The introduction of the fast Fourier transform algorithm has led to the development of the Fourier transform image coding technique whereby the two-dimensional Fourier transform of an image is transmitted over a channel rather than the image itself. This devlopement has further led to a related image coding technique in which an image is transformed by a Hadamard matrix operator. The Hadamard matrix is a square array of plus and minus ones whose rows and columns are orthogonal to one another. A high-speed computational algorithm, similar to the fast Fourier transform algorithm, which performs the Hadamard transformation has been developed. Since only real number additions and subtractions are required with the Hadamard transform, an order of magnitude speed advantage is possible compared to the complex number Fourier transform. Transmitting the Hadamard transform of an image rather than the spatial representation of the image provides a potential toleration to channel errors and the possibility of reduced bandwidth transmission.

Hadamard transform image coding

Picture processing: 1972

Several nonlinear filtering methods have been developed for two-dimensional image signals obtained from an Anger Scintillation camera. The first method termed "pruning" is implemented in the frequency domain and filters out certain components of the two-dimensional amplitude spectrum. The other methods called "E-filtering" and "P-filtering" utilize a nonlinear transformation of the spatial coordinates which is proportional to the two-dimensional signal amplitude and energy spectrum, respectively. These nonlinear filtering techniques have been applied to phantom scintigrams as well as brain and heart scans of normal and abnormal patients. In addition, the detection of boundaries using the hexagonal Golay transform on filtered and unfiltered images is included. Finally, these filtering and edge detection techniques were for the first time applied to a new technique for the detection of acute myocardial infarcts.

Digital Image Processing Applied to Scintillation Images from Biomedical Systems

A technique for obtaining the digital Walsh-Hadamard transform of two-dimensional discrete pictures in real time is presented. A considerable reduction in total time for the transformation of a picture results since the process of scanning and digitizing one picture line overlaps with computations on the previous line. This is accomplished by operating on each picture line as it is generated from the scanner, instead of starting the transformation operation after the entire picture has been scanned. Two theorems are presented which decompose a picture by row or by column to facilitate real-time transformation. Based on these theorems, a hardware machine for real-time parallel Walsh-Hadamard transformation is introduced.

Real-Time Walsh-Hadamard Transformation

An entropy concept similar to that of thermodynamics and information theory is applied to locate coherent propagation planes which are interesting from an optical viewpoint. It is shown that for entropy to be stationary, intensity stationarity is a sufficient condition. Talbot self-images and Gaussian beam waists are found to satisfy entropy stationarity. Paraxial operator equations are found for other extreme entropy solutions which are not stationary in intensity.

An Entropy Approach to Light Propagation

Two-dimensional functions are investigated with respect to their information content. Entropy relations of two functions, related by a finite Fourier transformation, are shown to be identical. The proof demonstrates the invariant nature of a Fourier scene and a space scene as sources of a communication system.

H.C. Andrews

Papers

Entropy considerations in the frequency domain