noise power spectra of images from digital mammography...

Noise power spectra of images from digital mammography detectorsMark B. Williams,a) Peter A. Mangiafico, and Piero U. SimoniDepartment of Radiology, University of Virginia, Charlottesville, Virginia 22908

~Received 9 September 1998; accepted for publication 12 April 1999!

Noise characterization through estimation of the noise power spectrum~NPS! is a central compo-nent of the evaluation of digital x-ray systems. We begin with a brief review of the fundamentals ofNPS theory and measurement, derive explicit expressions for calculation of the one- and two-dimensional~1D and 2D!NPS, and discuss some of the considerations and tradeoffs when theseconcepts are applied to digital systems. Measurements of the NPS of two detectors for digitalmammography are presented to illustrate some of the implications of the choices available. For bothsystems, two-dimensional noise power spectra obtained over a range of input fluence exhibit pro-nounced asymmetry between the orthogonal frequency dimensions. The 2D spectra of both systemsalso demonstrate dominant structures both on and off the primary frequency axes indicative ofperiodic noise components. Although the two systems share many common noise characteristics,there are significant differences, including markedly different dark-noise magnitudes, differences inNPS shape as a function of both spatial frequency and exposure, and differences in the natures ofthe residual fixed pattern noise following flat fielding corrections. For low x-ray exposures, quan-tum noise-limited operation may be possible only at low spatial frequency. Depending on themethod of obtaining the 1D NPS~i.e., synthetic slit scanning or slice extraction from the 2D NPS!,on-axis periodic structures can be misleadingly smoothed or missed entirely. Our measurementsindicate that for these systems, 1D spectra useful for the purpose of detective quantum efficiencycalculation may be obtained from thin cuts through the central portion of the calculated 2D NPS.On the other hand, low-frequency spectral values do not converge to an asymptotic value withincreasing slit length when 1D spectra are generated using the scanned synthetic slit method.Aliasing can contribute significantly to the digital NPS, especially near the Nyquist frequency.Calculation of the theoretical presampling NPS and explicit inclusion of aliased noise power showsgood agreement with measured values. ©1999 American Association of Physicists in Medicine.@S0094-2405~99!00707-5#

Key words: noise power spectrum, digital mammography, detector, image analysis

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I. INTRODUCTION

The signal-to-noise ratio imposes the fundamental limitatto object perceptibility in a digital radiograph because imacontrast can be manipulated during the display of digitaacquired radiographic images. Thus, noise characterizaplays an increasingly central role in the evaluation of systperformance in medical imaging. Well-developed methoused for noise characterization in film-based systems arerently being reassessed and reformulated to accommoboth the different nature of the noise and to take advantof the analytical tools available with digital processing. Aindication of both the current interest level and lack of cosensus regarding noise characterization is the recent elishment of an AAPM Task Group~No. 16!for Standards forNoise Power Spectrum Analysis.

The noise power spectrum~NPS! is a spectral decomposition of the variance. As such, the NPS of a digital radgraphic image provides an estimate of the spatial frequedependence of the pixel-to-pixel fluctuations present inimage. Such fluctuations are due to the shot~quantum!noisein the x-ray quanta incident on the detector, and any nointroduced by the series of conversions and transmissionquanta in the cascaded stages between detector inputoutput. Examples of the latter are gain variances in the c

1279 Med. Phys. 26 „7…, July 1999 0094-2405/99/26„7…/1

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version of x-ray quanta to light quanta in a phosphor~or toelectron–hole pairs in a solid-state detector!, statistical fltuations in the transmission of optical quanta between a stillator and a photodetector, and additive noise sources sas preamplifier noise. The NPS is a much more compdescription of image noise than is quantification of integra~total! noise via simple measurement of the rms pixel flutuations, because it gives information on the distributionfrequency space of the noise power. An understanding offrequency content of image noise can provide insight regaing its clinical impact. For example, in mammography, ecess high-frequency noise may render the detection of micalcifications impossible. If desired, the total variance canobtained by integrating the NPS over spatial frequency.

In this paper we discuss aspects of NPS estimationapplied to digital radiographic systems. We begin withgeneral review of pertinent definitions and relations, and dcuss the most important considerations when applying sptral estimation techniques to digital radiographic systemMathematical derivations sufficient to provide the readwith tools for quantitative application of the concepts agiven, with more detailed development available in the cireferences. These techniques are then applied to the ana

1279279/15/$15.00 © 1999 Am. Assoc. Phys. Med.

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1280 Williams, Mangiafico, and Simoni: Noise power spectra of images 1280

of the noise properties of two developmental detectorsfull-field digital mammography using charge coupled dvices ~CCDs!.

II. BACKGROUND

Linear systems analysis can be usefully applied toradiographic system whose transfer characteristic betwinput and output islinear ~or linearizable!andstationary~thetransfer properties are independent of location at the insurface!.1,2 In this formalism, signal and noise are typicaldecomposed into their Fourier components~although otherbases can be used!, and input and output relations are epressed in terms of the frequency space Fourier amplituDigital radiographic systems such as the one consideredusually exhibit linear responses to the incident x-ray flueover several orders of magnitude. The following analysissumes that the detector is operated within its linear ranFor the digital mammography systems whose noise chaterization is reported here, care was taken to first determthe linear range of operation for each. The range of exsures used for the NPS estimates in this paper fall wwithin the boundaries of linear operation. In a digital detetor the analog~presampling!signal and noise propagatethrough the detector are sampled by the digital matrixdiscrete points. Therefore, the system response is not strshift invariant ~unless shifts are by an integer numberpixels!. The effects of digital sampling on the NPS are dscribed below.

A random process such as the noise fluctuations inimage is considered~wide-sense!statistically stationary, ifits autocorrelation function~see below!is independent of theparticular data sample~e.g., the particular location in thimage!used to obtain it. This is only strictly true in a radiographic image if~a! noise samples are obtained from sptially uniform exposures, and~b! the detector adds no spatially fixed correlated noise. Dobbins has shown thatpresence of a deterministic signal in the image data useestimate the NPS has no effect on the resultant speestimates.3 One other important assumption concerningnoise is whether or not it isergodic. The noise in an image iergodic if expectation values~averages!obtained from datasamples at various locations in the image are equivalenensemble averages obtained from repeated measuremender identical conditions at a single location. Ergodicity implies stationarity, but not necessarily vice versa. The assution of ergodicity of the noise in radiographic images maunder conditions of uniform irradiation is typically madand permits averaging of spectra obtained from several aof the image. Strictly speaking, even under conditions

Medical Physics, Vol. 26, No. 7, July 1999

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nominally uniform irradiation, factors such as the heel efferesult in deviation from true ergodicity. For this reason, itoften best to estimate the NPS from a common, limited aof a series of images, rather than from a large portion osingle image. In the work described here, data used fogiven spectral estimate were obtained from a series ofages, but using an area located at a fixed distance fromchest wall edge.

