Direct background-oriented schlieren tomography using radial basis functions

Abstract

Background-oriented schlieren tomography (BOST) is effective for flow field measurement; however, different from general computed tomography (CT), the BOST utilizes the deflection of rays passing through an inhomogeneous field for measurement. It is sensitive to the refractive index gradient. Therefore, an additional integration step is typically employed to obtain the refractive index. In this article, a calculation method of projection matrix is proposed based on the radial basis function (RBF). The 3D distribution of the refractive index can be reconstructed directly. This method was first verified by numerical simulation. Then, the 3D instantaneous refractive index field above a candle flame was measured. The reprojection error was calculated by ray tracing. The results illustrate the accuracy and stability of the proposed method. This research provides a new and complete solution for the 3D instantaneous flow field (refractive index, density, or temperature) measurement.

A quantitative reconstruction method for background-oriented schlieren by reducing the weight matrix

“…The discretization of the BOS projection model and the computation of the weight matrix based on the radial basis function 11 are briefly described in this section. The purpose of this step is to convert the Eq.…”

“…If there are K rays sampled from all projections, the sizes of W , c and ε are 3K N  , 1 N  and 3 1 K  are determined accordingly. For example, the size of W used in Ref 11 is 839160 12800000  . Although the weight matrix contains many zero elements and can be stored as a sparse matrix, it is still a relatively large matrix 11 .…”

“…For example, the size of W used in Ref 11 is 839160 12800000  . Although the weight matrix contains many zero elements and can be stored as a sparse matrix, it is still a relatively large matrix 11 .…”

“…However, directly obtaining the 3D displacement using Equation ( 3) is not possible because there are only two equations with three unknowns. In previous studies, the 3D displacement was obtained by decomposing the 2D image displacement into the world coordinate system 7,11 However, the accuracy of the 3D displacement may be influenced by both the decomposition process and the errors resulting from camera calibration. Consequently, the calculation of the weight matrix will have an impact on the time and memory consumption as well as the accuracy of the reconstruction results.…”

“…This method achieved a significant speedup of 62.5% compared to traditional BOST. In 2022, Cai et al proposed a direct reconstruction method using radial basis functions 11 . Typically, establishing a system of linear equations based on the BOS projection model requires extensive computation for the weight matrix, which can be time-consuming and result in a large matrix.…”