Characterization of the imaging properties of x-ray focal spots

Characterization of the imaging propertiesof x-ray focal spots

Martin Trefler and Joel E. Gray

The imaging characteristics of diagnostic x-ray systems are dependent, in part, upon the size, shape, and in-

tensity distribution of the x-ray source. These properties, however, vary with such parameters as kVp (tube

voltage), mA (tube current), and, because of the steep anode face, field position. We have investigated the

characterization of the focal spot using the image of a random array of small radiopaque objects. The Fouri-

er transform of this image is, to a good approximation, the modulation transfer function (MTF) of the focal

spot. We have compared the accuracy and reliability of this technique with both pinhole and star imaging

techniques. Focal spot variations with kVp, mA, and field position have been measured, and the use of this

technique with microfocus x-ray sources has been demonstrated.

1. IntroductionIn diagnostic radiology, an image of a physiological

object is formed by a shadow casting procedure. Thex-ray source is produced by a beam of high energyelectrons impinging on a heavy metal target which isoriented at an angle both to the electron beam and therecording medium as shown in Fig. 1. The intensitydistribution in the image is given by

(x,y) = h(x,y) * s(x,y), (1)

where h(x,y) is the two-dimensional point spreadfunction (PSF) of the source, s(x,y) is the objecttransmission characteristic, and * indicates the convo-lution operation. It is clear that if h(x,y) were a two-dimensional 5-function, an ideal image would result andthat any deviation of h (x,y) from this ideal PSF impliespossible image degradation.

In order that a source can be evaluated with respectto a given diagnostic task or, alternatively, the focal spotartifacts in a radiograph can be separated from physi-ological information, it is necessary to characterize thefocal spot experimentally. However, it must be notedthat the function h(x,y) varies with accelerating voltage(kVp), tube current (mA), and, due to the steep anodeangle, position in the imaging field.'

Several techniques for the characterization of x-rayfocal spots have been described in the literature. TheNEMA standard for measurement of focal spot sizestates that a pinhole camera should be used for focalspots larger than 0.3 mm and a failure of resolution

The authors are with University of Toronto, Radiological ResearchLaboratories, Medical Sciences Building, Toronto, Ontario M5S1A8.

Received 19 April 1976.

technique for focal spots smaller than 0.3 mm.2 Thepinhole method provides a picture of the focal spot (areproduction of the point spread function) and is auseful guide for heat load estimates. It does not, how-ever, give the imaging characteristics directly-thepinhole image must be Fourier transformed to providethis information. Furthermore, because of the smallarea of the pinhole and relatively low sensitivity of thefilm used, this technique requires multiple exposureswhich can be very time consuming and results in tubedamage if proper care is not taken.

Failure of resolution techniques involve the imagingof a radiating star pattern. The spatial frequency atwhich resolution failure occurs is determined by themeasurement of the image. The size of a theoreticalhomogeneous focal spot which would produce a similarfailure of resolution is then used as the effective focalspot size. The measurement to be made in this methodis difficult because of frequency tripling effects and lowcontrast in the area of resolution failure.3 Furthermore,its extension to use with microfocus x-ray tubes isquestionable because of the difficulty in fabrication ofhigh frequency, radiopaque targets.

Both pinhole and star techniques characterize theimaging properties at only one point in the imagingfield. Consequently, multiple images must be made inorder to examine field variations.

Braun and Stoner have recently investigated a tech-nique wherein a radiopaque zone plate is imaged by thefocal spot.4 This image is optically transformed toprovide a reproduction of the point spread function.This technique may be difficult to realize because of theproblem of constructing zone plates of sufficientthickness to attenuate x-rays but having the finestructure necessary to characterize small focal spots.

December 1976 / Vol. 15, No. 12 / APPLIED OPTICS 3099

in the star image, frequency tripling occurs near theseA j points of failure of resolution making accurate mea-

C surement quite difficult. These measurements are thenconverted to focal spot sizes by comparing the frequencyat which resolution fails to the frequency of resolution

/ i OBJECT failure for a focal spot with a uniform intensity distri-

Fig. 1. Geometry of typical x-ray imaging system showing thecathode (C), anode (A), and object and image planes. Note the steep

anode angle (usually 10-20° from the central ray).

Groh et al. have proposed and demonstrated a tech-nique which utilizes a monodisperse random distribu-tion of lead particles. 5 The degraded image of thisdistribution is optically Fourier transformed to providethe two-dimensional MTF.