In a cascaded system such as a digital mammographytector, the output of one stage constitutes the input tofollowing stage. In a linear, shift invariant cascaded systeeach of the independent stages are also linear and shifvariant. At the output of any given stage of a cascadedtector system, the minimum possible value of the NPS is tset by the ~uncorrelated!Poisson variance of the outpuquanta, which is equal to the number of quanta. Buildingthe work of Rossmann4 and Kemperman and Trabka,5 Rab-bani, Shaw, and Van Metter have developed expressionsthe propagation of the NPS through cascaded linsystems.6 Using moment generating functions, they shothat noise generated during a stochastic amplification proin one stage~such as the conversion of a single absorbx-ray photon to many light photons! is effectively passed assignal to the subsequent stage. Furthermore, they showin a stochastic scattering process, while correlated comnents of the NPS are reduced by the square of the modulatransfer function~MTF! associated with the blurring procesuncorrelated Poisson noise is transferred independentlthe MTF. Thisstochastictype of blur has a different effecon the NPS thandeterministicblur, in which both correlatedand uncorrelated components of the NPS of the precedstage are attenuated by the square of the blur MTF.7 Deter-ministic blur does not degrade the signal-to-noise ratiocause the signal is also modulated by the blur MTF. Visiphoton scatter in a phosphor is an example of stochastic bthe effect of the integrating aperture of the individual pixein a digital system is an example of deterministic blur.

A. Direct and indirect methods of NPS calculation

The NPS is defined as the Fourier transform~FT! of theautocorrelation function, defined in one dimension as8

C~x!5 limL→`

1

L E2L/2

L/2

p~x1t!p* ~t!dt, ~1!

wherep(x) is the value~pixel value for a digital image! ofthe one-dimensional image~1D! at positionx, andp* (x) isits complex conjugate. Sincep(x) is real,p(x)5p* (x). TheFourier transform ofC(x) is then

S~u!5E2`

`

C~x!e22p ixudx5 limL→`

1

L E2L/2

L/2

p* ~t!e12p i tuE2`

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p~x1t!e22p ixudx@e22p i tu#dt

5 limL→`

1

L E2L/2

L/2

p* ~t!e12p i tuE2`

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p~s!e22p isudsdt5 limL→`

1

L E2L/2

L/2

p* ~t!e12p i tudtP~u!. ~2!

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Inspection of Eqs.~1! and ~2! shows that

S~u![FT$C~x!%5 limL→`

1

LuP~u!u2. ~3!

Thus, the NPS may be calculated either from the Foutransform of the autocorrelation function~the indirectmethod!, or from the square of the modulus of the Foutransform of the data itself~the direct method!. Note thatsince the datap(x) are real, thatP(2u)5P* (u). That is,the NPS at a given negative frequency is equal in magnitto the NPS at the corresponding positive frequency. Exteing the expression of Eq.~3! to two dimensions:

S~u,v !5FT$C~x,y!%5 limX,Y→`

1

XYuP~u,v !u2. ~4!

With the advent of the fast Fourier transform~FFT! and fastcomputers, indirect calculation of the NPS via the autocolation function has largely been replaced by the dirmethod. This paper will discuss only the latter method.

If the stationary random process being characterizedthe NPS is also ergodic, as is typically the case for radgraphic image noise, then the spectral estimate is foundensemble averaging. That is, the final NPS is the averagspectra obtained from a series of uniform irradiation imagThen, Eq.~4! becomes

S~u,v !5 limX,Y→`

K 1

XYU E2X/2

X/2 E2Y/2

Y/2

p~x,y!

3e22p i ~ux1vy!dxdyU2L , ~5!

where the pointed brackets denote ensemble averaging.

B. One-dimensional NPS

The detective quantum efficiency~DQE! is perhaps thebest single descriptor of radiographic detector performanIt quantifies how well the detector is able to transfer tsignal-to-noise ratio~SNR! inherent in the x-ray fluence at itinput to the resulting image. For most two-dimensional~2D!digital imaging detectors, the DQE is a function of at leathree independent variables: its two spatial coordinates,the x-ray exposure. It is a function of a single spatial codinate for the special case of detectors with an isoplanresponse. The DQE can be written:

DQE~u,v,w!5~wg!2C~u,v !MTF2~u,v !

S~u,v,w!SNRin2 , ~6!

wherew represents the magnitude of the input x-ray fluenat the detector surface;g relates changes in the zerofrequency output signal to that at the input~i.e., it is thesystem gain!;C(u,v) is the Fourier decomposition of thinput signal, normalized to unity at maximum amplitudMTF(u,v) is the detector modulation transfer function; aS(u,v,w) is the output noise power spectrum.9 AlthoughS(u,v,w) is a function of the input fluencew, for notationalsimplicity it will be referred to hereafter simply asS(u,v).


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For a monochromatic input x-ray spectrum, or in the casea photon-counting x-ray detector, SNRin is given by the mag-nitude of the input fluencew.

In part because illustration of DQE(u,v,w) requires afour-dimensional medium, DQEs of 2D imaging systemstypically presented one spatial frequency coordinate at a tfor clarity. By definition,

MTF~u,v !5u2D FT$psf~x,y!%u, ~7!

where psf(x,y) is the spatial response of the detector topoint x-ray input, or point spread function, and 2D FT reresents a two-dimensional Fourier transform. The 1D MTFtypically measured using a narrow slit or edge,10 and is

MTF~u!5FT$ lsf~x!%5FTH d

dx@esf~x!#J , ~8!

where lsf(x)5*2`` psf(x,y)dy is the line spread function an

esf(x) is the edge spread function. Then,

E2`

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lsf~x!e22p ixudx5U E2`

`

$psf~x,y!dy%e22p ixudxU5u2D FT$psf~x,y!%uv50

5MTF~u,0!. ~9!

Equations~8! and ~9! indicate that evaluation of FT$ lsf(x)%yields MTF(u,0), and thusthe appropriate expression of thNPS for calculation of the single coordinate DQE is S(u,0).Similar arguments pertain to MTF(y).11

Up to now, most characterizations ofS(u,0) @or S(0,v)#of digital radiographic systems have been made usingsynthesized slit method.12–15 In the original development othe method in the analog~film-based!context, the opticalaperture of the microdensitometer used to measure theoptical density formed the slit. The methodology has sinbeen applied to digital systems by forming an effectivefrom rectangular groups of contiguous pixels. In either caa long, narrow slit is scanned in steps in one direction othe image. At each step the average intensity in the slit~av-erage transmitted light for a scanning microdensitometeraverage pixel value for a digital system!, is recorded. In thisway, a one-dimensional data series is generated. The squmodulus of the Fourier transform of the series, after scaligives the measured 1D NPS. In the limit of an infinitely lon1D data series in thex direction, the resulting NPS is16,17

SSS~u!5E2`

`

S~u,v !uT~u,v !u2dv, ~10!

whereT(u,v) is the optical transfer function~OTF! of theslit, and whereSSS(u) is the 1D NPS measured via thscanned slit method. If the slit is rectangular, wix-dimensionw andy-dimensionL, then its point spread function is

h~x,y!5S 1

wrect

x

wD S 1

Lrect

y

L D , ~11!

and therefore,T(u,v) is

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T~u,v !5FT$h~x,y!%5sinc~wu!sinc~Lv !, ~12!

where the sinc function is defined as

sinc~x!5sinpx

px. ~13!

