See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/19572599 The noise power spectrum of CT images. Phys Med Biol ARTICLE in PHYSICS IN MEDICINE AND BIOLOGY · JUNE 1987 Impact Factor: 2.76 · DOI: 10.1088/0031-9155/32/5/003 · Source: PubMed CITATIONS 103 READS 457 2 AUTHORS: Marie Foley Kijewski Harvard University 67 PUBLICATIONS 899 CITATIONS SEE PROFILE Philip F Judy Partners HealthCare 141 PUBLICATIONS 2,406 CITATIONS SEE PROFILE Available from: Philip F Judy Retrieved on: 23 September 2015
Phys. Med. Biol., 1987, Vol. 32, No 5, 565-575. Printed in the UK
The noise power spectrum of CT images
Marie Foley Kijewski and Philip F Judy Department of Radiology, Harvard Medical School and Brigham and Women’s Hospital, Boston, Massachusetts 02115, USA
Received 22 July 1986
Abstract. An expression for the noise power spectrum of images reconstructed by the discrete filtered backprojection algorithm has been derived. The formulation explicitly includes sampling within the projections, angular sampling, and the two-dimensional sampling implicit in the discrete representation of the image. The effects of interpolation are also considered. Noise power spectra predicted by this analysis differ from those predicted using continuous theory in two respects: they are rotationally asymmetric, and they do not approach zero at zero frequency. Both of these properties can be attributed to two-dimensional aliasing due to pixel sampling. The predictions were confirmed by measurement of noise power spectra of both simulated images and images from a commer- cial x-ray transmission CT scanner.
1. Introduction
The quality of computed tomographic (CT) images is usually evaluated in terms of spatial resolution and level of noise. A single-parameter measure of image noise, however, such as the standard deviation over an area, is inadequate to predict the utility of an image for a given task. Riederer et al (1978) showed that the random variation in CT number at one point of an image is not independent of the random variation at other points. These spatial correlations can be fully described by either the autocorrelation function or its Fourier transform, the noise power spectrum ( NPS)
(Dainty and Shaw 1974). Riederer er al (1978) used the continuous convolution-backprojection model to
predict the image NPS from the projection NPS, under the assumption that the noise in the projections is uncorrelated (white). The image NPS was analysed in light of decision theory by Wagner et a1 (1979) and by Hanson (1979), also using continuous variables. These authors concluded that the shape of the NPS is determined by the reconstruction algorithm, specifically the convolution filter, and that, consequently, the image NPS is rotationally symmetric and proportional to frequency at low frequencies.
For discrete backprojection, the reconstruction algorithm includes sampling and interpolation, as well as filtering and backprojection. Faulkner and Moores (1984) derived a formula which represented the NPS for a discrete reconstruction process. Although they considered sampling within the projections and angular sampling, they neglected both the interpolation and the two-dimensional sampling intrinsic to discrete backprojection. The formula which they derived for the NPS resulting from discrete convolution-backprojection was equivalent to those derived previously for continuous
convolution-backprojection, and, therefore, the NPS which they predicted were also rotationally symmetric and proportional to frequency at low frequencies. We show here that the formulation of Faulkner and Moores (1984) does not describe the NPS
of images reconstructed by discrete convolution-backprojection, and that the spectra which they predict, particularly for the standard ramp filter, are inaccurate.
In 5 2 we derive an expression for the NPS of an image reconstructed by the discrete convolution-backprojection algorithm. We include the effects on the image NPS of all sampling operations, including the two-dimensional sampling caused by the discrete two-dimensional representation of the image. We also explicitly incorporate the effects of the interpolation required during backprojection. Our formulation predicts that aliasing of noise destroys the rotational symmetry of the image NPS. A second implica- tion is that because noise is aliased into all portions of the spectrum, the ramp-like low-frequency behaviour predicted by Riederer er a1 (1978) and Faulkner and Moores (1984) will be altered and, furthermore, the zero-frequency (DC) component will be non-zero. In 0 3 we compare our predictions with measured NPS from both simulated images and images from a commercial cr scanner.