In this paper, we will present data comparing theaccuracy of this technique with pinhole and star tech-niques. We will show that the method can be easilyused to study kVp, mA and field variations of the focalspot using moderate exposures and machine processedfilm. Finally, we will indicate its extension to thecharacterization of microfocus x-ray tubes.

II. Experimental TechniqueWe have used the three techniques mentioned above

to characterize a series of seventeen x-ray focal spots.These focal spots range (as specified by the manufac-turer) from 0.3 mm to 2.0 mm.

A. Pinhole ImagingIn this technique, the magnified image of the focal (-)

spot is recorded using a 0.03-mm pinhole on dental x-rayfilm. A typical pinhole image is shown in Fig. 2(a).The size of the focal spot is then determined, using a ; 0;iat i 0 0: freticle and 6X magnification in directions parallel to the anode-cathode axis (Fig. 1) and perpendicular to thisaxis. Measurements are taken, as specified in theNEMA standard, to the point at which no furtherdensity is observable. 2 In each case, the pinhole isplaced close to the central ray of the system. The exactpositioning of this pinhole in the central ray is difficultto accomplish accurately, and with tubes having a steepanode face (<150) errors of a few millimeters will causea large change in the pinhole image.

B. Star ImagingThe magnified image of a radiating star target is

produced near the center of the x-ray field. A typical __image obtained with the focal spot of Fig. 2(a) is shown (c)in Fig. 2(c). The distance, along a diameter in this Fig. 2. (a) Pinhole image of a typical 1.0-mm x-ray focal spot usingimage, between points at which failure of resolution a 0.030-mm pinhole, (b) contact radiograph of a star pattern, and (c)occurs is measured and converted to a spatial frequency. a magnified radiograph of this star pattern taken with the 1.0-mmIt should be mentioned that, because of odd harmonics focal spot.

3100 APPLIED OPTICS / Vol. 15, No. 12 / December 1976

Fig. 3. (a) Contact radiograph of100-gm lead random object dis-tribution, (b) magnified radio-graph of the random distribution,and (c) pinhole image of an x-rayfocal spot used for (b). (Note thereproductions of the focal spot in

Figure 3b.)

bution. The size of this equivalent focal spot is givenby

a = M0/k[ (M. - 1)], (2)

where a is the size of the equivalent focal spot, tO is thespatial frequency at which resolution fails, and M, is themagnification of the star pattern.

C. Random ArrayA monodisperse layer of particles of lead shot ap-

proximately 100 Am in size is imaged, with magnifica-tion, using nonscreen radiographic film (Kodak RP/M,machine processed). It should be noted that the filmmay exhibit irregularities in the thickness of the basematerial and the emulsion. Also, some irregularitiesmay be introduced into the emulsion by the film pro-cessor. These are manifested by a line through theorigin in the Fourier plane, but this is easily recognizableand can be ignored. Figure 3 shows both a contact anddegraded image of the random array. The magnifiedimage of this array is then Fourier transformed with acoherent optical processor, shown schematically in Fig.4, consisting of a He-Ne laser, beam diverger, and col-limating and transform lenses. The transform is re-corded directly on 35-mm film (in a lensless 35-mmcamera). Figure 5 shows this image for the focal spotof Fig. 3.

In order to understand the significance of the imageof Fig. 5, it is necessary to consider briefly the mathe-matical background of the technique. If s(x) is thex-ray transmission function of the random particle array(we will consider a one-dimensional array for simplic-ity), the image of this array is given by

i(x) = h(x) * s(x),

If the array particle size is small compared to the PSF,s (x) is the sum of a series of one-dimensional b-functionsand S () is a constant. Since the photographic film inthe transform plane (Fig. 4) records the intensity of theincident light, it can be seen that the intensity distri-bution of Fig. 5 is proportional to the MTF of theimaging system disregarding film nonlinearities. TheMTF can be obtained directly by scanning the imageplane with a photomultiplier. However, we wish onlyto characterize the imaging properties of the system,and we are mainly interested in the zeros of the MTF(particularly the first zero). If the particle size cannotbe considered a true b-function, the particle spectrumS(Q) will modulate the desired MTF as recorded on thefilm. However, as we are only interested in the zeros,some distortion of the actual MTF by the particlespectrum is acceptable. We have arbitrarily chosen theparticle size to be always less than 1/2 the minimum ex-pected dimension of the focal spot. This ensures thatthe zeros we measure are indeed the zeros of the MTFand not of the particle frequency spectrum.

