有限角度下X射线成像重建问题的研究
|
摘要
X射线成像技术在医学诊断和工业无损检测中取得了革命性的进展和突破。然而在实际应用中,很多情况下并不能采集到完全角度下的投影数据。例如,在实际X射线扫描物体过程中,由于X射线剂量的限制或成像系统设计的限制、或者X射线穿过物体高密度区域时都可能导致一部分投影数据的丢失。在牙科、外科、胸部以及乳腺等X射线成像过程中,投影数据只在有限角度范围内才能被采集到。利用有限角度下的投影数据进行图像重建被称为X射线成像有限角度重建。在有限角度重建中,运用解析算法进行不完全投影数据的重建将会导致伪像的产生。本文主要研究待检测目标在有限角度范围内的投影数据迭代重建。论文的主要工作如下：
(1)本文在结合总变分方法优点的基础上,把一种基于乘性正则化的最小化目标函数引入到有限角度重建中,该目标函数将TV (Total Variation)函数作为其一个乘法因子,并利用一般可微函数的共轭梯度法对目标函数进行迭代求解。该算法能够在迭代过程中自适应地调整参数,克服了最小化函数的权值参数不易确定的缺陷。实验结果表明该方法能够较好地应用于扇束CT (Computed Tomography)的有限角度重建。
(2)本文提出了基于非二次惩罚函数的双约束目标函数方法来解决CT的有限角度重建。在常规的CT图像重建中,通常采用带有二次惩罚函数的正则化方法。然而,这种方法会使重建图像中拥有比较重要信息的边缘区域出现模糊趋势。因此,为了重建出具有更好边缘的图像,我们利用交替算法来求解基于非二次惩罚函数的双约束目标函数。通过验证该交替迭代算法的收敛性保证了重建的有效性和鲁棒性。
(3)本文在结合非二次正则化的基础上,提出一种基于总变分的目标函数方法来处理CT有限角度重建。该方法具有非二次正则化优点的同时,能够在迭代过程中自适应的调整参数,从而能够重建出良好边缘的图像。实验结果表明该方法能够很好地处理锥束CT的有限角度重建。(4)跟CT成像模式不同,数字合成X射线体层成像技术在最近几年的研究已经取得了很大进展。然而,由于这些重建算法需要大量的计算时间,从而制约了数字合成X射线体层成像技术在实际中的运用。为此,本文引入一种新的重建算法—基于正则化的自适应小波-伽辽金重建算法。该算法融合了伽辽金方法的计算简洁性和小波内在的多尺度特性,更好地适应了待重建图像的求解。仿真实验结果表明,与ART重建算法相比,自适应小波-伽辽金重建算法能在不损害重建质量前提下加快收敛,从而大大地节省了计算时间。
X-ray imaging technique has made a revolutionary impact on medical diagnosis and industrial non-destructive testing. It is not always possible to acquire projection data through a complete angular range in CT and tomosynthesis in real applications. Some examples would be X-ray dose limitations, or imaging system design constraints when imaging a moving object, or X-rays being obstructed when passing through high-density region of objects, any of which could result in loss of some projections. When projection data are only available in a limited angular range, as occurs in a number of applications in dental radiology, surgical imaging, thoracic imaging, mammography, etc. The conventional and most commonly used method for reconstruction from tomographic projections is the analytical reconstruction technique which is not so adaptable to incomplete projection data and results in poor reconstructions with severe artifacts in limited angle cases. One approach of overcoming the insufficient projections is to reconstruct the object by making use of iterative method. This dissertation is focused on image reconstructions when projection data is insufficient in limited angle range. Our results can be summarized into the following:
(1) Considering the advantages of the total variation(TV) method, the paper introduced an iterative image reconstruction algorithm based on multiplicative regularization method.The method obtains the advantages of the TV method by introducing the TV as a multiplicative factor in the cost function, and it can also self-adaptively adjust the regularization parameter during the iterative process. Experimental results show that the proposed algorithm works effectively.
(2) We propose the double constraint method to overcome data insufficiency based on non-quadratic penalties method. A classical method for solving limited angle tomogramphy is regularization with a quadratic penalty function in CT. However, this regularization method has a tendency of smoothing those sharp edges in solutions that often carry important information. The proposed method provides us a robust and efficient reconstruction by showing the convergence of the alternating minimization method. The results demonstrate that the reconstruction strategy has a comparable performance.
(3) Considering the adavantages of the non-quadratic penalties method, a new TV objective function minimization is proposed for treatment of the limited angle tomography in this paper. The objective function minimization provided us a robust and efficient reconstruction without artificial parameters in iterative processes, by includeing the advantage of the non-quadratic penalties method. The results demonstrate that the reconstruction strategy can give good edge-preserving reconstructions.
(4) Some research groups have been studing tomosynthesis reconstruction algorithm as well as its applications in recent years. The long calculation requirements of tomosynthesis reconstruction algorithm, however, limit its commercialization. In this paper, an adaptive wavelet galerkin method is introduced to reconstruct images in tomosynthesis. The method combined the numerical simplicity of Galerkin method with the inherent scale multiplicity characteristic of wavelet which is more adaptable to the resolution of the reconstructed image. Compared to the ART reconstruction, the results demonstrate that the reconstruction strategy has comparable performance with the improvement of the convergence speed and the reduction of computational time.
(1)本文在结合总变分方法优点的基础上,把一种基于乘性正则化的最小化目标函数引入到有限角度重建中,该目标函数将TV (Total Variation)函数作为其一个乘法因子,并利用一般可微函数的共轭梯度法对目标函数进行迭代求解。该算法能够在迭代过程中自适应地调整参数,克服了最小化函数的权值参数不易确定的缺陷。实验结果表明该方法能够较好地应用于扇束CT (Computed Tomography)的有限角度重建。
(2)本文提出了基于非二次惩罚函数的双约束目标函数方法来解决CT的有限角度重建。在常规的CT图像重建中,通常采用带有二次惩罚函数的正则化方法。然而,这种方法会使重建图像中拥有比较重要信息的边缘区域出现模糊趋势。因此,为了重建出具有更好边缘的图像,我们利用交替算法来求解基于非二次惩罚函数的双约束目标函数。通过验证该交替迭代算法的收敛性保证了重建的有效性和鲁棒性。
(3)本文在结合非二次正则化的基础上,提出一种基于总变分的目标函数方法来处理CT有限角度重建。该方法具有非二次正则化优点的同时,能够在迭代过程中自适应的调整参数,从而能够重建出良好边缘的图像。实验结果表明该方法能够很好地处理锥束CT的有限角度重建。(4)跟CT成像模式不同,数字合成X射线体层成像技术在最近几年的研究已经取得了很大进展。然而,由于这些重建算法需要大量的计算时间,从而制约了数字合成X射线体层成像技术在实际中的运用。为此,本文引入一种新的重建算法—基于正则化的自适应小波-伽辽金重建算法。该算法融合了伽辽金方法的计算简洁性和小波内在的多尺度特性,更好地适应了待重建图像的求解。仿真实验结果表明,与ART重建算法相比,自适应小波-伽辽金重建算法能在不损害重建质量前提下加快收敛,从而大大地节省了计算时间。
X-ray imaging technique has made a revolutionary impact on medical diagnosis and industrial non-destructive testing. It is not always possible to acquire projection data through a complete angular range in CT and tomosynthesis in real applications. Some examples would be X-ray dose limitations, or imaging system design constraints when imaging a moving object, or X-rays being obstructed when passing through high-density region of objects, any of which could result in loss of some projections. When projection data are only available in a limited angular range, as occurs in a number of applications in dental radiology, surgical imaging, thoracic imaging, mammography, etc. The conventional and most commonly used method for reconstruction from tomographic projections is the analytical reconstruction technique which is not so adaptable to incomplete projection data and results in poor reconstructions with severe artifacts in limited angle cases. One approach of overcoming the insufficient projections is to reconstruct the object by making use of iterative method. This dissertation is focused on image reconstructions when projection data is insufficient in limited angle range. Our results can be summarized into the following:
(1) Considering the advantages of the total variation(TV) method, the paper introduced an iterative image reconstruction algorithm based on multiplicative regularization method.The method obtains the advantages of the TV method by introducing the TV as a multiplicative factor in the cost function, and it can also self-adaptively adjust the regularization parameter during the iterative process. Experimental results show that the proposed algorithm works effectively.
