Homomorphic filtering

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Homomorphic filtering is a generalized technique for signal and image processing, involving a nonlinear mapping to a different domain in which linear filter techniques are applied, followed by mapping back to the original domain. This concept was developed in the 1960s by Thomas Stockham, Alan V. Oppenheim, and Ronald W. Schafer at MIT.

Image enhancement[edit]

Homomorphic filter is sometimes used for image enhancement. It simultaneously normalizes the brightness across an image and increases contrast. Here homomorphic filtering is used to remove multiplicative noise. Illumination and reflectance are not separable, but their approximate locations in the frequency domain may be located. Since illumination and reflectance combine multiplicatively, the components are made additive by taking the logarithm of the image intensity, so that these multiplicative components of the image can be separated linearly in the frequency domain. Illumination variations can be thought of as a multiplicative noise, and can be reduced by filtering in the log domain.

To make the illumination of an image more even, the high-frequency components are increased and low-frequency components are decreased, because the high-frequency components are assumed to represent mostly the reflectance in the scene (the amount of light reflected off the object in the scene), whereas the low-frequency components are assumed to represent mostly the illumination in the scene. That is, high-pass filtering is used to suppress low frequencies and amplify high frequencies, in the log-intensity domain.[1]

Audio and speech analysis[edit]

Homomorphic filtering is used in the log-spectral domain to separate filter effects from excitation effects, for example in the computation of the cepstrum as a sound representation; enhancements in the log spectral domain can improve sound intelligibility, for example in hearing aids.[2]


  1. ^ Douglas B. Williams and Vijay Madisetti (1999). Digital signal processing handbook. CRC Press. ISBN 0-8493-2135-2. 
  2. ^ Alex Waibel and Kai-Fu Lee (1990). Readings in Speech Recognition. Morgan Kaufmann. ISBN 1-55860-124-4. 

A.V. Oppenheim, R.W. Schafer, T.G. Stockham "Nonlinear Filtering of Multiplied and Convolved Signals" Proceedings of the IEEE Volume 56 No. 8 August 1968 pages 1264-1291

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