John von Neumann’s Space-Frequency Orthogonal Transforms

Among the invertible orthogonal transforms employed to perform the analysis and synthesis of 2D signals (especially images), the ones defined by means of John von Neumann’s cardinal sinus are extremely interesting. Their definitions rely on transforms similar to those employed to process time-varying 1D signals. This article deals with the extension of John von Neumann’s transforms from 1D to 2D. The approach follows the manner in which the 2D Discrete Fourier Transform was obtained and has the great advantage of preserving the orthogonality property as well as the invertibility. As an important consequence, the numerical procedures to compute the direct and inverse John von Neumann’s 2D transforms can be designed to be efficient thanks to 1D corresponding algorithms. After describing the two numerical procedures, this article focuses on the analysis of their performance after running them on some real-life images. One black and white and one colored image were selected to prove the transforms’ effectiveness. The results show that the 2D John von Neumann’s Transforms are good competitors for other orthogonal transforms in terms of compression intrinsic capacity and image recovery.