HOMOMORPHIC FILTERING

Illumination-Reflection MODEL

Images are denoted by two dimensional light intensity functions of the form f(x,y).

The value of f at spatial coordinates(x,y) gives the intensity of the image at that point.

f(x,y) must be nonzero and finite 0<f(x,y)<∞

The function f(x,y) may be characterized by two components

1) Illumination 2) Reflection

Homomorphic Filtering

The digital images are created from optical image that consist of two primary components: The lighting component The reflectance component

The lighting component results from the lighting condition present when the image is captured. Can change as the lighting condition change.

Homomorphic Filtering

The reflectance component results from the way the objects in the image reflect light. Determined by the intrinsic properties of the object

itself. Normally do not change.

In many applications, it is useful to enhance the reflectance component, while reducing the contribution from the lighting component.

Homomorphic Filtering

Basis: illumination-reflectance model

Homomorphic filtering is a frequency domain filtering process that compresses the brightness (from the lighting condition) while enhancing the contrast (from the reflectance properties of the object).

Derivation

The two functions combine as a product to form f(x,y):

f(x,y)=i(x,y)r(x,y) 0<i(x,y)<∞and 0<r(x,y)<1

Cont..

The fourier transform of the product of two functions is not seperable :

Ŧ{f(x,y)} ≠Ŧ {i(x,y)}Ŧ{r(x,y)}

Suppose if we define z(x,y)=ln f(x,y) =ln[ i(x,y) r(x,y)] =ln i(x,y)+ln r(x,y)

Cont..

Then Ŧ{ z(x,y)} =Ŧ {ln(f(x,y)} = Ŧ{ln i(x,y)}+Ŧ{ln

r(x,y)}

Z(u,v) =I(u,v) +R(u,v)Where I(u,v) = Fourier transform of ln i(x,y) R(u,v)= Fourier transform of ln r(x,y)

Cont..

Now if we process Z(u,v)by means of a filter function H(u,v) then

S(u,v)= H(u,v)Z(u,v) =H(u,v)[I(u,v) +R(u,v)] = H(u,v) I(u,v) + H(u,v) R(u,v)In time domain, s(x,y) =Ŧ-1{S(u,v)} =Ŧ-1 {H(u,v) I(u,v)} + Ŧ-1{H(u,v)

R(u,v)}

Cont..

By letting i’(x,y) = Ŧ-1 {H(u,v) I(u,v)} And r’(x,y) = Ŧ-1{H(u,v) R(u,v)},Now s(x,y) can be expressed as s(x,y) = i’(x,y)+r’(x,y)

Cont..

Enhanced image g(x,y) is g(x,y)=es(x,y)

= e[ i’(x,y)+r’(x,y)]

=e i’(x,y). er’(x,y)

=i(x,y) r(x,y)

Block Diagram

f(u,v) H(u,v)F(u,v)

f(x,y)

g(x,y)Input image

enhanced image

Pre -processin

g

Post -processin

g

Fourier transform

Filter function H(u,v)

Inverse Fourier

transform

Homomorphic filtering for image enhancement

f(x,y)

g(x,y)

ln DFT H(u,v)

Inverse DFT exp

Filter function

original image image processed by homomorphic

filtering

Applications

Application of homomorphic filtering results in

Sharper image Increase in contrast Increase in dynamic range

compression