Equation~10! becomes

SSS~u!5sinc2~wu!E2`

`

S~u,v !sinc2~Lv !dv. ~14!

if the slit width is chosen sufficiently narrow, thesinc(wu);1 within the range ofu over whichSSS(u) is tobe evaluated. In any case, correction of the measured NPthis factor is straightforward. In a digital system, a synthsized slit of 13NL pixels constitutes sampling the presampling NPS~the NPS smoothed by the nonzero pixel apertubut prior to sampling at discrete intervals given by the pispacing!, using a line ofNL delta functions, at an interval inthe scan direction equal to the center-to-center pixel spacDx. The NL sampled values are then averaged. Thsinc(wu)→sinc(0)51, and L5NLDy, where Dy is thepixel size perpendicular to the scan direction.

As L→`, sinc2(Lv) becomes sufficiently narrow thaS(u,v) is approximately constant over the range ofv ~nearv50! in which sinc2(Lv) is appreciably nonzero, and

SSS~u!5E2`

`

S~u,v !sinc2~Lv !dv

>S~u,0!E2`

`

sinc2~Lv !dv5S~u,0!

L. ~15!

Thus, the desired NPS isS(u,0)5LSSS(u).Substantial effort has been devoted to understanding

criteria for ‘‘sufficiently largeL.’’ Sandrik and Wagner studied the effect of varying slit length on the 1D NPS of radigraphic screen–film systems.18 They found that the magnitude of the low-frequency NPS is underestimated if thelength is insufficient, but that the low-frequency componeapproach a limiting value as the slit length is increased. Tslit length required for low-frequency NPS values within 5of the plateau value was dependent on the particular screfilm combination, but ranged between a value equal tolength of the scan and one nearly twice as long. KoedooStrackee, and Venema developed an expression to numcally approximate the integral in Eq.~14!.19 The technique isapplicable only to systems with rotational symmetry.

An alternate to the scanned slit approach is to extraslice along one of the primary spatial frequency axes frthe 2D NPS. However, because these slices contain npower from the orthogonal dimension near to and at zfrequency, low-frequency trends in the data can contribusignificant portion of the noise power unless these trendsfirst removed. For this reason, the limited number of studemploying this approach have used ‘‘thick’’ slices madeextracting NPS data near to, but not directly on, the primaxes.20,21 As will be demonstrated below, in some digitdetectors this approach can overlook structures that o


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only at zero frequency in the orthogonal dimension~i.e., theyare constant in the orthogonal spatial direction, such astripe or band!.

C. Digital noise power spectrum

In the above expressions, the 1D and 2D NPS are wrias functions of a continuous variablep(x) that describes thepoint-to-point fluctuations in the image. In a digital systethis continuous function of the spatial coordinates is sampat regular intervals,Dx. Thus,p(x) is evaluated at a set odiscrete locations,x5nDx, n50,1,2,...,N, whereX5NDx.Correspondingly, functions of the spatial frequency variau are now evaluated at discrete frequencies given byu5kDu, k50,61,62,... . The maximum spatial frequencsampled, the Nyquist frequency, isuN[1/(2Dx). To experi-mentally estimate the digital NPS, the above continuouspressions must be rewritten in terms of discrete variabWith the preceding definitions, Eq.~3! can be written

Sd~u!5S~kDu!

5 limNDx→`

1

NDx UDx (n50

N21

p~nDx!e22p i ~kDu!~nDx!U2

5 limNDx→`

Dx

N U (n50

N21

p~nDx!e22p i ~kDu!~nDx!U2

, ~16!

whereN is the number of samples in the intervalX and thevariable u now takes on the discrete values 0,61/X,62/X,... . Similarly, Eq. ~4! for the 2D NPS becomes

Sd~u,v !5S~ j Du,kDv !

5 limNDx→`MDy→`

Dx

N

Dy

M U (n50

N21

(m50

M21

p~nDx,mDy!

3e22p i @~ j Du!~nDx!1~kDv !~mDy!#U2

. ~17!

The experimentally determined estimate of the digital Ndiffers from the underlying analog spectrum in the followinways:

~a! The noise has been convolved with the rect~boxcar!function given by the pixel aperture~deterministic blurring!.

~b! The noise data has been sampled at discrete intergiven by the pixel center-to-center spacing~equivalent tomultiplication by a Comb function!.

~c! The record~image data!used to compute the estimahas been truncated from its original infinite length to a lengiven in each dimension by the number of points timessampling interval~equivalent to multiplication by a 2D boxcar function of dimensionsNDx3MDy!.

The result of~a! is the multiplication of the analog NPSwith the absolute square of the 2D sinc function that isFourier transform of the pixel aperture function. The resuloften referred to as thepresamplingNPS:

Spre~u,v !5S~u,v !uab sinc~au!sinc~bv !u, ~18!

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whereS(u,v) is the analog NPS, anda andb are the pixelaperture dimensions.

The result of~b! is that the presampling NPS is convolvewith the transform of the sampling Comb functioIII„u ,v;(1/Dx)(1/Dy)…:

Sd~u,v !5Spre~u,v ! ^ ^ III S u,v;1

Dx

1

DyD , ~19!

whereDx andDy are the sampling intervals in thex andydimensions, and̂ ^ denotes two-dimensional convolutionNote that this Comb function consists of an infinite arraydelta functions separated in each dimension in frequespace by two times the respective Nyquist frequencies;UN

51/(2Dx), vN51/(2Dy). Sd(u,v) is, therefore, an infinite2D array of replications ofSpre(u,v), with the origin of eachSpre separated by 2uN in the u direction and 2vN in the vdirection. This replication means that noise beyond the Nquist frequency overlaps noise below the Nyquist frequeof adjacent replications. This overlap, oraliasing, becomesincreasingly severe as the spatial sampling interval iscreased, that is, asSpre is increasingly undersampled. Thpresampling NPS cannot be directly measured usingsampling techniques such as those commonly employemeasure the presampling MTF, because the phases oFourier components of the image noise are random.22

Using digitized screen–film images, Giger, Doi, and Mehave studied the effects on the digital NPS of the sizes ofintegrating pixel aperture and the sampling interval.23 Theirresults show that aliasing of high-frequency noise~due, forexample, to the high-frequency components of x-ray qutum noise!above the Nyquist frequency can form a signicant component of the measured digital NPS. Large integing aperture size~i.e., larger than the sampling interva!tends to reduce the effect of this aliasing by smoothinghigh-frequency x-ray quantum noise, at the expense of stial resolution. Aperture sizes less than the sampling distaincrease the overall noise level, and particularly so at hfrequencies. Dobbins has recently discussed the impacundersampling on the interpretation of the MTF of digisystems.3

The result of~c! is that the NPS is also convolved with thsquare of the transform of the data windoww(x,y):

Sd,w~u,v !5Sd~u,v ! ^ ^ uW~u,v !u2, ~20!

whereSd(u,v) is the measured NPS, and the frequency wdow, W(u,v) is the Fourier transform of the data window(x,y).24

If the data series is simply truncated afterN3M points,corresponding to multiplication of the infinite data sequenby a unity height box of dimensionsNDx3MDy, then theNPS is convolved with a frequency window with modulus

uW~u,v !u5NDxMDysin~pNDxu!sin~pMDyv !

p2NDxuMDyv. ~21!

