Conclusion#
Filtering involves the manipulation of frequencies within an image, either by enhancing or attenuating them. It finds application in several image processing tasks, such as image acquisition, where a blurred photo is a (low-pass) filtered version of the original scene.
In this chapter, we have seen two mathematical tools used to filter images and analyze their properties. The first one is convolution, which is the mathematical operation used to compute the result of filtering in the usual spatial domain. The second one is the Fourier transform, which gives an image’s representation in the frequency domain. Filtering in this domaine comes to a multiplication. Besides, the Fourier transform is used to analyze the frequency content of an image, as we will see in Denoising and Deconvolution.
A question remains: is there an inverse convolution operator, like division is the inverse of multiplication? The answer is generally no. Practically, the quantification in the image, the presence of noise and the lack of knowledge of the PSF make the problem hard, justifying a specific section to deal with.
References#
B. Jähne, Digital Image Processing, Springer, 2005.