Homomorphic Filtering Algorithm

This algorithm is very similar to Horn's algorithm except the low spatial frequency illumination is separated from the high frequency reflectance by Fourier high-pass filtering. In general a high-pass filter is used to separate and suppress low frequency components while still passing the high frequency components in the signal, if the two types of signals are additive, i.e., the actual signal is the sum of the two types of signals. However, in this illumination/reflection problem low-frequency illumination is multiplied, instead of added, to the high-frequency reflectance. To still be able to use the usual high-pass filter, the logarithm operation is needed to convert the multiplication to addition. Specifically, the steps of this algorithm are

• Take logarithm of the input light signal:
  

  $$\stackrel{\triangle}{=} log\; L(x,y)=log \; [R(x,y) I(x,y)]$$ L'(x,y)

  $$log\; R(x,y)+log\; I(x,y)\stackrel{\triangle}{=}R'(x,y)+I'(x,y)$$ = (1)

  
  

• Carry 2D Fourier transform of the signal L'(x,y)=R'(x,y) + I'(x,y)
  
  $$L(u,v)\stackrel{\triangle}{=}{\cal F}\;L'(x,y) ={\cal F}\;R'(x,y)]+{\cal F}\;I'(x,y)\stackrel{\triangle}{=}R(u,v)+I(u,v)$$
  
  
  where R(u,v), I(u,v) and L(u,v) are the Fourier spectra of the corresponding spatial signals R'(x,y), I'(x,y) and L'(x,y), respectively.

• Suppress low frequency components in Fourier domain
  
  H(u,v)L(u,v)=H(u,v)R(u,v)+H(u,v)I(u,v)
  
  
  where H(u,v) is a filter in the frequency domain whose entries corresponding to the low frequencies are smaller than 1 (suppression of low-frequency components, the illumination) while the rest entries are 1 to keep the high-frequency components in the signal (mostly the reflectance) unchanged.

• Take inverse Fourier transform
  

  $$\stackrel{\triangle}{=} {\cal F}^{-1}[H(u,v)L(u,v)] ={\cal F}^{-1} [H(u,v)R(u,v)]+{\cal F}^{-1}[H(u,v)I(u,v)]$$ L'(x,y)

  $$\stackrel{\triangle}{=}$$ R'(x,y)+I'(x,y)

  
  

• Take exponential operation
  

  $$\stackrel{\triangle}{=}$$ L(x,y) exp[L'(x,y)] =exp[R'(x,y)+I'(x,y)]

  $$exp[R'(x,y)]exp[I'(x,y)] \stackrel{\triangle}{=}R(x,y)I(x,y)$$ =

  
  

After this so called homomorphic filtering process, I(x,y), the processed illumination should be drastically reduced due to the high-pass filtering effect, while the reflectance R(x,y) after this procedure should still be very close to the original reflectance. That is, color constancy results as the color of the surface is not affected much by the color illumination.