(2) We propose the double constraint method to overcome data insufficiency based on non-quadratic penalties method. A classical method for solving limited angle tomogramphy is regularization with a quadratic penalty function in CT. However, this regularization method has a tendency of smoothing those sharp edges in solutions that often carry important information. The proposed method provides us a robust and efficient reconstruction by showing the convergence of the alternating minimization method. The results demonstrate that the reconstruction strategy has a comparable performance.
(3) Considering the adavantages of the non-quadratic penalties method, a new TV objective function minimization is proposed for treatment of the limited angle tomography in this paper. The objective function minimization provided us a robust and efficient reconstruction without artificial parameters in iterative processes, by includeing the advantage of the non-quadratic penalties method. The results demonstrate that the reconstruction strategy can give good edge-preserving reconstructions.
(4) Some research groups have been studing tomosynthesis reconstruction algorithm as well as its applications in recent years. The long calculation requirements of tomosynthesis reconstruction algorithm, however, limit its commercialization. In this paper, an adaptive wavelet galerkin method is introduced to reconstruct images in tomosynthesis. The method combined the numerical simplicity of Galerkin method with the inherent scale multiplicity characteristic of wavelet which is more adaptable to the resolution of the reconstructed image. Compared to the ART reconstruction, the results demonstrate that the reconstruction strategy has comparable performance with the improvement of the convergence speed and the reduction of computational time.
引文
[1]Jiang Hsich.计算机断层成像技术原理、设计、伪像和进展[M].北京：科学出版社,2006.
[2]Feldkamp L A, Davis L C, Kress J W. Practical cone-beam algorithm[J]. J. Opt. Soc. AM. A,1984,1(6):612-619.
[3]Novikov R G. An inversion formula for the attenuated X-ray transformation [J]. Ark. Mat. 2002,40:145-167.
[4]Faridani A, Finch D V, Ritman E L, and Smith K T. Local tomography II[J]. SIAM J. Appl. Math.,1997,57(4):1095-1127.
[5]Farrokhi F R, Liu K J R, Berenstein C A, Walnut D. Localized wavelet based computerized tomography[C]. International Conference on Image Processing,1995,2:445-448.
[6]Farrokhi F R, Liu K J R, Berenstein C A, Walnut D. Wavelet-based multiresolution local tomography [J]. IEEE Trans. on Imag. Proc.,1997,6(10):1412-1430.
[7]Bhatia M, Karl W C, Willsky A S. A wavelet-based method for multiscale tomographic reconstruction [J]. IEEE Trans. on Med. Imag.,1996,15(1):92-101.
[8]Bottema M, Moran B, Suvorova S. An application of wavelets in tomography [J]. Digital Signal Processing,1998,8:244-254.
[9]Zhao S Y, Wang G. Feldkamp-type cone-beam tomography in the wavelet framework[J]. IEEE Trans. on Med. Imag.,2000,19(9):922-929.
[10]Clackdoyle R, Guo J Y, Noo F. Quantitative reconstruction from truncated projections in classical tomography [J]. IEEE Trans. on Nucl. Sci.,2004,51(5):2570-2578.
[11]Nassi M, Brody W R, Medoff B P, Macovski A. Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography[J]. IEEE Trans. on Biomed. Eng.,1982,29(5):333-341.
[12]Snyder D L, Sullivan J A 0, Murphy R J et al. Maximum likelihood methods for recons-tructing an image in a region of interest for transmission tomography[C]. Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2003,1:312-315.
[13]Tam K C, Perez M V. Tomographical imaging with limited-angle input [J]. J. Opt. Soc. Am.,1981,71 (5):582-592.
[14]Gerchberg R W. Super-resolution through error energy reduction [J]. J. Modern Optics, 1974,21 (9):709-720.
[15]Papoulis A. A new method in image restoration[R]. Joint Services Technical Activity Report 39,1973.
[16]Papoulis A. A new algorithm in spectral analysis and band-limited extrapolation[J]. IEEE Trans. on Circuits Syst.,1975, CAS-22 (9):735-742.
[17]Salomon B G, Ur H. Accelerated iterative band-limited extrapolation algorithms[J]. IEEE Signal Proc. Lett.,2004,11(11):871-874.
[18]Qi J, Huesman R H. Theoretical study of penalized likelihood image reconstruction for region of interest quantification [J]. IEEE Trans. on Med. Imag.,2006,25(5): 640-648.
[19]Hamelin B, Goussard Y, Dussault J P. Penalized likelihood region of interest CT reconst-ruction by local object supersampling[C].29th IEEE EMBS.,2007,739-742.
[20]Inouye T. Image reconstruction with limited angle projection data[J]. IEEE Trans. on Nucl. Sci.,1979,26(2):2665-2669.
[21]Candes E J, Romberg J K. Practical signal recovery from random projections[C]. Computa-tional Imaging III, Proc. SPIE,2005,5674:56741-56749.
[22]Rantala M, Vanska S, Jarvenpaa S, et al. Wavelet-based reconstruction for limited-angle X-ray tomography [J]. IEEE Trans. on Med. Imag.,2006,25(2):210-217.
[23]Natterer F. The mathematics of computerized tomography[M]. John Wiley & Sons Ltd, Chichester,1986.
[24]Natterer F, Wubbeling F. Mathematical methods in image reconstruction [M]. SIAM Mono-graphs on Mathematical Modeling and Computation, Philadelphia, PA,2001.
[25]Rangayyan R, Dhawan A P, Gordon R. Algorithms for limited-view computed tomography [J]. Appl. Opt.,1985,24(23):4000-4012.
[26]Lan C Q, Xiong W. An iterative method of ultrasonic reflection mode tomography[J]. IEEE Trans. on Med. Imag.,1994,13(2):419-425.
[27]Byrne C, Scares E, Pan T S et al. Accelerating the EM algorithm using rescaled block iterative methods[C]. IEEE Nuclear Science Symposium,1996,3:1752-1756.
[28]Verhoeven D. Limited-data computed tomography algorithms for the physical sciences [J]. Appl. Opt.,1993,32(20):3736-3754.
[29]Dhawan A P, Rangayyan R M, Gordon R. Image restoration by wiener deconvolution in limited-view computed tomography [J]. Appl. Opt.,1985,24(23):4013-4020.
[30]Andersen H A. Algebraic reconstruction in CT from limited views[J]. IEEE Trans. on Med. Imag.,1989,8(1):50-55.
[31]Byrne C. Block-iterative interior point optimization methods for image reconstruction from limited data[J]. Inverse Problems,2000,16:1405-1419.
[32]Popa C, Zdunek R. Kaczmarz extended algorithm for tomographic image reconstruction from limited-data[J]. Math. Comput. Simul.,2004,65(6):579-598.
[33]Hstau Y L. A gradually unmasking method for limited data tomography[C]. IEEE Interna-tional Symposium on 4th Biomedical Imaging,2007, pp:820-823.
[34]Hanson K M, Wecksung G W.. Bayesian approach to limited-angle reconstruction in computed tomography [J]. J. Opt. Soc. Am.,1983,73(11):1501-1509.
[35]Sauer K, James S, Klifa K. Bayesian estimation of 3-D objects from few radiographs[J]. IEEE Trans. on Nucl. Sci.,1994,41(5):1780-1790.
[36]Siltanen S, Kolehmainen V, et al. Statistical inversion for medical x-ray tomography With few radiographs:I. General theory [J]. Phys. Med. Biol.,2003,48(10):1437-1463.
[37]Kolehmainen V, Siltanen S, et al. Statistical inversion for medical X-ray tomography With few radiographs:II. Application to dental radiology [J]. Phys. Med. Biol.,2003, 48(10):1465-1490.
[38]Milanfar P, Karl W C, Willsky A S. A moment-based variational approach to tomographic reconstruction [J]. IEEE Trans. on Imag. Proc.,1996,5(3):459-470.
[39]Lassas M, Siltanen S. Can one use total variation prior for edge-preserving bayesian inversion [J]. Inverse Problems,2004,20:1537-1563.
[40]Kybic J, Blu T, and Unser M. Variational approach to tomographic reconstruction[C]. Proc. of SPIE on Medical Imaging,2001,4322:30-39.
[41]Dempster A P, Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm[J]. J. Roy. Stat. Soc. Ser. B,1977,39(1):1-38.
[42]Shepn L A, Vardi Y. Maximum likelihood reconstruction for emission tomography [J]. IEEE Trans. on Med. Imag.,1982, MI-1 (2):113-122.
[43]Green P J. Bayesian reconstruction from emission tomography data using a modified EM algorithm[J]. IEEE Trans. on Med. Imag.,1990,9(1):84-93.
