An Optimization and Parametric Study of a Schlieren Motion Estimation Method

Abstract

Schlieren imaging is a widely used technique for flow visualization in turbulence and combustion investigations due to its high sensitivity, flexibility and easiness in use. With the development of digital imaging and image processing techniques, it is possible to retrieve velocity measurements using time-resolved schlieren imaging sequences. In this paper, an optimization and parametric study has been conducted on a newly proposed schlieren motion estimation (SME) algorithm, based on the high speed schlieren images of a jet flow and the transient ignition process of impinging flames. The SME algorithm is optimized using a graduated non-convexity (GNC) computing scheme, which employs a three stage strategy by linearly combining a convex quadratic function and a slightly non-convex generalized Charbonnier function. The Euler–Lagrange equations have been derived, while the penalty function was separated so that penalty functions can be changed conveniently. Parametric investigations have been conducted to discuss the influence of weight parameters, while the suitable ranges have been obtained after intensive calculations. Comprehensive comparisons have been made between the SME and GNC-SME methods, which indicates that the GNC scheme can preserve the boundary well and avoid local divergence and over-smoothness at the same time. The suitable weight parameter range is also broadened by using the GNC technique. The better robustness of GNC-SME method makes it more adaptive to various applications.

Appendix

A. Unknown variables separation.

Equation (14) is valid under the conditions of

$$\mathop {{\text{lim}}}\limits_{x \to 0} \frac{\varphi ^{\prime}\left( x \right)}{x} = \mathop {{\text{lim}}}\limits_{{x_{0} \to 0}} \frac{{\varphi ^{\prime}\left( {x_{0} } \right)}}{{x_{0} }} = {\text{const}}.$$

(A1)

For the quadratic penalty function $\varphi_{1} \left( x \right) = x^{2}$, we may derive that

$$\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\varphi^{\prime}_{1} \left( x \right)}}{x} = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{2x}{x} = 2.$$

(A2)

For the generalized Charbonnier function $\varphi_{2} \left( x \right) = \left( {\sigma^{2} + x^{2} } \right)^{a}$, we have

$$\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\varphi^{\prime}_{2} \left( x \right)}}{x} = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{2ax\left( {\sigma^{2} + x^{2} } \right)^{a - 1} }}{x} = \mathop {{\text{lim}}}\limits_{x \to 0} 2a\left( {\sigma^{2} + x^{2} } \right)^{a - 1} = 2a\sigma^{a - 1} = {\text{const}}.$$

(A3)

From Eqs. (A2) and (A3), it can be seen that both the quadratic and generalized Charbonnier penalty functions satisfy the conditions defined in Eq. (A1). Thus the unknown variables can be seperated from the two penalty functions.

B. Derivative order decrease.

For the Eq. (17), firstly, we write the Taylor expanstions of $\varphi^{\prime\prime}\left( x \right)$ and $\frac{{\varphi^{\prime}\left( x \right)}}{x}$ at x = 0, which are as follows:

$$\begin{gathered} \varphi^{\prime}\left( x \right) = \varphi^{\prime}\left( 0 \right) + \frac{{\varphi^{\prime\prime}\left( 0 \right)}}{1!}x + \frac{{\varphi^{\left( 3 \right)} \left( 0 \right)}}{2!}x^{2} + \cdots + \frac{{\varphi^{{\left( {n + 2} \right)}} \left( 0 \right)}}{{\left( {n + 1} \right)!}}x^{n + 1} \hfill \\ \varphi^{\prime\prime}\left( x \right) = \varphi^{\prime\prime}\left( 0 \right) + \frac{{\varphi^{\left( 3 \right)} \left( 0 \right)}}{1!}x + \frac{{\varphi^{\left( 4 \right)} \left( 0 \right)}}{2!}x^{2} + \cdots + \frac{{\varphi^{{\left( {n + 2} \right)}} \left( 0 \right)}}{\left( n \right)!}x^{n} \hfill \\ \frac{{\varphi^{\prime}\left( x \right)}}{x} = \frac{{\varphi^{\prime}\left( 0 \right)}}{x} + \varphi^{\prime\prime}\left( 0 \right) + \frac{{\varphi^{\left( 3 \right)} \left( 0 \right)}}{1!}\frac{x}{2} + \cdots + \frac{{\varphi^{{\left( {n + 2} \right)}} \left( 0 \right)}}{\left( n \right)!}\frac{{x^{n} }}{n + 1} \hfill \\ \end{gathered}$$

(B1)

Here we have $\varphi^{\prime}\left( 0 \right) = 0$. As for quadratic penalty function,$\varphi_{1}^{^{\prime}} \left( x \right) = \left. {2x} \right|_{x = 0} = 0$; for generalized Charbonnier penalty function, $\varphi_{2}^{^{\prime}} \left( x \right) = \left. {2ax\left( {\sigma^{2} + x^{2} } \right)^{a - 1} } \right|_{x = 0} = 0$. By neglecting the higher order terms, we have

$$\mathop {{\text{lim}}}\limits_{x \to 0} \left[ {\varphi ^{\prime\prime}\left( x \right) - \frac{{\varphi^{\prime}\left( x \right)}}{x}} \right] = \frac{{\varphi^{\left( 3 \right)} \left( 0 \right)}}{2}x$$

(B2)

If the penalty function satisfies

$$\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\varphi^{\left( 3 \right)} \left( 0 \right)}}{2}x = 0$$

(B3)

then the relationship in Eq. (17) can be satisfied. In the following, we will prove that Eq. (B2) is valid for the two penalty functions we use.

For the quatratic penalty function $\varphi_{1} \left( x \right) = x^{2}$, it is easy to find that

$$\mathop {{\text{lim}}}\limits_{x \to 0} \varphi_{1\left( x \right)}^{\left( 3 \right)} = 0$$

(B4)

For the generalized Charbonnier function $\varphi_{2} \left( x \right) = \left( {\sigma^{2} + x^{2} } \right)^{a}$, it can be written as

$$\mathop {{\text{lim}}}\limits_{x \to 0} \varphi_{2\left( x \right)}^{\left( 3 \right)} = \mathop {{\text{lim}}}\limits_{x \to 0} \left[ {12a\left( {a - 1} \right)x\left( {\sigma^{2} + x^{2} } \right)^{a - 2} + 8a\left( {a - 1} \right)\left( {a - 2} \right)x^{3} \left( {\sigma^{2} + x^{2} } \right)^{a - 3} } \right] = 0$$

(B5)

Up to present, we have proved that Eq. (17) is satisfied for both penalty functions in the current GNC-SME scheme.