances from center. Of the three phantom configurations, Water (no bowtie) is the least stationary (largest variation in noise) consistent with hypothesis. I.5 Normalized SD of noise at.4 mas = a (.95.9 water -water + bowtie air Radius (cm) Figure 8. Standard deviation of the noise as a function of radius (distance from reconstruction center). The standard deviation is normalized to 1 at a radius of 4 cm for all three imaging conditions. The variability with radius is consistent with a rank order that Air is more stationary than Water+Bowtie, which in turn is more stationary than Water (without bowtie). We can define simple measures of relative stationarity by computing the standard deviation of the noise as a function of radius and dividing by the average standard deviation: Κ σ stdev = ( σ ) σ where σ is the sample standard deviation in voxel values for a given region of interest (ROI). The term stdev(σ) is the standard deviation in the noise e.g., exemplified in the variation with radius shown in Fig. 8. For a constant standard deviation across the image, therefore, the quantity K σ should be zero. Further, we hypothesize smaller values of K σ for images that are more stationary specifically, K air σ < K water+bowtie σ < K water σ. Similarly, a measure of stationarity based on the off-diagonal elements of the Fourier covariance matrix may be defined as: Κ 1 NPS = 1 ODE x NPS were the superscript 1 refers to the 1 st off-diagonal element. The numerator is based on the correlations with the nearest frequency in the x-direction, and the denominator to the variance as defined by the integral over the NPS. We compare the degree of stationarity of the three experimental configurations based on these simple measures of relative stationarity. As shown in Table II, the results are consistent with hypothesis specifically, the K metrics are smallest for the Air scan reconstructions (most stationary) and largest for the Water (without bowtie) reconstructions (least stationary). (3) (4) Proc. of SPIE Vol Q-8 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

9 (a) Spatial Domain: K σ (Noise Non-Stationarity) Exposure Level (mas / projection) Phantom Configuration.4 mas.8-1 mas 1.6- mas 3.-4 mas Water Water + Bowtie Air (b) Fourier Domain: K 1 NPS (NPS Non-Stationarity) Exposure Level (mas / projection) Phantom Configuration.4 mas.8-1 mas 1.6- mas 3.-4 mas Water Water + Bowtie Air Table II. Summary of (a) spatial and (b) Fourier domain (non-) stationarity metrics as defined in Eqs (3) and (4), respectively. Both spatial and Fourier metrics should be zero for stationary images. We observe overall consistency with hypothesis regarding the extent to which experimental conditions affect stationarity viz., air data are more stationary than water data, and the bowtie filter helps somewhat to restore the degree of stationarity. 5. DISCUSSION AND CONCLUSIONS These results explore for the first time the complex correlations in the Fourier components of CBCT data. The analysis suggests that as the non-stationarity increases so do the off-diagonal elements of the DFT covariance matrix; hence, ignoring such effects by characterizing noise solely in terms of the NPS would lead to significant errors or misinterpretation. The spatial domain analog of the current investigation is moving from looking solely at the variance across the image to looking at the correlations between pixels. Since the NPS is the variance of the Fourier coefficients, we are looking to do the same in the Fourier domain by understanding the complex correlations between different frequencies. In light of the non-negligible magnitude of off-diagonal terms under certain experimental conditions, we need to develop further a proper interpretation and intuition as to their meaning and impact on the practical evaluation of noise properties of CT reconstructions. In particular, two questions beckon: (i) what level of non-stationarity is tolerable such that NPS is a reliable, meaningful metric of image noise?; and (ii) to what extent (and in what manner) do the observed non-stationarities affect detectability? Future work also involves refinement of the experimental and analysis techniques in estimating elements of the DFT covariance matrix. The variability in the correlations in the data associated with multiple realizations of the noise volumes needs to be better understood to avoid the effects of structured artifacts (e.g., rings and streaks). Furthermore, possible effects of arising from the size of the subvolumes and associated spectral leakage need to be more fully investigated and minimized in order to obtain accurate estimates of the DFT covariance matrix. The measures of relative stationarity defined in this paper were for the most part heuristic. While the results (Table II) agreed overall with the hypothesis regarding experimental conditions and degree of stationarity, we still need to understand their variability and proper interpretation. The single data point in Table II that did not agree with hypothesis (Air data at.4 mas) is believed to be an outlier. Future work will include more fundamental metrics of stationarity in terms of the difference in detectability estimates arising from the NPS alone versus the full DFT covariance matrix as well as statistics related to testing stationarity as a hypothesis test. The hypothesis underlying these experiments was that implementing a bowtie filter would improve stationarity for the water reconstructions, which was indeed observed, but to a somewhat smaller extent than ideal. There are several Proc. of SPIE Vol Q-9 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use:

10 reasons why the improvement in stationarity was not as much as might be expected. (i.) An ideal bowtie filter would modulate the beam such that the (raw, uncorrected) projection image is completely flat. Without the bowtie, we measured approximately 1:1 modulation between the center of the beam and the edge (bare-beam). With the bowtie, the modulation dropped to 5:1, which is less than the 1:1 ideal. One might expect only a slight improvement in stationarity given this (fairly weak) bowtie. Future work will include a more aggressive bowtie that gives stronger (nearly 1:1) modulation. (ii.) It is worth noting that even for a perfect (1:1 modulation) bowtie filter, there is still an important potential source of non-stationarity: hardening of the beam at the periphery (more than at the center). While such a bowtie may perfectly modulate the mean signal, one must consider how the filter has modulated the energy of the beam - - and therefore, the DQE of the detector. Even with the mean signal perfectly modulated, if the beam incident on the detector is 'harder' (higher mean energy) at the periphery than at the center, the DQE() is likely lower at the periphery than at the center, and one would expect higher quantum noise at the periphery. The converse is also true. Therefore, there are underlying factors in modulating the mean signal that affect the noise and, hence, the stationarity. Such should be investigated in future work, including modeling of the x-ray beam and use of bowtie filters of various material types. It is also worth noting that the metrics of relative stationarity were designed to characterize the degree of stationarity but do not necessarily tell if the noise is higher or lower in one area of the image than another. The local NPS does provide such information and conveys further how the noise correlation differs. Future work will study different metrics related to the DFT covariance matrix to better understand how to characterize both the degree of non-stationarity as well as the variations in noise correlation. In summary, the degree of stationarity (as characterized by the off-diagonal elements of the DFT covariance matrix with respect to the NPS) depends on the experimental imaging conditions, and conditions can be selected that improve the stationarity assumption. Our work shows that the relative magnitude of off-diagonal elements conventionally ignored in NPS analysis warrants investigation of their effect on object detectability and that ignoring such terms may lead to an inaccurate characterization of the image noise. ACKNOWLEDGEMENTS This work was performed in a research consortium between the University Health Network (Toronto ON), California State University (Fullerton CA), and Stanford University (Palo Alto CA). The work was supported by the National Institutes of Health, R1-CA REFERENCES 1. Barrett, H.H., Objective assessment of image quality: Effects of quantum noise and object variability, J. Opt. Soc. Am. A 7, (199).. Hanson, K.M., "The characteristics of computed-tomographic reconstruction noise and their effect on detectability," IEEE Trans. Nucl. Sci. NS-5, (1978). 3. Siewerdsen, J.H. and Jaffray, D.A., Optimization of x-ray imaging geometry (with specific application to flat-panel cone-beam computed tomography, Med. Phys. 7(8), (). 4. Barrett, H.H. and Myers K.J., Foundations of image science (Wiley-Interscience Hoboken, N.J. 4) 5. Riederer, S.J., Pelc, N.J., and Chesler, D.A., The Noise Power Spectrum in Computed X-ray Tomography, Phys. Med. Biol., 3 (3), (1978). 6. Siewerdsen, J.H., Cunningham, I.A. and Jaffray, D.A., A framework for noise-power spectrum analysis of multidimensional images, Med. Phys. 9(11), (). 7. Kay, S.M., Fundamentals of Statistical Signal Processing (Prentice-Hall, Upper Saddel River, NJ. 1993) 8. Gagne, R.M., Myers, K.J., and Quin P.W., Effect of shift invariance and stationary assumptions on simple detection tasks: Spatial and spatial frequency domains, Proc. SPIE 43, (1). 9. Pineda, A.R. and Barrett, H.H., Figures of merit for detectors in digital radiography. I. Flat background and deterministic blurring, Med. Phys. 31() (4) 1. Barrett H.H., Myers K.J., Gallas B., Clarkson E. and Zhang H., Megalopinakophobia: Its symptoms and cures, Proc. SPIE 43, (1). Proc. of SPIE Vol Q-1 Downloaded from SPIE Digital Library on 13 Nov 9 to Terms of Use: