Vectorized transforms in scalar processors

We disclose a generalized approach to creating efficient implementations of linear, orthogonal transforms, with specific examples discussed for the 8 x 8 DCT used in image compression. We connect this with a method for performing signed, parallel processing in scalar, off-the-shelf processors for integer transforms. Uniform data precision may be used, but is not required for the method. The coefficients resulting from the new algorithm converge more quickly than the approximation made to the coefficients. Furthermore, the new algorithm allows more control of the specific representation chosen for the coefficients, as is detailed below. The methods described were designed for addressing this need with two\´s-complement arithmetic. Data that can be processed in parallel, because of the algorithm structure, are packed in a "vector" format, described, into registers. Many signed arithmetic operations can be performed on these vectors, including addition, subtraction, multiplication by scalars, shifting, and others. When the parallel processing is completed, the vectors can be unpacked into scalar values for storage or subsequent processing. The importance of these methods lies in their handling of carries and borrows in the packed vector format. The generalized method is described. Notation is given at the beginning to establish consistency through the article. We discuss a generalized approach to integer transforms, using the DCT as a specific example. Then we detail the vector format, which allows vector computation in scalar processors of parallelizable algorithms. The IDCT is used as a numerical example in the discussion of the vector format. The results were developed for high-end printers (e.g., more than 100 pages per minute), where image compression and decompression must be performed in real time, either in FPGAs, or in embedded processors; however, the methods are applicable to a broad range of signal processing systems