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RECONSTRUCTION OF THE FRESNEL—CODED IMAGES BY DIGITAL COMPUTER

T. F. Budinger and B. Macdonald

Donner Laboratory and Lawrence Berkeley Laboratory, University of California, Berkeley, California

An algorithm for the digital reconstruction because of this spread function overlap. This fact has of the Fresnel-coded gamma camera image con been recognized by Barrett and De Meester (11), sists of the spatial Fourier transform of the who show that the SNR is reduced by a factor of hologram after applying a quadratic phase shift. M 1/2where M is the number of resolution elements Background subtraction is implemented by use because all photons from each resolution element of positive and zone plates. Tomo contribute to the noise but only those photons origi graphic planes are reconstructed in 27 sec from nating from one resolution element contribute to the a single zone-plate shadow image by successively signal. In the limit of a very large source this leads varying a parameter in the reconstruction al to a small disadvantage in time required for gorithrn on a small computer system. the zone plate over the pinhole. Even an improve ment in SNR by digital computer reconstruction cannot give results in less exposure time than the This paper presents an algorithm for the digital for extended objects. Moreover, Levy reconstruction of the Fresnel zone-plate coded image (12) argues that the gain in SNR of the Fresnel zone and also experimental results with the gamma cam plate over that of the pinhole is proportional to the era showing the tomographic potential of coded ratio of the intensity of the signal at each point to imaging, the signal-to-noise ratio (SNR) the average intensity in the whole image. There is comparison between zone-plate imaging and pinhole real doubt, therefore, that the efficiency of the basic imaging, and the use of positive and negative zone pinhole camera could be improved upon by Fresnel plates for background subtraction. coding or other multi-aperture systems for simple The transmission efficiency is in the range of only two-dimensional imaging. Although there is no ex 0.05 % or less for a conventional parallel-hole or posure time advantage of the Fresnel zone plate over pinhole collimator. New techniques of coding such the exposure time for a single pinhole aperture for as multiple random pinhole-coded aperture imaging extended objects, there is an inherent tomographic (1—4) and Fresnel zone-plate coding (5—10) have potential of the Fresnel-coded image ( 13) as well been suggested for gamma source imaging because as the multiple random pinhole-coded image (14). the geometric efficiency of these schemes is 1,000 To realize this potential for the Fresnel-coded image times or more greater than that of collimators in use and to simplify reconstruction of the hologram with today. In nuclear medicine studies, however, the out photographic transfer and optical bench manip sources are relatively large and therefore the effective ulation, we developed a digital computer algorithm efficiency of coded is considerably less than that will allow rapid reconstruction with most of the their geometric efficiency. Consider an aperture (col available new series of nuclear medicine computer limator) with N pinholes. The SNR will beimproved systems interfaced to a gamma camera (15). by @/Ntimes that of a single pinhole only if the image corresponding to each hole is separated from THEORY images of all other holes. A basic problem of coded aperture imaging in nuclear medicine is the fact that Fresnel zone-plate imaging is done in two parts. inherent noise increases as the side lobes of the First, the zone plate is used between the object and spread function from one point source superimpose on the spread function of another point source. The Received Dec. 19, 1973; revision accepted Nov. 18, 1974. advantage of increased collimator transmission for For reprints contact: Thomas F. Budinger, Lawrence signal is lessened by the increase in inherent noise Berkeley Laboratory, Berkeley, Calif. -94720.

Volume 16, Number 4 309 BUDINGER AND MACDONALD

I(x,y) = exp (!@S@2±Y!))

