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. Source Code The source code and the online demo are accessible at the IPOL web part of this article 1. The C99 implementation of the code that we provide is the one which has been peer reviewed and accepted by IPOL. It contains two modules : one performs the standard filtering algorithm (Algorithm 1) presented in Section 3 and the other performs the comparison detailed in Section 4. The comparison module computes for nine filters the filtered images obtained by applying the three variant methods. For each filter a pairwise comparison of the results is done by computing the (absolute) difference image, the maximum difference and the mean difference. It creates fifty-five images: twenty-seven filtered images, twenty-seven difference images and the modulus of the DFT input in logarithmic scale. See Section 4 and the code documentation for additional information.
[1]
William T. Freeman,et al.
Presented at: 2nd Annual IEEE International Conference on Image
,
1995
.
[2]
Lionel Moisan,et al.
The Shannon Total Variation
,
2017,
Journal of Mathematical Imaging and Vision.
[3]
B. Galerne,et al.
The Heeger & Bergen Pyramid Based Texture Synthesis Algorithm
,
2014,
Image Process. Line.
[4]
Steven G. Johnson,et al.
The Design and Implementation of FFTW3
,
2005,
Proceedings of the IEEE.
[5]
Eero P. Simoncelli,et al.
A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients
,
2000,
International Journal of Computer Vision.
[6]
Pascal Getreuer,et al.
A Survey of Gaussian Convolution Algorithms
,
2013,
Image Process. Line.
[7]
D. Hubel,et al.
Receptive fields, binocular interaction and functional architecture in the cat's visual cortex
,
1962,
The Journal of physiology.
[8]
L. Schwartz.
Théorie des distributions
,
1966
.
[9]
Ives Rey-Otero,et al.
Computing an Exact Gaussian Scale-Space
,
2016,
Image Process. Line.
[10]
Y. Katznelson.
An Introduction to Harmonic Analysis: Interpolation of Linear Operators
,
1968
.
[11]
Antonia Papandreou-Suppappola,et al.
Applications in Time-Frequency Signal Processing
,
2002
.
[12]
P. Lennie.
Receptive fields
,
2003,
Current Biology.
[13]
C J Isham,et al.
Methods of Modern Mathematical Physics, Vol 1: Functional Analysis
,
1972
.