高速高精度解析式图像重建算法研究
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摘要
CT (Computed Tomography),即计算机断层成像技术,是医学和工业中的一种优质无损诊断技术,其本质是利用投影数据重建物体断层图像。高分辨率重建、大物体重建和三维重建的工程需求使得投影数据量激增,使重建出的CT图像的数据量也激增,从而重建速度就成为一个瓶颈。
精密部件的检测对图像重建提出了更高的精度要求,如何直接重建出满足工程目标的CT图像也是CT应用中亟待解决的关键问题。本文致力于重建精度的提高、重建速度的提高以及目标重建等三个方面的研究。
论文首先研究了滤波反投影(FBP)算法的离散实现公式,利用信号处理理论及定积分的定义,推导出了它应该包含投影采样间隔和角度采样间隔两个参数,并作为整个公式的因子。利用定积分数值计算方法,并根据反投影的特点,提出了基于辛普森公式的精确反投影方法。将插值方法应用于反投影的过程,仿真结果表明线性插值比样条插值效果更好,并给出了定性解释。
在分析斜变滤波器特点的基础上,提出了一种斜变滤波器的设计原则,分析了频域加窗和时域加窗对重建的影响,确定了斜变滤波器单位脉冲响应的长度选择方法和一种理想斜变滤波器单位脉冲响应的中心值计算方法。仿真结果表明,频域中指数窗性能最优而时域中矩形窗性能最优,单位脉冲响应的长度为(?)-1时,CT图像无空气伪像。
根据投影信号和滤波器单位脉冲响应都为长序列的特点,提出了基于算术傅里叶变换(AFT)的滤波过程的加速方法,以及基于快速哈达玛变换(FHT)的滤波过程的加速方法。仿真结果表明,AFT滤波算法比FFT滤波算法快30%,而FHT滤波算法比FFT滤波算法快近50%。
本文研究了反投影过程的三种加速方法：对每个投影预插值到更精细的间隔,反投影时采用最快的零阶插值方法；采用被动式射线驱动法反投影；以空间换时间的策略来预存储三角函数值而避免反投影时复杂的三角运算。在此基础上,提出了一种组合反投影加速方法。仿真实验表明,比传统方法快4倍多。
在“特征重建”概念的基础上,提出了“目标重建”的概念,并设计了目标重建的三种实现方法：近似法、空间域精确法、频域精确法。实际实验表明：近似法重建效果一般,而精确法重建效果良好,且可以有效避免非白噪声对CT图像后处理的影响。
论文从成像理论的角度,以解析式图像重建算法为研究对象,研究了加速方法、精度提高方法以及目标重建的方法,研究结果对解析式图像重建算法具有很好的推动作用。
CT (Computed Tomography) is an excellent non-destructive diagnosis technology in medical and industrial science. Its essence is to reconstruct an object's tomographic image using its projection data. Due to the engineering demands of high resolution reconstruction, big object reconstruction and three-dimensional reconstruction, the data of CT images become larger and larger. So the reconstruction speed becomes a bottleneck.
CT testing for precise products needs a very high precision and how to reconstruct a CT image directly is also a key and challenging task in CT engineering. This dissertation focuses on reconstruction speedup, precision improving and objective reconstruction.
Firstly, the discretization formulas of FBP algorithm, proved by using signal processing theory and definite integral's definition, were studied, in which there are two key parameters, projection sampling interval and angle sampling interval. Numerical calculating method of definite integral was studied and a precise back projection method based on the Sampson formula was proposed. Simulation proved that the linear interpolation method is better than the spline interpolation. And a qualitative explanation was also given.
On the base of analyzing the characteristics of the ramp filter, a design principle of the ramp filter was proposed. The impact of spectrum window-adding and time domain window-adding on reconstruction were analyzed. A length selection method for the unit impulse response of the ramp filter was proposed. A novel calculation method for the central value of the unit impulse response of the ideal ramp filter was adopted. The simulation results demonstrate that the exponent window is the best for spectrum domain and the rectangle window is the best for time domain. There is not air artifact in CT image when the length of the unit impulse response is 2(?)N-1.
According to the characteristic that the projection and the unit impulse response are both long sequences, arithmetic Fourier transform (AFT) and fast Hadamard transform (FHT) were proposed respectively aiming to speedup methods for filtering process. Simulation results shows that the AFT-filtering method is faster than the FFT-filtering method about 30% and the FHT-filtering method is faster than the FFT-filtering method about 50%.
Three speedup methods were studied. One is that the projection was interpolated to smaller intervals and the zero-order interpolation method was adopted for back projection. And one is that a passive ray-driven method was used for back projection. And the last one is that based on the strategy of "time advantage based on space occupancy", the values of the trigonometric functions were stored in advance to avoid complicated trigonometric calculations. Then a hybrid speedup method was proposed. Simulation results show that the method is faster than the traditional method for more than 4 times.
The conception of objective reconstruction was proposed based on analyzing the conception of characteristic reconstruction. Three implementation methods were achieved, i.e., approximate method, precise method of space domain and precise method of spectrum domain. Practical experiments illustrate that the effect of the approximate method is not satisfactory and the others are effective, which can avoid effectively the impact of the not-white noise on CT image processing.