A variety of data windows have been studied in masignal processing contexts, with the optimum choice fo


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given situation depending on the characteristics of the pticular spectrum being characterized. Selection of a data wdow is based on the shape and width of the resultingquency window. In general, there is a compromise betwthe width of the central lobe and the magnitude and extenany side lobes. A broad central lobe reduces spectral restion, and significant side lobes produceleakage, whereinnoise power from nearby frequencies are mixed in the cvolution process. In the case of power spectra exhibitstrong periodic components, as is the case in many digradiographic systems, minimization of side lobe magnituis often necessary for distinguishing small amplitude featuadjacent to large amplitude peaks.25,26

D. Experimental estimation of the noise powerspectrum

1. Uncertainty in the spectral estimates

Because noise in a radiographic image arises fromchastic ~random!processes, any realization~image! of thenoise associated with those processes is merely a sachosen from the ensemble of all possible samples. Ththere is an inherent statistical uncertainty in the represetion of the underlying~true! population variance by thasample. In this regard, Fourier decomposition of image no~such as is performed in determination of the NPS! is differ-ent from the decomposition of signal~such as is performedin determination of the modulation transfer function!, whereamplitudes and phases are known and fixed. The variancthe spectral components of the measured NPS is indepenof the number of data points~e.g., the number of pixels! inthe sample used to calculate it, and for mean-zero, normdistributed noise, results in a coefficient of variation~COV!of approximately unity for each spectral estimate,S(u,v)except at zero frequency and the Nyquist frequency, whthe COV is 21/2.27 The variance~uncertainty!in the estimatecan be reduced by sectioning the entireNDx3MDy noiserecord into a number of contiguous blocks, and averagingspectra obtained from each~ensemble averaging!. The COof the final estimate decreases as the inverse square rothe number of determinations. However, the spectralresolu-tion is inversely proportional to the number of data poinused to generate each NPS. Thus, for a fixed amount of dthere is a tradeoff between frequency resolution and variain the final NPS estimate. For example, ifN data points,spaced at an intervalDx, are used to calculate a 1D NPSthen the spectral resolution and COV are, respectively,27

D f 51

NDx, COV>1.0. ~22!

If theseN points are divided inton successive sections, thethe frequency resolution and COV become

D f 51

S N

n DDx

5n

NDx, COV>

1.0

An. ~23!

onents


Medical Physics, Vo

TABLE I. Summary of the various components of the NPS and methods for isolating each. The NPS compare~a! random~stochastic!,~b! nonstochastic but varying from one image to another, and~c! nonstochastic andfixed from image to image~fixed pattern noise!.

NPS Method Components present

Stot(u,v) Average spectra from individualimages

Stochastic, varying nonstochastic,fixed pattern

Sdiff(u,v) Average 0.5 times spectra fromdifference images

Stochastic, varying nonstochastic

Savg(u,v) Spectrum from average of manyimages

Varying nonstochastic, fixed pattern

Sf.p.(u,v) Stot(u,v)2Sdiff(u,v) Fixed pattern

Sv.n.(u,v) Savg(u,v)2Sf.p.(u,v) Varying nonstochastic

Sstoch(u,v) Stot(u,v)2Savg(u,v) Stochastic

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2. Stochastic and nonstochastic components

Image noise can be categorized as~a! random~stochas-tic!, ~b! nonstochastic but varying from one image to aother, or ~c! nonstochastic and fixed from image to ima~fixed pattern noise!. For a complete description of imanoise, each component should be analyzed.

Fixed pattern noise usually arises from spatial variatioin detector sensitivity. Although theoretically fixed pattenoise should be removed during routine postacquisitionage processing~flat fielding!, the removal may be incompleor its completeness may be signal dependent. Note that sflat fielding typically involves division of the raw image bynormalized, highly averaged raw flood image, random noin areas of low gain~average output signal per incidentray! is amplified in the flat fielding process, while thatareas of high gain is reduced. Nonstochastic noise thapresent intermittently or varies in location from imageimage will not be reliably removed during flat fielding. Aexample of this type of noise is 60 Hz power line pickduring detector readout.

The NPS obtained by averaging the spectra of msingle uniform irradiation images,Stot(u,v), contains compo-nents from all three types of noise. The NPS using theference of two images obtained under identical conditionsuniform irradiation~and divided by 2!, Sdiff(u,v), containsvariance from stochastic and varying nonstochastic comnents, but none from fixed pattern noise. The magnitudethe noise power from fixed pattern noise can be determifrom Sf.p.(u,v)5Stot(u,v)2Sdiff(u,v). Note that the NPS obtained from a highly averaged image,Savg(u,v), differs fromSf.p.(u,v) by an amount equal to the noise power due to atime-varying nonstochastic components. The stochastic Ncomponents can be found by subtractingSavg(u,v) fromStot(u,v).28 Table I summarizes the methods for isolatispectral components due to these three types of noise.

III. METHODS

A. Data acquisition

Both mammographic systems evaluated utilize molybnum target x-ray tubes, with molybdenum or rhodium filtr

l. 26, No. 7, July 1999

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tion. Data were acquired using Mo/Mo spectra and a variof kVp and mAs settings. A block of acrylic 3.8 cm thicand sufficiently large to cover the entire detector surface wplaced on the breast support to create realistic spectral cposition and scatter conditions. In selecting areas of theage for estimating noise power spectra, care was takeavoid regions near the image periphery, where the scacontribution is nonuniform. Detector entrance exposuwere calculated by raising the acrylic block slightly abothe detector surface, and using a thin ion chamber to meathe exposure at its exit surface. Corrections were thenplied to account for the slight difference between the locatof the chamber and the detector entrance surface, relativthe focal spot. For each selected kVp and mAs, eight imawere obtained. Data used for spectral estimates were tafrom ;5 cm35 cm regions, located centrally in the imagThe data regions were divided into the maximum numbercontiguous sections consistent with the NPS estimation tenique to be used~e.g.,N3L pixel sections for a scan in thx direction, orN3M pixels for a 2D NPS calculation!. Spec-tra were calculated from each section of each of the eimages, and were then averaged. The final ensemaveraged NPS estimates were thus averages of;300–1200spectra, resulting in standard errors of;3%–6%.

B. Subtraction of background trends

Low-frequency background trends, such as those fromx-ray source heel effect, are often present in the data.though such trends may have significant Fourier compononly at frequencies below the lowest measurable frequein the power spectrum (1/NDx), leakage into low-frequencyspectral components can artificially inflate the low-frequenNPS.~As discussed in Sec. II C, leakage is the spreadingspectral power from one frequency to adjacent frequencand is a result of the nonzero width of the frequency wdow, which is the Fourier transform of the finite data widow.! For this reason, noise power spectra are frequeobtained by first subtracting one uniform exposure imafrom another made with the same exposure, and dividingNPS, resulting from the subtracted image, by 2. The facto

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2 accounts for the fact that the variance in the subtracimage gets approximately equal contributions from thecorrelated noise in the two original images. Alternatelydifference image may be obtained by subtracting a higaveraged uniform exposure image from a single image.random noise in the highly averaged image is negligiblelarge number of images is used to obtain it.

