11-08-2012, 03:42 PM
Multiplicative Homomorphic Processing and its Application to Image Enhancement
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Introduction:
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.
Consequences of the nonlinear behavior of the logarithmic function:
In this work, the logarithmic function is considered to be in the natural basis, log x = loge x.
An important drawback of the multiplicative homomorphic processing is that, even if the two
components x(n) and y(n) rely in different bands in the frequency domain, the sequences log x(n) and
log y(n) can present considerable overlap between their spectra. The objective of this section is to
give a perspective on the shortcomings of working in the context of nonlinear systems.
Multiplicative Homomorphic Processing:
The application of multiplicative homomorphic processing to image enhancement was
inspired in a ECE253a homework, where the goal was to enhance an image through contrast
boosting. In this case, the image "butterfly", shown in Figure 4.1, has a non uniform illumination
pattern, and the classical technique of global histogram equalization does not achieve good results.
Conclusions
It is clear that it is not easy to design a multiplicative homomorphic system. In most of the
literature, the authors comment that the design of the filter requires "tuning" [1-5]. Maybe, the
greatest weakness of the homomorphic processing is the necessity of spending considerable effort
"tuning" the algorithm. In [3], the authors address the alternative of an optimal design for the linear
system, but the technique is exclusive for use in convolutional homomorphic systems.
This work was very illustrative because allowed to deal, in a small extent, with non linear
signal processing. Besides, there was considerable improvement in the intuition and experience
related to 2D signal processing, mainly in terms of spectrum representations and its properties. In an
educational perspective, the investigation of homomorphic processing applied to image processing
was really important and worth.