Images can be characterized by a two dimensional spacial function of the form *f(x,y)* where x,y are the spacial coordinates and **f** is the intensity value proportional to the the radiated energy. Hence

* 0<f(x,y)<∞*

The function ** ‘f’** can be decomposed into

**and**

*i(x,y)***where :**

*r(x,y)***is the measure of the amount of**

*‘i’***‘illumination’**.

**is the measure of**

*‘r’***‘reflectance’**.

Therefore the function **f(x,y) = i(x,y)*r(x,y)** such that:

** 0<i(x,y)<∞** and

**where r=0(total absorption) and r=1(total reflectance).**

*0<r(x,y)<1*The intensity or gray level of a monochromatic image is given by *l*=f(x,y)

From the given conditions on ‘i’ and ‘r’ it can be concluded that ‘l’ lies within the range:

**L _{min}< l <L_{max}**

where:

**L _{min}=i_{min}r_{min}**

L_{max}=i_{max}r_{max}

The interval **[L _{min},L_{max}]** is called the

**‘gray scale’**. Generally the scale is shifted numerically to

**[0,L-1]**where

**is**

*l*=0*black*and

**is**

*l*=L-1*white*and the intermediate values are various shades of gray.