Multiplicative Homomorphic Processing and its Application to Image Enhancement

1Introduction: The generalized linear signal processing systems have been introduced and extensively studied by Oppenheim et al. [5] at the end of the 1960s. A generalized linear signal processing system consists of vector spaces for the input and output signal sets and a linear mapping, called the system transformation [5]. In their research, Oppenheim et al. have mainly focused on the implementation of multiplicative homomorphic and the convolutional homomorphic systems, that are designed for signal processing problems where the signals are combined by multiplication and convolution, respectively. The convolutional homomorphic systems are applied in speech processing, seismic signal processing, underwater acoustics, etc.. Basically, this technique is used to obtain a deconvolution. The convolutional systems are by far more used than the multiplicative systems. This works concentrate attention in the multiplicative homomorphic systems. The multiplicative homomorphic system possesses several physical connections with the human visual system. Stockham et al. [1] have largely studied its relationships with the multiplicative transmittance formation model, multiplicative reflectance image formation model and human brightness perception. Nevertheless, multiplicative homomorphic system approach also presents several drawbacks. Firstly, although it is based on physical image formation models and on a logarithmic homomorphism (which expresses physically the optical density and is psychophysically related to Fechner's human visual law) [1], it does not involve the intensity range constraints associated with many physical or practical situations, and with the saturation effects of human brightness perception [4]. The basic idea of multiplicative homomorphic processing is quite simple. If one sequence z(n) was obtained by the multiplication of two sequences x(n) and y(n), i.e., z(n)=x(n) y(n), one can take the logarithm of z(n) and use a linear system to filter log x(n) + log y(n). Although the principle is simple, there are a lot of problems associated to a successful implementation of this technique. This work is organized as follows. In section 2 some consequences of dealing with a non linear signal processing technique are highlighted. Section 3 discusses the interpretation of spectra of 2D signal, based on the Matlab tools. Section 4 presents a discussion and the results of applying multiplicative homomorphic processing to image enhancement. The final considerations are described in section 5.