The image subtraction technique has the drawback, hever, of also removing any uncorrected fixed pattern nooccurring at higher frequencies. Thus, this method maybe optimum for estimating the NPS of clinical images folowing uniformity correction~flat fielding!, because noisremoved in the subtraction process would neverthelesspresent in clinical images. We have, therefore, evaluatedmethods for removing low-frequency background trenfrom single images: subtraction of a low-pass-filtered versof the image, and subtraction of a low-order polynomial fitthe image. When using the first method, for simplicity,rectangular~boxcar! filter was used to filter the entire 2Ddata set. Subtraction of the resulting filtered image fromoriginal is equivalent to subtraction of a running local meaas has been employed for one-dimensional data by othe23

Convolution in coordinate space with a rectangular filterequivalent to multiplication in frequency space by a sfunction. The dimensions of the square filtration kernel wchosen sufficiently large that the first zero crossing ofresulting sinc function was at or below the minimum frquency sampled in the NPS. In the second method of baground trend subtraction, a two-dimensional polynomial wfirst fit to each uniform exposure image, and subtracted. Bfirst- and second-degree polynomials were evaluated.

In order to compare the above two methods of baground trend subtraction, a synthetic data set was creaThe data consist of a dc~average!pixel value, a normallydistributed random component, a periodic component~one ineach dimension!, a linear~ramp!component, and a secondorder component. The digital valuep( i , j ) of the pixel in thei th column andj th row was computed as

p~ i , j !5d1* @rand~ i , j !#1d2~ i , j !, ~24!

where the rand~!operator generates one of a set of normadistributed random numbers with average zero and standeviation one,d1 is a scaling factor, and

d2~ i , j !5a0 sin~ iup1du!1b0 sin~ j vp1dv!

1a1i 1b1 j 1a2i 21b2 j 21C0 , ~25!

whereup and vp are the spatial frequencies of the periodcomponents in thex and y directions, respectively, anddu

anddv are phase factors. The constantC0 sets the magnitudeof the dc component, and the constantsa0 , b0 , a1 , b1 , a2 ,and b2 set the magnitudes of the periodic, linear, and qdratic components in thex and y directions, respectivelyFigure 1 is a gray-scale plot of a portion of one of the sythetic data sets.

Figure 2 shows a semilog plot of four representativepower spectra generated from synthetic data. The speclabeled with diamonds is that of a data set with no ba


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ground trend (a15a25b15b250). The second spectrumlabeled with squares, is from the same data, but with linand quadratic components added. The third and fourth stra ~triangles and crosses!, are calculated following trendmoval via subtraction of a least-squares surface fit tosecond-order polynomial or by subtraction of a boxcfiltered version of the image, respectively. In this exampthe boxcar filter was 64364 pixels, the same size as thsections of data used to calculate the 2D NPS. The specobtained with no trend removal demonstrates substanleakage of low-frequency noise power into the frequenrange of the periodic component. Figure 2 shows that btrend removal techniques reduce the low-frequency nopower sufficiently that it is no longer apparent in the spetrum, even though the leakage is unchanged. The quadfit technique virtually restores the original NPS, a result nsurprising given the quadratic nature of the trend.

Figure 3 also shows the effects of different trend remotechniques, this time using flat-field image data obtainfrom one of the mammography detectors. The 2D spewere computed from the average of 648 1283128 pixel sec-tions. Thin slices were extracted along thev axis. Again,slices are shown from 2D spectra following no trendmoval, and following subtraction of either a fitted quadrasurface or a low-pass-filtered data set. In addition, two dferent size convolution kernels for low-pass filtering acompared. For a rectangular,n3m pixel convolution kernelof amplitude (nDxmDy)21, whereDx andDy are the pixelsize in thex and y dimensions, the trend-corrected NPSgiven by

Sd,w,b5Sd,w@12sinc~nDxu!sinc~mDyv !#. ~26!

In order that only noise power close to dc is removed, owould like the term sinc(nDxu)sinc(mDyv) to be appre-ciable only for very low frequency. Therefore, the quantiti

FIG. 1. Gray-scale image of a synthetic data set used to evaluate methobackground trend removal. For ease of visualization, the data set shcontains only sinusoidal, random, and dc components~no linear or quadraticcomponent!.

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FIG. 2. Comparison of background trend removal techniques, using a synthesized data set.~a! shows the NPS calculated from data with no background tre~diamonds!, and the same data but with linear and quadratic trends added~squares!.~b! compares the data with linear and quadratic trends added be~squares!and after subtraction of a two-dimensional second-order polynomial fit to the data~triangles!.~c! compares the data with linear and quadratic trenadded before~squares!and after subtraction of a low-pass-filtered version of the data~crosses!.

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nDx and mDy, corresponding to the reciprocal of the firzero crossings of the sinc functions in the two dimensiomust be sufficiently large compared to the lowest frequeof interest in the NPS. In Fig. 3,Sd,w,b is shown forn5m5100 and forn5m5200. Note that form5100, the firstzero crossing of sinc(mDyv) is at a slightly higher fre-quency than the lowest frequency (v51/(128Dy)) in thespectrum. The effect is a slight reduction in the noise powat the lowest frequency, relative to that shown form5200.Because of this, in general, we have found the best resultmaking the size of the smoothing kernel somewhat larthan the size of the blocks used to calculate the individualspectra.

It should be noted that 2D quadratic fitting carries a snificantly higher cost in processing time than doessmoothing. This is true even if the fits are performed onrelatively smaller blocks individually, rather than on the fudata set. For example, on a Sun Sparcstation 10 in our lratory, trend removal using 2D fits to 800 1283128 blockstakes approximately 4 h ofprocessing time, whereas onlyfew minutes are required for 2D smoothing of the saamount of data.

C. Units and scaling

The noise power spectrum is more precisely termedpower spectral density, that is, the noise power per ufrequency.8 The quantityS(u,v)dudv is the contribution tothe variance from noise components with spatial frequenbetweenu and u1du, andv and v1dv. S(u,v) thus has


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dimensions of inverse spatial frequency squared. The corscaling of the NPS can be verified by observing thatvolume ~area!under the 2D~1D! NPS equals the total variance. Thus, integration over frequency of spectra estimausing mean-zero sections from uniform illumination imag

FIG. 3. Comparison of background trend removal techniques, appliedimages obtained with one of the digital mammography systems, obtausing nominally uniform irradiation. The graph shows four 1D power sptra, S(0,v), obtained from thin cuts from four different 2D spectra. The 2spectra were calculated following~a! no trend removal~none!,~b! subtrac-tion of a fitted quadratic surface~fit!, ~c! subtraction of a low-pass-filteredversion of the data using a 1003100 pixel convolution filter, and~d! sameas ~c!, but using a 2003200 pixel filter.

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should numerically equal the mean-square deviations ftheir average of the pixel values in those sections.

