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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/pixel2. 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.

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Abbreviations

a :

parameter in the cubic interpolation

C :

particle density (particle/pixel2)

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):

Pth 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)

γ2 :

parameter used in the second-order Gaussian interpolation

γ6 :

parameter used in the sixth-order Gaussian interpolation

Δx :

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

Δy :

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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Authors and Affiliations

  1. Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, South Korea

    Byoung Jae Kim & Hyung Jin Sung

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  1. Byoung Jae Kim
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  2. Hyung Jin Sung
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Correspondence to Hyung Jin Sung.

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Kim, B.J., Sung, H.J. A further assessment of interpolation schemes for window deformation in PIV. Exp Fluids 41, 499–511 (2006). https://doi.org/10.1007/s00348-006-0177-y

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