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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 52, NO. 3, JUNE 2005

Application of Expectation Maximization Algorithms for Image Resolution Improvement in a Small Animal PET System Pietro Antich, Member, IEEE, Robert Parkey, Serguei Seliounine, Member, IEEE, Nikolai Slavine, Edward Tsyganov, Member, IEEE, and Alexander Zinchenko

Abstract—Modern positron emission tomography (PET) systems, offering high counting rate capabilities, high sensitivity, and near-submillimeter coordinate resolution, require fast image reconstruction software that can operate on list-mode data and take into account most of finite resolution effects such as photon scattering, positron range in tissue, and detector features. It has already been demonstrated that the expectation maximization (EM) method with extended system matrix modeling looks very attractive for image resolution recovery in PET imaging studies. In this paper, the performance of EM-based algorithms (in particular, their ability to improve the image resolution) is evaluated for a small animal PET imager with several phantoms. The achievement of a substantial decrease in processing time using an EM deblurring procedure is shown, as is an approach to successfully treat what are essentially nonspace-invariant resolution effects within a shift-invariant model. Index Terms—Iterative image reconstruction, positron emission tomography (PET), small animal imaging.

I. INTRODUCTION

Fig. 1.

S

MALL animal PET imaging offers a unique opportunity for in vivo studies of animal models of human diseases as well as for the development of new radioactive tracers with specific desirable properties (e.g., [1]–[4]). In order to successfully perform these tasks, the imaging apparatus should have the highest possible spatial resolution, which is currently can only be achievable in dedicated small animal PET scanners [5]–[11]. It is equally important that image reconstruction algorithms be chosen which are capable of preserving and even further enhancing the hardware advantages. It has been shown that iterative reconstruction methods, with accurate detection system modeling, demonstrate the most promising results (e.g., [12]–[14]). In PET systems with list-mode data acquisition it is also advantageous to use reconstruction methods working directly with list-mode data, in order to avoid any information loss [15]–[19]. The purpose of this study is to evaluate the recently proposed list-mode EM image reconstruction algorithm with a convolution model [20], [21], which we consider as the most appropriate Manuscript received December 5, 2003; revised November 4, 2004. This work was supported in part by the Cancer Imaging Program (an NCI Pre-ICMIC) 1P 20. The authors are with the University of Texas Southwestern Medical Center at Dallas, TX 75390 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TNS.2005.851479

Detectors and gantry of small animal PET.

choice for a small animal PET device built at the University of Texas Southwestern Medical Center at Dallas. We also present some results of our attempts to further develop the EM-based image resolution recovery approach, in order to make it more suitable for practical applications. II. METHODS A. Experimental Setup A small animal PET imaging device (Fig. 1) has been developed at the University of Texas Southwestern Medical Center at Dallas, using scintillating 1-mm-round BCF-10 fibers and a small admixture of CsF microcrystals between the fibers [22]. For a 511 keV photon in plastic, the photoabsorption is small and Compton scatter interactions are dominant. The scattered electrons typically give up their energy within a fiber diameter, but waveshifting produces light in proximal fibers. The imager uses the twofold coincident detection of a single event in two orthogonal fibers to detect the location and the energy transferred at a point within the detector. Two sets of fibers of 60 cm in length each were used to construct two alternating, mutually orthogonal sets of 14 planar arrays of 135 fibers each. In this detector, the planar fiber arrays are arranged along two axes and and stacked along the third ( ). Scintillating light from the fibers is detected by two ( and directions) Hamamatsu R-2486 position sensitive photomultiplier tubes (PSPMTs). A

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ANTICH et al.:IMAGE RESOLUTION IMPROVEMENT IN A SMALL ANIMAL PET SYSTEM

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of the probability In the simplest approach, the element matrix is taken as the intersection length of LOR with voxel . However, in this case all resolution effects (positron range, photon acollinearity, and intrinsic detector resolution) are neglected. To account for these effects, which can be measured and considered known, the following approach can be used. The can be factorized as matrix of probabilities

