RETRACTED ARTICLE: A robust vector field correction method via a mixture statistical model of PIV signal

Abstract

Outlier (spurious vector) is a common problem in practical velocity field measurement using particle image velocimetry technology (PIV), and it should be validated and replaced by a reliable value. One of the most challenging problems is to correctly label the outliers under the circumstance that measurement noise exists or the flow becomes turbulent. Moreover, the outlier’s cluster occurrence makes it difficult to pick out all the outliers. Most of current methods validate and correct the outliers using local statistical models in a single pass. In this work, a vector field correction (VFC) method is proposed directly from a mixture statistical model of PIV signal. Actually, this problem is formulated as a maximum a posteriori (MAP) estimation of a Bayesian model with hidden/latent variables, labeling the outliers in the original field. The solution of this MAP estimation, i.e., the outlier set and the restored flow field, is optimized iteratively using an expectation–maximization algorithm. We illustrated this VFC method on two kinds of synthetic velocity fields and two kinds of experimental data and demonstrated that it is robust to a very large number of outliers (even up to 60 %). Besides, the proposed VFC method has high accuracy and excellent compatibility for clustered outliers, compared with the state-of-the-art methods. Our VFC algorithm is computationally efficient, and corresponding Matlab code is provided for others to use it. In addition, our approach is general and can be seamlessly extended to three-dimensional-three-component (3D3C) PIV data.

Appendix: The derivation of EM algorithm

It is very difficult to find the closed-form solution to this minimization problem (Eq. 9). As a typical iterative method, the expectation–maximization framework alternates constructing an upper bound and minimizing this bound, and thus provides a numerical solution. Here, we use Jensen’s inequality to construct the upper bound. $(\mathsf {E}(\varphi (X))\le \varphi (\mathsf {E} (X))$, where $\varphi$ is a concave function, and $\mathsf {E}$ is the expectation in probability theory.)

$$\begin{aligned} E(\varvec{\theta })= & {} -\hbox {ln}(p(\mathbf f ))-\sum _{i=1}^N \hbox {ln}\left( \sum _{z_i}p(\mathbf v _i,z_i|\mathbf x _i,\varvec{\theta })\right) \nonumber \\= & {} -\hbox {ln}(p(\mathbf f ))-\sum _{i=1}^N \hbox {ln}\left( \sum _{z_i}\left( p\left( z_i|\mathbf x _i,\mathbf v _i,\varvec{\theta } \right) \cdot \frac{p(\mathbf v _i,z_i|\mathbf x _i, \varvec{\theta })}{p(z_i|\mathbf x _i,\mathbf v _i, \varvec{\theta })}\right) \right) \nonumber \\\le & {} -\hbox {ln}(p(\mathbf f ))-\sum _{i=1}^N \sum _{z_i}\left( p \left( z_i|\mathbf x _i,\mathbf v _i,\varvec{\theta } \right) \cdot \hbox {ln}\left( \frac{p\left( \mathbf v _i,z_i|\mathbf x _i, \varvec{\theta } \right) }{p\left( z_i|\mathbf x _i, \mathbf v _i, \varvec{\theta }\right) } \right) \right) \nonumber \\= & {} -\mathscr {Q}( \varvec{\theta }) \end{aligned}$$

(19)

where $E(\varvec{\theta })$ is the negative log posterior function and $\mathscr {Q}( \varvec{\theta })$ denotes the complete-data log posterior. We just change the position of sum operation and ln function to make the formula tractable. When a certain condition has been reached, the inequality will become equal. The condition is

$$\begin{aligned} \frac{p(\mathbf v _i,z_i|\mathbf x _i,\varvec{\theta })}{p(z_i|\mathbf x _i,\mathbf v _i,\varvec{\theta })} = \hbox {const for}\, z_i=1\, \hbox {and}\,z_i=0 \end{aligned}$$

(20)

i.e.,

$$\begin{aligned} \frac{p(\mathbf v _i,z_i=1|\mathbf x _i, \varvec{\theta })}{p(z_i=1|\mathbf x _i, \mathbf v _i,\varvec{\theta })}&= \frac{p(\mathbf v _i,z_i=0|\mathbf x _i, \varvec{\theta })}{p(z_i=0|\mathbf x _i, \mathbf v _i,\varvec{\theta })}\nonumber \\&= \frac{{p(\mathbf v _i,z_i=0|\mathbf x _i,\varvec{\theta })} + p(\mathbf v _i,z_i=1|\mathbf x _i,\varvec{\theta })}{{p(z_i=0|\mathbf x _i,\mathbf v _i,\varvec{\theta })} + {p(z_i=1|\mathbf x _i,\mathbf v _i,\varvec{\theta })}} \nonumber \\&= {p(\mathbf v _i|\mathbf x _i,\varvec{\theta })} \end{aligned}$$

(21)

The expectation step (E-step) result is thus got from this condition. The Eq. 11 is the specific form with probability substitution.

$$\begin{aligned} p(z_i=1|\mathbf x _i,\mathbf v _i,\varvec{\theta }) = \frac{p(\mathbf v _i,z_i=1|\mathbf x _i,\varvec{\theta })}{p(\mathbf v _i|\mathbf x _i,\varvec{\theta })} \end{aligned}$$

(22)

As the $p(z_i|\mathbf x _i,\mathbf v _i,\varvec{\theta })$ is determined in E-step, the complete-data log posterior should be expressed more precisely with symbol $\varvec{\theta }^\mathrm{old}$.

$$\begin{aligned} \mathscr {Q}\left( \varvec{\theta },\varvec{\theta }^\mathrm{old}\right)&= \hbox {ln}(p(\mathbf f )) \nonumber \\&+\sum _{i=1}^N \sum _{z_i}\left( p \left( z_i|\mathbf x _i,\mathbf v _i,\varvec{\theta }^\mathrm{old} \right) \cdot \hbox {ln}\left( \frac{p\left( \mathbf v _i,z_i|\mathbf x _i, \varvec{\theta }\right) }{p\left( z_i|\mathbf x _i, \mathbf v _i, \varvec{\theta }^\mathrm{old} \right) } \right) \right) \end{aligned}$$

(23)

The Eq. 10 is the expansion of this equation, and we omit some terms that are independent of $\varvec{\theta }$. For a complete derivative process, refer to the author’s personal research Web site.