[44]Zhang J, Anastasio M A, Pan X, et al. Weighted EM reconstruction algorithms for thermoa-coustic tomography [J]. IEEE Trans. on Med. Imag.,2005,24(6):817-820.
[45]Chen C M, Lee S Y. On parallelizing the EM agorithm for PET image reconstruction[J]. IEEE Trans. on Parallel and Distributed Systems,1994,5(8):860-873.
[46]Figueiredo M A, Nowak R D. An EM algorithm for wavelet-based image restoration[J]. IEEE Trans. on Imag. Proc.,2003,12(8):906-916.
[47]Louis A K. Incomplete data problems in X-ray computerized tomography I. singular value decomposition of the limited angle transform[J]. Numer. Math,1986,48:251-262.
[48]Louis A K, Rieder A. Incomplete data problems in X-ray computerized tomography II. truncated projection and region of interest tomography[J]. Numer. Math,1989, 56:371-383.
[49]Quinto E T. Singularities of the X-ray transform and limited data tomography in R2and R3 [J].SIAM J. Math. Anal.,1993,24(5):1215-1225.
[50]Candes E, Donoho D L. Ridgelets:a key to higher-dimensional intermittency[J]. Phil. Trans. R. Soc. Lond. A.,1999,357(1760):2495-2509.
[51]Meyer Y. Wavelets and operators[M]. Cambridge, U.K.:Cambridge Univ. Press,1992.
[52]Mallat S G. A theory for multiresolution signal decomposition:the wavelet representat-ion [J]. IEEE Trans. on Pattern Anal. Mach. Intell.,1989,11(7):674-693.
[53]Abramovich F, Sapatinas T, Silverman B W. Wavelet thresholding via a bayesian approach [J]. J. Roy. Statist. Soc. B,1998,60:725-749.
[54]Delanay A H, Bresler Y. Multiresolution tomographic reconstruction using wavelets [J]. IEEE Trans. on Imag. Proc.,1995,4(6):799-813.
[55]Strahlen K. Local vector tomography by use of wavelets[C]. IEEE Internation conf. on Imag. Proc.,2000,2:617-620.
[56]Olson T, DeStefano J. Wavelet localization of the Radon transforms [J]. IEEE Trans. on signal Proc.,1994,42(8):2055-2067.
[57]Zhao S, Welland G, Wang G. Wavelet sampling and localization schemes for the Radon transform in two dimensions [J]. SIAM J. Appl. Math.,1997,57(6):1749-1762.
[58]Sahiner B, Yagle A E. Limited angle tomography using wavelets[C]. IEEE Conf. on Nucl. Sci. Symposium and Medical Imaging,1993,3:1912-1916.
[59]Frese T, Bouman C A, Sauer K. Adaptive wavelet graph model for bayesian tomographic reconstruction [J]. IEEE Trans. on Imag. Proc.,2002,11 (7):756-770.
[60]Bhatia M, Karl W C, Willsky A S. A wavelet-based method for multiscale tomographic reconstruction [J]. IEEE Trans. on Med. Imag.,1996,15(1):92-101.
[61]Buccigrossi R W, Simoncelli E P. Image compression via joint statistical character-ization in the wavelet domain [J]. IEEE Trans. on Imag. Proc.,1999,8(12):1688-1701.
[62]Nam Y L, Lucier B J. Wavelet methods for inverting the Radon transform with noisy data [J]. IEEE Trans. on Imag. Proc.,2001,10(1):79-94.
[63]Choi H, Baraniuk R G. Multiple wavelet basis image denoising using Besov ball projectio-ns[J]. IEEE Signal Proc. Lett.,2004,11 (9):717-720.
[64]Wang G, Zhang J, Pan G W. Solution of inverse problems in image processing by wavelet expansion [J].IEEE Trans, on Imag. Proc.,1995,4(5):579-593.
[65]Berkner K, Gormish M J, Schwartz E L. Multiscale sharpening and smoothing in Besov spaces with applications to image enhancement [J]. Appl. Computational Harmonic Anal. 2001,11(1):2-31.
[66]Muller P, Vidakovic B. Bayesian inference in wavelet-based models [M]. Springer-Verlag, 1999,141:24-56.
[67]Vidakovic B. Statistical modeling by wavelets[M]. Series Probability and Mathematical Statistics. New York:Wiley,1999.
[68]Grunbaum F A. A study of fourier space methods for limited-angle image reconstruction [J]. Numer. Funct. Anal. Optim.,1980,2(1):31-42.
[69]Tam K C, Perez M V. Limited-angle 3D reconstructions using fourier transform iterations and radon transform iterations[J]. In International Optical Computing Conf.,1980, 142-148.
[70]Shkolnisky Y, Averbuch A.3D fourier based discrete radon transform [J]. Appl. Comput. Harmon. Anal.,2003,15(1):33-69.
[71]Hanson K M, Cunningham G S, Jennings G R, Wolf D R. Tomographic reconstruction based on flexible geometric models[C]. IEEE International Conf. on Imag. Proc.,1994, 2:145-147.
[72]Battle X L, Cunningham G S, Hanson K M.3D tomographic reconstruction using geometrical models[C]. Proceedings of SPIE, Medical Imaging:Image Processing,1997,3034:346-357.
[73]Kadoury S, Cheriet F, Labelle H. Personalized X-ray 3-D reconstruction of the scoliotic spine from hybrid statistical and image based models[J]. IEEE Trans. on Med.Imag. 2009,28 (9):1422-1435.
[74]Feng H, Karl W C, Castanon D A. A curve evolution approach to object-based tomographic reconstruction [J]. IEEE Trans. on Imag. Proc.,2003,12(1):44-57.
[75]Yu D F, Fessler J A. Edge-preserving tomographic reconstruction with non-local regular-ization[J]. IEEE Trans. on Med. Imag.,2002,21(2):159-173.
[76]Kolehmainen V, Lassas M, Siltanen S. Limited data X-ray tomography using nonlinear evolution equations [J]. SIAM J. Sci. Comput.,2008,30(3):1413-1429.
[77]Nguyen C D, Hoppe H W. Amorphous surface growth via a level set approach[J]. Nonlinear Anal.,2007,66:704-722.
[78]Osher S, Fedkiw R P. Level set methods and dynamic implicit surfaces [M]. New York, Sprin-Ger,2003.
[79]Osher S, Santosa F. Level set methods for optimization problems involving geometry and constraints Ⅰ. Frequencies of a two-density inhomogeneous drum[J]. J. Comput. Phys. 2001,171:272-288.
[80]Kenneth M H, Gregory S C, Robert J M. Uncertainties in tomographic reconstructions based on deformable models[C]. Proc. of SPIE in Medical Image,1997,3034:276-286.
[81]Charbonnier P, Blanc L, Barlaud M. An adaptive reconstruction method involving discont-inuities[J]. Proc. of ICASSP, Minneapolis, MN,1993, V:491-494.
[82]Delaney A H, Bresler Y. Globally convergent edge-preserving regularized reconstru-uction:An application to limited-angle tomography[J]. IEEE Trans. on Imag. Proc.,1998, 7(2):204-221.
[83]Charbonnier P, Blanc L. Deterministic edge-preserving regularization in computed imag-ing[J]. IEEE Trans. on Imag. Proc.,1997,6(2):298-311.
[84]Bert W R, Dianne P L. Residual periodograms for choosing regularization parameters for ill-posed problems[J]. Inverse Problems,2008,24:1-30.
[85]Hansen P C. Analysis of discrete ill-posed problems by means of the L-curve[J]. SIAM Rev.,1992,34(4):561-580.
[86]Morozov A V. Methods for solving incorrectly posed problems[M]. New York:Springer, 1984.
[87]Golub G H, Heath M, Wahba G. Generalized cross-validation as a method for choosing a good ridge parameter[J]. Technometrics,1979,21(2):215-223.
[88]Jianli X, Jun Z. An improved model function method for choosing regularization paramete-rs in linear inverse problems[J]. Inverse problems,2002,18:631-643.
[89]Sidky E Y, Kao C M, Pan X C. Accurate image reconstruction from few-views and limited angle data in divergent-beam CT[J]. J.X-Ray Sci. Technol.,2006,14:119-139.
[90]Sidky E Y, Pan X C. Image reconstruction in circular cone-beam computed tomography by constrained total variation minimization [J]. Physics in Medicine and Biology,2008,53: 4777-4807.
[91]Zou Yu, Pan X C. Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT[J]. Physics in Medicine and Biology,2006,49:941-959.
[92]Zou Yu, Pan X C, Sidky E Y. Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan[J]. Physics in Medicine and Biology, 2005,50:13-27.