TABLE I PERFORMANCE CHARACTERISTICS OF THE SYSTEM

(3)

single-ended readout scheme is used, where the and interaction positions in a detector are determined from coincident detection in the two PSPMTs. The precision of the detection of the interaction point depends on PSPMT performance and software filters. Acquired data are recorded in list-mode. At present we have completed two of the four detectors necgeometry. Each planar detector is posiessary for closed tioned to measure one of the two 511 keV annihilation photons. By requiring a coincidence between the two detectors (i.e., four PSPMTs), the position of an electron–positron interaction can be reconstructed. The two detectors can be rotated around the central axis to approximate a truncated cylindrical detector. The performance of the system is shown in Table I. The object spatial resolution is unchanged across the entire field of view (FOV), while the sensitivity varies between 40% and 100% of the cencm FOV. tral maximum over a B. List-Mode EM Algorithm With a Convolution Model The list-mode EM algorithm was developed from the standard projection data-based maximum likelihood expectation maximization (ML-EM) algorithm [23] and is given by

The matrix accounts for finite resolution effects, is the matrix of intersection lengths as mentioned previously, and the diagonal matrix takes into account geometric sensitivity variations. Then the expected count is expressed as (4) and the list-mode EM algorithm becomes

(5) The expression (5) can be rewritten in vector form (6) where denotes an element-by-element multiplication of vecand contain the new and old image estimates, tors, and contains the sensitivity correction factors. contains the multiplicative image correction values, given by

FP with

(1)

where is the expected integral count in the line of response (at the iteration), is the (LOR) if the intensity was probability of an emission from voxel being detected along LOR , is the number of measured events, and is the number of all possible system LORs. is the number of voxels in the image. The algorithm takes into account the fact that the measured list-mode data is implicitly 1 for each acquired LOR. Depending on the number of events, direct iterative reconstruction from list-mode data can be burdensome and computationally expensive. If the data is abundant, the ML-EM algorithm can be extended to incorporate subsets [one-pass listmode (OPL-EM)] [20], as follows: (2)

where only a subset update. Each subset of list-mode events.

of the

(7)

total LORs is used in each contains equal numbers

where FP is an operator that forwardprojects vector along LOR to give a scalar value, and is an operator that backprojects a scalar value along LOR into a three-dimensional (3D) image. If the system can be considered as space-invariant, the blurcan be taken as a shift-invariant kernel . ring component Then the OPL-EM algorithm becomes

FP

(8)

Thus, the algorithm OPL-EM has two parameters: the resolution kernel, , and number of subsets (equal to the number of image updates) to use, . Incorporation of the system model into the EM procedure proposed in [20] is a mathematically elegant deconvolution mechanism taking into account the spatial resolution due to finite detector accuracy, positron range in tissue, etc. As will be seen subsequently, this step is of primary importance for our utilization of the algorithm. To emphasize this fact and distinguish the approach from the “conventional” ML-EM method with only the matrix of intersection lengths taken into account

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(EM), we call it expectation maximization with deconvolution (EMD). C. Application of EM Algorithm for Image Deblurring As will be shown hereafter, the EMD method can be successfully used for resolution recovery in PET images. However, introduction of two array convolutions for each image update increases the processing time as compared with the EM method even if the “one-pass” mode is implemented. Combined with the fact that the EMD reconstruction procedure requires much more iterations to converge, it imposes significant limitations for practical applications of the method. In order to overcome the problem, the EM image deblurring procedure (EMdeb) similar to the one proposed for computed tomography (CT) [24] was developed. It takes the image reconstructed after EM as the observation and applies an iterative procedure similar to that described previously, as follows: (9) is the image obtained after EM and is the rewhere construction kernel parameter (identical to the one in (8)). In this scheme, list-mode data are used only at the initial stage for EM reconstruction (a few iterations are usually enough for convergence). The main iteration procedure (9) works with already binned data, so one can expect to have significant processing time savings, especially for high volumes of list-mode data. D. Double Compton Scattering Model As was mentioned previously, the annihilation photons are registered in our PET imager mainly through their Compton interactions with the active material of the detectors. This means that there is some probability for the photon to have a second interaction, with the consequential production of multiple signals in one detector. For those events, there is a nonnegligible probability of selecting signals from the second scattering instead of the first one, or of choosing and coordinates from different interactions. The result is faulty reconstruction of photon lines and can be seen as a “cross-like” pattern for a point-like source (Fig. 2) making the Gaussian resolution kernel not the optimal choice for . However, it is reasonable to try to treat this effect in the same way as any other (e.g., scattering [25]) and incorporate it into the system model of the EMD reconstruction algorithm. E. Phantoms To evaluate the reconstruction methods the phantoms listed hereafter were used. 1) Hot spot mathematical phantom: The hot spot mathematical phantom was simulated as a point source centered inside a cylinder with a diameter of 40 mm and a height of 40 mm. The point source activity was 2.5% of the total cylinder activity uniformly distributed over its volume. All the resolution effects were approximated by one cumulative value and simulated by Gaussian smearing of the ends of recorded photon lines with fullwidth at half-maximum (FWHM) of 4.2 mm, giving