Col imating TransformLens * Lens

Laser BeamDiverger

Sample Fourier TransformPlane Plane

Fig. 4. Schematic of optical processor.

(3)

where h (x) is the impulse response function of the sys-tem or point spread function (PSF) and the asterisk (*)denotes the convolution operation. This equation canbe Fourier transformed to give

I(Q) = H)SQ), (4)

where is a spatial frequency (in cycles/mm), and theFourier transform is defined as

I(Q) = | i(x) exp(-j2irtx)dx. (5)

0 2Cc

(a) (b)Fig. 5. (a) Two-dimensional MTF of focal spot in Fig. 3 and (b)

stylized one-dimensional profile through the origin of (a).


b..

1.

M

0.9

0.8N

. 0.7

0.A

0.5

0.6t

E

0 0.5N

' o.4'U

0.:

_ Pinhole, Star

Rand. Dist.

Perpendicular toAnode-Cathode Axis0.3 n Focal Spot

50 100 150MA

FOCAL SPOT SIZE AS

A FUNCTION OF mA

(a)

Parallel toAnode-Cathode Axis0.3 n Focal Spot

i_

- Pinhole

- Star-- Rand. Dist.

I-

0 50 100 150mA

FOCAL SPOT SIZE AS

A FUNCTION OF mA(b)

Fig. 6. Focal spot size measured (a) parallel to the anode-cathodeaxis and (b) perpendicular to the anode-cathode axis as a function

of mA.

array techniques measure a different quantity than doesthe pinhole technique. With the pinhole the charac-terization is mainly that of the heat loading propertiesof the anode. The star and random array methods,through the use of a focal spot model, attempt to char-acterize the imaging capabilities of the x-ray tube.

Since this technique measures the modulus of thetransfer function and not the phase, it ignores someeffects of focal spot asymmetries. These effects havebeen discussed in detail in a previous publication.6 Theshifts in the zeros of the MTF will, of course, be evident,but other effects such as skewing of the edge gradientprofile will not be predicted by this technique as thisrequires the entire optical transfer function.

111. ResultsSince this is a new technique, it is imperative that its

use be validated by comparing accuracy and repeat-ability relative to the presently available methods inaddition to possible benefits in terms of additional in-formation provided. In a series of studies we havefound that the repeatability of the random distributiontechnique ranges from 1% to 6%, with the average beingapproximately 3%. This compares favorably with thestar and pinhole techniques.

In measuring seventeen focal spots variability of thethree techniques is the same at a 95% confidence levelwhether the measurements are made parallel or per-pendicular to the anode-cathode axis.

It is also imperative that this technique be capableof measuring small changes in the imaging propertiesof a focal spot. We have utilized, as a test, the fact thatas tube current is increased the focal spot tends to in-crease in size. Figure 6 shows the growth of a nominal0.3-mm focal spot with increasing current. The randomdistribution and star techniques track quite well overthe entire range of values. It should be noted that asmall normalizing correction may have to be applied tothe star measurements since, due to the frequency tri-pling effect at the zero crossings, it is not always obviousbetween which points in the star image the measure-ments of D must be made.

It was of some concern whether the star and randomarray techniques would track as the kVp of the x-ray

The interpretation of Fig. 5 is straightforward sincethis is a reproduction of the two-dimensional MTF.The one-dimensional MTF, in any direction, is obtainedfrom a cross section through the origin. Since thisfunction must be symmetrical about the origin, thedistance D between the two first order zeros is inverselyproportional to twice the spatial frequency at whichresolution fails. Geometrical considerations of theimaging and transform systems show that the effectivefocal spot size (i.e., the size of a uniform focal spot whichleads to the same resolving power) is given by

a = (2Xf)/(MpD), (6)

where X is the wavelength of the laser, f is the focallength of the transform lens, and Mp = M-1 is themagnification of the point spread function.

It should be emphasized that the star and random

E-

4

1 la

. C

X a

1.5

X.4

- Pinhole

'- StarRand. DIst.

1.3h

I I I

40 60 80kVp

FOCAL SPOT SIZE ASA FUNCTION OF kVp

Fig. 7. Focal spot size measured perpendicular to the anode-cathodeaxis as a function of kVp.


