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In this paper, a systematic method for developing a binary version of a given transform by using the Walsh-Hadamard transform (WHT) is proposed. The resulting transform approximates the underlying transform very well, while maintaining all the advantages and properties of WHT. The method is successfully applied for developing a binary discrete cosine transform (BDCT) and a binary discrete Hartley transform (BDHT). It is shown that the resulting BDCT corresponds to the well-known sequency-ordered WHT, whereas the BDHT can be considered as a new Hartley-ordered WHT. Specifically, the properties of the proposed Hartley-ordering are discussed and a shift-copy scheme is proposed for a simple and direct generation of the Hartley-ordering functions. For software and hardware implementation purposes, a unified structure for the computation of the WHT, BDCT, and BDHT is proposed by establishing an elegant relationship between the three transform matrices. In addition, a spiral-ordering is proposed to graphically obtain the BDHT from the BDCT and vice versa. The application of these binary transforms in image compression, encryption and spectral analysis clearly shows the ability of the BDCT (BDHT) in approximating the DCT (DHT) very well.