The Radon transform fits badly Single Photon Emission Tomography (SPECT). However, Thin Holes Collimator (THC) and Radon model are widely used. The CACAO project has been proposed to enhance the quality of SPECT images. CACAO is a short hand notation for computer aided collimation tomography. The main idea of this project is to use collimators with much larger holes to increase the sensitivity, and slightly longer holes to increase the spatial resolution. The acquisition sequence includes a translation. The Radon projection is replaced by a 2D sum. A dedicated reconstruction algorithm has been developed. If the physical advantage of the project in terms of sensitivity and spatial resolution is generally admitted, a question remains unanswered: Would the ill-posedness of the inverse problem ruin the quality of the reconstructed images? In this article, a representation of the 2D direct problem matrix is derived. This allows us to compare the two inverse problems (CACAO versus THC). The condition number was used for this comparison. We studied the variation of these condition numbers with several parameters. For a proper set of parameters, the CACAO inverse problem may appear easier to solve and more accurately than the THC one.
References
[1]
Anger, H. (1957) A New Instrument for Mapping the Distribution of Radio-Active Isotopes. Biology and Medicine Quaterly Report UCRL 3653, 38.
[2]
Radon, J. (1917) Uber die bestimmung von funktionen durch ihre integralwerte langs gewisser mannigfaltigkeiten. Berichte uber die Verhandlungen Gesellshaft der Wissenschaften zu Leipzig. Journal of Mathematical Physics, 69, 262-277.
[3]
Meng, L.J. and Clinthorne, N.H. (2004) A Modified Uniform Cramer-Rao Bound for Multiple Pinhole Aperture Design. IEEE Transactions on Medical Imaging, 23, 896-902. https://doi.org/10.1109/TMI.2004.828356
[4]
Zhang, B. and Zeng, G.L. (2007) High-Sensitivity Spect Imaging Using Large Collimator Holes and Geometric Blurring Compensation. IEEE Nuclear Science Symposium Conference Record, 6, 4279-4284.
[5]
Lodge, M.A., Webb, S., Flower, M.A., et al. (1996) A Prototype Rotating Slat Collimator for Spect. IEEE Transactions on Medical Imaging, 15, 500-511.
[6]
Jeanguillaume, C., Quartuccio, M., Begot, S., et al. (2002) Emission Tomography with a Large-Hole Collimator (Cacao): A Possible New Way to Improve the Radionuclide Imaging. Journal of Computer Assisted Tomography, 26, 1057-1062. https://doi.org/10.1097/00004728-200211000-00036
[7]
Jeanguillaume, C., Quartuccio, M., Begot, S., et al. (1996) Gamma camera a collimation assistee par ordinateur (Cacao). Premiere partie: principes de base et methode de reconstruction. RBM, 18, 198-206. https://doi.org/10.1016/S0222-0776(97)89224-3
[8]
Jeanguillaume, C., Douiri, A., Quartuccio, M., et al. (2001) Cacao a Collimation Means Well Suited for Pixellated Detectors. IEEE Nuclear Science Symposium Conference Record, 4, 2291-2294.
[9]
Jeanguillaume, C., Quartuccio, M., Begot, S., et al. (1996) Gamma camera a collimation assistee par ordinateur (Cacao). Deuxieme partie: une nouvelle approche dans l’etude des performances en tomographie d’emission. RBM, 18, 207-215. https://doi.org/10.1016/S0222-0776(97)89225-5
[10]
Strang, G. (2009) Introduction to Linear Algebra. 4th Edition, Wellesley-Cambridge Press, Wellesley, MA.
[11]
Barrett, H. and Myers, K. (2003) Foundations of Image Science. John Wiley & Sons, Hoboken New Jersey.
[12]
Seo, Y., Wong, K.H., Sun, M., Franc, B.L., Hawkins, R.A. and Hasegawa, B.H. (2005) Correction of Photon Attenuation and Collimator Response for a Body-Contouring Spect/Ct Imaging System. The Journal of Nuclear Medicine, 46, 868-877.
[13]
Bertero, M. and Boccacci, P. (1998) Introduction to Inverse Problems in Imaging. IOP Publishing Ltd., Philadelphia, PA. https://doi.org/10.1887/0750304359
[14]
Sun, X., Ma, T. and Jin, Y. (2004) Svd Reconstruction Algorithm in 3d Spect Imaging. Nuclear Science Symposium and Medical Imaging Conference IEEE Conference Record, 6, 3527-3530.
[15]
Usman, M., Hero, A.O., Fessler, J.A. and Rogers, W.L. (1993) Bias-Variance Tradeoffs Analysis Using Univorm Cr Bound for a Spect System. IEEE Nuclear Science Symposium and Medical Imaging Conference Report, 1, 1463-1038.
