How to Apply a Filter Defined in the Frequency Domain by a Continuous Function
Thibaud Briand, Jonathan Vacher
Thibaud Briand, and Jonathan Vacher, How to Apply a Filter Defined in the Frequency Domain by a Continuous Function, Image Processing On Line, 6 (2016), pp. 183–211.

Communicated by Loïc Simon, Sandra Doucet
Demo edited by Nelson Monzón


We propose algorithms for filtering real-valued images, when the filter is provided as a continuous function defined in the Nyquist frequency domain. This problem is ambiguous because images are discrete entities and there is no unique way to define the filtering. We provide a theoretical framework designed to analyse the classical and computationally efficient filtering implementations based on discrete Fourier transforms (DFT). In this framework, the filtering is interpreted as the convolution of a distribution, standing for the filter, with a trigonometric polynomial interpolator of the image. The various plausible interpolations and choices of the distribution lead to three equally licit algorithms which can be seen as method variants of the same standard filtering algorithm. In general none should be preferred to the others and the choice depends on the application. In practice, the method differences, which come from the boundary DFT coefficients, are not visible to the naked eye. We demonstrate that claim on several experimental configurations by varying the input image and the considered filter. In some cases however, we discuss how the choice of the variant may affect fundamental properties of the filtering.