Image Enhancement Frequency Domain

Resmi N.G.Reference:

Digital Signal ProcessingRafael C. GonzalezRichard E. Woods

� Frequency Domain Methods� Basics of filtering in frequency domain� Basic Filters and Properties

� Notch filter� Lowpass Filter� Highpass Filter

� Smoothing Frequency Domain Filters� Ideal Lowpass Filters� Butterworth Lowpass Filters� Gaussian Lowpass Filters� Gaussian Lowpass Filters

� Sharpening Frequency Domain Filters� Ideal Highpass Filters� Butterworth Highpass Filters� Gaussian Highpass Filters� Enhancement using The Laplacian� Unsharp Masking� High Boost Filtering� High-Frequency Emphasis Filtering

� Homomorphic Filtering

3/20/2012 CS04 804B Image Processing - Module2 2

Basics of Filtering in Frequency Domain

1. Multiply the input image by (-1)x+y to center the transform.

2. Compute the DFT, F(u,v) of the resulting image.

3. Multiply F(u,v) by a filter function H(u,v) to obtain G (u,v).3. Multiply F(u,v) by a filter function H(u,v) to obtain G (u,v).

4.Compute the inverse DFT of G(u,v) to obtain g*(x,y).

5. Obtain the real part of g*(x,y).

6. Multiply the result by (-1)x+y to obtain g (x,y).

Basic Steps for Filtering in Frequency Domain

3/20/2012 4CS04 804B Image Processing - Module2

Basic Filters and Properties� Notch Filter

� It is a constant function with a hole at the origin.� Sets F(0,0) to zero.

� Lowpass Filter� It attenuates high frequencies and passes low frequencies.

� Highpass Filter� It attenuates low frequencies and passes high frequencies.

Smoothing Frequency Domain Filters

�Low Pass Filter (Smoothing Filter)� The result in the spatial domain is equivalent to that of

a smoothing filter as the blocked high frequenciesa smoothing filter as the blocked high frequenciescorrespond to sharp intensity changes, i.e. to the fine-scale details and noise in the spatial domain image.

High Pass Filter(Sharpening Filter)� A highpass filter attenuates the low-frequency

components without disturbing the high frequencyinformation in the Fourier Transform.

� It yields edge enhancement or edge detection in thespatial domain, because edges contain many highfrequencies. Areas of constant gray level consistmainly of low frequencies and are thereforesuppressed.

Band Pass Filter� A bandpass filter attenuates very low and very high

frequencies, but retains a middle range band offrequencies. Bandpass filtering can be used to enhanceedges (suppressing low frequencies) while reducing theedges (suppressing low frequencies) while reducing thenoise(attenuating high frequencies).

� Bandpass filter is a combination of both lowpass andhighpass filters. These filters attenuate all frequenciesbelow a specific frequency and above a specific frequency,while retaining the frequencies between the two cut-offs.

Ideal Low Pass Filters

1 ( , )( , )

0 ( , )

Transfer Function

if D u v DH u v

if D u v D

D is a specified non negativequantity

≤= >

( ) ( )

( , ) 2 2

D is a specified non negativequantity

D(u,v)is thedistance from point (u,v)to theoriginof

the frequency rectangle.

NMD u v u v

= − + −

� Ideal – because all frequencies inside a circle of radius D0are passed without any attenuation, whereas allfrequencies outside the circle are completely attenuated.

� The point of transition between H(u,v) = 1 and H(u,v) = 0is called the cut-off frequency.

Ideal Low pass Filter

� Produces “Ringing” effect.� Cannot be realized in electronic components.� Cannot be realized in electronic components.� Not very Practical

Butterworth Low Pass Filters� The transfer function of a BLPF of order n, and with cut-

off frequency at a distance D0 from the origin, is definedas

1( , )

( , )nH u v

D u v=

( ) ( )

( , )1

D u vD

= − + − +

� Provides a smooth transition between low and highfrequencies.

� Butterworth filter of order 1 has neither ringing nornegative values.

� BLPF of order 2 has mild ringing and small negativevalues.

� Reduced ringing effect than ILPF.

Gaussian Low Pass Filters2

2( , )

2( , )D u v

H u v e

D(u,v)is thedistance fromtheoriginof the Fourier

Transform.