[16]
Fessler, J.A. and Booth, S.D. (1996) Conjugate-Gradient Preconditioning Methods for Shift-Variant Pet Image Reconstruction. IEEE Transactions on Image Processing, 8, 688-699. https://doi.org/10.1109/83.760336
[17]
La, V. and Grangeat, P. (1998) Minimal Residual Cone-Beam Reconstruction with Attenuation Correction in Spect. Physics in Medicine & Biology, 43, 715-727. https://doi.org/10.1088/0031-9155/43/4/002
[18]
Zeng, G.L. and Gagnon, D. (2004) Image Reconstruction Algorithm for a Spect System with a Convergent Rotating Slat Collimator. IEEE Transactions on Nuclear Science, 51, 142-148. https://doi.org/10.1109/TNS.2003.823042
[19]
Borwein, J.M. and Sun, W. (1997) The Stability Analysis of Dynamic Spect Systems. Numerische Mathematik, 77, 283. https://doi.org/10.1007/s002110050287
[20]
Jorgensen, A.N. and Zeng, G.L. (2008) Svd-Based Evaluation of Multiplexing in Multipinhole Spect Systems. International Journal of Biomedical Imaging, 2008, Article ID: 769195. https://doi.org/10.1155/2008/769195
[21]
Tikhonov, A.N. (1963) Solution of Incorrectly Formulated Problems and the Regularization Method. Soviet Mathematics Doklady, 4, 1035-1038.
[22]
Phillips, D.L. (1962) A Technique for the Numerical Solution of Certain Integral Equations of the First Kind. ACM, 9, 84-97. https://doi.org/10.1145/321105.321114
[23]
Hansen, C. (1987) The Truncated Svd as a Method for Regularization. BIT Numerical Mathematics, 27, 534-553. https://doi.org/10.1007/BF01937276
[24]
Barrett, H.H., Yao, J., Rolland, P. and Myers, K.J. (1993) Model Observers for Assessment of Image Quality. Proceedings of the National Academy of Sciences of the United States of America, 90, 9758-9765. https://doi.org/10.1073/pnas.90.21.9758
[25]
Fessler, J.A. (1996) Mean and Variance of Implicitly Defined Biased Estimators (such as Peanlized Maximum Likelihood): Applications to Tomography. IEEE Transactions on Image Processing, 5, 493-506. https://doi.org/10.1109/83.491322
[26]
Qi, J. (2000) Resolution and Noise Properties of Map Reconstruction for Fully 3d Pet. IEEE Transactions on Medical Imaging, 19, 493-506. https://doi.org/10.1109/42.870259
[27]
Cao, N., Huesman, R.H., Moses, W.W. and Qi, J. (2010) Detection Performance Analysis for Time-of-Flight Pet. Physics in Medicine and Biology, 55, 6931-6949. https://doi.org/10.1088/0031-9155/55/22/021
[28]
Soares, E.J., Byrne, C.L., Glick, S.J., Appledorn, C.R. and King, M.A. (1991) Implementation and Evaluation of an Analytical Solution to the Photon Attenuation and Nonstationary Resolution Reconstruction Problem in Spect. IEEE Nuclear Science Symposium Conference Record, 2-9 November 1991, 1789-1796. https://doi.org/10.1109/NSSMIC.1991.259222
[29]
Beekman, F.J., Eijkman, E.G.J., Viergever, M.A., Borm, G.F. and Slijpen, E.T.P. (1993) Object Shape Dependent Psf Model for Spect Imaging. IEEE Transactions on Nuclear Science, 40, 31-39.
[30]
Elfakhri, G., Buvat, I., Benali, H., Todd-Prokropek, A. and DiPaola, R. (2000) Relative Impact of Scatter, Collimator Response, Attenuation, and Finite Spatial Resolution Corrections in Cardiac Spect. The Journal of Nuclear Medicine, 41, 1400-1408.
[31]
Smith, M.F., Floyd, C.E. and Jaszczak, R.J. (1992) Reconstruction of Spect Images Using Generalized Matrix Inverses. IEEE Transactions on Medical Imaging, 11, 165-175. https://doi.org/10.1109/42.141640
[32]
He, Z., Li, W., Knoll, G.F., Wehe, D.K. and Stahle, C.M. (2000) Measurement of Material Univormity Using 3-d Position Sensitive Cdznte Gamma-Ray Spectrometers. Nuclear Instruments and Methods in Physics Research A, 441, 459-467. https://doi.org/10.1016/S0168-9002(99)00860-8
[33]
Salcin, E., Barrett, H.H., Barber, H.B., Takeda, S., Watanabe, S., Takahashi, T. and Furenlid, L.R. (2014) Fisher Information Analysis of Depth-of-Interaction Estimation in Double-Sided Strip Detectors. IEEE Transactions on Nuclear Science, 61, 1243-1251. https://doi.org/10.1109/TNS.2014.2317454
[34]
Montemont, G., Lux, S., Monnet, O., Stanchina, S. and Verger, L. (2012) Evaluation of a Czt Gamma-Ray Detection Module Concept for Spect. IEEE Nuclear Science Symposium and Medical Imaging Congress Conference Report, 27 October-3 November 2012, 4091-4097. https://doi.org/10.1109/NSSMIC.2012.6551935