Noise power spectra are sometimes presented normaby the square of the large-area~average!output signal.20,21

This normalization has the effect of compensating for diffence in gain between two systems which are being cpared, and of simplifying the expression for the DQE. Wthis normalization, Eq.~6! becomes

DEQ~u,v,w!5C~u,v !MTF2~u,v !

S̃~u,v,w!SNRin2

5C~u,v !

SNRin2

NEQ~u,v,w!,

~27!

where S̃(u,v,w)[S(u,v,w)/(wg)2 is the normalized NPSand NEQ(u,v,w) is the noise equivalent quanta.

IV. RESULTS

A. Two-dimensional NPS

Neglecting digitization noise, the additive noise in tNPS obtained from a CCD-based digital mammographytector can be written as

sa25s t

21s r21Sp~u,v !, ~28!

wheres t is the CCD thermal~Johnson!noise,s r is the readnoise, and Sp(u,v) is the variance due to frequencydependent periodic noise. These noise contributionssummed in quadrature because they are independentrespect to each other. Periodic noise can arise from captively coupled clock pulses, electromagnetic interferenfrom power supplies, or 60 Hz pickup due to ground looThe location in the image of the noise due to such periointerference is not fixed from frame to frame, becausephase~in time! is not fixed with respect to detector readouIt is, therefore, not removed during the uniformly correctioNote that spikes from periodic additive noise are apparenthe v axis ~the central, vertical axis in each of the plotsFig. 4!, but not on theu axis. As is shown below, these samstructures are apparent in the 1d spectra obtained by sning a synthetic slit in they direction.

2D spectra obtained from both developmental systemshibit strong periodic structures over all exposure rantested. These structures take the form of sharp peaks inspectra, and are due to periodic additive noise. Figurshows gray-scale plots of the 2D spectra of one digital mamographic system at four exposure levels. Gridlinesspaced at 1 mm21. With increasing x-ray exposure, the peodic noise peaks are decreasingly apparent as their matude becomes progressively less relative to that of the xquantum noise contribution at low frequency.

B. Effect of varying slit length on 1D NPS

We have studied the effect of varying the length of tsynthetic scanning slit used for the estimation ofS(u,0) orS(0,v). Figure 5 shows 1D spectra obtained from unifoillumination images using an exposure of 38 mR at thetector surface. Each of the spectra shown in the plot wgenerated by stepping a synthetic slit withx and y dimen-


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sions of 13L pixels, over the data in thex direction in stepsof one pixel. The longest slit length shown (L5512 pixels)corresponds to a slit approximately 20 mm long. At eastep, the pixels in the slit were averaged. The resultingelement series was then Fourier transformed and scalefollows to give the measured 1D NPS. Combining Eqs.~15!and ~16! we have

FIG. 4. ~a! 2D NPS at 9 mR exposure to the detector surface. Gridlinesspaced by 1 mm21 ~b! 2D NPS at 18 mR exposure to the detector surfaGridlines are spaced by 1 mm21. ~c! 2D NPS at 38 mR exposure to thdetector surface. Gridlines are spaced by 1 mm21. ~d! 2D NPS at 76 mRexposure to the detector surface. Gridlines are spaced by 1 mm21.

FIG. 5. 1D noise power spectra obtained from the same data set, butvarious scanning slit lengths. Spectral components above approximatelcycles per millimeter are insensitive to changes in slit length for slit lengof 64 pixels or greater. However, a slit length of 4 pixels results in undestimation of the noise power out to;3 cycles per millimeter.

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of the pixel values in the synthetic slit. Note that from E~17!, a section of the digital 2D NPS along theu axis, cal-culated from aN3M section of the image, is

S~kDu,0!5 lim NDx→`MDy→`

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Thus, for example, a cut along theu axis from a 1283128 2D NPS is identical to the 1D NPS obtained by scning a 13128 pixel synthetic slit over 128 pixels in thexdirection. In all cases, we verified that the 1D NPS obtainfrom the scanned slit method with the slit length equal tosize of the data block used to generate the 2D NPS,identically equal to the section from the 2D spectrum alothe u or v axis.

As shown in Fig. 5, the very low-frequency values of t1D NPS did not converge to a plateau value for slit lengof up to 512 pixels, which is four times the scanned distanThis behavior is similar to that observed by Sandrik aWagner for the Kodak Hi-Plus/XRP screen–film system.18 Inthat study, measured NPS values at 0.39 cycles/mm coued to increase with increasing slit length up toL55.9 mmfor scan a distance of 2.5 mm, while convergence to ateau value was observed for other screen–film systemsimilar frequencies.

C. Slices from the 2D NPS

Notably, in the context of the evaluation of storage phphor systems, 1D spectral estimates have been obtaineextracting thick cuts from the 2D NPS. The cuts are maparallel to the primary axis of interest, but do not include taxis, so as to avoid low-frequency trending effects. We hevaluated this approach for analysis of the spectra ofdigital mammography systems. Figure 6 is a plot compara thin cut ~a single-frequency-bin wide!along the primary


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(v) axis of the 2D NPS with a thick cut generated by suming the two four-bin-wide sections immediately on eithside of, and adjacent to, thev axis. The thick slices ad-equately represent the on-axis slice except at low frequewhere the NPS decreases rapidly with increasingu. Ofgreater significance is the omission of the periodic nostructures near spatial frequencies of 3 and 6 cycles/mThese represent noise power from fluctuations in they direc-tion ~left to right direction of the breast support! that areconstant in the anterior-chest direction. Such noise maypear as horizontal stripes in the mammographic image.

As would be expected, thicker slices are poorer appromations to the on-axis cuts, as the falloff in the 2D NPSthe dimension orthogonal to the slices causes the avespectral values to be reduced. This produces an effect simto that resulting from utilization of a scanning slit of insuficient length.

D. NPS of the developmental mammographicsystems

Spectral estimates were obtained from images acquusing two developmental digital mammography systemscated at the University of Virginia. Both systems contabutted arrays of CCD-based modules, but they differ intype of x-ray converter and in the details of the optical copling to the CCDs. The pixel size is;37 mm for system 1and;47 mm for system 2. The NPS of uniform illuminatioimages from both digital mammographic systems was msured over a range of detector entrance surface exposfrom 9 to 78 mR. For each exposure, 2D noise power speestimates were made as described in Sec. III A. Figures~a!and 7~b!are semilog plots of cuts along theu andv axes ofboth systems at four exposure levels. The spectra havenormalized by the respective large-area output sigsquared. Immediately apparent from Fig. 7 is the fact thatnoise behavior of system 1 is different in thex andy dimen-sions, with they dimension exhibiting significantly greate

FIG. 6. Slices from 2D NPS of digital mammographic system. The solid lis a slice one-bin wide, along thev axis. The dotted line is the sum of twofour-bin sections taken from either side of thev axis. Differences betweenthe two occur primarily at low frequency and because of an-axis perionoise components.

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FIG. 7. Noise power spectra of the two developmental digital mammography systems studied. Spectra are shown at four exposures for each sysspectra of system 1, the magnitude ofS(0,v) exceeds that ofS(u,0) at high frequency for all exposures, even though their magnitudes at low frequencequal.