2. Theory
In this section, we derive an expression for the NPS which includes the effects o f interpolation and of all sampling operations. We use parallel geometry and assume that the noise in the raw projection data is stationary, additive and uncorrelated (white). The projection NPS is constant, with independent components confined to frequencies below the Nyquist frequency, which is determined by the projection sampling distance. Because the projections are discrete, however, the spectrum is replicated at intervals of twice the projection Nyquist frequency (Barrett and Swindell 1981). This can be expressed by (see figure l ( a ) )
u2 * N ( f ) = p c rect[a(f-2kffl)] =- 2 rect(af- k)
u2 x
k=-* P
where a is the projection sampling distance, ffl = 1/2a is the projection Nyquist frequency, U' is the variance, P is the number of projection elements, and rect( . . . ) is the rectangle function (Gaskill 1978),
rect(x) = if 1x1 <t otherwise.
Filtering the projections multiplies their power spectrum by the squared magnitude of the filter frequency response. The convolution-backprojection algorithm filters the projections by a ramp filter which may be apodised by a function H ( f ) . The NPS then becomes (see figure l ( b ) )
The points of the projection at which data are required for backprojection will not, in general, correspond to points at which data were measured. The projection values between the measured points must be estimated; interpolation converts the discrete
The noise power spectrum of CT images 567
l i
Spat la l f r e q u e n c y i x p r o l e c t l o n N y q u l s t f r e q u e n c y I
Figure 1. ( a ) NPS of discrete projection consisting of white noise (equation ( l ) ) . The main portion of the spectrum, shown by full curves, is replicated at f=2kfn , where k is an integer. The projection Nyquist frequency, f,, equals 1/2a, where a is the projection sampling distance. The dots on either side of the plot indicate that the main portion of the spectrum is replicated ad injnifurn. ( b ) NPS of discrete projection consisting of ramp-filtered white noise (equation ( 3 ) in text, H ( f ) = 1). (c ) NPS of continuous projection formed from spectrum shown in ( b ) by nearest-neighbour interpolation (equation (4) in text, H ( f ) = 1, G ( f ) = a sinc(af)). ( d ) NPS of continuous projection formed from spectrum shown in ( b ) by linear interpolation (equation (4) in text, H ( f ) = 1, G ( f ) = a sinc2(af)). Note: these four noise power spectra are scaled differently in order to show detail.
projection to a continuous one whose power spectrum is (figures I ( c ) and ( d ) )
where G(f ) is the frequency space representation of the interpolating function. It is important to emphasise that although the raw projections are discrete, their
power spectrum is continuous. Furthermore, because of the discrete nature of the projections and the consequent replication in frequency space, the spectrum extends to infinite frequencies regardless of the extent to which the apodisation filter rolls off the power within each replication. Both of these points were recognised by Joseph et ul (1980), although the presence of supra-Nyquist frequency components was attributed to the interpo1,ating function. These components are present in the raw and filtered projections; the interpolating function, in fact, apodises the high-frequency components
568 M F Kijewski and P F Judy
(see figure 1). Interpolation, furthermore, is the only mechanism for removing very- high-frequency noise power components. The NPS is not apodised, as is the signal spectrum, by the physical aperture.
According to the well known projection-slice theorem (Deans 1983), backprojecting a projection at a given angle superimposes its power spectrum along a spoke through the frequency space origin at that angle. The (two-dimensional) NPS of an image formed by backprojection of L projections over T radians is
where fi = (f;+f;)"*, fx and f , are rectilinear frequency space coordinates, ( T / L ) is a normalisation factor and S [ f x sin(Tl/L) -x, c o s ( ~ l / L ) ] is a two-dimensional delta function (Gaskill 1978) which lies along a spoke in frequency space.
Two characteristics of this image NPS should be pointed out. Firstly, the NPS
extends, in general, to infinite frequencies; this follows from the fact that its components, the projection spectra, extend to infinite frequencies. Secondly, the NPS is continuous in radial frequency and discrete in angular frequency, i.e. it lies along spokes in frequency space which are separated by gaps.
