摘要
计算机层析成像(Computed Tomography,简称为CT)技术,利用围绕被成像物体在不同视角下的X射线测量值(投影)能得到物体内部的二维或三维结构图像。它以无损、直观、准确的方式显示物体的内部结构、材质分布和缺陷等信息,已广泛应用于医学诊断、工业制造、军工、安检等领域,被喻为最佳的现代无损检测技术之一。锥束X射线CT,通过被检测物体旋转(或射线源和面板探测器绕被检测物体旋转),射线束在每个旋转角度下都能覆盖被检测物的一段,通过采集射线穿过物体后的衰减信号,获得重建所需的投影数据,从而重建物体被扫描部分的三维内部信息图像。和二维CT相比,锥束CT由于采用高密度面板探测器,具有扫描速度快(一周扫描可重建上千层图像)、X射线利用率高、重建图像轴向分辨率和水平分辨率一致等优点。与X射线源轨迹为圆周的锥束CT相比,螺旋锥束CT能以一种自然的方式连续检测诸如人、管道等长物体,且其扫描轨迹满足精确重建的数据完备性条件。随着面板探测器制造技术的进步,其耐辐射和抗干扰能力的提高,基于面板探测器的锥束CT成像技术已取得了突破,正逐渐应用于工业CT中,是工业CT技术发展的方向之一。
基于视场区域半覆盖的螺旋锥束CT扫描技术,利用与传统视场区域全覆盖的螺旋锥束CT同样的探测器尺寸可以将视场区域扩大近一倍,且机械实现简单、扫描速度与传统锥束CT相当。重建时,利用推广的标准螺旋锥束FDK算法(简称偏心FDK算法)进行重建。由于FDK算法本身的优点以及偏心FDK算法不需要对投影数据重排和插值,该算法的计算效率高,与全覆盖螺旋锥束FDK算法相比,能将重建时间减少近一倍。但略显不足的是半覆盖螺旋锥束CT的投影数据沿面板探测器的行方向是截断的,而斜坡滤波器又是非局部的,使得重建图像受截断误差影响较大。针对该问题,将二维局部重建的思想推广到三维的半覆盖螺旋锥束CT,研究了基于局部化重建滤波器的半覆盖螺旋锥束CT的改进偏心FDK算法,实验结果表明,该改进的重建算法减弱了偏心FDK算法的截断误差,且对噪声具有抑制作用,提高了重建图像的质量,且重建时间更少。
由于受现场检测条件的限制(例如在役管道的检测,其离地面太近或相邻有管道或其他物体阻碍等)或出于射线辐射剂量以及扫描成像时间的考虑,锥束扫描旋转的角度有时不能达到重建所需的最小角度(比如圆周FDK算法所需的PI+扇角),就会产生有限角锥束CT。当被检测物直径太大或中心铁质部件穿不透,而我们所关心的是其外壳部分(比如管道等),此时可将面阵探测器偏置,使每个旋转角度下的锥束只覆盖被检测物外壳部分,就会产生外部锥束CT。有限角锥束CT和外部锥束CT是两类典型的投影数据截断重建问题且在实际的应用中具有重要意义。一个工业部件(或一个人体器官)往往由一种或几种材料组成,而同种材料具有相同或近似的衰减系数,这一特征在CT图像上的表现就是图像的灰度值为近似分段常数(Piecewise Constant) (部分存在灰度渐变的图像除外),从而其梯度图像是稀疏的,符合压缩感知(Compressed Sensing)理论的假设。近几年,作为压缩感知原理在CT重建中的应用,基于图像全变差最小化(Total Variation Minimization,TVM)的正则化迭代算法对于求解投影角度采样稀疏型的重建问题取得了令人惊喜的成果。但是,TVM迭代重建算法用于有限角锥束图像重建问题时,稍显不足的是其重建结果在轮廓附近存在灰度渐变的伪影。而TVM迭代重建算法用于外部锥束重建时只能得到模糊的轮廓和扭曲的重建图像。针对TVM重建算法对有限角锥束问题的这一不足,我们推广了2D滑坡修正算法而得到3D滑坡修正算法,并将其加入到TVM算法中,得到一种带滑坡修正的锥束CT的TVM(Cone-Beam Slide-Corrected TVM,CBSC-TVM)重建算法。针对TVM重建算法对管状物的外部锥束重建只能得到模糊的边缘和扭曲的重建图像的这一不足,为了提高重建图像的质量,我们推广了基于2D C-V (Chan-Vese)图像分割活动轮廓模型的2D子区域平均化修正算法得到基于3D C-V图像分割活动轮廓模型的3D子区域平均化修正算法。在修正的过程中,先用3D C-V图像分割活动轮廓模型将重建图像分割成不同的子区域,再用各个子区域内的平均灰度值来代替该子区域内各点的灰度值,使其成为分片常数。将该3D子区域平均化修正算法加入到锥束CT的TVM迭代重建算法中,得到了带子区域平均化修正的外部锥束CT的TVM(Cone-Beam Subregion-Averaged TVM,CBSA-TVM)重建算法。实验结果验证了CBSC-TVM重建算法对有限角锥束CT和CBSA-TVM重建算法对外部锥束CT的有效性,能改善重建图像的质量。在CT的一些实际应用中,我们往往需要探测被检测物体的轮廓或边缘,而不仅仅是得到高质量的物体切片图像或三维的体图像,比如工业上物体的缺陷识别和测量、基于工业CT的逆向设计(即从CT图像获得计算机辅助诊断(Computer Aided Design, CAD)图形)应用于工业制造等;诊断医学应用上,往往探测和突出器官、组织、感兴趣区域(如肿瘤)的边界或轮廓是基本的要求。现有的一般方法是先CT扫描、重建,然后再在重建的CT图像上探测轮廓的两步方法。此方法不仅耗时,而且经常在后处理重建的CT图像时陷入困境(原因是任何实际的投影数据或多或少都含有噪声,且即使在投影数据上的白噪声,经过重建以后可能转化为非白噪声)。针对无损检测的应用需求和现有方法的缺陷,在研究了小波与其Radon变换投影之间的关系、利用反投影构造三维非张量积母小波及其在图像边缘提取中的应用一些理论基础上,研究了一种基于小波分析和单切片重组(Single-slice Rebinning , SSRB)算法的从螺旋锥束CT扫描数据直接重建物体轮廓特征的算法。
实验结果表明,该算法不仅能很好地重建出被检测物的轮廓,达到基于CT的应用需要,而且针对噪声投影数据,该算法的重建结果要好于两步方法,且需要的计算时间消耗也明显少于两步方法。
Computed Tomography (CT) is a technique that obtains inner two-dimensional (2D) cross-sectional or three-dimensional (3D) structural images of an object using multiple X-ray measurements (projections) taken at different angles around the scanned object. It can show the interior structures, defects and material components of the inspected object with a nondestructive, distinct and accurate manner. And it has been widely used in many domains such as diagnosis medicine, manufacturing industry, military and security check etc., which is considered as one of the best modern nondestructive technique. For cone-beam X-ray CT, the scanned objected is rotated around the rotation center (or the X-ray source and planar detector are rotated around the scanned object), the cone shaped X-ray beam can cover a section of the inspected object at each view angle and collect the attenuated X-ray signals by planar detector to obtain the projection data, then, reconstruction algorithm is used to reconstruct the 3D inner image of the scanned section of the object. Compared with 2D CT, Cone-beam CT, acquired data by use of high density planar detector, can shorten the scanning time and make use of the x-ray more effectively, and the longitude resolution of the reconstruction image is higher. It can obtain more than thousand images with one turn scanning. Helical cone-beam CT not only has the merits of cone-beam CT but also can solve the problem of detecting long object such as humans, pipelines etc. as a natural way, and the scanning loci satisfy the conditions for exact reconstruction. With the developments of manufacturing technology of planar detectors, the improvements of the radiation resistance and anti-disturbance performance in flat-panel detectors, planar detector-based cone-beam CT made a breakthrough in CT imaging in terms of large 3D volume reconstruction and isotropic resolution, and it is gradually used to industrial CT which is one development direction of industrial CT.
