it523: digital image processingcourses.daiict.ac.in/pluginfile.php/28248/mod_resource/... ·...

IT523: Digital Image Processing

Chapter 4: Frequency Domain Enhancement

Frequency domain filtering methodology

Figure: Frequency domain filtering

IT523 DIP: Lecture 8 2/28

Padding during convolution

Figure: Ciruclar and linear convolution

Notch filtering

Figure: Periodic noise

Notch filtering

H(u, v) =0 (u, v) = (p, q)

=1 otherwise.

Figure: Notch filteringIT523 DIP: Lecture 8 5/28

Low pass and High pass filtering

Figure: Low and High pass filtering

Ideal low pass filter

Figure: Ideal low pass fitler. (top) Frequency response (bottom) Impulseresponse

Ideal low pass filter

Figure: Ideal Low pass filter output

Butterworth filter

Hn(u, v) = 11+(D(u,v)/D0)2n

, where D(u, v) is the Euclidean

distance from origin and D0 is the cut-off frequency.

Figure: Butterworth low pass fitler. (top) Frequency response (bottom)Impulse response

Butterworth low pass filter

Figure: Butterworth Low pass filter output

Gaussian low pass filter

Hn(u, v) = exp(−D2

u0,v0(u, v)/2D20

), where

Du0,v0(u, v) = (u − u0)2 + (v − v0)2 is the Euclidean distancefrom (u0, v0) and D0 is the cut-off frequency.

Figure: Gaussian low pass fitler. (top) Frequency response (bottom)Impulse response

Gaussian low pass filter

Figure: Gaussian Low pass filter output

Applications of Low pass filters

Figure: Applications of low pass filter (top) Character recognition(bottom) Removing horizontal scan lines.

High pass filters

Hhp(u, v) = 1− Hlp(u, v).

Figure: High pass filter frequency responses.

High pass filters

Figure: High pass filter impulse responses.

Ideal High pass filters

Figure: Ideal High pass filter outputs.

Gaussian High pass filters

Figure: Gaussian High pass filter outputs.

Laplacian filter

One-dimensional signals:

dx2f (x)

)= −ω2F (ω).

Two-dimensional signals:

∂x2f (x , y) + ∂2

∂y2 f (x , y))

= −(u2 + v2)F (u, v).

Therefore the Laplacian filter’s frequency response is

Hlap(u, v) = −(u2 + v2)

Laplacian filter

dx2f (x)

)= −ω2F (ω).

∂x2f (x , y) + ∂2

∂y2 f (x , y))

= −(u2 + v2)F (u, v).

Hlap(u, v) = −(u2 + v2)

Laplacian filter

dx2f (x)

)= −ω2F (ω).

∂x2f (x , y) + ∂2

∂y2 f (x , y))

= −(u2 + v2)F (u, v).

Hlap(u, v) = −(u2 + v2)

Laplacian filter

Figure: Laplacian frequency response and impulse response

Laplacian filter

Figure: Laplacian filter output

Phase of Image DFT

Fourier transform is a complex valued function:

F (f (x , y)) =F (u, v)

F (u, v) =R(u, v) + jI (u, v)

Phase: φ(u, v) = tan−1(

I (u,v)R(u,v)

Phase of Image DFT

Fourier transform is a complex valued function:

F (f (x , y)) =F (u, v)

F (u, v) =R(u, v) + jI (u, v)

Phase: φ(u, v) = tan−1(

I (u,v)R(u,v)

Figure: Phase is more crucial than DFT magnitude.

Effect of Translation and rotation

Figure: Effect of translation and rotation on DFT magnitude and phase.

Homomorphic filtering

Image formation model: f (x , y) = i(x , y)r(x , y)

We are interested in r(x , y), not in the illumination i(x , y).

Unfortunately f is a multiplicative combination of i and r . SoF (u, v) 6= I (u, v)R(u, v).

Use Logarithm:z(x , y) = ln(f (x , y)) = ln(i(x , y)) + ln(r(x , y)).

Take Fourier Transform: Z (u, v) = Fi (u, v) + Fr (u, v), whereFi (u, v) = F{ln(i(x , y))} and Fr (u, v) = F{ln(r(x , y))}.

Apply a filter H(u, v) to eliminate Fi (u, v):

S(u, v) = H(u, v)Z (u, v) = H(u, v)Fi (u, v) + H(u, v)Fr (u, v)

Output image:

s(x , y) =F−1{S(u, v)}=F−1{H(u, v)Fi (u, v) + H(u, v)Fr (u, v)}=F−1{H(u, v)Fi (u, v)}+ F−1{H(u, v)Fr (u, v)}=i1(x , y) + r1(x , y)

Output image:

Inverting the logarithm using exponential:

g(x , y) = exp(s(x , y))

= exp(i1(x , y) + r1(x , y))

= exp(i1(x , y)) exp(r1(x , y))

g(x , y) =i0(x , y)r0(x , y)

Assumption about Illumination

Illumination component typically has low spatial variationscompared to reflectance component, for example aroundedges.

Figure: Homomorphic filter response.

Homomorphic filtering output

Figure: Eliminating Illuminance component.

it523: digital image processingcourses.daiict.ac.in/pluginfile.php/28248/mod_resource/... ·...

Documents