We propose a nonuniform frequency sampling method for 2D FIR filter design based on the concept of the nonuniform discrete Fourier transform (NDFT). The NDFT of a 2D sequence is defined as a sequence of samples of its z-transform taken at distinct points located arbitrarily in the (z$- 1$/, z$-2$/) space. In our design method, we determine the independent filter coefficients by taking samples of the desired frequency response at points located nonuniformly on the unit bi-disc, and then solving the linear equations given by the NDFT formulation. The choice of sample values and locations depends on the shape of the 2D filter. Best results are obtained when samples ar placed on contour lines that match the desired passband shape. The proposed method produces nonseparable filters with good passband shapes and low peak ripples. In this paper, we consider the design of square- and diamond-shaped filters. Extensive comparisons with filters designed by other methods demonstrate the effectiveness of the proposed method. We also investigate the performances of the filters designed by applying them as prefilters and postfilters to schemes for rectangular and quincunx downsampling of images. Examples show that the filters designed by our method produce output images which are sharper and have a higher PSNR, as compared with other filters.
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