High-resolution velocimetry from tracer particle fields using a wavelet-based optical flow method

Abstract

A wavelet-based optical flow method for high-resolution velocimetry based on tracer particle images is presented. The current optical flow estimation method (WOF-X) is designed for improvements in processing experimental images by implementing wavelet transforms with the lifting method and symmetric boundary conditions. This approach leads to speed and accuracy improvements over the existing wavelet-based methods. The current method also exploits the properties of fluid flows and uses the known behavior of turbulent energy spectra to semi-automatically tune a regularization parameter that has been primarily determined empirically in the previous optical flow algorithms. As an initial step in evaluating the WOF-X method, synthetic particle images from a 2D DNS of isotropic turbulence are processed and the results are compared to a typical correlation-based PIV algorithm and previous optical flow methods. The WOF-X method produces a dense velocity estimation, resulting in an order-of-magnitude increase in velocity vector resolution compared to the traditional correlation-based PIV processing. Results also show an improvement in velocity estimation by more than a factor of two. The increases in resolution and accuracy of the velocity field lead to significant improvements in the calculation of velocity gradient-dependent properties such as vorticity. In addition to the DNS results, the WOF-X method is evaluated in a series of two-dimensional vortex flow simulations to determine optimal experimental design parameters. Recommendations for optimal conditions for tracer particle seed density and inter-frame particle displacement are presented. The WOF-X method produces minimal error at larger particle displacements and lower relative error over a larger velocity dynamic range as compared to correlation-based processing.

Appendices

Appendix 1: Wavelet transforms and signal decomposition

A wavelet is a mathematical function that can be used to divide a given function f(x) into different scale components. A wavelet transform is the representation of f(x) through a linear combination of a set of basis functions that are translations and dilations of a fast-decaying function known as the mother wavelet, $\psi$, along with an associated scaling function, $\varphi$. Specifically, a wavelet transform of f(x) first computes the set of inner products of f(x) with a wavelet atom $\psi _{j,n}$ at scales j and positions n, resulting in a set of detail coefficients, $d_j [n] = \langle f, \psi _{j,n} \rangle$. Wavelet atoms are defined by scaling and translating the mother wavelet $\psi$ as $\psi _{j,n} = \frac{1}{2^j} \psi \left( \frac{x-2^j n}{2^j} \right)$. Associated with the “high-pass” mother wavelet is a scaling function $\varphi$ that gives a low-pass representation of the function f(x). Thus, at each scale j and position n, approximation coefficients are computed as $a_j [n] = \langle f, \varphi _{j,n} \rangle$, where $\varphi _{j,n} = \frac{1}{2^j} \varphi \left( \frac{x-2^j n}{2^j} \right)$. The wavelet transform at each scale j produces a set of detail coefficients $d_j$ and approximation coefficients $a_j$, both of which are contained within $\varTheta$ as shown below.

In signal processing, one has a discrete signal $f \left[ x_i \right]$ with a length of $2^F$. The wavelet transforms are applied at increasingly coarser scales, starting from the finest scale $\left( j = F \right)$ down to a predetermined coarse scale $j_0<F$. At each subsequent scale, the approximation coefficients are divided into coarser approximations and details; that is, the transform at each scale $j-1$ operates on the coarse approximation from the next finest scale $a_j$ to produce $d_{j-1}$ and $a_{j-1}$. Thus, a discrete wavelet transform (DWT) applied to $f \left[ x_i \right]$ forms a multiscale representation of the signal, where the detail coefficients from all scales $d_{j_0}, d_{j_0+1},..., d_{F-1}, d_F$ and the remaining coarse approximation $a_C$ are stored as:

$$\begin{aligned} \varTheta \left[ n_i \right] = \left[ a_{j_0}, d_{j_0}, d_{j_0+1}, \ldots , d_{F-1}, d_F \right] , \end{aligned}$$

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which also has a length of $2^F$ like the original signal $f \left[ x_k \right]$. The exact mathematical details of the lifting DWT used by WOF-X are given in Mallat (2009) and Daubechies and Sweldens (1998). If the wavelets and scaling functions form an orthonormal basis as is the case of the biorthogonal 9–7 wavelets used by WOF-X, then the wavelet transform can be inverted and $f \left[ x_k \right]$ is recovered exactly. Thus, it is concluded that the output from a wavelet transform is a set of wavelet coefficients that exactly describe the input signal in the wavelet basis.

1.1 Wavelet transforms of images

The above procedure can be applied to higher-dimensional signals (i.e. images) by applying the wavelet transform along each dimension isotropically at each scale, demonstrated schematically in Fig. 15.

