Some Mathematical Properties of a Scheme for Reducing the Bandwidth of Motion Pictures by Hadamard Smearing

Abstract

M. R. Schroeder recently proposed a scheme for compression of motion picture data by taking the difference of two successive frames and then smearing.1 The smearing is accomplished by a Hadamard matrix. If the Hadamard matrix is of a certain particularly well-understood type, then we show that if the input differential picture consists of a small odd number of large pulses of identical magnitudes (but arbitrary signs), then the output will consist of three components: (i) Large pulses of equal magnitude and the correct signs, matching each of the input pulses. (ii) One additional “stray” large pulse, of magnitude equal to the others, but located at a point where the input was zero. (iii) Scatlered pulses of amplitude low relative to the pulses of types i and ii, but so numerous that they consume (π − 2)/π of the total energy of the output differential picture. We give an explicit formula for the amplitude of each of these pulses. The problem of determining Ihe distributions of all possible outputs of the proposed system for other classes of inputs is shown to be equivalent to the unsolved problem of finding the weight enumerators for the cosets of the first order Reed-Muller codes.