The NPS described by equation (5) incorporates all sampling operations except the discrete representation of the image. Sampling the image at discrete points in a two-dimensional array corresponds to convolving its spectrum with the two- dimensional sampling function (Gaskill 1978):
where b is the (square) pixel dimension. The spectrum is replicated at points of a rectangular grid in frequency space; the spacing of these points is twice the pixel Nyquist frequency. Components at frequencies higher than can be represented in the pixel array are aliased, i.e. added to lower-frequency components (figure 2 ) . The aliased spectrum is
The discrete angular sampling, represented in equation (5) by the delta function, was not made explicit by Faulkner and Moores (1984), who assumed sufficient angular sampling and replaced the summation of delta functions by the spoke density. We emphasise the angular sampling operation both for generality and because of our concern with the high-frequency components of the spectrum. Even if the image is formed from what is generally considered a sufficient number of projections (Joseph and Schulz 1980), the supra-Nyquist-frequency components which are aliased onto the main portion of the spectrum can be undersampled in angle.
For illustration, we will consider the spectra which characterise images that have been reconstructed using an unwindowed ramp filter ( H ( f ) = l ) , and two interpolation methods: nearest-neighbour interpolation (G(f) = a sinc(af)) and linear interpolation (G( f ) = a sinc'(af)). A ramp filter was considered in previous publications on image NPS (Riederer et al 1978, Faulkner and Moores 1984). Nearest-neighbour and linear
The noise power spectrum of CT images 569
interpolation are the most commonly used methods. If the pixel dimension, 6, equals the projection sampling distance, a, then the NPS becomes
(8)
where f A [(fx - n / a ) 2 + (f, - m/a)']'" for nearest-neighbour interpolation and
for linear interpolation. The predictions of equations (8) and (9) will be compared with spectra from
simulated images which were reconstructed using a ramp filter with nearest-neighbour and linear interpolation, respectively. The results of equation (9) will also be compared with the spectrum of images, reconstructed using a ramp filter and linear interpolation, from a commercial x-ray transmission CT scanner.
3. Methods 3.1. Simulations
Two sets of 2000 images each were reconstructed from projections consisting of Gaussian-distributed random numbers. Each random number was generated by sum- ming twelve pseudorandom numbers from a uniform distribution and scaling the sum to yield a Gaussian distribution of zero mean and unit variance (James 1980). For each image, 200 projections, each containing 256 elements, were convolved with a ramp filter and backprojected, using nearest-neighbour interpolation for one set and linear interpolation for the other, onto a 256 by 256 array. The pixel sampling distance, the projection sampling distance, and the convolution filter sampling distance were all the same. The projections covered 180°, and parallel geometry was used.
To demonstrate two-dimensional aliasing, 2000 images were reconstructed by back- projecting a single filtered projection using linear interpolation.
3.2. Measurements
The Technicare Corporation Deltascan 2020 CT scanner is a fourth-generation fan-beam device, consisting of a rotating x-ray source and a stationary ring of solid-state bismuth germinate detectors. Images are reconstructed by the convolution-backprojection method, with several convolution filters available; we selected a ramp filter. Linear interpolation is implemented during backprojection. Sixty-six images of a 25 cm diameter water phantom were obtained at 120 kVp and 50 mA. Scanning time was 2 S
and slice thickness was 10 mm.
3.3. Calculation of power spectra
Two-dimensional NPS were calculated from both sets of 2000 simulated noise images, and also from 33 images which were obtained from the 66 water-phantom images by
570 M F Kijewski and P F Judy
subtracting sequential pairs. We used difference images to ensure that the estimate of the rips of the CT scanner was free of artefacts due to scatter, uncorrected beam hardening or other systematic errors. Spectra were estimated using a two-dimensional extension of the method proposed by Welch (1967), similar to that used by Hanson (1979). NPS of the simulated images were calculated from the central 128 x 128 pixels. Spectra of the Technicare 2020 images were calculated from the central 256 x 256 pixels. In order to reduce truncation errors, the sub-arrays were windowed by
and Fourier transformed. The NPS was estimated as the average over all samples of the squared modulus at each frequency.
