Homomorphic Filters

Abstract

In many applications signals are combined in a rather complicated way. Convolved signals are encountered in seismic signal processing, digital speech processing, digital echo removal and digital image restoration. Signals combined in a nonlinear way are encountered in digital signal processing for communication systems and in digital image filtering. Classical linear processing techniques are not so useful in those cases because the superposition property does not hold any more. Therefore, a special class of filters has been developed for the processing of convolved and nonlinearly related signals. They are called homomorphic filters. Their basic characteristic is that they use nonlinearities (mainly the logarithm) to transform convolved or nonlinearly related signals to additive signals and then to process them by linear filters. The output of the linear filter is transformed afterwards by the inverse nonlinearity. Homomorphic filtering has found many applications in digital image processing. It is recognized as one of the oldest nonlinear filtering techniques applied in this area. The main reason for its application is the need to filler multiplicative and signal-dependent noise, whose form was described in chapter 3. Linear filters fail to remove such types of noise effectively. Furthermore, the nonlinearity (logarithm) in the human vision system suggests the use of classical homomorphic filters. Homomorphic filtering can also be used in image enhancement. As we saw in chapter 3, object reflectance and source illumination contribute to the image formation in a multiplicative way. Ideally, the source illumination is constant over the entire image. However, in many practical cases, e.g., in outdoor scenes, source illumination is not constant over the entire scene. Therefore, it can be modeled as noise in the low spatial frequencies. If this noise is removed, the object reflectance is enhanced. Homomorphic filtering has found various practical applications, e.g., in satellite image processing and in the identification of fuzzy fingerprints. Homomorphie filtering has also found several applications in other areas, e.g., in speech processing and in geophysical signal processing. In the following, the theory and several applications of homomorphic filters will be given.