[93]Xia D, Yu L, Sidky E Y, et al. Noise properties of chord-image reconstruction[J]. IEEE Trans. on Med. Imag.,2007,26(10):1328-1344.
[94]Persson M, Bone D, Elmqvist H. Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography [J]. Phys. Med. Biol.,2001,46:853-866.
[95]Candes E J, Romberg J. The role of sparsity and incoherence for exactly reconstructing a signal from limited measurements [M]. Tech. Rep., Pasadena:California Inst. Technol. 2004.
[96]Rudin L, Oher S, Fatemi E. Nonlinear total variation based noise removal algorithms [J]. Phys.D,1992,60:259-268.
[97]Chan T F, Wong C K. Total variation blind decovolution[J]. IEEE Trans. on Imag. Proc. 1998,7(3):370-375.
[98]Sidky E Y, Kao C M, Pan X C. Effect of the data constraint on few-view, fan-beam CT image reconstruction by TV minimization[C]. IEEE Conferece on Nuclear Science Symposium Record,2006,4:2296-2298.
[99]Velikina J, Leng S, Chen G H. Limited view angle tomographic image reconstruction via total variation minimization[C]. Proc. of SPIE,2007,6510:651020_1-651020_12.
[100]Grant D. Tomosynthesis:a three dimensional radiographic imaging technique[J]. IEEE Trans. on Biomed. Eng.,1972,19(1):20-28.
[101]Stiel G M, Stiel L S G, Klotz E, Nienaber C A. Digital flashing tomosynthesis:a promis-ing technique for angiocardiographic screening[J]. IEEE Trans. on Med. Imag., 1993,12(2):314-321.
[102]Chakraborty D P, Yester M V, Barnes G T, Lakshminarayanan A V. Self-masking subtracti-on tomosynthesis [J]. Radiology,1984,150:225-229.
[103]Knutsson H E, Edholm P, Granlund G H, et al. Ectomography-a new radiographic reconstruc-ction method:Ⅰ. theory and error estimates[J]. IEEE Trans. on Biomed. Eng.,1980, 27(11):640-648.
[104]Petersson C U, Edholm P, Granlund G H, et al. Ectomography-a new radiographic reconstr-uction method:Ⅱ. computer simulated experiments [J]. IEEE. Trans. on Biomed. Eng.,1980, 27(11):649-655.
[105]Ghosh R D N, Kruger R A, Yih B, et al. Selective plane removal in limited angle tomogra-phic imaging[J]. Med. Phys.,1985,12(1):65-70.
[106]Kolitsi Z, Panayiotakis G, Pallikarakis N. A method for selective removal of out-of-plane structures in digital tomosynthesis[J]. Med. Phys.,1993,20(1):47-50.
[107]Badea C, Kolitsi Z, Pallikarakis N. A wavelet-based method for removal of out-of-plane structures in digital tomosynthesis [J]. Comput. Med. Imaging and Graphics,1998, 22:309-315.
[108]Dobbins J T, Godfrey D J. Digital X-ray tomosynthesis:current state of the art and clinical potential[J]. Phys. Med. Biol.,2003,48:R65-R106.
[109]Wu T, Moore R H, Rafferty E A, Kopans D B. A comparison of reconstruction algorithms for breast tomosynthesis [J]. Med. Phys.,2004,31 (9):2636-2647.
[110]Zhang Y, Chan H P, Sahiner B, et al. A comparison study of limited-angle cone-beam reconstruction methods for breast tomosynthesis [J]. Med. Phys.,2006,33(10):3781-3795.
[111]Lauritsch G, Haerer W. A theoretical framework for filtered backprojection in tomos-ynthesis[C]. Proc. SPIE.,1998,3338:1127-1137.
[112]Stevens G M, Fahrig R, Pelc N J. Flitered backprojection for modifying the impulse response of circular tomosynthesis [J]. Med. Phys.,2001,28(3):372-380.
[113]Matsuo H, Iwata A, Horiba I, Suzumura N. Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering[J]. IEEE Trans. on Med. Imag.,1993, 12(2):307-313.
[114]Warp R J, Godfrey D G, Lynch J A. Application of matrix inverse tomosynthesis[C]. Proc. SPIE,2000,3977:376-383.
[115]Godfrey D J, Warp R J, Dobbins J T. Optimization of matrix inverse tomosynthesis [C]. Proc. SPIE,2001,4320:696-704.
[116]Orman J, Mertelmeier T, Haerer W. Adaptation of image quality using various filter setups in the backprojection approach for digital breast tomosynythesis[J]. TWDM LNCS,2006,4046:175-182.
[117]Claus B E H, Eberhard J W, Schmitz A, et al. Generalized filtered back-projection reconstruction in breast tomosynthesis[J]. TWDM LNCS,2006,4046:167-174.
[118]Chen Y, Lo J Y, Baker J A, Dobbins J T. Gaussian frequency blending algorithm with matrix inversion tomsynthesis (MITS) and filtered back projection (FBP) for better digital breast tomosynthesis reconstruction[C]. Proc. SPIE,2006,6142:1-9.
[119]Ruttimann U E, Groenhuis R A, Webber J. Restoration of digital multiplane tomosyn-thesis by a constrained iteration method[J]. IEEE Trans. on Med. Imag.,1984,3(3): 141-148.
[120]Wang B, Barnera K, Leeb D. Algebraic tomosynthesis reconstruction[C]. Proc. SPIE, 2004,5370:711-718.
[121]Gordon R, Bender R,Herman G T. Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and x-ray photography[C]. J. Theor. Biol.,1970, 29:471-481.
[122]Bleuet P, Guillemaud R, Desbat L, et al. An adapted fan volume sampling scheme for 3D Algebraic reconstruction in linear tomosynthesis [J]. IEEE Trans. on Nucl. Sci.,2002, 49(5):2366-2372.
[123]Zhang Y, Chan H P, Sahiner B, et al. Tomosynthesis reconstruction using the simulta-neous algebraic reconstruction technique (SART) on breast phantom data[C]. Proc. SPIE, 2006,6142:614249_1-614249_9.
[124]Wojciech C.3D simultaneous algebraic reconstruction technique for cone-beam project-ions [D]. Greek:Univ. of Patras,2001.
[125]Chen P, Barner K. Maximum likelihood reconstruction for tomosynthesis[C].IEEE Conf. on 29th Annual Northeast Bioengineering,2003,59-60.
[126]Das M, Gifford H 0, Connor J, Glick S. Penalized maximum likelihood reconstruction for improved microcalcification detection in breast x-ray tomosynthesis[J]. IEEE Trans. on Med. Imag.,2010, PP(99):1-15.
[127]Pei C, Barner K E. Three-dimensional multi-resolution statistical reconstruction for tomosynthesis[C]. IEEE International Sym. on Biomed. Imag.,2004,1:559-562.
[128]Anna K J, Markus K, Thomas M. Regularization parameter selection in maximum a posteri-ori iterative reconstruction for digital breast tomosynthesis[J]. Lecture Notes in Computer Science,2010,6136:548-555.
[129]Kolehmainen V, Vanne A, Arvenp S, Jarvenpaa, S, et al. Bayesian inversion method for 3D dental X-ray imaging[J]. Elektrotechnik & informationstechnik,2007,124/7/8: 248-253.
[130]Kolehmainen V, Vanne A, Samuli S, et al. Parallelized bayesian inverseion for three dimensional dental X-ray imaging [J]. IEEE Trans. on Med. Imag.,2006,25(2):218-228.
[131]Sidky E Y, Resier I S, Nishikawa R, Pan X C. Image reconstruction in digital breast tomosynthesis by total variation minimization[C]. Proc. SPIE,2007,6510:1-6.
[132]温俊海,贾中宁,程敬之.利用Vandermonde矩阵快速重建断层像的层析摄影合成方法[J].电子学报,1999,27(4)：71-74.
[133]温俊海,卜凡亮,程敬之.有限角投影下的快速重建方法[J].西安交通大学学报,2000,34(4)：33-36.
[134]卜凡亮,温俊海,程敬之.基于tomosynthesis的图像重建方法[J].中国医疗器械杂志,2000,24(3)：129-132.
[135]王巍,刘传亚,卢传友.数字合成X线体层成像原理与研究进展[J].医学影像学杂志,2005,15(7)：606-609.
[136]张利军,刘文勇,王田苗,胡磊.基于C型臂的tomosynthesis快速重建方法[J].北京航空航天大学学报,2006,32(9)：1113-1116.
[137]高峰,牛憨笨.光学CT中的图像重建算法[J].光学学报,1996,·16(4)：494-499.