We did not remove the low-frequency components from the data before calculating the spectrum, as did Hanson (1979). The purpose of such an operation would have been to remove from the spectrum any variation which is constant from image to image, such as that due to uncorrected beam hardening. These systematic errors are not present in simulated images, and were eliminated from the Technicare images by using difference images for the estimation. Any remaining variation is truly random, and, thus, properly part of the NPS.
4. Results and discussion
The two-dimensional aliasing effects predicted by our analysis (figure 2) were demon- strated in the NPS of images reconstructed from a single simulated projection (figure 3). The projection NPS has contributed to the image NPS along a spoke at the projection angle; however, components of the projection NPS at frequencies higher than can be represented by the pixel array have been aliased onto other portions of the spectrum. The apodisation due to linear interpolation is also apparent; the departure from the ramp-like spectrum predicted by continuous theory is marked. The complete image
f x
Figure 2. Illustration of two-dimensional aliasing. The contribution to the image NPS from a single projection at angle p is shown. The projection contributes to the spectrum along a spoke through the origin at angle p . The two-dimensional sampling implied by the discrete representation of the image causes the spectrum to be replicated at f,,f, = m / b , n / b ; m and n are integers, b is the pixel dimension. For clarity, only the main spectrum and three replications are shown. The projection NPS is as described in figure l ( d ) . The pixel dimension is equal to the projection sampling distance. Dark shading indicates both primary and aliased contributions to the image NPS.
The noise power spectrum of CT images 57 1
Figure 3. Two-dimensional image NPS calculated from 2000 images reconstructed from a single simulated noise projection at 10". Zero frequency is at the centre of the plane. The image was reconstructed using a ramp filter and linear interpolation. The pixel dimension was equal to the projection sampling distance.
NPS is composed of contributions from all projections; the agreement between the predicted spectra and those calculated from the simulated images is excellent (figure 4). The major features predicted by our formulation, i.e. the apodisation due to linear interpolation, the rotational asymmetry and the additional low-frequency components,
Figure4. ( a ) Image NPS predicted by equation (8) (ramp filter, nearest-neighbour interpolation). ( b ) Image NPS predicted by equation (9) (ramp filter, linear interpolation). ( c ) Image NPS calculated Prom 2000 images, each reconstructed from 200 simulated noise projections, using a ramp filter and nearest-neighbour interpola- tion. ( d ) Image NPS calculated from 2000 images, each reconstructed from 200 simulated noise projections, using a ramp filter and linear interpolation.
572 M F Kijewski and P F Judy
Figure 5. Image noise power spectrum of commercial CT scanner (ramp filter, linear interpolation).
are also present in the NPS of images from the Technicare 2020 (figure 5 ) , although it does not agree exactly with the predictions. The discrepancies in the low-frequency components probably reflect deviations of the noise in the Technicare projections from the assumed model, i.e. stationary white noise. The minor discrepancies in the high- frequency components are not surprising, since the images were reconstructed in a fan-beam geometry rather than in the parallel-beam geometry assumed for the analysis.
The rotational asymmetry can be seen in a plot of noise power against angle for a constant radial frequency (figure 6). The NPS calculated from the simulated images agree closely with the predictions; the spectrum calculated from the Technicare images agrees qualitatively. The implication of a NPS which is not rotationally symmetric is that the detectability of an object, or the precision of an estimate of the average CT
number over an area, will depend on the orientation of the object or area with respect to the pixels.
The low-frequency portion of the NPS (figure 7) departs from the ramp function which would describe it in the absence of two-dimensional aliasing; furthermore, the zero-frequency (DC) component is non-zero. The increase in low-frequency noise is
Figure 6. ( a ) Noise power plotted against polar angle at Nyquist frequency (along a semi-circle in frequency space) for ramp filter, linear interpolation. The data points are from the NPS measured from simulated images and the full curve indicates predictions of equation (9). The corresponding complete spectra are shown in figures 4(b) and 4 ( d ) . The variation with angle is a result of two-dimensional aliasing due to pixel sampling; the broken line shows predicted NPS without aliasing. ( b ) Noise power plotted against polar angle at Nyquist frequency, images from commercial CT scanner. The complete spectrum is shown in figure 5.
