Rotation invariant separable functions are Gaussian

In digital image analysis, edge detection, line detection, texture classification, and so forth are basic to image understanding and interpretation. Since a typical image is void of predetermined directions of edge, line, or texture, rotation invariant filters are important for their detection. In designing such filters the concept of rotation invariant separable function is often used. Here it is shown that every rotation invariant separable real valued function of two variables is either Gaussian or identically zero. This justifies the use of Gaussian filters in image processing. Furthermore, while deriving this result, the authors obtained the general solutions of two functional equations considered by Swiatak in 1975. Swiatak found the solutions when one of the unknown functions is continuous at zero. No regularity assumptions were made and all the general solutions that contain the solution obtained in [Aequationes Math., 12 (1975), pp. 39–64] were determined.