[138]庄天戈.CT原理与算法[M].上海：上海交通大学出版社,1992.
[139]Van Den Berg P M, Van Broekhoven A L, and Abubakar A. Extended contrast source inversion[J]. Inverse Problems,1999,15:1325-1344.
[140]Abubakar A, Van Den Berg P M. Multiplicative regularization technique for location and shape reconstructions of homogeneous objects[C]. IEEE Antennas and Propagation Society International Symposium,2002,1:292-295.
[141]Abubakar A, Van Den Berg P M, Habashy T M, and Braunisch H. A multiplicative regulariza-tion approach for deblurring problems [J]. IEEE Trans. on Imag. Proc.,2004,13(11): 1524-1532.
[142]Chan T F, Osher S, Shen J H. The digital TV filter and nonlinear denoising [J]. IEEE Trans. on Imag.Proc.,2001,10(2):231-241.
[143]Beilei W, Barner K. Fast, accurate and memory-saving ART based tomosynthesis [C]. IEEE Conference on 29th Bioengineering,2003, pp:61-62.
[144]Lewitt R M. Multidimensional digital image representations using generalized Kaiser Bessel window functions [J]. J. Opt. Soc. Am. A,1990,7(10):1834-1846.
[145]Zbijewski W, Beekman F J. Comparison of methods for suppressing edge and aliasing artifacts in iterative X-ray CT reconstruction [J]. Phys. Med. Biol.,2006,51:1877-1889.
[146]Wright S J, Nowak R D, Figueiredo M A T. Sparse reconstruction by separable approximat-ion [J]. IEEE Trans. on Signal Proc.,2009,57(7):2479-2493.
[147]Nesterov Y. Gradient methods for minimizing composite objective function[R]. Center for operations research and econometrics (CORE), Catholic Univ. Louvain, Belgium, CORE Discussion Paper2007,2007.
[148]Bioucas D J, Figueiredo M. A new twist:two-step iterative shrinkage/thresholding algorithms for image restoration[J]. IEEE Trans. on Imag. Proc.,2007,16(12):2992-3004.
[149]Combettes P L, Wajs V R.. Signal recovery by proximal forward backward splitting[J]. Multiscale Model. Simul.,2005,4(4):1168-1200.
[150]Rockafellar R T, Wets R. Variational analysis [M]. Berlin, Germany:Springer Verlag, 1998.
[151]Bertsekas D, Nedic A, Ozdaglar E. Convex analysis and optimization[M]. Athena Scie-ntific, Belmont, MA,2003.
[152]Wang Z, Dowling E M. Stochastic conjugate gradient constant modulus blind equalizer for wireless communications[C]. IEEE International Conference on Converging Technologies for Tomorrow's Applications,1996,2:832-836.
[153]Yu G, Zhao Y, Wei Z. A descent nonlinear conjugate gradient method for large-scale unconstrained optimization[J]. Appl. Math. Comp.,2007,187:636-643.
[154]Gonglin Y. Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems[J]. Optimization Letters,2009, 3:11-21.
[155]Beilei W, Barner K, Denny L Algebraic tomosynthesis reconstruction[C]. Proc. SPIE on Med. Imag.,2004,5370:711-718.
[156]Gordon G, Bender R,Herman G T. Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography [J]. J. Theor. Biol.,1970,29: 471-481.
[157]Ge W, Ming J. Ordered-subset simultaneous algebraic reconstruction techniques (OS-SART)[J]. J.X-Ray Sci. Technol.,2004,12:169-177.
[158]Mueller K, Yagel R. Rapid 3-D cone beam reconstruction with the simultaneous algebr-aic technique using 2-D texture mapping hardware [J]. IEEE Trans. on Med. Imag., 2000,19(12):1227-1237.
[159]Guan H, Gordon R. Computed tomography using algebraic reconstruction techniques (ARTs) with different projection access schemes:a comparison study under practical situations [J]. Phys. Med. Biol.,1996,41:1727-1743.
[160]Ge W, Schweiger G D. Vannier M W. An iterative algorithm for X-ray CT fluoroscopy [J]. IEEE Trans. on Med. Imag.,1998,17(5):853-856.
[161]Ming J, Ge W. Convergence of the simultaneous algebraic reconstruction technique (SART) [J]. IEEE Trans. on Imag. Proc.,2003,12(8):957-961.
[162]Tikhonov A N, Arsenin V Y. Solutions of ill-posed problems[M]. New York:John Wiley, 1977.
[163]Dahlke P, Weinreich I. Wavelet-Galerkin methods:an adapted biorthogonal wavelet Basis [J]. Constr. Approx.,1993,9:237-262.
[164]Delves L M, Mohamed J L. Computational methods for integral equations[M]. Cambridge University Press,1985.
[165]Maleknejad K, Lotfi T. Numerical expansion methods for solving integral equation by interpolation and gauss quadrature rules [J]. Appl. Math. Comp.,2005,168:111-124.
[2]Feldkamp L A, Davis L C, Kress J W. Practical cone-beam algorithm[J]. J. Opt. Soc. AM. A,1984,1(6):612-619.
[3]Novikov R G. An inversion formula for the attenuated X-ray transformation [J]. Ark. Mat. 2002,40:145-167.
[4]Faridani A, Finch D V, Ritman E L, and Smith K T. Local tomography II[J]. SIAM J. Appl. Math.,1997,57(4):1095-1127.
[5]Farrokhi F R, Liu K J R, Berenstein C A, Walnut D. Localized wavelet based computerized tomography[C]. International Conference on Image Processing,1995,2:445-448.
[6]Farrokhi F R, Liu K J R, Berenstein C A, Walnut D. Wavelet-based multiresolution local tomography [J]. IEEE Trans. on Imag. Proc.,1997,6(10):1412-1430.
[7]Bhatia M, Karl W C, Willsky A S. A wavelet-based method for multiscale tomographic reconstruction [J]. IEEE Trans. on Med. Imag.,1996,15(1):92-101.
[8]Bottema M, Moran B, Suvorova S. An application of wavelets in tomography [J]. Digital Signal Processing,1998,8:244-254.
[9]Zhao S Y, Wang G. Feldkamp-type cone-beam tomography in the wavelet framework[J]. IEEE Trans. on Med. Imag.,2000,19(9):922-929.
[10]Clackdoyle R, Guo J Y, Noo F. Quantitative reconstruction from truncated projections in classical tomography [J]. IEEE Trans. on Nucl. Sci.,2004,51(5):2570-2578.
[11]Nassi M, Brody W R, Medoff B P, Macovski A. Iterative reconstruction-reprojection: an algorithm for limited data cardiac-computed tomography[J]. IEEE Trans. on Biomed. Eng.,1982,29(5):333-341.
[12]Snyder D L, Sullivan J A 0, Murphy R J et al. Maximum likelihood methods for recons-tructing an image in a region of interest for transmission tomography[C]. Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2003,1:312-315.
[13]Tam K C, Perez M V. Tomographical imaging with limited-angle input [J]. J. Opt. Soc. Am.,1981,71 (5):582-592.
[14]Gerchberg R W. Super-resolution through error energy reduction [J]. J. Modern Optics, 1974,21 (9):709-720.
[15]Papoulis A. A new method in image restoration[R]. Joint Services Technical Activity Report 39,1973.
[16]Papoulis A. A new algorithm in spectral analysis and band-limited extrapolation[J]. IEEE Trans. on Circuits Syst.,1975, CAS-22 (9):735-742.
[17]Salomon B G, Ur H. Accelerated iterative band-limited extrapolation algorithms[J]. IEEE Signal Proc. Lett.,2004,11(11):871-874.
[18]Qi J, Huesman R H. Theoretical study of penalized likelihood image reconstruction for region of interest quantification [J]. IEEE Trans. on Med. Imag.,2006,25(5): 640-648.
[19]Hamelin B, Goussard Y, Dussault J P. Penalized likelihood region of interest CT reconst-ruction by local object supersampling[C].29th IEEE EMBS.,2007,739-742.
[20]Inouye T. Image reconstruction with limited angle projection data[J]. IEEE Trans. on Nucl. Sci.,1979,26(2):2665-2669.
[21]Candes E J, Romberg J K. Practical signal recovery from random projections[C]. Computa-tional Imaging III, Proc. SPIE,2005,5674:56741-56749.
[22]Rantala M, Vanska S, Jarvenpaa S, et al. Wavelet-based reconstruction for limited-angle X-ray tomography [J]. IEEE Trans. on Med. Imag.,2006,25(2):210-217.
