利用秩和条件数分析透射式超声CT的数据完备性

为了优化探头设计、估计重构图像误差,提出了一种利用秩和条件数分析透射式超声CT数据完备性的方法。首先分析了透射式超声CT的原理,透射式超声CT的测量方程可以表示成线性方程组,进而透射式超声CT的重构问题可以转化为线性方程组的求解问题。对于所选用的设计方案,描述线性方程组的可解性和解的稳定性的秩和条件数可以用来评估其数据完备性、估计重构图像误差。数学分析和试验结果表明,只有在测量方程组满秩时才可以求得唯一解,在测量误差和介质不均匀性相同时,重构误差近似正比与方程组的条件数,因此可以使用测量方程组的秩和条件数来衡量设计方案的数据完备性,指导实际装置的设计。 

Analysis of data adequacy in transmission-mode ultrasound computed tomography using rank and condition number

In order to optimize the data acquisition plan design and estimate the final error ranges of reconstruction images, a data adequacy analysis method of transmission-mode ultrasound computed tomography (TUCT) using rank and condition number is proposed. By analyzing the principle of TUCT, measurement equations are presented by linear equations set, which transforms the reconstruction problem of TUCT into the problem of solving linear equations set. The rank and condition number which indicate solvability and stability of the linear equations set are used to analysis the data adequacy, and estimates the error range of reconstruction images. Theoretical analysis and experiment show that the reconstruction problem is solvable if and only if the linear equations set is full rank and the final error ranges of reconstruction images are approximatively proportional to the condition numbers of the coefficient matrixes if measurement error and medium nonhomogeneity are identical. Thus the rank and condition number can be used to analysis the data adequacy of the collected data and evaluate the final errors, which would help optimizing equipment design.