Using the same detector size of traditional full-covered CT, the helical cone-beam CT scanning based on field of view (FOV) half-covered can almost double the FOV, whose mechanism is simple and the scanning efficiency is the same as that of traditional helical cone-beam CT. During reconstruction, the extended helical cone-beam FDK algorithm (called half-covered helical FDK for short) is developed. There is no need for rebinning and interpolation of the projection data of this algorithm, so it can avoid the error from rebinning and interpolation. Compared with full-cover helical cone-beam CT, it only takes approximately half of the time for reconstruction as that of standard helical cone-beam FDK algorithm. So the computational efficiency of this algorithm is high. Only a little more than a half cross-section of an object needs to be illuminated with x-rays at every view angle, and the projection data of the half-cover helical cone-beam CT is transverse, but the ramp filter is non-local, so the reconstruction image has truncation error. Regarding this problem, this paper extends the idea of 2D local reconstruction to 3D half-covered helical cone-beam CT, and develops an improved half-covered helical cone-beam CT reconstruction algorithm based on localized reconstruction filter. Experimental results indicate that the presented algorithm well solves the truncation error of the half-covered helical FDK algorithm, improves the quality of the reconstruction image. And for the noise projection data, the presented algorithm can suppress noise and get better results. Moreover, the reconstruction time is much less.
Owing to the limitation of on-the-scene inspecting conditions (for instance, the inspection of in-service pipeline, the pipeline is too near to ground or there are some other pipelines or objects next to the inspected pipeline to hinder the inspection etc.) or consideration of the X-ray radiation dosage and the scanning time, sometime the rotation angle of cone-beam can’t reach the required minimum angle for reconstruction (for example, the required minimum angle of circular FDK algorithm is PI plus horizontal cone-angle ), then, it will occur the limited angle cone-beam CT. When the diameter of scanned object is too large or the central iron material can’t be penetrated by X-ray, and the outer is our interesting part (such as pipelines etc.), then it can offset the detector and let the cone shaped X-ray beam only covering the outer part of the scanned object, and under this condition, it will occur the exterior cone-beam CT. The limited angle and exterior cone-beam CT are two typically truncated reconstruction problems and have significant meanings in practical applications. One industrial part (or one human organ) is always consist of one kind or several kinds of materials, the same material of casting (or the same tissue of human body) has identical or similar attenuation coefficient. So, the CT image is approximately piecewise-constant(except the grayscales of some images are gradual) and its gradient image is sparse which conforms with the assumption of compressed sensing. In recent years, as the application of CS in CT reconstruction, the total variation minimization (TVM) based regularization iterative reconstruction algorithms have achieved approval results for few-views problems. These algorithms are also effective for limited angle cone-beam problems, except there are still gradually changed artifacts in the nearby regions of contours. And the TVM based algorithms are used to exterior cone-beam CT which only can obtain reconstructed images with blurred and distorted contours. For the drawback of TVM based algorithms to limited angle cone-beam problems, in this paper, the 2D slide correction algorithm is generalized to the case of 3D. And by introducing the 3D slide correction method into the TVM regularization algorithms, we get a new iterative algorithm for limited angle cone-beam CT, which is referred to as cone-beam slide-corrected TVM (CBSC-TVM). For the drawback of TVM based algorithms to pipe-like objects exterior cone-beam problems, and in order to improving the quality of reconstructed image, the 2D subregion averaged correction algorithm which is based on 2D C-V (Chan-Vese) model, is generalized to 3D case to obtain a 3D C-V model based 3D subregion averaged correction algorithm. In the process of correction, it divides the 3D preliminary reconstruction image into some subregions by using 3D C-V active contours model, and then averages the grayscale values of all voxels in each subregion respectively. By introducing the 3D subregion averaged correction method into the TVM regularization algorithm, we get a new algorithm for exterior cone-beam problem, which is referred to as cone-beam subregion-averaged TVM (CBSA-TVM). Experiment results verify that CBSC-TVM to limited angle cone-beam CT and CBSA-TVM to exterior cone-beam CT can improve the quality of reconstructed images.