[23]Natterer F. The mathematics of computerized tomography[M]. John Wiley & Sons Ltd, Chichester,1986.
[24]Natterer F, Wubbeling F. Mathematical methods in image reconstruction [M]. SIAM Mono-graphs on Mathematical Modeling and Computation, Philadelphia, PA,2001.
[25]Rangayyan R, Dhawan A P, Gordon R. Algorithms for limited-view computed tomography [J]. Appl. Opt.,1985,24(23):4000-4012.
[26]Lan C Q, Xiong W. An iterative method of ultrasonic reflection mode tomography[J]. IEEE Trans. on Med. Imag.,1994,13(2):419-425.
[27]Byrne C, Scares E, Pan T S et al. Accelerating the EM algorithm using rescaled block iterative methods[C]. IEEE Nuclear Science Symposium,1996,3:1752-1756.
[28]Verhoeven D. Limited-data computed tomography algorithms for the physical sciences [J]. Appl. Opt.,1993,32(20):3736-3754.
[29]Dhawan A P, Rangayyan R M, Gordon R. Image restoration by wiener deconvolution in limited-view computed tomography [J]. Appl. Opt.,1985,24(23):4013-4020.
[30]Andersen H A. Algebraic reconstruction in CT from limited views[J]. IEEE Trans. on Med. Imag.,1989,8(1):50-55.
[31]Byrne C. Block-iterative interior point optimization methods for image reconstruction from limited data[J]. Inverse Problems,2000,16:1405-1419.
[32]Popa C, Zdunek R. Kaczmarz extended algorithm for tomographic image reconstruction from limited-data[J]. Math. Comput. Simul.,2004,65(6):579-598.
[33]Hstau Y L. A gradually unmasking method for limited data tomography[C]. IEEE Interna-tional Symposium on 4th Biomedical Imaging,2007, pp:820-823.
[34]Hanson K M, Wecksung G W.. Bayesian approach to limited-angle reconstruction in computed tomography [J]. J. Opt. Soc. Am.,1983,73(11):1501-1509.
[35]Sauer K, James S, Klifa K. Bayesian estimation of 3-D objects from few radiographs[J]. IEEE Trans. on Nucl. Sci.,1994,41(5):1780-1790.
[36]Siltanen S, Kolehmainen V, et al. Statistical inversion for medical x-ray tomography With few radiographs:I. General theory [J]. Phys. Med. Biol.,2003,48(10):1437-1463.
[37]Kolehmainen V, Siltanen S, et al. Statistical inversion for medical X-ray tomography With few radiographs:II. Application to dental radiology [J]. Phys. Med. Biol.,2003, 48(10):1465-1490.
[38]Milanfar P, Karl W C, Willsky A S. A moment-based variational approach to tomographic reconstruction [J]. IEEE Trans. on Imag. Proc.,1996,5(3):459-470.
[39]Lassas M, Siltanen S. Can one use total variation prior for edge-preserving bayesian inversion [J]. Inverse Problems,2004,20:1537-1563.
[40]Kybic J, Blu T, and Unser M. Variational approach to tomographic reconstruction[C]. Proc. of SPIE on Medical Imaging,2001,4322:30-39.
[41]Dempster A P, Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm[J]. J. Roy. Stat. Soc. Ser. B,1977,39(1):1-38.
[42]Shepn L A, Vardi Y. Maximum likelihood reconstruction for emission tomography [J]. IEEE Trans. on Med. Imag.,1982, MI-1 (2):113-122.
[43]Green P J. Bayesian reconstruction from emission tomography data using a modified EM algorithm[J]. IEEE Trans. on Med. Imag.,1990,9(1):84-93.
[44]Zhang J, Anastasio M A, Pan X, et al. Weighted EM reconstruction algorithms for thermoa-coustic tomography [J]. IEEE Trans. on Med. Imag.,2005,24(6):817-820.
[45]Chen C M, Lee S Y. On parallelizing the EM agorithm for PET image reconstruction[J]. IEEE Trans. on Parallel and Distributed Systems,1994,5(8):860-873.
[46]Figueiredo M A, Nowak R D. An EM algorithm for wavelet-based image restoration[J]. IEEE Trans. on Imag. Proc.,2003,12(8):906-916.
[47]Louis A K. Incomplete data problems in X-ray computerized tomography I. singular value decomposition of the limited angle transform[J]. Numer. Math,1986,48:251-262.
[48]Louis A K, Rieder A. Incomplete data problems in X-ray computerized tomography II. truncated projection and region of interest tomography[J]. Numer. Math,1989, 56:371-383.
[49]Quinto E T. Singularities of the X-ray transform and limited data tomography in R2and R3 [J].SIAM J. Math. Anal.,1993,24(5):1215-1225.
[50]Candes E, Donoho D L. Ridgelets:a key to higher-dimensional intermittency[J]. Phil. Trans. R. Soc. Lond. A.,1999,357(1760):2495-2509.
[51]Meyer Y. Wavelets and operators[M]. Cambridge, U.K.:Cambridge Univ. Press,1992.
[52]Mallat S G. A theory for multiresolution signal decomposition:the wavelet representat-ion [J]. IEEE Trans. on Pattern Anal. Mach. Intell.,1989,11(7):674-693.
[53]Abramovich F, Sapatinas T, Silverman B W. Wavelet thresholding via a bayesian approach [J]. J. Roy. Statist. Soc. B,1998,60:725-749.
[54]Delanay A H, Bresler Y. Multiresolution tomographic reconstruction using wavelets [J]. IEEE Trans. on Imag. Proc.,1995,4(6):799-813.
[55]Strahlen K. Local vector tomography by use of wavelets[C]. IEEE Internation conf. on Imag. Proc.,2000,2:617-620.
[56]Olson T, DeStefano J. Wavelet localization of the Radon transforms [J]. IEEE Trans. on signal Proc.,1994,42(8):2055-2067.
[57]Zhao S, Welland G, Wang G. Wavelet sampling and localization schemes for the Radon transform in two dimensions [J]. SIAM J. Appl. Math.,1997,57(6):1749-1762.
[58]Sahiner B, Yagle A E. Limited angle tomography using wavelets[C]. IEEE Conf. on Nucl. Sci. Symposium and Medical Imaging,1993,3:1912-1916.
[59]Frese T, Bouman C A, Sauer K. Adaptive wavelet graph model for bayesian tomographic reconstruction [J]. IEEE Trans. on Imag. Proc.,2002,11 (7):756-770.
[60]Bhatia M, Karl W C, Willsky A S. A wavelet-based method for multiscale tomographic reconstruction [J]. IEEE Trans. on Med. Imag.,1996,15(1):92-101.
[61]Buccigrossi R W, Simoncelli E P. Image compression via joint statistical character-ization in the wavelet domain [J]. IEEE Trans. on Imag. Proc.,1999,8(12):1688-1701.
[62]Nam Y L, Lucier B J. Wavelet methods for inverting the Radon transform with noisy data [J]. IEEE Trans. on Imag. Proc.,2001,10(1):79-94.
[63]Choi H, Baraniuk R G. Multiple wavelet basis image denoising using Besov ball projectio-ns[J]. IEEE Signal Proc. Lett.,2004,11 (9):717-720.
[64]Wang G, Zhang J, Pan G W. Solution of inverse problems in image processing by wavelet expansion [J].IEEE Trans, on Imag. Proc.,1995,4(5):579-593.
[65]Berkner K, Gormish M J, Schwartz E L. Multiscale sharpening and smoothing in Besov spaces with applications to image enhancement [J]. Appl. Computational Harmonic Anal. 2001,11(1):2-31.
[66]Muller P, Vidakovic B. Bayesian inference in wavelet-based models [M]. Springer-Verlag, 1999,141:24-56.
[67]Vidakovic B. Statistical modeling by wavelets[M]. Series Probability and Mathematical Statistics. New York:Wiley,1999.
[68]Grunbaum F A. A study of fourier space methods for limited-angle image reconstruction [J]. Numer. Funct. Anal. Optim.,1980,2(1):31-42.
[69]Tam K C, Perez M V. Limited-angle 3D reconstructions using fourier transform iterations and radon transform iterations[J]. In International Optical Computing Conf.,1980, 142-148.
[70]Shkolnisky Y, Averbuch A.3D fourier based discrete radon transform [J]. Appl. Comput. Harmon. Anal.,2003,15(1):33-69.
[71]Hanson K M, Cunningham G S, Jennings G R, Wolf D R. Tomographic reconstruction based on flexible geometric models[C]. IEEE International Conf. on Imag. Proc.,1994, 2:145-147.