( , )2

D u vD

Transform.

is ameasureof the spread of theGaussiancurve

When D

H u v e

whereD is thecut off frequency

( , ) 0, ( , ) 1

( , ) , ( , ) 0.607D

WhenD u v H u v

WhenD u v D H u v e e−

= = = =

Gaussian Low Pass Filters

� Very smooth filter function.� Inverse DFT of the Gaussian lowpass filter is Gaussian.� No “Ringing” effect.� No “Ringing” effect.

Applications of Low Pass Filters� In the field of machine perception

� Character Recognition

� In printing and publishing industry.� In printing and publishing industry.� Cosmetic processing prior to printing

� For processing satellite and aerial images.

Sharpening Frequency Domain Filters

� Ideal High Pass Filters� Transfer Function of high pass filter is given by

( , ) 1 ( , )hp lpH u v H u v= −

� That is, when low pass filter attenuates frequencies, high pass filter passes them and vice versa.

lpH u v is thetransfer functionof corresponding

low pass filter

� Opposite of ideal lowpass filter.

� Sets to zero all frequencies inside a circle of radius D0while all frequencies outside the circle are passed without

0 ( , )( , )

1 ( , )

if D u v DH u v

if D u v D

≤= >

while all frequencies outside the circle are passed withoutattenuation.

� Not physically realizable with electronic components.

� Produces ringing effect.

D0 = 15,30,80

Butterworth High Pass Filter

1( , )

nH u vD

� Represents a transition between the sharpness of IHPF andthe total smoothness of Gaussian filter.

D0 = 15,30,80

Gaussian High Pass Filter2

( , )2( , ) 1

D u vDH u v e

D0 = 15,30,80

Enhancement using The Laplacian

( )( ) ( )

( , ) ( , )

d f xju F u

d f x y d f x y

2 22 2

( , ) ( , )( ) ( , ) ( ) ( , )

( ) ( , )

( , ) ( , )( , ).

d f x y d f x yju F u v jv F u v

u v F u v

d f x y d f x yis the Laplacianof f x y

ℑ + = +

= − +

( ) ( )

( , ) ( ) ( , )

, ( , ) ( ).

( , ) ( )2 2

f x y u v F u v

Laplaciancanbeimplemented in the frequency domain

using the filter H u v u v

NMH u v u v shifted

∴ℑ ∇ = − +

= − +

= − − + −

The Laplacian filtered imageinthe spatial domain is

obtained by comput

( ) ( )222 1

( , ) ( , ) :

( , ) ( , )2 2

ing theinverse FourierTransform

of H u v F u v

NMf x y u v F u v− ∇ = ℑ − − + −

Unsharp Masking and High Boost Filtering� High pass filters eliminate the zero frequency component

of their Fourier transforms and hence average backgroundintensity reduces to near black.

� Solution: Add a portion of the image back to the filteredresult.

� Enhancement using Laplacian adds the entire image backto the filtered result.

� Unsharp masking consists of generating a sharp image by subtracting a blurred version of an image from itself.

� That is, obtaining a highpass-filtered image by subtracting from the image a lowpass-filtered version of itself.

( , ) ( , ) ( , )f x y f x y f x y= −

� High-boost filtering generalizes this by multiplying f(x,y) by a constant A≥1.

( , ) ( , ) ( , )hp lpf x y f x y f x y= −

( , ) ( , ) ( , )hp lpf x y Af x y f x y= −

� High-boost filtering thus increases the contribution made by the image to the overall enhanced result.

� When A=1, high-boost filtering reduces to regular highpass filtering.

High Frequency Emphasis Filtering� To increase the contribution made by high-frequency

components of an image.

� Multiply a highpass filter function by a constant and addan offset so that the zero frequency term is not eliminatedby the filter.by the filter.

� Filter transfer function is given by

� Where a ≥0 and b>a.

( , ) ( , )hfe hpH u v a bH u v= +

Module 2 Assignment� Explain the following point operations:

� Contrast Stretching� Range Compression� Image Clipping� Image Clipping

� Explain Homomorphic Filtering.� Explain Convolution and Correlation Theorems.

Thank YouThank You

Image Enhancement Frequency Domain

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