Si.H(x1,y1)exp(_@ir(Xi@+y12)) x1y1 A

exp(_i2@-@+ YYI))dxdy (6)

I ‘rIx2@v2I Note the operations of Eq. 6 involve merely the Fou Phase shift e e,2 H(x.y) / rier transform of the shadowgram times a phase shift Fourier transfer —e@ IMAGE function that is quadratic in the distance in the shadowgram plane from the optic axis. FIG. 1. Schemeof digitalreconstructionof imagefromgamma camera. It is well known that a Fresnel zone plate acts as a , and thus the image I(x,y) is in when the distance f in Eq. 6 is chosen as the detector to form a coded shadow H(x1,y, ) of the of the shadow zone plate object on the detector (Fig. 1 ) . This shadowgram is R2 f=±@--1 (7) then placed in a coherent beam of light of A A, and an image I(x,y) of the object is formed on a where R, is the radius of the shadow of the first zone screen using the and focusing properties and A is the wavelength. The radius of the first zone's of the individual zone-plate shadows that make up shadow depends on the distance s1 between the ob the shadowgram. The image produced, according to ject plane and the zone plate and the distance s2 be the Huygens-Fresnel diffraction principle, is tween the zone plate and the camera (Fig. I). 1(x,y)= ffH(x,y,) A(x,y;x,,y,)dx1dy, (1) R1=R1(l+@@) (8) where where R is the true radius of the zone plate. Sub I /i2@-@ @ A(x,y;x,y,):o:.AfexP@ (2) stitution of Eq. 8 for the term Af in Eq. 6 gives the final algorithm: f being the distance between the shadowgram plane and the image plane and Qthe path length traversed 1(x,y)* = cET2[H(xi,yi) exp(@T@?@-@2)] (9) by a light ray between points in these two planes. where @J2denotes a two-dimensional spatial Fourier Q= (f@+ (x—x1)2+ (y—y1)2)112= transform that is easily implemented with a computer (x — x )2 ( _ )2 1/2 using the fast Fourier transform (FFT). @ f(l + + y (3 ) For the process described here with the on-axis zone plate, the virtual image, due to the dual sign Assume that the square root term in Eq. 3 is ade in Eq. 7, is also present and superimposed on the quately approximated by the first two terms in its bi true image. More important is a large DC component nomial expansion. This is the Fresnel approximation caused by object intensity and the zone-plate trans and in effect is a replacement of the spherical Huy mission both being non-negative. This background gens' wavelet by quadratic surfaces (16,17) . If we can be effectively eliminated by a method that does drop the constant phase term exp(i2@rf/A)/iAf, Eq. 2 not involve the use of an off-axis zone plate with becomes its attendant loss of resolution. The technique con sists of subtracting the shadowgram obtained using A(x,y;x,,y,) = a negative zone plate from that obtained using a 1i2'zr 2 positive zone plate (Fig. 2 ) ( 18) . A similar proce exp@-@-@((x—x1)2+(y—y1) } (4) dure is applicable for random pinhole encoding (19). The basis for the method can be seen from an exami and Eq. 1 becomes nation of the equations that govern image formation with the zone-plate system. The shadowgram H in I(x,y) = ffH(x1,y1) x1y1 Eq. 1 is just the convolution of the object intensity 0 with the transmission of the zone plate, either 1 + Z fi@r@(x—x1)2+(y_y1)2}\@ d exp@ xl I @1y1 (5) for a positive zone plate or 1 —Z for a negative zone plate. Thus, a positive zone plate shadowgram is By regrouping terms after expanding the exponential, * I(x,y) is a complex amplitude. The image recorded is we have the modulus squared.