Fig. 2. Projected angle of detectors.

XY -view of a point-like source at true Z for one rotation

the point spread function with of 1.2 mm in the median plane between detectors. FDG point-like phantom: The point-like phantom 2) was prepared by filling a 1.5-mm-diameter semi-spherical hole in a thin plastic plate with an FDG solution. FDG line phantom: The line phantom was made of 3) an FDG-filled tube with 0.95 mm inner and 1.05 mm outer diameters and a length of 55 mm. This experimental event sample was also used to numerically create a two-line phantom by adding coordinate shifts to end points of the photon lines for a randomly selected event subsample. 4) Mouse: An 18F (NaF)-injected mouse with total activity of 700 Ci was used for a bone scan. For all phantoms the list-mode data were simulated/collected with a distance between detectors of 200 mm and for eight rotation angles covering the whole 180 . F. Evaluation The following figures of merits were considered. 1) Resolution: For the hot spot mathematical phantom the resolution was defined as an average of the axial and transverse values obtained from a two-dimensional (2D) Gaussian fit of the reconstructed activity distribution around the point source location. For the line source, the resolution was obtained from the Gaussian fit of the line transverse profile. 2) Noise: For the hot spot mathematical phantom the image noise was taken as the root mean square (RMS) of the voxel activity, normalized to the mean value in the uniform portion of the cylinder. For the line source, the longitudinal line profile was fitted to a second-degree polynomial function (to account for slowly varying activity along the tube length) using the least-squares minimization algorithm with unit weights. The RMS deviation of voxel activity from the function was normalized to the average activity along the tube.

ANTICH et al.:IMAGE RESOLUTION IMPROVEMENT IN A SMALL ANIMAL PET SYSTEM

Fig. 3. FWHM of the reconstructed point source inside the hot spot mathematical phantom versus iteration number of the EM method (full circles and solid line), EMD (open circles and dashed line), and EMdeb (triangles and dotted line).

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Fig. 4. Noise versus resolution for the mathematical phantom: EM (full circles), EMD (open circles), EMdeb (triangles), and EMdeb after preconditioning (squares).

III. RESULTS AND DISCUSSION All the results reported here reflect a shift-invariant approach, i.e., the system model was introduced through the convolution procedure based on fast Fourier transform. The shift-variant approach will be the subject of future studies. in (3) was obtained from The geometric sensitivity matrix calculations. Since the amount of data for all the studied phantoms was in (2) was chosen to quite moderate, the number of subsets be 1, except for the double Compton scattering model study (see hereafter). in (9) used for deblurThe EM reconstructed image ring was obtained after ten iterations of EM, corresponding to its full convergence (see hereafter). A. Comparison of Reconstruction Methods Fig. 3 shows the FWHM of the reconstructed point source inside the mathematical phantom as a function of the iteration number. The full simulated sample (4 million background events and 100 000 events from the point source), combining data for all rotation angles, was used for each image update. voxels of 0.5 mm in The FOV was subdivided into 1.2 mm was size. The Gaussian resolution kernel with used for the EMD and EMdeb methods, approximating the point spread function of the system. One can see that EM converges rapidly on the resolution value given by the imaging apparatus. Conversely, both of the reconstruction methods with resolution recovery show their ability to improve the image. The relationship between resolution and image noise is shown in Fig. 4. It is obvious that EMD and EMdeb have better noise characteristics than EM at the resolution level, corresponding to the EM convergence value, due to built-in regularization. Obviously, a postprocessing filter would decrease the EM noise level, but at the expense of resolution. Since the EMdeb method is based on the image given by EM, one might expect it to show somewhat inferior noise properties than EMD, and

