90 puted f( : d more the sametime accurateresults for the integration t the stat, conditioy, with (Arg) < 2/3 FWHM. The limit on the errors pertinent to g(z’) given not only by intrinsic properties of the in- [Pr 3, (6) ~ ee e peaks jf xzz;tAxe near the extremities of the interval (a,b) of the actual distribution can be obtained by taking readings g(x) beyond the interval (a,b) and set- ia) ae = 6 strument(efficiency, natural background, count- ing time, etc.), but also by the shape of the curve y(e). This means that if y(e’) belongs to a cer- Ing provg tain family of curves, it may be possible to ob' tain optimal results on f(z) even if the errors a(x), superimposed on g(x’) are rather large ‘rein, It j estimate (e.g., in Figure 70 for the FWHM = 8.5 em, the rough value deduced from the experiment is e” = 5,000; but we would obtain comparable accuracy for the computed f(x) if the errors a(z’) were large enough to give a value of e° ~~ 25-10*, in which case we choose y ~~ 20-10°, The interval h between two consecutive readings on g(x) can influence strongly the shape of f(z) and consequently the net (integrated) activities | activity? r) and th, S Ate. 4 Ze shan, Cys; good results can be obtained in practice with h < 1/3 FWHM. arated bt obtain af, (e) Improvement in the shape of the computed f(z) ting f(z) = 0 outside this interval. The authors wish to acknowledge the technical help of Mr. Richard F. Selman.” REFERENCES 1. Marinelli, L. D., Clemente, G. F., Abu-Shumays, I. K., and Steingraber, O. J. Argonne National Laboratory Radiological Physies Division Annual Report, July 1967- June 1968. ANL-7489, pp. 1-12. 2. Marinelli, L. D., Clemente, G. F., Abu-Shumays, I. K., and Steingraber, O. J. Radiology 92, 167 (1969). 3. Iinuma, T. A. and Nagai, T. Phys. Med. Biol, 12, 501-509 (1967). 4, DiPaolo, R., Albaréde, P., Patau, J. P., and Tubiana, M. Compt. Rend, 263, (Series D) 1160-1163 (1966). 5. Phillips, D. L. /. Assoc. Comput. Mach. 9, 84-97 (1962). 6. Monahan, J. E. Seintillation Spectroscopy of Gamma Radiation, Ed. 8S. M. Shafroth. Gordon and Breach, New York, 1968, pp. 371-428. 7. Marinelli, L. D. Proc. XI Int. Congr. Radiology, Excerpta Medica Int. Congr. Series 105, 1291-1301 (1966). * Central Shops Department. REGULARIZATION UNFOLDING FOR TWO DIMENSIONS: PROGRESS REPORT LOK, Abu-Shumays* A solution to a Fredholm equation of the first kind, relevant 4 two-dimensional distribution of radioactivity is being sucht by smoothing the experimental data by the use of approprittely weighted smoothing matrices. The kernel is as- In recent work@: ® we have successfully adapted the regularization (smoothing) techniques for unfolding where K is a known response kernel, g(x,y’) the measured spectrum and e(xy)t is the experimental and statistical error in the measurement of the spectra. We assume that K = Ky(2,2')Ko(2,y’), i.e., that the z and y components of the kernel are separable. This is justifiable, for example, whenever the kernel can be approximated by a Gaussian lincar speetra,® to determine unknown distributions of _ rudioactivity per unit length of a one-dimensional phantom model, We report here an extension of the method K(2,2',y,y) = C Exp [—-[(@ — 2)’ + (y — y')"}/4o°}. , to two dimensions, which meets both eriteria of sim- For numerical calculations, the domain of the distribution f(x,y} and the range of the spectra g(x,y’) are covered with a hypothetical grid suitable for mathe- tu sinned to be Gaussian. ' plicity of form and considerable economy of computatron time. ‘Thedistribution f(z,y) of radioactivity in two dimen| slons is governed by the following Fredholm equations of the first kind de A dy'K(2,2',y, ¥ f(xy) = g(a’y’) + e(x'y’), Applied Mathematics Division. (1) matical treatment. (For simplicity, we select a uniform grid with size up to 100 x 100.) If we neglect the error term, Eq. (1) is then transformed to the matrix equation AFB = G, (3) +It is well known that without exact knowledge of the magnitude of the error term, Eq. (1) is a mathematically illposed problem forthe distribution f(z’y’).