Non-Separable Two-Dimensional Hadamard Transform via a Discrete Hadamard Slice Theorem

In this letter, we demonstrate how characteristics of a permutation of the Hadamard transform (HT), known as the binary discrete Hartley transform (BDHT), can be leveraged to develop a Hadamard slice theorem (HST). In doing so, we establish an orthogonal binary two dimensional (2D) transform consisting of square wave basis functions. Typically, the 2D-HT is separated into a series of one dimensional (1D) transforms of the columns and rows of an image. This process yields binary, checkerboard pattern basis functions similar in appearance to the 2D separable discrete Hartley transform (SDHT). Instead, basis functions of our proposed non-separable 2D Hartley-Hadamard transform (HHT) are analogous to sinusoids of the non-separable 2D-discrete Hartley transform (DHT). This new transform closely mimics the 2D-DHT, while preserving the benefits and characteristics of the HT (such as being multiplication free). To our knowledge, this is the first non-separable 2D-HT, with the accompanying HST being the first to relate projections of the discrete Radon transform (DRT) to the 2D-HT of an image.