For the analytical image reconstruction, speedup methods, precision development methods and objective reconstruction methods were studied in detail. The research results may provide an effective promotion on present analytical image reconstruction methods.
The dissertation consists of eight chapters, with 58 figures,25 tables, and 110 references.
精密部件的检测对图像重建提出了更高的精度要求,如何直接重建出满足工程目标的CT图像也是CT应用中亟待解决的关键问题。本文致力于重建精度的提高、重建速度的提高以及目标重建等三个方面的研究。
论文首先研究了滤波反投影(FBP)算法的离散实现公式,利用信号处理理论及定积分的定义,推导出了它应该包含投影采样间隔和角度采样间隔两个参数,并作为整个公式的因子。利用定积分数值计算方法,并根据反投影的特点,提出了基于辛普森公式的精确反投影方法。将插值方法应用于反投影的过程,仿真结果表明线性插值比样条插值效果更好,并给出了定性解释。
在分析斜变滤波器特点的基础上,提出了一种斜变滤波器的设计原则,分析了频域加窗和时域加窗对重建的影响,确定了斜变滤波器单位脉冲响应的长度选择方法和一种理想斜变滤波器单位脉冲响应的中心值计算方法。仿真结果表明,频域中指数窗性能最优而时域中矩形窗性能最优,单位脉冲响应的长度为(?)-1时,CT图像无空气伪像。
根据投影信号和滤波器单位脉冲响应都为长序列的特点,提出了基于算术傅里叶变换(AFT)的滤波过程的加速方法,以及基于快速哈达玛变换(FHT)的滤波过程的加速方法。仿真结果表明,AFT滤波算法比FFT滤波算法快30%,而FHT滤波算法比FFT滤波算法快近50%。
本文研究了反投影过程的三种加速方法：对每个投影预插值到更精细的间隔,反投影时采用最快的零阶插值方法；采用被动式射线驱动法反投影；以空间换时间的策略来预存储三角函数值而避免反投影时复杂的三角运算。在此基础上,提出了一种组合反投影加速方法。仿真实验表明,比传统方法快4倍多。
在“特征重建”概念的基础上,提出了“目标重建”的概念,并设计了目标重建的三种实现方法：近似法、空间域精确法、频域精确法。实际实验表明：近似法重建效果一般,而精确法重建效果良好,且可以有效避免非白噪声对CT图像后处理的影响。
论文从成像理论的角度,以解析式图像重建算法为研究对象,研究了加速方法、精度提高方法以及目标重建的方法,研究结果对解析式图像重建算法具有很好的推动作用。
CT (Computed Tomography) is an excellent non-destructive diagnosis technology in medical and industrial science. Its essence is to reconstruct an object's tomographic image using its projection data. Due to the engineering demands of high resolution reconstruction, big object reconstruction and three-dimensional reconstruction, the data of CT images become larger and larger. So the reconstruction speed becomes a bottleneck.
CT testing for precise products needs a very high precision and how to reconstruct a CT image directly is also a key and challenging task in CT engineering. This dissertation focuses on reconstruction speedup, precision improving and objective reconstruction.
Firstly, the discretization formulas of FBP algorithm, proved by using signal processing theory and definite integral's definition, were studied, in which there are two key parameters, projection sampling interval and angle sampling interval. Numerical calculating method of definite integral was studied and a precise back projection method based on the Sampson formula was proposed. Simulation proved that the linear interpolation method is better than the spline interpolation. And a qualitative explanation was also given.
On the base of analyzing the characteristics of the ramp filter, a design principle of the ramp filter was proposed. The impact of spectrum window-adding and time domain window-adding on reconstruction were analyzed. A length selection method for the unit impulse response of the ramp filter was proposed. A novel calculation method for the central value of the unit impulse response of the ideal ramp filter was adopted. The simulation results demonstrate that the exponent window is the best for spectrum domain and the rectangle window is the best for time domain. There is not air artifact in CT image when the length of the unit impulse response is 2(?)N-1.
According to the characteristic that the projection and the unit impulse response are both long sequences, arithmetic Fourier transform (AFT) and fast Hadamard transform (FHT) were proposed respectively aiming to speedup methods for filtering process. Simulation results shows that the AFT-filtering method is faster than the FFT-filtering method about 30% and the FHT-filtering method is faster than the FFT-filtering method about 50%.
Three speedup methods were studied. One is that the projection was interpolated to smaller intervals and the zero-order interpolation method was adopted for back projection. And one is that a passive ray-driven method was used for back projection. And the last one is that based on the strategy of "time advantage based on space occupancy", the values of the trigonometric functions were stored in advance to avoid complicated trigonometric calculations. Then a hybrid speedup method was proposed. Simulation results show that the method is faster than the traditional method for more than 4 times.
The conception of objective reconstruction was proposed based on analyzing the conception of characteristic reconstruction. Three implementation methods were achieved, i.e., approximate method, precise method of space domain and precise method of spectrum domain. Practical experiments illustrate that the effect of the approximate method is not satisfactory and the others are effective, which can avoid effectively the impact of the not-white noise on CT image processing.
For the analytical image reconstruction, speedup methods, precision development methods and objective reconstruction methods were studied in detail. The research results may provide an effective promotion on present analytical image reconstruction methods.
The dissertation consists of eight chapters, with 58 figures,25 tables, and 110 references.
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