Homomorphic filtering

in order to remove illumination effects we first need to transform the input image to frequency space using a Fourier transform. The one-dimensional continuous Fourier transform is defined as (Bronstein et al. 2001) 

This transform can be considered a coordinate transformation where spatial coordinates are transformed into frequencies. Any given curve can be described by adding sine and 

A straightforward implementation of the Fourier transform requires O (N2) operations, where n = nx x ny is the number of pixels of the input image. A number of optimizations can be applied to reduce the time needed to perform this transformation. First, the transformation is separable. Therefore, the transformation can be performed independently for the two coordinates. Second, if the image dimensions are a power of two then the Fourier transformation can be computed recursively, which gives a considerable speedup. This is known 

A matrix Tc is applied to the tristimulus data. The result will be the response of the cone receptors denoted by L, M, S that respond to light in the long, middle, and short part of the spectrum respectively. Since the response of the receptors is proportional to the logarithm of the intensity, the logarithm is applied at the next stage. After the logarithm is applied, the data is transformed by Tp such that one of the coordinate axes is the achromatic axis and the other two axes are the red-green and yellow-blue axes respectively. 

Instead of performing the homomorphic filtering operation independently on the three color bands, one can transform the image into the coordinate system used by the human visual system and then filter the image in this space. After filtering, the transformation into the coordinate system used by the human visual system can be undone. Note that this method works only if we assume that we have receptors that are similar to delta 

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Figure 7.24 Output images produced by homomorphic filtering. The images in the top row were computed using the first high emphasis filter. The image in the bottom row were computed using the second high emphasis filter.

Horn (1974)/Blake (1985) Moore et al. (1991) Retinex Moore et al. (1991) Extended Rahman et al. (1999) Homomorphic filtering Homomorphic filtering (HVS) Intrinsic (min. entropy)... 

Homomorphic filtering Described in Section 7.5. Zeros were removed from the input by transforming each channel with data in the range [0, 1] according to y = (255x + l)/256. We have used the homomorphic filter shown in Figure 7.23(b). Rescaling is done at the third percentile. (pwhite=0.03)... 

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