I _

l .- - s s -- s -

-w -

I l

I - I7 _ 10

Fig. 8. Variation of imaging properties of a focal spot with field position from anode end (upper left) to cathode end (lower right), a total distanceof 43 cm (17 in.) in the image plane.

tube is varied. As kVp is increased the small particlesof the random distribution are penetrated by the ra-diation to a greater extent resulting in an apparent sizechange as well as a lower contrast image. It is expectedthat, provided the particles are small compared to thepoint spread function, this effect should be negligible.Figure 7, showing the variation in size with kVp of anominal 1-mm focal spot, bears this out. It is evidentfrom this figure that the pinhole technique lacks theability to reflect the imaging characteristics of the focalspot. As the kVp increases space charge and electronrepulsion effects decrease in importance, causing thetwo bands of the focal spot to move together, and hencebetter imaging properties result.7 The over-all extentof the electron distribution, however, still reflects thegeometry of the focusing cup, and the pinhole imagesshow no change in extent.

It was mentioned above that a pinhole or star imagecharacterizes the focal spot at only one point in theimage field. From a single radiograph of the randomarray we have obtained a series of MTF's by trans-forming small portions of the radiograph through a2.54-cm (1-in.) circular aperture placed in the Fourierplane of the transform system. Figure 8 shows this

series moving from the anode end (upper left) to cath-ode end (lower right). This series clearly shows thechange in apparent focal spot size in the direction par-allel to the anode-cathode axis as well as the heel effect(loss of intensity toward the anode end).

Both the pinhole and star techniques present tech-nical difficulties when used to characterize small focalspots. The random array, however, can be extendedeasily to encompass use with such focal spots by simplyreducing the particle size in the array. To verify thiswe have used a 200-mesh Bi powder (Ventron, AlfaProducts, Danvers, Mass.) to construct a random array.Some difficulty was encountered as this powder con-tained the entire range of sizes up to the maximumpassed by the 200-mesh grid. This tended to reduce thecontrast in the radiograph considerably. This difficultywas overcome by separating the powder into a series ofsize ranges using a precipitation technique. Two arraysusing particles of approximately 40 gm and 20 ,4m wereconstructed. Both of these arrays contained largeempty spaces between the particles as a result of ourdifficulties in arriving at a compromise between amultilayer and a monolayer deposit. These deficiencesin the arrays lead to decreased diffraction efficiency in


Fig. 9. Fourier transforms of a 0.3-mm focal spot using (a) 40-jtmBi random distribution and (b) 20-jum Bi random distribution.

the transform process and necessitate the use of a highcontrast film for recording the Fourier transform.Kodak Contrast Process Pan film 4155 was used to en-compass the large size of the transform. Figure 9 showstwo transforms of our smallest focal spot (0.3 mmnominal) using each of the two Bi arrays. The mea-sured focal spot sizes were the same (0.45 mm) and alsoagreed with that measured (using much lower objectmagnification) with the 100-,um lead particle array.

IV. ConclusionsThe random object distribution technique for char-

acterizing the imaging properties of x-ray focal spots hasbeen evaluated. It has been found to compare favor-ably to the star and pinhole techniques in variabilityand in detecting the changes in focal spot size. For thepurposes of characterizing the imaging properties of afocal spot, the random array technique provides moreinformation than the pinhole or star techniques. Thetwo-dimensional MTF is produced directly without thenecessity. of secondary calculations. Variations inimaging performance over the entire field can be mea-sured from a single radiograph. The technique can beeasily extended for use with microfocus x-ray tubes. Inaddition, advantages of speed, reduced exposure tech-nique, and machine film processing are offered.

The authors thank H. U. Ross, Department of Met-allurgy and Materials Science, University of Torontofor his assistance in the separation of the Bi powder andH. Weiss of Philips Forschungslaboratorium, Hamburg,GmbH for the random lead object distribution.

This research was supported in part by Medical Re-search Council of Canada Grant MA-5180, James PickerFoundation, and Edward Christie Stevens Founda-tion.

References1. K. Doi, Am. J. Roentgenol. 94, 712 (1965).2. -, "Measurement of Dimensions of Focal Spots of Diagnostic

X-ray Tubes," National Electrical Manufacturers AssociationStandards Publication XR5 (1974).

3. J. E. Gray, M. P. Capp, and F. R. Whitehead, Invest. Radiol. 9,252(1974).

4. M. Braun and W. W. Stoner, Abstract: Med. Phys. 2, 173(1975).

5. G. Groh, E. Klotz, and H. Weiss, Appl. Opt. 12, 1693 (1973).6. J. E. Gray and M. Trefler, Med. Phys. in press.7. E. L. Chaney and W. R. Hendee, Med. Phys. 1, 141 (1974).

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Characterization of the imaging properties of x-ray focal spots