[72]Battle X L, Cunningham G S, Hanson K M.3D tomographic reconstruction using geometrical models[C]. Proceedings of SPIE, Medical Imaging:Image Processing,1997,3034:346-357.
[73]Kadoury S, Cheriet F, Labelle H. Personalized X-ray 3-D reconstruction of the scoliotic spine from hybrid statistical and image based models[J]. IEEE Trans. on Med.Imag. 2009,28 (9):1422-1435.
[74]Feng H, Karl W C, Castanon D A. A curve evolution approach to object-based tomographic reconstruction [J]. IEEE Trans. on Imag. Proc.,2003,12(1):44-57.
[75]Yu D F, Fessler J A. Edge-preserving tomographic reconstruction with non-local regular-ization[J]. IEEE Trans. on Med. Imag.,2002,21(2):159-173.
[76]Kolehmainen V, Lassas M, Siltanen S. Limited data X-ray tomography using nonlinear evolution equations [J]. SIAM J. Sci. Comput.,2008,30(3):1413-1429.
[77]Nguyen C D, Hoppe H W. Amorphous surface growth via a level set approach[J]. Nonlinear Anal.,2007,66:704-722.
[78]Osher S, Fedkiw R P. Level set methods and dynamic implicit surfaces [M]. New York, Sprin-Ger,2003.
[79]Osher S, Santosa F. Level set methods for optimization problems involving geometry and constraints Ⅰ. Frequencies of a two-density inhomogeneous drum[J]. J. Comput. Phys. 2001,171:272-288.
[80]Kenneth M H, Gregory S C, Robert J M. Uncertainties in tomographic reconstructions based on deformable models[C]. Proc. of SPIE in Medical Image,1997,3034:276-286.
[81]Charbonnier P, Blanc L, Barlaud M. An adaptive reconstruction method involving discont-inuities[J]. Proc. of ICASSP, Minneapolis, MN,1993, V:491-494.
[82]Delaney A H, Bresler Y. Globally convergent edge-preserving regularized reconstru-uction:An application to limited-angle tomography[J]. IEEE Trans. on Imag. Proc.,1998, 7(2):204-221.
[83]Charbonnier P, Blanc L. Deterministic edge-preserving regularization in computed imag-ing[J]. IEEE Trans. on Imag. Proc.,1997,6(2):298-311.
[84]Bert W R, Dianne P L. Residual periodograms for choosing regularization parameters for ill-posed problems[J]. Inverse Problems,2008,24:1-30.
[85]Hansen P C. Analysis of discrete ill-posed problems by means of the L-curve[J]. SIAM Rev.,1992,34(4):561-580.
[86]Morozov A V. Methods for solving incorrectly posed problems[M]. New York:Springer, 1984.
[87]Golub G H, Heath M, Wahba G. Generalized cross-validation as a method for choosing a good ridge parameter[J]. Technometrics,1979,21(2):215-223.
[88]Jianli X, Jun Z. An improved model function method for choosing regularization paramete-rs in linear inverse problems[J]. Inverse problems,2002,18:631-643.
[89]Sidky E Y, Kao C M, Pan X C. Accurate image reconstruction from few-views and limited angle data in divergent-beam CT[J]. J.X-Ray Sci. Technol.,2006,14:119-139.
[90]Sidky E Y, Pan X C. Image reconstruction in circular cone-beam computed tomography by constrained total variation minimization [J]. Physics in Medicine and Biology,2008,53: 4777-4807.
[91]Zou Yu, Pan X C. Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT[J]. Physics in Medicine and Biology,2006,49:941-959.
[92]Zou Yu, Pan X C, Sidky E Y. Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan[J]. Physics in Medicine and Biology, 2005,50:13-27.
[93]Xia D, Yu L, Sidky E Y, et al. Noise properties of chord-image reconstruction[J]. IEEE Trans. on Med. Imag.,2007,26(10):1328-1344.
[94]Persson M, Bone D, Elmqvist H. Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography [J]. Phys. Med. Biol.,2001,46:853-866.
[95]Candes E J, Romberg J. The role of sparsity and incoherence for exactly reconstructing a signal from limited measurements [M]. Tech. Rep., Pasadena:California Inst. Technol. 2004.
[96]Rudin L, Oher S, Fatemi E. Nonlinear total variation based noise removal algorithms [J]. Phys.D,1992,60:259-268.
[97]Chan T F, Wong C K. Total variation blind decovolution[J]. IEEE Trans. on Imag. Proc. 1998,7(3):370-375.
[98]Sidky E Y, Kao C M, Pan X C. Effect of the data constraint on few-view, fan-beam CT image reconstruction by TV minimization[C]. IEEE Conferece on Nuclear Science Symposium Record,2006,4:2296-2298.
[99]Velikina J, Leng S, Chen G H. Limited view angle tomographic image reconstruction via total variation minimization[C]. Proc. of SPIE,2007,6510:651020_1-651020_12.
[100]Grant D. Tomosynthesis:a three dimensional radiographic imaging technique[J]. IEEE Trans. on Biomed. Eng.,1972,19(1):20-28.
[101]Stiel G M, Stiel L S G, Klotz E, Nienaber C A. Digital flashing tomosynthesis:a promis-ing technique for angiocardiographic screening[J]. IEEE Trans. on Med. Imag., 1993,12(2):314-321.
[102]Chakraborty D P, Yester M V, Barnes G T, Lakshminarayanan A V. Self-masking subtracti-on tomosynthesis [J]. Radiology,1984,150:225-229.
[103]Knutsson H E, Edholm P, Granlund G H, et al. Ectomography-a new radiographic reconstruc-ction method:Ⅰ. theory and error estimates[J]. IEEE Trans. on Biomed. Eng.,1980, 27(11):640-648.
[104]Petersson C U, Edholm P, Granlund G H, et al. Ectomography-a new radiographic reconstr-uction method:Ⅱ. computer simulated experiments [J]. IEEE. Trans. on Biomed. Eng.,1980, 27(11):649-655.
[105]Ghosh R D N, Kruger R A, Yih B, et al. Selective plane removal in limited angle tomogra-phic imaging[J]. Med. Phys.,1985,12(1):65-70.
[106]Kolitsi Z, Panayiotakis G, Pallikarakis N. A method for selective removal of out-of-plane structures in digital tomosynthesis[J]. Med. Phys.,1993,20(1):47-50.
[107]Badea C, Kolitsi Z, Pallikarakis N. A wavelet-based method for removal of out-of-plane structures in digital tomosynthesis [J]. Comput. Med. Imaging and Graphics,1998, 22:309-315.
[108]Dobbins J T, Godfrey D J. Digital X-ray tomosynthesis:current state of the art and clinical potential[J]. Phys. Med. Biol.,2003,48:R65-R106.
[109]Wu T, Moore R H, Rafferty E A, Kopans D B. A comparison of reconstruction algorithms for breast tomosynthesis [J]. Med. Phys.,2004,31 (9):2636-2647.
[110]Zhang Y, Chan H P, Sahiner B, et al. A comparison study of limited-angle cone-beam reconstruction methods for breast tomosynthesis [J]. Med. Phys.,2006,33(10):3781-3795.
[111]Lauritsch G, Haerer W. A theoretical framework for filtered backprojection in tomos-ynthesis[C]. Proc. SPIE.,1998,3338:1127-1137.
[112]Stevens G M, Fahrig R, Pelc N J. Flitered backprojection for modifying the impulse response of circular tomosynthesis [J]. Med. Phys.,2001,28(3):372-380.
[113]Matsuo H, Iwata A, Horiba I, Suzumura N. Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering[J]. IEEE Trans. on Med. Imag.,1993, 12(2):307-313.
[114]Warp R J, Godfrey D G, Lynch J A. Application of matrix inverse tomosynthesis[C]. Proc. SPIE,2000,3977:376-383.
[115]Godfrey D J, Warp R J, Dobbins J T. Optimization of matrix inverse tomosynthesis [C]. Proc. SPIE,2001,4320:696-704.
[116]Orman J, Mertelmeier T, Haerer W. Adaptation of image quality using various filter setups in the backprojection approach for digital breast tomosynythesis[J]. TWDM LNCS,2006,4046:175-182.
[117]Claus B E H, Eberhard J W, Schmitz A, et al. Generalized filtered back-projection reconstruction in breast tomosynthesis[J]. TWDM LNCS,2006,4046:167-174.
[118]Chen Y, Lo J Y, Baker J A, Dobbins J T. Gaussian frequency blending algorithm with matrix inversion tomsynthesis (MITS) and filtered back projection (FBP) for better digital breast tomosynthesis reconstruction[C]. Proc. SPIE,2006,6142:1-9.