In some practical applications of CT, detecting contours (or edges) of different contrast image regions of scanned object are the essential and necessary requirements, and high-quality imaging is not necessary to achieve the final goal of detecting and characterizing objects within the reconstructed images. For instance, in industrial applications, including image segmentation and object defect recognition and measurement, and industrial CT based inverse design which obtain the computer aided design (CAD) graph used in industrial manufacturing; in medicine applications, detection and highlighting of boundaries of organs, tissues, focus regions (such as tumors) are also the essential and necessary requirements. The main stream method is a two-step process, the image is reconstructed from projections first, and then it is followed by an edge detection operation on the reconstructed image. This method is time-consuming and post-processing of a reconstructed image is often difficult(because any practical CT projection data more or less have some noises, and even when the noise in the projection is white, the noise in the reconstructed image always may be nonwhite). For the application requirement of nondestructive testing and the drawback of the two-step method, on the basis of studying the relationship of wavelet and its Radon transform projection, and using back-projecting method for constructing 3D non-tensor product wavelet and its application to image edge detection, an algorithm for reconstructing object contours directly from helical cone-beam projections based on wavelet analysis and single-slice rebinning (SSRB) method is presented. Experiment results show the usefulness and high efficiency of the presented algorithm. For some cases of noisy projection data, the edge detection outcome by our algorithm is obvious better than that of two-step approach. Moreover, the run-time is much less than that of two-step approach.
引文
[1]曾更生.医学图像重建[M].北京:高等教育出版社, 2010.
[2] Hsieh J. Computed tomography:principle, design, artifacts and recent advances[M].北京:科学出版社, 2006: 1-71.
[3] Germaneau A, Doumalin P and DupréJ C. Comparision between X-ray micro-computed tomography and optical scanning tomography for full 3D strain measurement by digital volume correlation[J]. NDE&E Int., 2008, 41: 407-415.
[4] Zhang Y and Ning R. Investigation of image noise in cone-beam CT imaging due to photon counting statistics with the Feldkamp algorithm by computer simulation[J]. Journal of X-Ray Science and Technology, 2008,16: 143-158.
[5]陈平,潘晋孝,刘宾. CT图像边缘退化模型的建立及其在图像尺寸测量中的应用[J].光学精密工程, 2009, 17(9): 2269-2275.
[6]刘露,刘宛予,楚春雨,等.胸部CT图像中孤立性肺结节良恶性快速分类[J].学精密工程, 2009, 17(8): 2060-2068.
[7] Defrise M, Noo F, Clackdoyle R, et al. Truncated Hilbert transform and image reconstruction from limited tomographic data [J]. Inverse Problems, 2006, 22:1037-1053.
[8]邹晓兵,曾理.锥束螺旋CT半覆盖扫描重建[J].光学精密工程, 2010, 18(2): 434-442.
[9] Fu J, Lu H N. Research of large field of view scan mode for industrial CT[J]. Chinese Journal of Aeronautics, 2003, 16(1): 59-64.
[10]傅健,路宏年.工业CT半扫描成像技术[J].北京航空航天大学学报, 2005, 31(9) :966-969.
[11] Leng S, Zhuang T L, Nett B E, et al. Exact fan-beam image reconstruction algorithm for truncated projection data acquired from an asymmetric half-size detector[J]. Physics in Medicine Biology, 2005, 50: 1805-1820.
[12] Li L, Chen Z Q, Zhang L, et al. A new cone-beam X-ray CT system with a reduced size planar detector[J]. High Energy Physics and Nuclear Physics, 2006, 30(8): 812-817.
[13]李亮,陈志强,张丽,等.一种用于小体积偏置探测器锥束CT系统的反投影滤波重建算法[J]. CT理论与应用研究, 2007, 16(1): 1-9.
[14] Courdurier M, Noo F, Defrise M, et al. Solving the interior problem of computed tomography using a priori knowledge[J]. Inverse Problem, 2008, 24:1-27.
[15] Ye Y B, Yu H Y, Wei Y, et al. A general local reconstruction approach based on a truncated Hilbert transform[J]. International Journal of Biomedical Imaging, 2007:1-8.
[16] Ye Y B, Yu H Y, and Wang G. Exact interior reconstruction from truncated limited-angle projection data[J]. International Journal of Biomedical Imaging, 2008:1-6.
