# A further assessment of interpolation schemes for window deformation in PIV

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## Abstract

We have evaluated the performances of the following seven interpolation schemes used for window deformation in particle image velocimetry (PIV): the linear, quadratic, B-spline, cubic, sinc, Lagrange, and Gaussian interpolations. Artificially generated images comprised particles of diameter in a range 1.1 ≤ *d* _{p} ≤ 10.0 pixel were investigated. Three particle diameters were selected for detailed evaluation: *d* _{p} = 2.2, 3.3, and 4.4 pixel with a constant particle concentration 0.02 particle/pixel^{2}. Two flow patterns were considered: uniform and shear flow. The mean and random errors, and the computation times of the interpolation schemes were determined and compared.

## Keywords

Particle Image Velocimetry Random Error Shear Flow Interpolation Scheme Uniform Flow## List of symbols

*a*parameter in the cubic interpolation

*C*particle density (particle/pixel

^{2})*d*arbitrary value between 0 and 1 (pixel)

*d*_{p}particle diameter (pixel)

*f*spatial frequency (pixel

^{−1})*f*(*x*,*y*)intensity data interpolated from the original image

*f*_{1}(*x*,*y*)intensity data of the first image

*f*_{2}(*x*,*y*)intensity data of the second image

*G*^{0}(*x*,*b*)Gaussian function

*G*^{P}(*x*,*b*)*P*th derivative of the Gaussian function*h*(*x*)one-dimensional impulse response function of a reconstruction filter

*h*^{2D}(*x*,*y*)two-dimensional impulse response function of a reconstruction filter

*H*(*f*)Fourier transform of the one-dimensional impulse response function of a reconstruction filter

*i*integer horizontal position in the image (pixel)

*j*integer vertical position in the image (pixel)

*k*iteration number

*M*total number of vectors

*N*kernel size of an interpolation (pixel)

*U*horizontal displacement in the uniform flow (pixel)

- \(\bar{U}\)
mean of the measured displacements in the uniform flow (pixel)

*U*_{exact}exact displacement on the image for the uniform flow (pixel)

*U*_{c}horizontal displacement in the shear flow (pixel)

- \(\bar{U}_{c}\)
mean of the measured displacements in the shear flow (pixel)

*U*_{c,exact}exact displacement on the image for the shear flow (pixel)

- \(\vec{V}(i,j)\)
velocity vector at the (

*i*,*j*) location (pixel, pixel)*W*size of a square interrogation window (pixel)

- γ
_{2} parameter used in the second-order Gaussian interpolation

- γ
_{6} parameter used in the sixth-order Gaussian interpolation

- Δ
*x* horizontal value to be determined through cross-correlation (pixel)

- Δ
*y* vertical value to be determined through cross-correlation (pixel)

- σ
random error (pixel)

- ω
shear rate (pixel/pixel)

- ω (
*i*,*j*) two-dimentional Gaussian windowing mask

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