[119]Ruttimann U E, Groenhuis R A, Webber J. Restoration of digital multiplane tomosyn-thesis by a constrained iteration method[J]. IEEE Trans. on Med. Imag.,1984,3(3): 141-148.
[120]Wang B, Barnera K, Leeb D. Algebraic tomosynthesis reconstruction[C]. Proc. SPIE, 2004,5370:711-718.
[121]Gordon R, Bender R,Herman G T. Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and x-ray photography[C]. J. Theor. Biol.,1970, 29:471-481.
[122]Bleuet P, Guillemaud R, Desbat L, et al. An adapted fan volume sampling scheme for 3D Algebraic reconstruction in linear tomosynthesis [J]. IEEE Trans. on Nucl. Sci.,2002, 49(5):2366-2372.
[123]Zhang Y, Chan H P, Sahiner B, et al. Tomosynthesis reconstruction using the simulta-neous algebraic reconstruction technique (SART) on breast phantom data[C]. Proc. SPIE, 2006,6142:614249_1-614249_9.
[124]Wojciech C.3D simultaneous algebraic reconstruction technique for cone-beam project-ions [D]. Greek:Univ. of Patras,2001.
[125]Chen P, Barner K. Maximum likelihood reconstruction for tomosynthesis[C].IEEE Conf. on 29th Annual Northeast Bioengineering,2003,59-60.
[126]Das M, Gifford H 0, Connor J, Glick S. Penalized maximum likelihood reconstruction for improved microcalcification detection in breast x-ray tomosynthesis[J]. IEEE Trans. on Med. Imag.,2010, PP(99):1-15.
[127]Pei C, Barner K E. Three-dimensional multi-resolution statistical reconstruction for tomosynthesis[C]. IEEE International Sym. on Biomed. Imag.,2004,1:559-562.
[128]Anna K J, Markus K, Thomas M. Regularization parameter selection in maximum a posteri-ori iterative reconstruction for digital breast tomosynthesis[J]. Lecture Notes in Computer Science,2010,6136:548-555.
[129]Kolehmainen V, Vanne A, Arvenp S, Jarvenpaa, S, et al. Bayesian inversion method for 3D dental X-ray imaging[J]. Elektrotechnik & informationstechnik,2007,124/7/8: 248-253.
[130]Kolehmainen V, Vanne A, Samuli S, et al. Parallelized bayesian inverseion for three dimensional dental X-ray imaging [J]. IEEE Trans. on Med. Imag.,2006,25(2):218-228.
[131]Sidky E Y, Resier I S, Nishikawa R, Pan X C. Image reconstruction in digital breast tomosynthesis by total variation minimization[C]. Proc. SPIE,2007,6510:1-6.
[132]温俊海,贾中宁,程敬之.利用Vandermonde矩阵快速重建断层像的层析摄影合成方法[J].电子学报,1999,27(4)：71-74.
[133]温俊海,卜凡亮,程敬之.有限角投影下的快速重建方法[J].西安交通大学学报,2000,34(4)：33-36.
[134]卜凡亮,温俊海,程敬之.基于tomosynthesis的图像重建方法[J].中国医疗器械杂志,2000,24(3)：129-132.
[135]王巍,刘传亚,卢传友.数字合成X线体层成像原理与研究进展[J].医学影像学杂志,2005,15(7)：606-609.
[136]张利军,刘文勇,王田苗,胡磊.基于C型臂的tomosynthesis快速重建方法[J].北京航空航天大学学报,2006,32(9)：1113-1116.
[137]高峰,牛憨笨.光学CT中的图像重建算法[J].光学学报,1996,·16(4)：494-499.
[138]庄天戈.CT原理与算法[M].上海：上海交通大学出版社,1992.
[139]Van Den Berg P M, Van Broekhoven A L, and Abubakar A. Extended contrast source inversion[J]. Inverse Problems,1999,15:1325-1344.
[140]Abubakar A, Van Den Berg P M. Multiplicative regularization technique for location and shape reconstructions of homogeneous objects[C]. IEEE Antennas and Propagation Society International Symposium,2002,1:292-295.
[141]Abubakar A, Van Den Berg P M, Habashy T M, and Braunisch H. A multiplicative regulariza-tion approach for deblurring problems [J]. IEEE Trans. on Imag. Proc.,2004,13(11): 1524-1532.
[142]Chan T F, Osher S, Shen J H. The digital TV filter and nonlinear denoising [J]. IEEE Trans. on Imag.Proc.,2001,10(2):231-241.
[143]Beilei W, Barner K. Fast, accurate and memory-saving ART based tomosynthesis [C]. IEEE Conference on 29th Bioengineering,2003, pp:61-62.
[144]Lewitt R M. Multidimensional digital image representations using generalized Kaiser Bessel window functions [J]. J. Opt. Soc. Am. A,1990,7(10):1834-1846.
[145]Zbijewski W, Beekman F J. Comparison of methods for suppressing edge and aliasing artifacts in iterative X-ray CT reconstruction [J]. Phys. Med. Biol.,2006,51:1877-1889.
[146]Wright S J, Nowak R D, Figueiredo M A T. Sparse reconstruction by separable approximat-ion [J]. IEEE Trans. on Signal Proc.,2009,57(7):2479-2493.
[147]Nesterov Y. Gradient methods for minimizing composite objective function[R]. Center for operations research and econometrics (CORE), Catholic Univ. Louvain, Belgium, CORE Discussion Paper2007,2007.
[148]Bioucas D J, Figueiredo M. A new twist:two-step iterative shrinkage/thresholding algorithms for image restoration[J]. IEEE Trans. on Imag. Proc.,2007,16(12):2992-3004.
[149]Combettes P L, Wajs V R.. Signal recovery by proximal forward backward splitting[J]. Multiscale Model. Simul.,2005,4(4):1168-1200.
[150]Rockafellar R T, Wets R. Variational analysis [M]. Berlin, Germany:Springer Verlag, 1998.
[151]Bertsekas D, Nedic A, Ozdaglar E. Convex analysis and optimization[M]. Athena Scie-ntific, Belmont, MA,2003.
[152]Wang Z, Dowling E M. Stochastic conjugate gradient constant modulus blind equalizer for wireless communications[C]. IEEE International Conference on Converging Technologies for Tomorrow's Applications,1996,2:832-836.
[153]Yu G, Zhao Y, Wei Z. A descent nonlinear conjugate gradient method for large-scale unconstrained optimization[J]. Appl. Math. Comp.,2007,187:636-643.
[154]Gonglin Y. Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems[J]. Optimization Letters,2009, 3:11-21.
[155]Beilei W, Barner K, Denny L Algebraic tomosynthesis reconstruction[C]. Proc. SPIE on Med. Imag.,2004,5370:711-718.
[156]Gordon G, Bender R,Herman G T. Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography [J]. J. Theor. Biol.,1970,29: 471-481.
[157]Ge W, Ming J. Ordered-subset simultaneous algebraic reconstruction techniques (OS-SART)[J]. J.X-Ray Sci. Technol.,2004,12:169-177.
[158]Mueller K, Yagel R. Rapid 3-D cone beam reconstruction with the simultaneous algebr-aic technique using 2-D texture mapping hardware [J]. IEEE Trans. on Med. Imag., 2000,19(12):1227-1237.
[159]Guan H, Gordon R. Computed tomography using algebraic reconstruction techniques (ARTs) with different projection access schemes:a comparison study under practical situations [J]. Phys. Med. Biol.,1996,41:1727-1743.
[160]Ge W, Schweiger G D. Vannier M W. An iterative algorithm for X-ray CT fluoroscopy [J]. IEEE Trans. on Med. Imag.,1998,17(5):853-856.
[161]Ming J, Ge W. Convergence of the simultaneous algebraic reconstruction technique (SART) [J]. IEEE Trans. on Imag. Proc.,2003,12(8):957-961.
[162]Tikhonov A N, Arsenin V Y. Solutions of ill-posed problems[M]. New York:John Wiley, 1977.
[163]Dahlke P, Weinreich I. Wavelet-Galerkin methods:an adapted biorthogonal wavelet Basis [J]. Constr. Approx.,1993,9:237-262.
[164]Delves L M, Mohamed J L. Computational methods for integral equations[M]. Cambridge University Press,1985.
[165]Maleknejad K, Lotfi T. Numerical expansion methods for solving integral equation by interpolation and gauss quadrature rules [J]. Appl. Math. Comp.,2005,168:111-124.