Ultrasonic tomographic reconstruction considering wood anisotropy

We proposed an approach to solve the inverse problem for tree imaging, using a method based on ultrasound travel-time tomography and adapted to the anisotropy of wood material. The proposed inversion procedure was developed between 2015-2018. The method considers the dependency of the wave velocity on the angle of propagation and the wood elastic constants, as described by the Christoffel equation. This dependency leads to curved trajectories (or rays), that are followed by the wave from the source to the receivers, affecting the time-of-flight (TOF) measurements. A first inversion is performed considering an hypothesis of straight-line rays, using an algebraic reconstruction algorithm (SIRT), adapting the matrix formulation to obtain the slowness for every pixel as a polynomial approximation of the Christoffel equation. Then, the obtained set of slowness functions are used to perform the direct (or forward) problem, simulating the wave propagation using a raytracing approach. This simulation allows to estimate the curved trajectories and synthetic TOF values, that are then compared to the experimental measurements. This process is repeated iteratively until convergence. From the final iteration, the polynomial approximations of the Christoffel equations for each pixel are used to obtain the elastic constants via a non-linear regression. Finally, the parametric image is presented. The data folder contains the time-of-flight measurements (in seconds) and the sensors positions (in meters) from an experimental setting using pine samples. A photography of the cross-section is included. TOF values are organized as a matrix (sinogram), where columns correspond to each source, and receivers to each row. Sensors positions are presented as a column vector, specifying the [X,Y] coordinates. Three different configurations were tested: (1) a healthy case, (2) a centered defect case and (3) an off-centered defect case. Defects were simulated by drilling a circular hole of diameter 7.6 cm. The forward folder contains the code to perform the raytracing simulation (direct problem), used in the iterative inversion schema. The main file is rayPix‧m, that performs the raytracing. The inverse folder contains the code to perform the image reconstruction (inverse problem). The main file is optimFinal‧m, where the iterative schema was implemented. To perform an inversion, the directives are in mainInv‧m

本研究提出一种针对木材层析成像的逆问题求解方案，采用基于超声走时层析成像的方法，并适配木材材料的各向异性特性。所提出的反演流程开发于2015年至2018年间。该方法考虑了波速随传播角度及木材弹性常数的变化关系，该关系由克里斯托费尔方程（Christoffel equation）描述。此种变化关系会导致波的传播轨迹（或称射线路径）呈弯曲形态，声波从发射源传播至接收传感器的过程中会遵循此类弯曲轨迹，进而对飞行时间（time-of-flight, TOF）测量结果产生影响。首先，基于直射线假设开展首次反演，采用同步迭代重建技术（Simultaneous Iterative Reconstruction Technique, SIRT），通过调整矩阵公式，将每个像素的慢度以克里斯托费尔方程的多项式近似形式求解得到。随后，利用所得到的慢度函数集开展正演（或称直接）问题求解，采用射线路径追踪方法模拟声波传播过程。通过该模拟可估算出弯曲射线路径与合成飞行时间数值，并将其与实验测量结果进行对比。上述流程将迭代重复直至收敛。在最终迭代步骤中，通过非线性回归方法，利用每个像素对应的克里斯托费尔方程多项式近似结果求解得到木材弹性常数。最终生成参数化成像结果。数据文件夹包含针对松木样本的实验场景中采集的飞行时间测量数据（单位：秒）与传感器位置数据（单位：米），同时附带样本横截面的实拍照片。飞行时间数据以矩阵（正弦图，sinogram）形式组织，其中列对应每个发射源，行对应各个接收传感器。传感器位置以列向量形式给出，标注有各传感器的[X,Y]坐标。共测试了三种不同的实验配置：(1) 无缺陷健康样本；(2) 中心带缺陷样本；(3) 偏心带缺陷样本。缺陷通过钻取直径7.6厘米的圆孔进行模拟。正演文件夹包含用于射线路径追踪模拟（正演问题）的代码，该代码应用于迭代反演流程中。其主文件为rayPix.m，用于执行射线路径追踪。反演文件夹包含用于图像重建（反演问题）的代码，其主文件为optimFinal.m，该文件实现了迭代反演流程。如需执行反演操作，可参考mainInv.m中的操作说明。