The noise power spectrum of cr images 573
Spat la l frequency l x Nyqulst frequency)
4.8
3.6
2.4
1.2
0 0.4 0.8 Spat la l frequency l X Nyqulst frequency)
1.2 0 0.4 0.8 Spat la l frequency l X Nyqulst frequency)
1.2
I - > 1 Figure 7. ( a ) Image NPS along frequency-space
- I
diagonal, ramp filter, nearest-neighbour interpola- tion. The data points show measurements from simu- lated images and the full curve represents the NPS
as predicted by equation (8). The broken curve indi- cates the predicted NPS in the absence of two-
k l 4 CL b t * 5 ~ l .*
. -. . .. dimensional aliasing. The corresponding complete spectra are shown in figures 4( a ) and 4(c). (b) Image NPS along frequency space diagonal, ramp filter, linear interpolation. The data points show measure- ments from simulated images and the full curve represents the NPS as predicted by equation (9). The broken curve indicates the predicted NPS in the
1 x Nyquist frequency1 sponding complete spectra are shown in figures 4(b) and 4(d). ( c ) Noise power along frequency space diagonal, images from commercial CT scanner. The complete spectrum is shown in figure 5.
.. L , , , *b.:, , l L" 3 L .'
0 0.4 0.6 1.2 absence of two-dimensional aliasing. The corre-
greater for nearest-neighbour interpolation (figure 7 ( a ) ) than for linear interpolation (figure 7 ( b ) ) because linear interpolation apodises the projection NPS to a greater extent. The source of the zero-frequency component can be understood by considering a projection at 45" to the axes. Those replications of this projection which originate at ( l / b, l / b ) and ( - l / b, - l / b ) intersect the frequency space origin at frequency a/ b in the replicated projection, i.e. the noise power present at frequency &'/b is aliased onto zero frequency. When the pixel dimension equals the projection sampling dist- ance, the contribution to zero-frequency noise power in the image NPS is from frequency 2 a f n = 2.8fn in the NPS of these diagonal projections; significant noise power is present at this frequency when a ramp filter and either nearest-neighbour or linear interpolation are used (figures l (c) and l ( d ) ) .
The question of the low-frequency components of the NPS is far from academic. As pointed out by Hanson (1979), the detectability of large, low-contrast objects is mainly determined by the low-frequency portion of the NPS. Our finding that the low-frequency components are greater than previously believed may help to explain the discrepancy between the performance of human observers and that of the ideal detector for large, low-contrast objects (Kijewski er a1 1983). These findings may also
574 M F Kijewski and P F Judy
have implications for quantitative CT. Zero-frequency (DC) components due to aliasing will shift the values of the reconstructed quantities, leading to biases in quantitation.
Another implication of our findings is that two-dimensional aliasing destroys the linearity of the convolution-backprojection algorithm; convolving the projections before backprojection is not equivalent to convolving the backprojected image with a two-dimensional filter. This means that the practice, sometimes used in SPECT imaging, of reconstructing the image with a ramp filter, then filtering the image with various smoothing functions, can lead to systematic errors in quantitation.