[17] Ye Y B, Yu H Y, and Wang G. Exact interior reconstruction with cone-beam CT[J]. International Journal of Biomedical Imaging, 2007:1-5.
[18] Kudo H, Courdurier M , Noo F, et al. Tiny a priori knowledge solves the interior problem in computed tomography[J]. Physics in Medical and Biology, 2008, 53: 2207-2231.
[19] Yu H Y and Wang G. Compressed sensing based interior tomography[J]. Physics in Medical and Biology, 2009. 54:2791-2805.
[20] Zeng G and Gullberg G T. Exact iterative reconstruction for the interior problem[J]. Physics in Medical and Biology, 2009. 54.
[21] Yang J S, Yu H Y, Jiang M, et al. High-order total variation minimization for interior tomography[J]. Inverse Problems, 2010. 26:5805-5814.
[22] Natterer F. The Mathematics of Computerized Tomography[M]. New York: Wiley, 1986.
[23] Sch?fer D, Grass M. Cone-beam filtered backprojection for circular X-ray tomography with off-center detector[C]. Proceeding of 10th Fully 3D Meeting and 2nd HPIR workshop, Beijing, China, 2009, 86-89.
[24] Wang G. X-ray micro-CT with a displaced detector array[J]. Medical Physics, 2002, 29 (7): 1634-1636.
[25] Kunzea K and Dennerlein F. Cone beam reconstruction with displaced flat panel detector[C]. Proceeding of 10th Fully 3D Meeting and 2nd HPIR workshop, Beijing, China, 2009, 138-141.
[26]刘宝东,曾理.管壁长弦投影数据截断问题的迭代重建算法[J].仪器仪表学报, 2011,32(1): 52-57.
[27]邹晓兵.大视场螺旋锥束工业CT的扫描方法与重建算法研究[D].重庆:重庆大学, 2010.
[28] Gonzalez A B, Mahesh M, Kim K, et al. Projected cancer risks from computed tomographic scans performed in the United States in 2007[J]. Arch. Intern. Med., 2009, 169: 2071-2077.
[29] Smith-Bindman R, Lipson J , Marcus R, et al. Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer[J]. Arch. Intern. Med., 2009, 169: 2078-2086.
[30]段黎明,邱猛,吴朝明.面向逆向工程的工业CT图像预处理系统开发[J].强激光与离子束, 2008, 20(4): 666-670.
[31]陈慧能,杨树彬,崔卫东.工业CT在火工品逆向工程的应用研究[J].火工品, 2006, 3: 36-38.
[32] Liu S, Ma W. Motif analysis for automatic segmentation of CT surface contours into individual features[J]. Computer-Aided Design, 2001, 33(12): 1091-1109.
[33]冯裕强,宁汝新.基于ICT切片图像的实体模型重建[J].计算机辅助设计与图像图形学报, 2003, 15(8): 944-948.
[34] Riederer S T, Pelec N J and Cheder D A. The noise power spectrum in computed X-ray tomography[J]. Phys. Med. Biol., 1978, 23, 446-454.
[35] Chang C Y. Contextual-based Hopfield neural network for medical image edge detection[J]. Optical Engineering, 2006, 45(3): 037006.
[36] Lin W C, Wang J W and Lin S Y. Contrast enhancement of x-ray image based on singular value selection[J]. Optical Engineering, 2010, 49(4): 047005.
[37] Munk W and Wunsch C. Ocean acoustic tomography: A scheme for large scale monitoring[J]. Deep. Sea Res., 1979, 26: 123-161.
[38] Bulcke J V and Boone M. High-resolution X-ray imaging and analysis of coatings on and in wood[J]. J. Coat. Technol. Res., 2010, 7(2): 271-277.
[39] Feldkamp L A, Davis L C, Kress J W. Practical cone-beam algorithm[J]. Journal of Optical Society of America, 1984, 1(A): 612-619.
[40] Kudo H, Satio T. Helical-scan computed tomography using cone-beam projections[C]. Nuclear Science Symposium and Medical Imaging Conference, 1991, 1958-1962.
[41] Wang G, Lin T H, Cheng P G, et al. A general cone-beam reconstruction algorithm[J]. IEEE Transactions on Image Processing, 1993, 12(3): 486-496.
[42] Silver M D. Initial evaluation of the performance of a CT add-on to a radiation therapy simulator[J]. Radiology, 1994, 193:227-228.
[43] La Riviere P J, Pan X, Gilland D, et al. Transmission image reconstruction and redundant information in SPECT with asymmetric fanbeam collimation[J]. IEEE Trans. Nucl. Sci., 2001, 48:1357-1363.
[44] Wang G. X-ray micro-CT with a displaced detector array[J]. Med. Phys., 2002, 29: 1634-1636.
[45] Parker D L. Optimal short scan convolution reconstruction for fan beam CT[J]. Med. Phys., 1982, 9:254-257.
[46] Cho P, Rudd A, and Johnson R. Cone-beam CT from width-truncated projections[J]. Computerized Medical Imaging and Graphics, 1996, 20:49-57.
[47] Gregor J, Gleason S S, Paulus M J. Cone beam X-ray computed tomography with an offset detector array[J]. IEEE Transactions on Image Processing, 2003, 2: 803-806.
[48] Vaz M S, Mersereau R, Spalla G, et al. CT Reconstruction from Truncated Scans[C].Proceeding of 10th Fully 3D Meeting and 2nd HPIR Workshop, 2009.