Fig. 15

Schematic of an isotropic 2D wavelet transform. The transform is performed in two steps at each scale. (1) Apply a 1D wavelet transform to each column in the image in the vertical direction to obtain vertical approximation and detail coefficients. (2) Apply a 1D wavelet transform to each row in the vertically transformed image. This gives new approximation coefficients $\underline{a}_{F-1}$ and detail coefficients $\underline{d}_{F-1}$ in the diagonal blocks, and sets of mixed coefficients in the off-diagonals. Step 3 is to repeat this procedure on the upper-leftmost block containing approximation coefficients $\underline{a}_j$ at each scale j until scale $j_0$ is reached. The inverse transform is performed by simply reversing these steps

For the wavelet-based optical flow described in the current manuscript, the generic signal $f \left[ x_{k_1}^1,x_{k_2}^2 \right]$ is replaced with the individual velocity components $v_i \left( \underline{x} \right)$ (which are two-dimensional). Truncated transforms are formed by setting all detail coefficients, including the mixed coefficients, above a specified scale L to zero. The truncated transform is then inverted to obtain a coarse approximation of the original signal $f_L \left[ x_{k_1}^1,x_{k_2}^2 \right]$.

1.2 Derivative regularization in the wavelet domain

This section outlines the construction of the regularization term $J_\mathrm{{R}}$ in Eq. 6 and its gradient. The proof that the following definition of $J_\mathrm{{R}}$ indeed penalizes the third derivative of the velocity field is described in Kadri-Harouna et al. (2013) and will not be given here. Kadri-Harouna et al. (2013) show that the regularization term $J_\mathrm{{R}}$ for penalization of the third derivative of a velocity field $\underline{v}$ is given by the following:

$$\begin{aligned} J_\mathrm{{R}} = \left( \underline{\psi }^V \right) ^T \cdot \left( \underline{\varLambda }^V \cdot \underline{\psi }^V \right) , \end{aligned}$$

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where $\underline{\psi }$ is the wavelet transform of the velocity field $\underline{v}$ and the superscript $^V$ indicates the vectorization operation. $\underline{\varLambda }$ is a matrix matching the dimensions of $\underline{\psi }$ whose elements are $4^{3j}$ (see Fig. 15) for every scale above $j_0$ and 0 for every element corresponding to $\underline{a}_{j_0}$. Its gradient with respect to $\underline{\psi }$ is easily found:

$$\begin{aligned} J_\mathrm{{R}}^\prime = \underline{\varLambda }^V \cdot \underline{\psi }^V. \end{aligned}$$

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Appendix 2: Synthetic particle image generation

Synthetic particle images are generated in a similar manner as described by Carlier and Wieneke (2005). Each particle i is centered at a location $\left( x_0^i,y_0^i\right)$ in the image domain, with diameter $d_i$ and a center brightness value $c_i$. The $d_i$ values are selected from a normal distribution with mean $\bar{d}$ and standard deviation $\sigma _d$. The center brightness is a simulation of out-of-plane displacement in a real-world experiment, and is given by $c_i = d_i^2 \exp \left( -b_i^2 \right)$, where $b_i$ is normally distributed with a mean of zero and a standard deviation $\sigma _b$. The $c_i$ values are then normalized, such that the largest particle in the domain has a value of $c_i = 1$. The intensity distribution of particle i is then given by the following:

$$\begin{aligned} I_i (x,y) = c_i \exp \left( - \frac{1}{2} \frac{ \left( x-x_0^i \right) ^2 + \left( y-y_0^i \right) ^2}{d_i^2} \right) . \end{aligned}$$

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A simulated digitized image can be created by considering the brightness at some pixel k. Its brightness is computed by summing the contribution of each particle and “digitizing” by integration over the pixel. The integration is done by convolving the intensity function of each particle with a rectangle function and the Dirac delta function as follows in Eq. 16:

$$\begin{aligned} I \left( x_k,y_k \right) = \underset{i}{\overset{N}{\sum }} I_i (x,y) *{\Pi } (x,y) *\delta \left( x-x_k,y-y_k \right) . \end{aligned}$$

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Carrying out the convolution for pixel width $\varDelta$, the resulting synthetic image is determined by the following:

$$\begin{aligned} \begin{aligned} I\left( x_k,y_k \right) =&\underset{i}{\overset{N}{\sum }} c_i d_i^2 \frac{\pi }{2} \left[ \text {erf} \left( \frac{x_k + \frac{\varDelta }{2} - x_i}{d_i \sqrt{2}} \right) - \text {erf} \left( \frac{x_k - \frac{\varDelta }{2} - x_i}{d_i \sqrt{2}} \right) \right] \\&\times \left[ \text {erf} \left( \frac{y_k + \frac{\varDelta }{2} - y_i}{d_i \sqrt{2}} \right) - \text {erf} \left( \frac{y_k - \frac{\varDelta }{2} - y_i}{d_i \sqrt{2}} \right) \right] \ . \end{aligned} \end{aligned}$$

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Pixels are synthetically saturated when their calculated pixel value $I \left( x_k,y_k \right)$ is greater than one. These pixel values are then set to a value of one.