Fig. 5. Noise versus resolution for the line source: EM (full circles), EMD (open circles), and EMdeb (squares).

this is reflected in the plot (EMD points lie lower for a small number of iterations). However, for larger iteration numbers, the EMdeb results become closer to EMD with the latter even showing tendency to exhibit greater noise. In an attempt to suppress the EMdeb noise, we applied a “preconditioning” procedure by smoothing the EM output image with a Gaussian 0.5 mm before applying EMdeb with the resolution filter 1.3 mm. One can see from the plot that it provided kernel a modest improvement (mostly for large iteration numbers). The line source image was reconstructed at a FOV of voxels of mm in size, with 1 mm in the axial direction (along the line). As with the previous (numerical) phantom, the data for all rotation angles were combined and the full event sample (about 900 thousand lines) was used for each image update. One can see from Fig. 5 that for this phantom the image noise was somewhat better for the case of EMdeb reconstruction, even without preconditioning.

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to double Compton scattering, and the background due to random coincidences its contribution

with (10)

with (11)

(12) and (13) where (14)

Fig. 6. FDG two-line phantom: (left) EM, (middle) EMD, and (right) EMdeb. Upper row: Axial view (central axial slice after extra smoothing with a Gaussian filter with  and  equal to pixel size (0.25 and 1 mm, respectively)]. Lower row: Transverse profile (X -projection of the central axial slice for Y from 40 to 85 mm).

To make a visual assessment of the image quality, a two-line phantom with a 2.5 mm center-to-center separation and equal activity in each line was numerically prepared from the one-line phantom as described previously. It was reconstructed by each of the three algorithms under study, with the number of iterations set at 10, 200, and 550, respectively, for EM, EMdeb, and EMD. These numbers were chosen based on the requirements of achieving the convergence value (in the case of EM) and of generating approximately the same noise level for EMD and EMdeb. The reconstructed images can be seen in Fig. 6, where axial slices and transverse profiles are presented. It is apparent that the EMdeb method achieves a comparable, or better, image quality than EMD after a substantially shorter processing time. The processing time per iteration scales as 1:4:5, respectively, for Emdeb, EM, and EMD, which gives 5:1:69 for the whole reconstruction (or 6:1:69 if the preliminary EM reconstruction is included for EMdeb). B. Double Compton Scattering Model The exact form of the kernel was found by fitting an expression, consisting of the following three parts, to the distribution describing the detector resoluin Fig. 2: the central peak tion, the “cross” with its relative contribution due

In summary, the peak is described by a double Gaussian function (in two orthogonal directions and ), the double Compton scattering term by an exponentially decaying expression in one direction (modified by a second-degree polynomial), and a Gaussian in the other direction, and the background is described by a modified exponentially decaying function. Here and are the detector local coordinates; therefore this description is valid only for the data collected for each rotation angle of the detectors rather than for a combined sample. The expressions above were therefore implemented for each rotation angle subsample separately, i.e., a separate point spread function was generated for each angle “on the fly,” and the image was updated after processing each subsample (similar to the ordered subset EM method [26] with the number of ). It should be noted that local and coordisubsets nates are practically equivalent and the expression parameters can be taken to be equivalent for both directions. The depth dependence of the point spread function (PSF) was taken to be 1.4 mm. Gaussian with The reconstruction results for the point-like phantom can be seen in Fig. 7. If only the peak of the distribution is included in the resolution kernel, the reconstructed image shows visible artifacts. However, if the whole distribution is modeled, the image is very much cleaner. Reconstruction results from the mouse bone scan are shown in Fig. 8 for EMD with the Gaussian kernel and EMD with the double Compton scattering model. The images were revoxels with a size constructed at a FOV of mm . The collected statistics included 5.5 million of list-mode events and the number of image updates was chosen to give a similar noise level for both models as was found from