[49]傅健,路宏年,龚磊.锥束射线三维大视场工业CT成像方法研究[J].光学技术, 2006, 32(20): 209-212.
[50] Noo F, Defrise M, Clackdoyle R, et al. Image reconstruction from fan-beam projections on less than a short-scan[J]. Phys. Med. Biol. ,2002,47:2525-2546.
[51] Grass M, Kohler T, Proksa R. 3D cone-beam CT reconstruction for circular trajectories[J]. Physics in Medicine and Biology, 2000, 45 (2) : 329 -347.
[52] Liu V, Lariviere N R, Wang G. X-ray micro-CT with a displaced detector array: Application to helical cone-beam reconstruction[J]. Med. Phys., 2003, 30(10): 2758-2761.
[53]邹晓兵,曾理.重排的半覆盖螺旋锥束CT的反投影滤波重建[J].光学精密工程, 2010, 18: 2077-2085.
[54] Noo F, Defrise M, Clackdoyle R. Single-slice rebinning method for helical cone-beam CT[J]. Physics in Medicine and Biology, 1999, 44: 561-570.
[55]陈炼,吴志芳,周立业.一种改进的锥束螺旋CT单层重排重建算法[J]. CT理论与应用研究, 2008, 17(1): 7-13.
[56] Li L, Chen Z Q, Zhang L, et al. A backprojection filtered image reconstruction algorithm for circular cone-beam CT[C]. Proceeding of SPIE Medical Imaging, 2006.
[57]傅健,路宏年.一种新颖的ICT扫描方式及其FBP重构算法[J].北京航空航天大学学报, 2003, 29(1): 9-12.
[58]郭艳艳,韩焱,王明泉.大型试件CT重构的一种方法[J].测试技术学报, 2002, 16(4): 305-307.
[59]傅健,路宏年,王宏钧.锥束准三代三维工业CT成像方法研究[J].兵工学报, 2005, 26(6): 776-779.
[60]魏东波,傅健,龚磊,等.大尺寸构件工业CT成像方法[J].北京航空航天大学学报, 2006, 32(12): 1477-1480.
[61] Fu J, Lu H N, Li B, et al. X-CT imaging method for large objects using double offset scan mode[J]. Nuclear Instruments and Methods in Physics Research A, 2007, 575: 519-523.
[62]赵飞,路宏年,孙翠丽.一种新的二维CT扫描方式及其重建算法[J].光学技术, 2006, 32(2): 284-291.
[63]赵飞,路宏年,孙翠丽. RT二维CT扫描的精确重建[J].光学技术, 2006, 32(4): 518-520.
[64] Chen M, Zhang H T, Zhang P. BPF-based reconstruction algorithm for multiple rotation–translation scan mode [J]. Progress in Natural Science, 2008, 18: 209-216.
[65]陈明.工业CT视野拓展方法及图像伪影校正研究[D].北京:首都师范大学, 2008.
[66]曾理,邹晓兵,卢艳平.窄角锥束CT的改进扫描方式[J].仪器仪表学报, 2007, 28(7):1190-1197.
[67] Zou X B, Zeng L, Wang J, et al. Improved scanning and reconstruction of large objects by cone-beam industrial computed tomography[C]. Proceeding of the ninth IASTED international conference on signal and image processing, 2007, 146-151.
[68]邹晓兵.锥束工业CT扫描方式与近似重建算法的改进[D].重庆:重庆大学, 2007.
[69]张朝宗,郭志平,张朋,等.工业CT技术和原理[M].北京:科学出版社, 2009.
[70]温俊海.有限角图像重建方法研究[D].西安:西安交通大学, 1999.
[71]孙杰.图象重建块迭代算法研究[D].北京:北京交通大学, 2007.
[72] Sidky E Y, Kao C, and Pan X C. Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT[J]. Journal of X-Ray Science and Technology, 2006, 14:119-139.
[73] Quinto E T. Local algorithms in exterior tomography[J]. Journal of Computational and Applied Mathematics, 2007, 199:141-148.
[74] Cormack A M. Representation of a function by its line integrals, with some radiological applications[J]. Journal of Applied Physics, 1963, 34:2722-2727.
[75] Bates R and Lewitt R. Image reconstruction from projections: I: general theoretical considerations, II: projection completion methods (theory) III: projection completion methods (computational examples)[J]. Journal of Applied Physics, 1978, 50 I: 19–33, II: 189–204, III: 269-278.
[76] Helgason S. The radon transform on Euclidean spaces, compact two-point homogeneous spaces, and Grassmann manifolds[J]. J Appl Phys, 1963, 34: 2722-2727.
[77] Lewitt R M and Bates R H. Image reconstruction from projections II. Projection completion methods[J]. Optik, 1978, 50: 189-204.
[78] Natterer F. Efficient implementation of‘optimal’algorithms in computerized tomography[J]. Math Methods Appl Sci, 1980, 2: 545-555.
[79] Quinto E T. Singular value decompositions and inversion methods for the exterior radon transform and a spherical transform[J]. Journal of Mathematical Analysis and Applications, 1983, 95:437-448.
[80] Quinto E T. Tomographic reconstructions from incomplete data-numerical inversion of the exterior radon transform[J]. Inverse Problems, 1988, 4:867-876.
[81] Quinto E T. Singularities of the x-ray transform and limited data tomography in R2 and R3[J]. SIAM Journal on Mathematical Analysis, 1993, 24:1215-1225.
[82] Courdurier M, Noo F and Defrise M, et al. Solving the interior problem of computed tomography using a priori knowledge[J]. Inverse Problem, 2008, 24: 1-28.