Our results and conclusions differ significantly from those of Faulkner and Moores (1984), who also investigated the NPS of images formed by discrete backprojection. The discrepancies can be explained by the fact that they considered neither two- dimensional pixel sampling nor interpolatyon. They predicted that the NPS of an image reconstructed with a ramp filter would be proportional to frequency, and reported that they had obtained from simulated images a spectrum which agreed with their prediction. Although no details of their NPS calculation were supplied, they presumably used a method analogous to the one by which they calculated autocorrelation functions, i.e. they assumed rotational symmetry and performed one-dimensional calculations on the rows and columns of the image. Such a calculation would yield only that portion of the spectrum which lies along the frequency axes, and cannot reveal rotational asym- metry. The frequency axes correspond to projection angles at which the data are backprojected parallel to the rows and columns of the pixels; if the pixel dimension equals the projection sampling distance, then these angles are unique in that interpola- tion is unnecessary. Had they calculated a true two-dimensional spectrum, Faulkner and Moores would have found that their prediction was true only along the axes. Furthermore, had they off set the projections with respect to the pixel array by a fraction of a pixel, their one-dimensional calculation would have given a very different result; Parker et a1 (1983) have demonstrated dramatic effects of resampling with sub-pixel translations.
The derivation of Faulkner and Moores (1984) contains several conceptual errors. Their equation (4) implies that the projection power spectrum is discrete, when it is, in fact, continuous. This led to expressions for the image NPS (their equations ( 5 ) - ( 7 ) ) which represented it as being discrete in radial frequency. As we showed in § 2, this spectrum is continuous in radial frequency and discrete in angular frequency. Their equation (8), which implies that the spectrum is discrete in rectangular frequency coordinates, is inconsistent with the preceding development. They also comment that if the projection sampling distance is too small, aliasing will be caused by oversampling. Aliasing, of course, can be caused only by undersampling.
Although the motivation for this work was to gain an understanding of the noise power spectrum, our formulation can also be applied to the signal spectrum if the apodisation by the physical aperture is included. Our analysis includes a thorough treatment of sampling in reconstruction, and can be used to predict the effects of sampling on objects as well as noise.
Acknowledgments
We wish to thank Dr Stefan Mueller and Dr Stephen Moore for many valuable discussions. We are grateful to Dr Norbert Pelc for his careful reading of the manuscript and useful suggestions. This research was supported by USPHS Grants CA 32813 and CA 40444.
The noise power spectrum of CT images 575
Resume
Spectre de puissance de bruit des images.
Les auteurs ont ttabli une expression du spectre de puissance de bruit pour des images reconstruites suivant un algorithme de rttroprojections filtrtes tchantillonntes. La formulation comprend explicitement I’ichantil- lonnage des projections ichantillonnage anguiaire et implicitement I’tchantillonnage bi-dimentionnel suivant la representation discrtte de I’image. Les effets d’interpolation ont considirt aussi. Le spectre de puissance de bruit obtenu par cette analyse difftre sur deux plans de ceux obtenus en utilisant la thtorie continue: ils ne prtsentent pas de symttrie circulaire et ne tendent pas vers ztro i la frtquence nulle. L‘ensemble de ces propriitis peut &re attribut au recouvrement bi-dimensionnel dfi i I’ichantillonnage des pixels. Les pridictions sont confirmtes par les mesures des spectres de puissance de bruit pour des images simultes et pour des images obtenues i partir de scanneurs X i transmission distributs commercialement.
Zusammenfassung
Das Rauschleistungsspektrum von CT-Bildern.
Entwickelt wurde ein Ausdruck fur das Rauschleistungsspektrum von Bildern, die mit Hilfe eines Riickprojek- tionsalgorithmus mit diskreter Filterung rekonstruiert wurden. Die Formeln behandeln insbesondere die Abtastung innerhalb der Projektionen, die Winkelauswahl und die zweidimensionale Abtastung bei der diskreten Bilddarstellung. Der EinAuJ3 von Interpolationen wird ebenfalls beriicksichtigt. So vorhergesagte Rauschleistungsspektren unterscheiden sich von den nach der kontinuierlichen Theorie vorhergesagten Spektren in zwei Punkten: sie sind rotationsasymmetrisch und sie nahern sich nicht null bei der Frequenz null. Diese beiden Eigenschaften sind auf zweidimensionale Abtastdefekte aufgrund der Bildelementauswahl zuriickzufiihren. Die Vorhersagen werden bestatigt durch Messungen der Rauschleistungsspektren der simulierten Bilder und von Bildern eines kommerziell erhaltlichen Rontgentransmissions-CT-Scanners.
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