ANTICH et al.:IMAGE RESOLUTION IMPROVEMENT IN A SMALL ANIMAL PET SYSTEM

Fig. 7. Reconstructed XY -view of the point-like source: (upper panel) with Gaussian kernel with  = 1.2 mm; (lower panel) kernel is chosen to fit the distribution in Fig. 2. 48 updates with voxel size of 0:25 mm . X is the transaxial direction, and Y is the axial direction.

reconstructions of the FDG line phantom (200 and 360 for EMD with the Gaussian kernel and scattering model, respectively). One can see that the EMD model including scattering gives a visually better image with the improvements being a higher contrast, lower background noise level, and reduced cross-like artifact around the bladder. The processing time per image update was approximately the same for both methods due to the fact that only a subset of the data was used for each image update in the case of EMD with scattering. IV. CONCLUSION The presented results show that the list-mode expectation maximization with deconvolution (EMD) method improves the quality of positron emission tomography (PET) images when proper account is taken of the finite resolution effects, through incorporation of the extended system model into the expectation maximization (EM) procedure. The proposed EM deblurring (EMdeb) procedure can be used to improve image quality after “conventional” EM reconstruction on a much shorter time scale than EMD, making the image resolution recovery techniques more attractive for practical applications.

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Fig. 8. Maximum intensity coronal (left) and sagittal (right) projections from the F mouse bone scan. Upper images: EMD reconstruction with the Gaussian kernel with  = 1.2 mm. Lower: Kernel is chosen to fit the distribution in Fig. 2. A nonlinear gray scale was used to reveal the detail.

The use of a shift-invariant approach to treat what are essentially space-variant resolution effects has proven to be quite successful for our small animal PET imager. ACKNOWLEDGMENT The authors would like to thank Drs. G. Arbique, A. Constantinescu, R. Mason, R. McColl, and O. Oz for valuable help. REFERENCES [1] C. J. Marriott et al., “High-resolution PET imaging and quantitation of pharmaceutical biodistributions in a small animal using avalanche photodiode detectors,” J. Nucl. Med., vol. 35, pp. 1390–1396, 1994. [2] N. M. Spyrou, J. M. Sharaf, and S. Rajeswaran, “Developments in tomographic methods for biological trace element research,” Biol. Trace Elem. Res., vol. 43–45, pp. 55–63, 1994. [3] S. P. Hume and T. Jones, “Positron emission tomography (PET) methodology for small animals and its application in radiopharmaceutical preclinical investigation,” Nucl. Med. Biol., vol. 25, pp. 729–732, 1998. [4] P. Antich, A. Constantinescu, R. Mason, R. McColl, O. K. Oz, P. Kulkarni, S. Seliounine, N. Slavine, and E. Tsyganov, “FDG imaging of lung metastases in Copenhagen rats on a small animal PET system with position-encoding fiberoptic detectors,” J. Nucl. Med., vol. 43, no. 216 Suppl. S, p. 60, May 2002. [5] J. Seidel, W. R. Gandler, and M. V. Green, “A very high resolution single-slice animal PET scanner based on direct detection of coincidence line endpoints,” J. Nucl. Med., vol. 35, no. Suppl. S, p. 40, May 1994. [6] B. McKee, A. Dickson, and D. Howse, “Performance of QPET, a highresolution 3D PET imaging system for small volumes,” IEEE Trans. Med. Imag., vol. 13, no. 1, pp. 176–185, Mar. 1994.

