A gardener once defined a weed as a plant growing where he does not want it to. Noise is a little like that. Noise may be something extraneous to the object or something intrinsic to it that is too detailed to be useful In either case, it is often more effective to remove the noise before we recognize the pattern than to learn to recognize the pattern with the noise. The problem is that removing the noise can distort the image.
Isolated points of noise (“Salt and pepper noise”) is easy to remove by morphological analysis (P. Soille, Morphological Image Analysis: Principles and Applications, Springer-Verlag, 1999, ISBN 3-540-65671-5.
http://www.springer.de/cgi-bin/search_book.pl?isbn=3-540-65671).
For more general noise, wavelet denoising has become the most popular approach. We like it not just because it gives good results but also because it allows us to eliminate fine detail if we choose (For example, see http://www.pearsonptg.com/book_detail/0,3771,0201634635,00.html ).
Wavelets are a wonderful generalization of Fourier analysis that very powerful and easy to analyze ("Wavelet Processing and Optics" (invited paper), Proc. of the IEEE 84(5):720-732, 1996 Y. Li, H. H. Szu, Y. Sheng, and H. J. Caulfield). One of the many nice features is that it is an invertible transformation. This means that we have a filtering opportunity in the transform plane just as we did in Fourier filtering. If we simply remove the wavelet coefficients that fall below some threshold condition, we can reconstruct the input as it would have been without those wavelets ever having been there. That is the simplified essence of wavelet denoising.
Both of the methods described above are nonlinear. If we want to do a simple linear operation, we can apply a so=called smoothing filter as a Fourier filter ["Optical Smoothing Filters," Appl. Opt., Vol. 13, 996 (1974) H. J. Caulfield].