310 JOURNAL OF NUCLEAR MEDICINE @ I@

FRESNEL-CODEDGAMMACAMERAIMAGES

POSITIVE NEGATIVE The zone-plate plane and camera plane are estab lished perpendicular to and centered on the “optical― aXiS (Fig. 1 ) . We used positive and negative zone plates with the center zone of 2.5-cm diam and only 00 three rings in order to match the resolution of the 1 POINT SOURCE gamma camera (Fig. 2) . The camera events from 9OmTcpoint sources were digitized in a 64 X 64 array using the Hewlett-Packard scintigraphic sys tem 5407. The gains of the system were set to give 5 POINT SOURCE 0.4 cm per picture element. Ten tomographic planes were established by varying R, of Eq. 9 in accord FIG. 2. PositiveandnegativeFresnelzoneplates. ance with Eq. 8. For the results of Figs. 2 and 3, the distance from the zone plate to camera was 60 cm H÷ =0@ (1 +Z) (10) and the distancebetweenthe zone plate and object planes was varied between I S and 25 cm in I -cm @ where denotes the convolution operation. The increments. image, from Eq. 1 is After the digital phase shift is performed, the com I@=H@ *A=0* 1 *A+0*Z*A (11) plex array is Fourier-transformed using a FFT al gorithm which requires about 25 sec for a 64 X 64 The term 0 * 1 * A represents the large component complex array. The result for each plane is stored of background. If one takes an exposure with a posi as an image frame and returned for viewing on a tive zone plate and obtains the shadowgram matrix CRT from a disk file of image frames. The total H+ = 0 @‘( 1 + Z) and a subsequent exposure with operation takes 27 sec for each plane. @ a negative zone plate, H = 0 * ( Z) , then the Positive-negative zone-plate subtraction technique shadowgram matrix, which is the difference of these for background elimination was accomplished by ac two matrices, yields an image that has no background. quiring two shadowgrams, one with the positive zone plate and the other using the negative zone plate. I=(H+_H_)*A=0*2Z*A (12) Both were accurately centered. Distance from the The tomographic potential of the Fresnel-coded zone plate to the camera was 73 cm and from zone zone-plate aperture is implemented by performing a plate to the sources was 62 cm. The result was ob series of reconstructions varying the parameter R, tamed by subtracting the two shadowgrams and using corresponding to various focal planes given by Eq. 8. the difference shadowgram to reconstruct the image. A given reconstruction plane s, is related to R1 as For comparison, a 3-mm pinhole collimator was used with the distance between the pinhole aperture and R0s2 camera of I 6 cm and the distance between aperture 51 — R, — (13) and sources of 13 cm.

Point saurces Scintillation camera Lead FresnelZone Plate image

@ r-@ REJRUC@ON

F1G.3. Reconstructionof fivesources of @mTcseparated by 1.5 cm in plane 90 cm from camera, 30 cm from zone-plate (first zone R0 = 1.25 cm). Background subtracted

Volume 16, Number 4 311 @ ‘@ -

BUDINGER AND MACDONALD

FIG. 4. Imagesfrompositive(A)and negative (B) zone plate reconstruction and @ Ar-si Br- CF-7_ _@@_ results of image subtraction (C).

@ A @,, S@i : - j B

@ ..@ 4.@ - - FIG. 5. Statisticalcomparisonbetween @-- .) ;.@- @C(., . .@,,.- . -@ - -@-@“ ,, zone-plate (A) and 3-mm aperture pinhole (B) images for identical data collection times (30 5cc) with five point sources; dis. tance from center is 3 cm.

Scintillation camera Lead Fresnel Zone Plate ir B

Point sources

U@RECONS@TION FIG. 6. Tomographicimagesof two point sources.

RESULTS AND DISCUSSION the zone-plate field of view is much smaller than that for the single pinhole. The field of view can be in The reconstruction of five small ( 2 mm ) sources creased by the use of a large multiwire chamber of RftmTc separated by about 2.0 cm is shown in (18) in which case the lowered intrinsic efficiency Fig. 3. Background subtraction shown below the of the gas proportional camera and the SNR for the center image was performed by thresholding all in Fresnel technique over the simple pinhole must be tensities less than one-fourth the maximum intensity. considered in assessing the relative efficiencies. A better method of background subtraction is to use Quite apart from the efficiency question, Fresnel the positive and negative zone-plate system (Fig. 4). or other coded aperture systems have a potential A comparison of the positive/negative zone-plate value for tomography in nuclear medicine. The tomo image with the image using a pinhole collimator ,is@ graphic itbiity of this system is demonstrated in Fig. given in Fig. 5. Using the zone plates 49,00@ counts 6 where the Fresnel hologram of two point sources were detected from the five sources in 25 sec and was processed. The blurring of the sources on planes the calculated ( 11 ) SNR was 40 : I with an expected adjacent to the focal plane can be quantitatively eval lateral resolution of 5.5 mm FWHM. For the pin uated to give a measure of the tomographic capa hole, 4,890 counts were collected from the five bility of a particular zone-plate camera system. sources in 30 sec and the expected SNR is 3 1 : 1 with@ a lateralresolutionof 5 mm. ACKNOWLEDGMENTS . This objective means of background subtraction This work was performed under the support of the United makes easier a comparison between zone-plate and States Atomic Energy Commission, with the assistance of William Hogan and John Harpootlian. We appreciate the pinhole imaging. A complete analysis of the efficiency interest and encouraging comments of Robert M. Glaeser of a zone plate relative to a pinhole must take into and George W. Stroke.