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[7] P. Bloomfield et al., “The design and physical characteristics of a small positron emission tomograph,” Phys. Med. Biol., vol. 40, pp. 1105–1126, 1995. [8] R. Lecomte et al., “Initial results from the Sherbrooke avalanche photodiode positron tomograph,” IEEE Trans. Nucl. Sci., vol. 43, no. 3, pp. 1952–1957, Jun. 1996. [9] S. R. Cherry et al., “MicroPET: A high resolution PET scanner for imaging small animals,” IEEE Trans. Nucl. Sci., vol. 44, no. 3, pp. 1161–1166, Jun. 1997. [10] A. P. Jeavons, R. A. Chandler, and C. A. R. Dettmar, “A 3D HIDAC-PET camera with sub-millimeter resolution for imaging small animals,” IEEE Trans. Nucl. Sci., vol. 46, no. 3, pp. 468–473, Jun. 1999. [11] N. C. Rouze and G. D. Hutchins, “Design and characterization of IndyPET-II: A high-resolution, high-sensitivity dedicated research scanner,” IEEE Trans. Nucl. Sci., vol. 50, no. 5, pp. 1491–1497, Oct. 2003. [12] E. U. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag., vol. 13, no. 4, pp. 687–701, Dec. 1994. [13] J. Qi, R. M. Leahy, S. R. Cherry, A. Chatziioannou, and T. H. Farquhar, “High-resolution 3D Bayesian image reconstruction using the microPET small-animal scanner,” Phys. Med. Biol., vol. 43, pp. 1001–1013, 1998. [14] R. Yao et al., “Performance characteristics of the 3-D OSEM algorithm in the reconstruction of small animal PET images,” IEEE Trans. Med. Imag., vol. 19, no. 8, pp. 798–804, Aug. 2000. [15] H. H. Barrett, T. White, and L. C. Parra, “List-mode likelihood,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 14, pp. 2914–2923, 1997. [16] L. Parra and H. H. Barrett, “List-mode likelihood: EM algorithm and image quality estimation demonstrated on 2-D PET,” IEEE Trans. Med. Imag., vol. 17, no. 2, pp. 228–235, Apr. 1998.

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[17] A. J. Reader, K. Erlandsson, M. A. Flower, and R. J. Ott, “Fast accurate iterative reconstruction for low-statistics positron volume imaging,” Phys. Med. Biol., vol. 43, pp. 835–846, 1998. [18] R. Huesman et al., “List-mode maximum likelihood reconstruction applied to positron emission mammography (PEM) with irregular sampling,” IEEE Trans. Med. Imag., vol. 19, pp. 532–537, 2000. [19] C. Byrne, “Likelihood maximization for list-mode emission tomographic image reconstruction,” IEEE Trans. Med. Imag., vol. 20, no. 10, pp. 1084–1092, Oct. 2001. [20] A. J. Reader et al., “One-pass list-mode EM algorithm for high resolution 3D PET image reconstruction into large arrays,” IEEE Trans. Nucl. Sci., vol. 49, no. , pp. 693–699, 2002. [21] A. J. Reader, P. J. Julyan, H. Williams, D. L. Hastings, and J. Zweit, “EM algorithm system modeling by image-space techniques for PET reconstruction,” IEEE Trans. Nucl. Sci., vol. 50, no. , pp. 1392–1397, 2003. [22] E. Tsyganov et al., “Performance of the small animal PET imager of the UT SWMC at Dallas,” IEEE Trans. Nucl. Sci., submitted for publication. [23] L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag., vol. MI-1, pp. 113–122, Oct. 1982. [24] M. Jiang, G. Wang, M. W. Skinner, J. T. Rubinstein, and M. W. Vannier, “Blind deblurring of spiral CT images,” IEEE Trans. Med. Imag., vol. 22, no. 7, pp. 837–845, Jul. 2003. [25] A. J. Reader, K. Erlandsson, M. A. Flower, and R. J. Ott, “Adaptive correction of scatter and random events for 3-D backprojected PET data,” IEEE Trans. Nucl. Sci., vol. 48, no. 4, pp. 1350–1356, Aug. 2001. [26] H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imag., vol. 13, no. 4, Dec. 1994.