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noise at high frequency. In both systems, they directioncorresponds to the parallel transfer direction of the CCAlso, while the magnitude of the system 2 NPS decreamonotonically with increasing exposure, that of system 1higher at 76 mR than at 18 or 36 mR. This unexpectedcrease in noise is caused by the appearance of uncorrefixed pattern noise in the images of system 1 at higher exsures.

Also apparent are the very different shapes of the speobtained from the two systems, with the spectra of systeleveling off at high frequency, while those of system 2 cotinue decreasing out to the Nyquist frequency. At all expsures tested, the normalized noise power of system 2 isthan 1026 at Nyquist. As is shown below, attenuation of thhigh-frequency noise in system 2 is primarily a result of tsmoothing effect of the spatial dewarping algorithm used

E. Comparison with pixel variance

Each of the 2D spectra presented in this study was igrated over spatial frequency in order to compare the resing value with the variance calculated from the pixel-to-pixfluctuations. The integrals were calculated over the rafrom 2 f N to 1 f N , wheref N is the Nyquist frequency. In alcases, differences between the integrals and the directlyculated total variance were less than 0.5%.

Figure 8 shows the total noise of the two digital mamography detectors over the range of exposures from 9 tmR. The noise was determined from the square roots ofintegrated volumes under the 2D spectra whose axial slare shown in Fig. 7. Figure 8 plots the log of the noise verthe log of the exposure. For both systems, straight lines hbeen fit to the experimentally determined data points, andcalculated slopes of the lines are shown in Fig. 8. In a sysfor which x-ray quantum noise is the dominant noise sourthe slope of such a plot is approximately 0.5. A slope lthan 0.5 indicates the presence of significant contributionthe noise power from sources other than x-ray noise.


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F. Effects of image correction

All detectors exhibit imaging characteristics that are iperfect but constant from image to image. This type of iperfection, including the presence of turned-on or dead pels, local or global spatial distortions, and spatiafluctuating sensitivity, can be corrected in software. Hoever, correction algorithms can have an appreciable effecthe noise power spectra, and care must be taken in this rewhen interpreting the spectra. Figure 9 shows two pairsspectral estimates calculated from the same set of unifx-ray exposures. In one pair, the raw data~prior to any cor-rections!were used. In the other case, the data were pcessed using the sequence of calibration steps typicallyplied to clinical data: masking of bad pixels, subtraction odark frame, geometric dewarping, and sensitivity nonunifmity correction. For both raw and processed data, specestimates were calculated from difference images obtaiby subtracting one image from another, pixel by pixel. Tmost apparent difference between the spectra of the rawcalibrated images is at high frequency, where the calibra

FIG. 8. Log noise vs log exposure for two digital mammographic detectThe noise was found from the square root of the integrated volume unde2D NPS. The plot shows the slope of straight lines fit to the measured d

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images exhibit much less noise power. The cause of thetenuation of high-frequency noise is the dewarping alrithm. During dewarping, the signal in the raw image isassigned to new locations in the dewarped image basedmatrix of displacement vectors stored in a look-up table.general, a signal from any given pixel in the warped imagmapped to a location that does not coincide with the ceof a pixel in the dewarped image. One method of handlthis is to perform a proximity-weighted distribution of thsignal among pixels whose boundaries fall within a half piof the corrected location. This has the effect of smearingthe signal content of each pixel in the raw image over sevpixels in the dewarped image, a smoothing operation.

Note that this method of dewarping is only one of mapotential approaches, each of which can have a differenfect on the final image. An important function of noise powspectral analysis is to identify the impact of various imagcorrection algorithms on the image noise. For this reasois useful to obtain noise power spectral estimates both fcorrected data and, when available, from raw data.

G. Effects of aliasing

Aliasing of noise components present above the Nyqfrequency is unavoidable because the only estimate ofNPS available from a digital system is that obtained folloing sampling by the digital matrix. In order to estimate teffect of aliasing in the measured NPS of the digital mamographic systems, cascaded linear systems analysisused to calculate the presampling NPS.6,29–33Details of thetheoretical development are presented elsewhere.34 For bothsystems, the presampling NPS can be written as

Spre~u,v !5G2F0

hAsuMTF~u,v !u21@~GF0!1s t

2#

3usinc~pxu!sinc~pyv !u21s r2, ~32!

whereG is the system gain in CCD electrons per incidex-ray photon;F0 is the incident x-ray fluence per pixel;h is

FIG. 9. Comparison of the NPS obtained from corrected vs raw image dThe reduction in the high-frequency noise in the corrected image speoccurs during dewarping. Of course, the detector MTF exhibits a simreduction at high frequency.


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the x-ray absorption efficiency;As is the Swank factor;MTF(u,v) is the overall detector MTF;s1 is the rms CCDthermal noise in electrons;px and py are the pixel sizes inthe x andy dimensions, respectively; ands r is the rms readnoise in electrons.G, F0 , h, andAs are each functions of thex-ray photon energy, and bars above symbols denweighted averages over the x-ray spectrum incident atdetector surface. Figure 10 shows an example of the alianoise in images obtained at 9 mR exposure using one ofdevelopmental systems. Plotted are the measured digitaspectra,Sd(u,0) andSd(0,v) obtained from differences oraw images; the calculated presampling spectrum,Spre; andthe calculated digital NPS,Sdigital . In calculating the digitalNPS, Spre values between one and two times the Nyqufrequency were added to those in the interval between 0the Nyquist frequency, according to the formula

Sdigital~v!5Spre~v!1Spre~2vN2v!2s r2, 0,v<vN .

~33!

The variance due to read noise is not aliased becauseadded after the digital sampling process, and is merelyoffset in the measured NPS. The calculated noise poabove twice the Nyquist frequency is negligible. The cloagreement in Fig. 10 between the measured NPS andcalculated NPS with aliasing shows that aliased noiseconstitute a significant fraction of the high-frequency nopower in the mammographic systems. Note, however,dewarping~c.f. Fig. 9! smoothes the high-frequency noisthereby greatly reducing the effects of aliasing.

V. DISCUSSION

Similarly to other types of radiographic systems, digidetectors for mammography can usefully be characterithrough Fourier analysis of their noise characteristics. Hoever, investigators preparing to estimate noise power spe

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FIG. 10. Comparison of measured and theoretical digital NPS. To calcuthe theoretical digital NPS, noise power in the theoretical presampling Nabove the Nyquist frequency was aliased into the frequency range betzero and Nyquist. The presampling NPS was calculated using casclinear systems analysis. Although the system gain, phosphor absorptioficiency, and Swank factor are all energy dependent, their spectrweighted values are approximately:G;3.5 e-/incident x ray,h;0.50, andAs;0.90.

