Two-dimensional FIR digital filters are very important in many image processing applications. They are used in image enhancement, image restoration as well as in seismic signal and motion picture processing. One serious problem with two-dimensional FIR filters is their high implementation complexity. It has been shown empirically that the filter support size is inversely proportional to the square of the filter's transition width. Thus, the implementation complexity of a sharp two-dimensional FIR filter would be prohibitively high.

The frequency response masking technique is an efficient method to realize sharp one-dimensional filters. This technique can synthesize sharp one-dimensional filters with a considerably lower complexity when compared to direct-form implementations. In this research, we extend the frequency response masking technique to the design of two-dimensional diamond-shaped filters. While the technique is general, we focused our attention on the very important diamond-shaped filter. Diamond-shaped filters are used for the conversion between signals sampled on the rectangular sampling grid and the quincunx sampling grid. Significant amount of complexity reduction can be obtained by using the new techniques. It is shown that when the transition width of the desired filter is very narrow, complexity savings of more than an order of magnitude can easily be achieved.

The frequency response masking filter design technique hinges on how the frequency spectrum can be divided into suitable complementary components. For the case of one-dimensional frequency response masking, this can be done by dividing the frequency spectrum into two components. There is no known simple rule that extends this procedure to the synthesis of any arbitrary-shaped two-dimensional filter. However, we discovered that by dividing the frequency spectrum into four suitably chosen complementary components and using appropriate masking filters, the frequency response masking technique can also be applied to the design of diamond-shaped filters.

 The block diagram of the frequency response masking (FRM) technique is illustrated in Figure 1. In Figure 1, X(W ) is the input, F_A(W ), F_B(W ), F_C(W ) and F_D(W ) are the two-dimensional band-edge shaping filters, A(W ), B(W ), C(W ) and D(W ) are the complementary components, and F_MA(W ), F_MB(W ), F_MC(W ) and F_MD(W ) are the masking filters. The input signal is filtered by the band-edge shaping filters to produce the complementary signal components which are then filtered by the masking filters. The outputs of the masking filters are summed to produce the final output F(W )X(W ).

We have applied the two-dimensional FRM technique by synthesizing a diamond-shaped filter with w _p = 0.5p and w _s = 0.52p . A direct-form diamond-shaped filter meeting the same specifications would require a filter support size of at least 129 by 129. Our technique produces a design using half the number of multipliers. Its frequency response is shown in Figure 2. Higher orders of complexity reduction is possible for the design of filters with sharper transition bands.