Homomorphic filtering
Encyclopedia
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
Thomas Stockham
Thomas Greenway Stockham was an American scientist who developed the first practical digital audio recording system, and pioneered techniques for digital audio recording and processing as well....

, Alan V. Oppenheim, and Ronald W. Schafer at MIT.

## Image enhancement

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
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

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 filter
High-pass filter
A high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...

ing is used to suppress low frequencies and amplify high frequencies, in the log-intensity domain.

## Audio and speech analysis

Homomorphic filtering is used in the log-spectral domain to separate filter effects from excitation effects, for example in the computation of the cepstrum
Cepstrum
A cepstrum is the result of taking the Fourier transform of the logarithm of the spectrum of a signal. There is a complex cepstrum, a real cepstrum, a power cepstrum, and phase cepstrum....

as a sound representation; enhancements in the log spectral domain can improve sound intelligibility, for example in hearing aids.