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of such system are confronted with a bewildering arrayprocedural considerations, lack of attention to which can lto misleading results. Limitations formerly imposed by ttradeoff between spectral resolution and uncertainty inspectral estimates are much less stringent than even a deago, since significant digital processing power is now avable to the majority of medical physicists. Thus, ensemaveraging over a large number of images obtained usingform x-ray exposure is feasible, thereby permitting faisubtle features of the NPS to be resolved. The necesresolution is determined by the proximity in Fourier spacerelevant features and should be determined on a systemsystem basis. For the digital systems characterized inpaper, 1283128 2D spectra, yielding 64 frequency binsthe single-sided 1D spectra, were sufficient to resolvefeatures of interest. Occasionally, we found it useful tocalculate particular spectra at higher resolutions. This mauseful, for example, for assurance that low-frequency strtures are, in fact, independent of background trend nopower residing in the lowest-frequency bin~the range be-tween zero and 1/NDx, whereNDx is the length of the dataseries!.

The noise in images from both the digital mammograpsystems has markedly nonisoplanatic properties, particulat low exposure where frequency-dependent system noimost predominant. Such noise can be due to digital clpulses that are radiatively or capacitively coupled to the alog data signals. Furthermore, the majority of this nopower resides away from the principle spatial frequenaxes, and thus, does not appear on the 1D NPS, whetheobtained from an extracted slice of the 2D NPS, or fromscanned slit. It is, therefore, important that the full 2D NPbe estimated, even if 1D spectra are subsequently usedSNR calculations.

The scanned slit method for obtaining 1D spectra yieldresults identical to those obtained from thin cuts through2D spectra, except for the susceptibility of the former aproach to slit-length-dependent variability in the lowfrequency components. The absence of convergence olow-frequency values ofS(u,0) reflects the peaked naturethev direction of the 2D NPS,S(u,v), nearv50. Since thevalue ofSss(u) obtained from scanning is scaled byL to getS(u,0) @c.f. Eq.~15!#, only for values ofu where the additionto the integral of Eq.~14! is proportional toL will the resultbe insensitive to changes inL. Thus, approximating the function sinc2(Lv) as a triangle of base width 2/L and unityheight centered atv50, S(u,0) obtained using a slit olength L1 is approximately equal to that obtained with slengthL2 only if S(u,v) is approximately constant in thevdirection betweenv52/L1 and v52/L2 . Note that exceptfor u50, whereS(0,v) is symmetric about theu axis, thisrequirement must also be met betweenv522/L1 and v522/L2 . Dainty and Shaw have recommended that thelength be at least equal to the inverse of the frequency rlution ~that is, at least equal to the scan distance!.17 Ourresults indicate that for the digital mammography systemlength significantly greater than the scan distance mayrequired for asymptotic convergence of the low-frequen


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NPS estimates. In practice, the most direct method of demining sufficiently largeL is probably inspection of crosssectional cuts through the 2D NPS, obtained at as higfrequency resolution as is practicable, bearing in mindguidelines of the preceding sentences.

For the noise data studied here, subtraction of a duplicdata set that has been convolved with a two-dimensionalfunction kernel is an effective and efficient method for tremoval of low-frequency trending effects. The effectiveneof the technique is indicated by the similarity between thslices extracted from the 2D NPS near the central axes,on-axis slices~c.f. Fig. 6!. 1D spectra obtained from thicslices parallel to the primary axis of interest may notrepresentative of the on-axis NPS, because the 2D NPScontain on-axis features not present in adjacent regionfrequency space. For this reason, thin on-axis cuts areommended.

Algorithms for correction of local detector imperfectiongeometric distortions, or sensitivity nonuniformity can eithamplify or attenuate components of the NPS. The bestmost straightforward way to quantify the noise prior to poacquisition processing is to obtain noise power spectral emates from the difference of raw images. Here, raw meansoon as possible after analog-to-digital conversion. Whevaluating commercial systems, it may not be possibleobtain data prior to image corrections without assistafrom the manufacturer.

It is our opinion that noise power spectral estimashould also be made with images processed using theware correction steps that are to be used in routine pracSuch data are always readily available, and provide the mrealistic assessment of noise components present in clinimages. This analysis involves simply calculating specestimates from a number of individual processed images,averaging spectra to reduce the uncertainty in the final emate. In order to distinguish spectral components that afrom fixed pattern noise that is not fully corrected, thespectra should be compared to those obtained from the sdata set, but subtracted from each other pairwise. The laspectra contain no contributions from fixed pattern noisince that noise is subtracted out~see Sec. II D 2!.

Calculation of an expression for the presampling NPScascaded linear systems analysis indicates that the shathe 1D NPS is governed by three terms. The first term arfrom x-ray quantum noise and the phosphor conversiontistics, and is modulated by the system MTF. The secoterm arises from the Poisson noise associated with the Celectrons generated by visible photons from the phosp~secondary quantum noise!and generated thermally. ThiPoisson noise is modulated only by the pixel sampling apture function. The third term is the frequency-independ~white! read noise. Comparison of the measured 1D digspectra with the theoretical presampling spectra shows gagreement for spatial frequencies up to;6 cycles/mm. Athigher frequencies, the digital NPS is increasingly largthan the theoretical presampling NPS with increasing fquency. This discrepancy is consistent with aliasing ofNPS components above the Nyquist frequency back

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those between zero frequency and the Nyquist frequeThe aliased noise consists primarily of secondary quannoise, because the x-ray quantum noise is greatly attenuby the system MTF above the Nyquist frequency.

VI. CONCLUSIONS

Fourier analysis has been used to characterize the nproperties of two developmental detectors for digital mamography. The noise power spectra of images from thesystems exhibit several common features. They containtures that are strongly dimension specific, and are oftencalized to relatively small portions of frequency space. Thalso contain predominant peaks indicative of periodic adtive noise, both on and off the primary frequency axes.the other hand, the noise properties of the two mammgraphic detectors differ significantly in some aspects. Overange of input exposures, the spectra of system 1 are chaterized by nearly constant noise power for frequencgreater than about 8 cycles/mm, while those of systemexhibit increasingly sharp falloff with increasing frequencCuts along the two primary axes of the 2D NPS of systemexhibit very different shapes, while those of system 2nearly identical along the two axes. At all frequencies,relative noise of system 2 decreases monotonically withcreasing x-ray exposure, while that of system 1 reacheminimum then begins to increase again. The overall noissystem 2 increases approximately as the square root onumber of x-ray quanta, indicative of x-ray quantum-limitbehavior, while that of system 1 increases more slowly,dicating that a significant portion of the noise power arisfrom sources other than x-ray noise.

The complex nature of the frequency distribution of tnoise power in images from digital mammographic systemakes analysis of the complete 2D spectra, rather thanthe 1D spectra, important. Straightforward removal of baground trending artifacts is possible via a simple filteriprocess. Cuts obtained along the primary axes of thespectra can provide valid estimates of the 1D NPS withthe use of synthetic scanned slit techniques. Spectralmates from difference images and averaged images caused to isolate stochastic, fixed pattern, and varying nonchastic components for identification of noise sources.

Aliasing can contribute significantly to noise power nethe Nyquist frequency. Cascaded linear systems analysisuseful tool for evaluation of the magnitude and sourcesaliased noise.

ACKNOWLEDGMENTS

The authors would like to thank Bob Gagne of the FDACenter for Devices and Radiological Health for helpful dcussions. This work was supported in part by NIH/NCl GraNo. RO1 CA69252.

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