An image is a 2-D function. A grey level image is given as f(x,y) where f is the grey level and x and y are the spatial coordinates. Images that are formed due to the fact that there is some amount of light falling on the object and some amount of light reflecting from the object. Light falling on the object is called illumination and light reflecting is called reflectance.

Since images that fall in the electromagnetic spectrum,

$0 \lt f(x, y) \lt ∞$

Based on this fact, the simple model for an image is given by

F(n1,n2) =i(n1,n2) x r(n1,n2) ………………….(1)

This model is known as illumination-reflectance model. This model can be used to address the problem of improving the quality of an image that has been acquired under poor illumination conditions. In the equation f(n1,n2) represents the image, i(n1,n2) represents illumination component and r(n1,n2) represents the reflectance component. For many images, the illumination is the primary contributor to the dynamic range and varies slowly in space, while the reflectance component r(n1,n2) represents the details of the object and varies rapidly in space. If the illumination and the reflectance component are taken separately, then the logarithmic of the input function f(n1,n2) is taken. Because f(n1,n2) is the product of i(n1,n2) with r(n1,n2), the log of f(n1,n2) separates the components as:

Ln[f(n1,n2)] =ln[i(n1,n2)] +ln[r(n1,n2)] …………………(2)

Taking Fourier transform on both sides,

F(k,l) =FI(k,l) +FR(k,l) …………………..(3)

Where FI(k,l) and FR(k,l) are the fourier transform of the illumination and reflectance components respectively. Then the desired filter function H(k,l) can be applied separately to the illumination and the reflectance component separately as:

F(k,l) x h(k,l) = FI(k,l) x H(k,l) + Fr(k,l) x H(k,l) …………………(4)

In order to visualize the image, inverse Fourier transform followed by exponential function is applied. First the inverse Fourier transform is applied as:

F’(n1,n2) = F-1[F(k,l) x H(k,l)] =F-1[F1(k,l) x H(k,l)] +F-1[FR(k,l) x H(k,l)]…….(5)

The desired enhanced image is obtained by taking the exponential operation:

g(n1,n2)= eƒ’(n1,n2) ……………….(6)

Here g(n1,n2) represents the enhanced version of the original image ƒ(n1,n2). The sequence of operation can be represented by the block diagram as: