Mobius Compression

In 1827 mathematician August Mobius introduced a predessor to liner algebra which he called the barycentric calculus. In Mobius scheme, an arbitrary vector x in n-dimensional Euclidean space is expressed as the center of mass (or barycenter) of N point masses located at positions w1, w2,...,wN, and having masses m1, m2,...,mN, respectively (i.e., x = summation of (mk)(wk) from k = 1 to N, where the summation of (mk) = 1 from k = 1 to N. In this project we shall explore the use of an adaptive neural network technique for selecting nearly orthogonal point mass positions embedded within a statistical population of data block vectors to dramatically improve the performance of vector quantization data compression systems for communications. The idea is to transmit not only the nearest matching codebook vector, but a small set of qualified codebook vectors and their corresponding masses. In this way the original data vector can be accurately reconstructed as the barycenter of these masses with much less error and much igher compression than fixed transform (e.g., JPEG), vector quantization, eigenimage, and principal componant analysis approaches. A modest computational burden and real-time operation are inherent in the method, unlike fractal compression. The goal of Phase I will be to demonstrate 40:1 compression of video images with an RMS pixel value distortion of less than 10% in the decompressed imagery. In Phase II, the goal will be 100:1 compression with an RMS error level below 5%.