## 3 Flow Velocity Measurement Method
### The Principle of Main Steps and Analysis  
#### 3.2.1 Image Deformation  
When the convex array probe of the ultrasound imaging device is used for detecting flow during the physical model test, the imaging of silt carry flow is a kind of conformal mapping from an arc-annular shape. Therefore, image deformation is necessary for the correction of flow imaging information. The imaging of the flow by using the ultrasound imaging device always is an annular conformal mapping within a certain angle, even though it uses the linear array probe. In order to create an accurate reconstruction of the velocity field, we converted the arc-annular shape into a rectangular shape using the convex array probe for a better feature match of overlapping sub-images.  
When we want to obtain the flow velocity, the displacement of imaging particle in the image is the key according to the normal PIV method, and the displacement can be detected by using a cross-correlation function under the assumption that the particles' motion in the interrogation window of the image is almost regular. However, this assumption is hardly valid in practice. The velocity profile vector shows an obvious change within the interrogation window in the silt carrying flow. The cross-correlation peak and different velocities produced by image pairs will become larger, as shown in Figure 2, which may be split into multiple cross-correlation peaks as a result of the large differences of particle velocity across the window during some extreme conditions. Therefore, the calculation of velocity profile can be affected when the velocity gradient is large and may suffer from a much higher velocity vector rejection rate. Because of this, we take an iterative window  
Figure 3: The main steps of the flow velocity measurement method based on the iterative multi-grid deformation technique.  
deformation technique to compensate for the effect of peak broadening and the gradient difference of in-plane velocity. This approach can improve results of image matching when two adjacent ultrasonic images are deformed iteratively according to the flow velocity field [33]. The main advantage of the image deformation is the added robustness and accuracy. It can improve the effect of sub-image match, especially for highly sheared flows when compared to the discrete window offset method. The basic principle of this process is shown in Figure 4.  
We use a spatial cross-correlation function for the deformed images in this paper compared with the standard cross-correlation function. The equation is defined as:  
\[R(m,n)\ =\ \sum_{i\ =\ 1}^{M}\sum_{j\ =\ 1}^{N}\widetilde{g}_{1}(i,j)\widetilde{ g}_{2}(i+m,j+n) \tag{1}\]  
while \(\widetilde{g}_{1}(i,j)\) and \(\widetilde{g}_{2}(i,j)\) denote the image intensity, which is reconstructed after image deformation by using the predicted deformation displacement field \(\Delta s(x)\) in a central difference process, shown as Formula (2):  
\[\left\{\begin{array}{rcl}\widetilde{g}_{1}(x)&=&g_{1}(x-\frac{\Delta s(x)}{ 2})\\ \widetilde{g}_{2}(x)&=&g_{2}(x+\frac{\Delta s(x)}{2})\end{array}\right. \tag{2}\]  
The deformation displacement field \(\Delta s(x)\) is a spatial distribution without a uniform condition. Therefore, smooth interpolation at each pixel in the ultrasonic image is required. In a steady condition of the flow, the first-order term of the Taylor series is enough for the image reconstruction by using local displacement, which can be shown as Formula (3):  
\[\Delta s_{1}(x)\ =\ \Delta s_{0}(x)+\
abla[\Delta s_{0}(x)]*(x-x_{0})+o(x-x_{0} )^{2} \tag{3}\]  
where \(x_{0}\) is the center position of the interrogation window. The smooth interpolation can be described as follows.