[83] Zeng L, Liu B D and Liu L H, et al. A new iterative reconstruction algorithm for 2D exterior fan-beam CT[J]. Journal of x-ray science and technology, 2010, 18: 267-277.
[84] Chan T F and Vese L A. Active Contours Without Edges[J]. IEEE Transactions on Image Processing, 2001, 10( 2): 266-277.
[85] Srinivasa N , Ramakrishnan K R and Rajgopal K. Detection of edges from projections[J]. IEEE transactions on medical imaging, 1992, 11(1): 76-80.
[86] Marr D and Hildreth E C. Theory of edge detection[J]. Proc. Roy. Soc. Lond. B., 1980. 207: 187-217.
[87]曾理,易东,陈廷槐.从平行束投影探测断面内边缘的小波方法[J].重庆大学学报, 1996, 19(6): 85-89.
[88]曾理,徐问之,陈廷槐.从扇束射线投影探测断面内边缘的小波方法[J].信号处理, 1998, 14(4): 353-357.
[89] Sastry C S, Mishra A K and Singh A. Detection of edges in CT images directly from projection data[C]. International Conference on Advances in Recent Technologies in Communication and Computing, 129-132, 2009.
[90]王黎明,严壮志,韩焱.基于特征重建的工业CT算法[J].测试技术学报, 2007, 21(1): 49-52.
[91]陈方林.直接重建目标特征的CT算法研究[D].山西:中北大学, 2009.
[92]李璟,李政.离散Radon变换卷积分配性及在从投影直接提取CT图像边缘的应用[J].核电子学与探测技术, 2004, 24(4): 359-362.
[93] Yan C H, Robert T W, BeaupréG S, et al. Reconstruction algorithm for polychromatic CT imaging: application to beam hardening correction[J]. IEEE Transactions on Medical Imaging, 2000, 19(1): 1-11.
[94]庄天戈. CT原理和算法[M].上海:上海交通大学出版社, 1992.
[95] Westenberg M, and Roerdink J, Frequency domain volume rendering by the wavelet x-ray transform[J]. IEEE Trans. on Image Process. 9(7) (2000) 1249-1261.
[96] Desbat L, Roux S, Grangeat P, et al. Sampling conditions of 3D parallel and fan-beam x-ray CT with application to helical tomography[J]. Physics in Medicine and Biology, 2004, 49: 2377-2390.
[97] Mallat S. A wavelet tour of signal processing [M].北京:机械工业出版社, 2003.
[98] Clack R, Defris M. Cone-beam reconstruction by the use of radon transform intermediate functions[J]. Journal of Optical Society America 1994, A, 11(2): 580-585.
[99] Tam K C. Computation of radon data from cone beam data in cone beam imaging[J]. Journal of Nondestructive Evaluation, 1998, 17(1):1-15.
[100] K?hler T, Bontus C, and Koken P. A new approach to handle redundant data in helical cone-beam CT[C]. The 8th International Meeting on Fully Three-dimensional Image Reconstruction in Radiology and Nuclear Medicine. Salt Lake City, Utah, USA, 2005. Utah:University of Utah,2005, 19-22.
[101]徐海军,魏东波,傅健,等.三维Radon变换的一种快速解析方法[J]. CT理论与应用研究, 2008, 17(2): 1-7.
[102] Kak A C, Slaney M. Principles of computerized tomographic imaging[M]. New York: IEEE Press, 1999.
[103]叶云长.计算机层析成像检测[M].北京:机械工业出版社, 2006.
[104] Wang G, Lee S W. Grangeat-type and Katsevich-type algorithm for Cone-beam CT[J]. CT理论与应用研究, 2003, 12(2): 45-55.
[105] Katsevich A, Basu S, and Hsieh J. Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography[J]. Physics in Medicine Biology, 2004, 49: 3089-3103.
[106]刘宝东.工业CT截断投影数据重建算法研究[D].重庆:重庆大学, 2010.
[107] Hsieh J. Computed tomography:principle, design, artifacts and recent advances[M].北京:科学出版社, 2006: 1-71.
[108] Crawford C R, King K. Computed tomography scanning with simultaneous patient translation[J]. Med. Phys., 1990, 17: 967-982.
[109] Kalender W A, King K. Physical performance scanning with simultaneous patient translation[J]. Med. Phys., 1991, 18: 910-915.
[110] Kalender W A. Computed tomography[M]. New York: Wiley, 2000.
[111] Kalender W A. X-ray computed tomography[J]. Physics in Medicine and Biology, 2006, 51: R29–R43.
[112] www.medical.toshiba.com [OL].
[113] Zhang Y. Three-dimensional image quality evaluation and improvement in flat-panel detector based cone-beam CT imaging[D]. New York: University of Rochester, 2009.
[114] Schwarzband G, Kiryati N. The point spread function of spiral CT[J]. Phys. Med. Biol., 2005, 50: 5307-5322.
[115] Colbeth R E, Mollov I P, Roos P G, et al. Flat panel CT detectors for Sub-second volumetric scanning[C]. Proc. SPIE Int. Soc. Opt. Eng., 2005, 5745: 387-398.
[116] Rau A W, Bakueva L, Rowlands J A. The x-ray time of flight method for investigation of ghosting in amorphous selenium-based flat panel medical x-ray imagers[J]. Med. Phys., 2005, 32: 3160-3177.
[117] Conover D, Ning R, Lu X, et al. Small animal imaging using a flat panel-based cone beam computed tomography (FPD-CBCT) imaging system[C]. Proc. SPIE, 2005, 5745: 307-318.
