You can do such things with the sequence FFT - cut-off all frequencies beyond the target image size's frequency limit - IFFT.The necessary 2D sequence can be separated into two 1D operations, one for x and one for y.

Or by convolution with the appropriate functions (depending on x1/x2 resp. y1/y2) in the spatial domain. Here it is wise to compute the transformation function coefficients (the number of coefficients is the smallest common multiple of x1 and x2 resp. y1 and y2) once, then apply it to the n rows resp. m columns).

The second variant (convolution) is valid for any numbers x1, x2, y1, y2 while the FFT usually is restricted to powers of 2 regarding the numbers of columns/rows - eventually requiring padding and resulting in some artifacts at the edges.

@U. Dreher Thank You Professor for your kind recommendation. I got your intense conclusion regarding Image Processing.

When performing a conventional 2D FFT in two 1D FFTs, the image reconstruction quality is somewhat degraded. Recently, Bagul et al. in "Color Image Compression Using Discrete Fractional Fourier Transform ", 2014 suggested a solution powered by discrete FRFT to enforce high visual quality decompressed images: http://www.ijeit.com/Vol%203/Issue%2010/IJEIT1412201404_20.pdf

FRFT has solid theoretical foundations:

• http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.703.6842&rep=rep1&type=pdf
• http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.297.5830&rep=rep1&type=pdf
• http://journal.library.iisc.ernet.in/archives/vol200501/paper2/11.pdf

FRFT application in color image encoding in association with wavelet transform: http://universe.bits-pilani.ac.in/uploads/Paper_4.pdf

Thank You Prof. @Stephane Breton for introducing me to new technology foundation Discrete Fractional Fourier Transform (DFFT) with various paper recommendations.