account the effective field of view as well as the REFERENCES relative exposure times, the SNR, and the lateral 1. DICKE RH : Scatter hole for x-rays and resolution. With the same detector and geometry, gamma rays.Astrop/zysI 153:Ll0l—Ll06,1968

312 JOURNAL OF NUCLEAR MEDICINE FRESNEL-CODED GAMMA CAMERA IMAGES

2. HAYAT OS: X-ray and Gamma Ray Imaging with 11. BARRETT HH, DE MEESTER GD: Quantum noise in Multiple Pinhole Cameras, Ph.D. thesis, State University of Fresnel zone-plate imaging. App! Opt I 3 : I 100—I109, 1974 New York at Stony Brook, 1971 12. LEvY 0: Comment on “Fresnelzone plate imaging in 3. GROH G, HAYAT OS, STROKE OW: X-ray and gamma nuclear medicine.―I Nuc! Med 15: 214—215,1974 ray imaging with multiple pinhole cameras using a pos 13. BARRETT HH, DE MEESTER GD, et al: Tomographic tenon image synthesis. App! Opt 11: 191—193, 1972 imaging with a Fresnel zone-plate system. In Tomograpliic 4. WOUTERSA, SIMON KM, HIRSCHBERGJO: Direct Imaging in Nuclear Medicine, Freedman OS, ed, New York, method of decoding multiple images. App! Opt 12: 1871— Society of Nuclear Medicine, 1973, pp 106—121 1873,1973 14. CHANG L-T, KAPLAN S. MACDONALD B, et at: A 5. BARRETT HH : Fresnel zone plate imaging in nuclear Method of Tomographic imaging Using a Multiple Pinhole medicine.JNuclMed 13: 382—385,1972 Coded Aperture. University of California Lawrence Berkeley 6. BARREi-rHH, DE MEESTERGD, WILSONDT: Recent Laboratory Report 3001, 1974 advances in Fresnel zone-plate imaging. In Medical Radio 15. BUDINGERTF: Clinical and research quantitative nu isotope Scintigraphy 1972, vol 1, Vienna, IAEA, 1973, pp clear medicine system. In Medical Radioisotope Scintigraphy 269—284 1972, vol 1, Vienna, IAEA, 1973, pp 501—555 7. ROGERSWL, HAN KS, JONESLW, et at: Application of a Fresnel zone plate to gamma-ray imaging. I Nucl Med 16. GOODMANJA : introduction to Fourier Optics, New 13:612—615,1972 York, McGraw-Hill, 1971, pp 57—60 8. BARRETT HH, WILSON DT, DE MEESTER GD, et al: 17. BORN M, WOLF E : Principles of Optics, 2nd rev ed, Oxford, Pergamon Press, 1964 Fresnel zone plate imaging in radiology and nuclear mcdi cine.Opt Eng 12:8—12,1973 18. MACDONALDB, CHANG L-T, PEREZ-MENDEZV. et al: 9. ROGERSWL, JONESLW, BEIERWALTESWH : Imag Gamma-ray imaging using a Fresnel zone-plate aperture, ing in nuclear medicine with incoherent . Opt multiwire proportional chamber detector, and computer re Eng 12:13—22,1973 construction. IEEE NS 2 1: 678—684,1974 10. TIPTON MD, DOWDY JE, CAULFIELD JH: Coded 19. BLAKE RL, BUREK AJ, FENIMORE E, et at: Solar aperture imaging with on-axis Fresnel zone plates. Opt Eng x-ray photography with multiplex pinhole camera. Rev Sci 12:166—168,1973 instrum45:513—516,1974

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