Evaluation of image pairs

Statistical evaluation technique is used to extract information of the displacement from the two single exposed grey level patterns acquired at and Statistical evaluation technique is suitable since individual particle image pair detection is not possible for high resolution PIV measurement. It is less sensitive to noise and image discretization errors. In this technique, the whole image is sampled with an appropriate step size. For each sampling location, a two dimensional grey level sample of certain shape and size is extracted from the source image. It is cross correlated with the corresponding sample from the second image.

In general, the cross-correlation function is given by

Here and are intensity values of the image pair. The cross correlation function produces a signal peak when the images align with each other, since the sum of the product of the pixel intensities will be larger than elsewhere.

In general, calculation of the correlation function is done in the Fourier space. For FFT analysis the data should be periodic. There are other issues such as aliasing and bias errors that need to be taken into consideration. Each subregion is transformed into the Fourier space via a Fourier transformation. The sub-windows are the spatially shifted and their Fourier transforms determined until the correlation is found. The correlation values are weighted accordingly to reduce bias error.

The cross correlation of two functions is equivalent to a complex conjugate multiplication of their Fourier transforms. In actual applications, the formulas used are

where and are the Fourier transforms of the image intensities and .