[118] Chen B, and Ning R. Cone beam volume CT Breast imaging: Feasibility Study[J]. Med. Phys., 2002, 29 (5): 755-770.
[119] Glick S J, Thacker S, Gong X, et al. Evaluating the impact of x-ray spectral shape on image quality in flat-panel CT breast imaging[J]. Med. Phys., 2007, 34(l): 5-24.
[120] Zeng K, Chen Z Q. Review of recent developments in cone-beam CT reconstruction algorithms for long-object problems[J]. Image Anal Stereol, 2004, 23: 83-87.
[121] Smith B D. Image reconstruction from cone-beam projections: Necessary and sufficient conditions and reconstruction methods[J]. IEEE Transactions on Medical Imaging, 1985, MI-4 (1): 14-25.
[122] Zeng K. Development and applications of cone-beam imaging techniques[D]. Iowa: University of Iowa, 2007.
[123] Kirillov A A. On a problem of I. M. Gel,fand[J]. Sov. Math. Dokl., 1961, 2: 268-277.
[124] Tuy H K. An inverse formula for cone-beam reconstruction[J]. SIAM J. Appl. Math., 1983, 43: 546-552.
[125] Simth B D, Chen J. Implementation, investigation, and improvement of a novel cone-beam reconstruction method [SPECT][J]. Medical Imaging, IEEE transactions on, 1992, 11(2): 260-266.
[126] Zeng G L, Clack R, Gullberg G T. Implementation of Tuy,s cone-beam inversion formula[J]. Phys. Med. Biol., 1994, 39(3): 493-507.
[127] Tam K C. Method and apparatus for converting cone-beam x-ray projection data to planar integral and reconstructing a three-dimensional computerized tomography (CT) image of an object[P]. US, 1995.
[128] Tam K C, Samarasekera S, Sauer F. Exact cone beam CT with a spiral scan[J]. Phys. Med. Biol., 1998, 43(4): 1015-1024.
[129] Kudo H, Noo F, Defrise M. Quasi-exact filtered backprojection algorithm for long-object problem in helical cone-beam tomography[J]. IEEE Trans. Med. Imaging, 2000, 19(9): 902-921.
[130] Katsevich A. Theoretically exact filtered backprojection type inversion algorithm for spiral CT [J]. SIAM J. APPL. MATH., 2002, 62(6): 2012–2026.
[131] Katsevich A. Analysis of an exact inversion algorithm for spiral cone-beam CT[J]. Physics in Medicine Biology, 2002, 47: 2583-2597.
[132] Katsevich A. An improved exact filtered backprojection algorithm for spiral computedtomography[J]. Advance in Applied Mathematics, 2004, 32: 681-697.
[133] Zou Y and Pan X C. Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT[J]. Physics in Medicine Biology, 2004, 49: 941-959.
[134] Zhao S Y, Yu H Y, Wang G. A unified framework for exact cone-beam reconstruction formulas[J]. Med. Phys., 2005, 32(6): 1712-1721.
[135] Ye Y B, Wang G. Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve[J]. Med. Phys., 2005, 32(1): 42-48.
[136] Yang H Q, Li M H, Koizumi K, et al. Exact cone beam reconstruction for a saddle trajectory[J]. Physics in Medicine and Biology, 2006, 51 1157-1172.
[137] Katsevich A. Image reconstruction for the circle-and-arc trajectory[J]. Phys. Med. Biol., 2005, 50(10): 2249-2265.
[138] Turbell H. Cone-beam reconstruction using filtered backprojection[D]. Link?pings universitet, Link?ping, Sweden, 2001.
[139] Hu H. An improved cone-beam reconstruction algorithm for the circular orbit[J]. scanning, 1996, 18: 572-581.
[140] Zeng K, Chen Z Q, Zhang L, et al. An error-reduction-based algorithm for cone-beam computed tomography[J]. Med. Phys., 2004, 31(12): 3206-3212.
[141] Tuy H K. 3D image reconstruction for helical partial cone beam scanners using wedge beam transform[P]. US, 2000.
[142] Tang X, Hsieh J. A filtered backprojection algorithm for cone beam reconstruction using rotational filtering under helical source trajectory[J]. Med. Phys., 2004, 31(11): 2949-2960.
[143] Yan M, Zhang C. Tilted plane Feldkamp type reconstruction algorithm for spiral cone beam CT[J]. Med. Phys., 2005, 32(11): 3455-3467.
[144] Tang N. Web-knowledge-based learning for ill-posed inverse problems[D]. California, University of California, 2006.
[145] Yendiki A and Fessler J A. A comparison of rotation- and blob-based system models for 3D SPECT with depth-dependent detector response [J]. Physics in Medical and Biology, 2004, 49: 2157-2168.
[146] Herman G T. Fundamentals of computerized tomography: image reconstruction from projections[M]. Springer, 2009: 119-122.
[147] Henson V E, Limber M A, Mccormick S F, et al. Multilevel Image Reconstruction with Natural Pixels[J]. SIAM Journal on Scientific Computing, 1995.
[148] Bevilacqua R, Bozzo E, Menchi O. Comparison of four natural pixel bases for SPECT imaging[J]. Journal of Computational and Applied Mathematics, 2007, 198: 361-377.
[149] Martin C. Voxel-based computational models of real human anatomy: a review[J]. Radiation and Environmental Biophysics, 2004, 42(4): 229-235.
