Naik Naren, Vasu R M, Ananthasayanam M R
Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208016, India.
Appl Opt. 2010 Feb 20;49(6):986-1000. doi: 10.1364/AO.49.000986.
The problem of reconstruction of a refractive-index distribution (RID) in optical refraction tomography (ORT) with optical path-length difference (OPD) data is solved using two adaptive-estimation-based extended-Kalman-filter (EKF) approaches. First, a basic single-resolution EKF (SR-EKF) is applied to a state variable model describing the tomographic process, to estimate the RID of an optically transparent refracting object from noisy OPD data. The initialization of the biases and covariances corresponding to the state and measurement noise is discussed. The state and measurement noise biases and covariances are adaptively estimated. An EKF is then applied to the wavelet-transformed state variable model to yield a wavelet-based multiresolution EKF (MR-EKF) solution approach. To numerically validate the adaptive EKF approaches, we evaluate them with benchmark studies of standard stationary cases, where comparative results with commonly used efficient deterministic approaches can be obtained. Detailed reconstruction studies for the SR-EKF and two versions of the MR-EKF (with Haar and Daubechies-4 wavelets) compare well with those obtained from a typically used variant of the (deterministic) algebraic reconstruction technique, the average correction per projection method, thus establishing the capability of the EKF for ORT. To the best of our knowledge, the present work contains unique reconstruction studies encompassing the use of EKF for ORT in single-resolution and multiresolution formulations, and also in the use of adaptive estimation of the EKF's noise covariances.
利用两种基于自适应估计的扩展卡尔曼滤波器（EKF）方法，解决了光学折射断层扫描（ORT）中基于光程差（OPD）数据重建折射率分布（RID）的问题。首先，将基本的单分辨率EKF（SR-EKF）应用于描述断层扫描过程的状态变量模型，以便从有噪声的OPD数据中估计光学透明折射物体的RID。讨论了与状态和测量噪声相对应的偏差和协方差的初始化。对状态和测量噪声偏差及协方差进行自适应估计。然后将EKF应用于小波变换后的状态变量模型，以产生基于小波的多分辨率EKF（MR-EKF）求解方法。为了从数值上验证自适应EKF方法，我们通过标准静态案例的基准研究对其进行评估，在这些研究中可以获得与常用的高效确定性方法的比较结果。对SR-EKF和两个版本的MR-EKF（使用哈尔小波和Daubechies-4小波）进行的详细重建研究，与从（确定性）代数重建技术的一个典型变体——平均每投影校正法获得的结果进行了很好的比较，从而确立了EKF用于ORT的能力。据我们所知，目前的工作包含了独特的重建研究，涵盖了在单分辨率和多分辨率公式中使用EKF进行ORT，以及对EKF噪声协方差的自适应估计。