[150] Wang G, YU H Y, and Man B D. An outlook on x-ray CT research and development[J]. Medical Physics, 2008, 35(3).
[151] Gorden R, 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.
[152] Gilbert P. Iterative methods for the three-dimensional reconstruction of an object from projections[J]. J. Theor. Biol., 1972, 36: 105-117.
[153] Jiang M and Wang G. Development of iterative algorithms for image reconstruction[J]. Journal of X-Ray Science and Technology, 2002, 10: 77-86.
[154] Andersen A H, Kak A C. Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm[J]. Ultrasonic Imaging, 1984, 6: 81-94.
[155] Byrne C L. Applied iterative methods[M]. Wellesley, 2008.
[156] Stefan W. Total variation regularization for linear ill-posed inverse problems: extension and application[D]. Arizona, Arizona State University, 2008.
[157] Demoment G. Image reconstruction and restoration: overview of common estimation structures and problems[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1989, 37: 2024-2036.
[158]刘继军.不适定问题的正则化方法及应用[M].北京:科学出版社, 2005.
[159] Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms[J]. Phys. D., 1992, 60: 259-268.
[160] Candes E, Romberg J, and Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency Information[J]. IEEE Transactions on Information Theory, 2006, 52(2): 489-509.
[161] Candes E, Romberg J, and Tao T, Stable signal recovery from incomplete and inaccurate measurements[J]. Communications on Pure and Applied Mathematics, 2006, 59(8): 1207-1223.
[162] Vogel C R. Computational methods for inverse problems[M]. SIAM, 2002.
[163] Salina F V, Mascarenhas N D A, Cruvinel P E. A comparison of POCS algorithms for tomographic reconstruction under noise and limited view[C]. Proceedings of the XV Brazilian Symposium on Computer Graphics and Image Processing, 2002.
[164] Sidky E Yand Pan X C. Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization[J]. Physics in Medical and Biology, 2008, 53:4777-4807.
[165] Zou X B, Zeng L and Li Z J. Dual helical cone-beam CT for inspecting large object[J]. Journal of X-Ray Science and Technology[J]. 2009,17: 233 -251.
[166]徐茂林,邱钧,范惠荣,等.用一种新滤波函数作CT图像局部重建[J].计算物理, 2004, 21(3): 362-368.
[167]范惠荣,徐茂林,邱钧,等.关于CBP算法的一种新型滤波函数和它的性质[J].电子学报, 2004, 32(2): 232-235.
[168]郑君里,应启珩,杨为理.信号与系统[M].北京:高等教育出版社,2000.
[169] Sun Y, Hou Y, and Hu J C. Reduction of artifacts induced by misaligned geometry in cone-beam CT[J]. IEEE Transactions on Biology Engineering, 2007, 54: 1461-1471.
[170]郑罡,王惠南.塔式多相水平集三维医学图像多目标分割算法[J].生物医学工程学杂志, 2008, 25(5): 989-993.
[171]向才兵.基于CV模型的工业CT图像裂纹检测算法研究[D].重庆:重庆大学, 2010.
[172] Daubechies I. Ten lectures on wavelets[M]. Philadelphia: SIAM Publication, 1992.
[173] Weber Z. The radon transform and its application of the processing of the VSP data[J]. Magyar Geofizika, 1994, 33(1): 22-35.
[174] Chui C. An introduction to wavelets[M]. USA: Academic Press, 1992.
[175]龙瑞麟.高维小波分析[M].北京:世界图书出版公司, 1995: 47-124.
[176]曾理,唐远炎,杨万年,等.二维母小波函数的投影[J].工程数学学报, 2000,17(2):8-12.
[177] Zeng L, Ma R and Huang J Y. The construction of 2D rotationally invariant wavelets and their application in image edge detection[J]. Int. J. Wavelets Multiresolut. Inf. Process., 2008, 6(1): 65-82.
[178] Yang J Y, Wang Y and Xu W L. Image and video denoising using adaptive dual-tree discrete wavelet packets[J], IEEE Trans. on Circuits and Syst. for Video Tech., 2009, 19(5): 642-655.
[179] Shi S, Zhang Y N and Hu Y N, A wavelet-based image edge detection and estimation method with adaptive scale selection[J]. Int. J. Wavelets Multiresolut. Inf. Process., 2010, 8(3): 385-404.
[180] Zao C B, Xin G Y, Chun F X, et al. Image denoising by using non-tensor product wavelets filter banks[C]. Proceedings of the Sixth International Conference on Machine Learning and Cybernetics, Hong Kong, China, 2007, 1734-1738.
[181] You X G, Zhang D, Chen Q H. Face representation by non-tensor product wavelets[C]. Proceedings of the 18th International Conference on Pattern Recognition, ICPR, 2006.
[182] Cohen A, Schlenker J M. Compactly supported bidimension wavelet bases with hexagonal symmetry[J]. Constr. Approx., 1993, 209-236.
[183] Belogay E, Wang Y. Arbitrary smooth orthogonal non-separable wavelets in R[J]. SIAM J. Math. Anal., 1999, 30: 678-697.
[184]曾理,杨万年,张世清.从一维母小波反投影二维母小波.工程数学学报[J], 1999,16(2):57-62.
[185] Mallat S and Hwang W L. Singularity detection and processing with wavelets[J]. IEEE Trans. Inform. Theory, 1992, 38(3): 617-643.
[186] Sahiner B and Yagle A E. Reconstruction from projection under time-frequency constraints[J]. IEEE transaction on medical imaging, 1995 14(2): 193-204.