Low Pass and High Pass Filters

7/29/2019 Low Pass and High Pass Filters

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Dr. D. J. Jackson Lectur e 10-1Electrical & Computer Engineering

Computer Vision &

Digital Image Processing

Frequency Domain Filters

Dr. D. J. Jackson Lecture 10-2Electrical & Computer Engineering

Butterworth lowpass filter

The transfer function of aButterworth lowpass filter (BLPF)of order n with cutoff frequency

D0 is given by

where D(u,v)=[u2+v2]1/2

For this smooth transition filter, acutoff frequency locus is chosensuch that D(u,v) is a certainpercentage of its maximum

Designed such that at D(u,v)=D0H(u,v)=0.50 (50% of its maximumvalue)

[ ] nDvuDvuH

2

0/),(1

1),(

+=

H(u,v)

D(u,v)/D0


Butterworth lowpass filter (continued)

Another transfer function ofa Butterworth lowpass filter(BLPF) of order n withcutoff frequency D0 is givenby

Designed such that atD(u,v)=D0

nDvuDvuH

20]/),(][12[1

1),(

+=

2

1),( =vuH

H(u,v)

D(u,v)/D0


MATLAB Butterworth lowpass filter

f unct i on [ g]=bl pf ( f , or der, cutof f );

% Usage [ g]=bl pf (f , order, cutof f );

F=f f t2(f ) ;

F=f f tshi f t( F);

[ umax vmax] =si ze( F) ;

f or u=1: umax

f or v=1: vmax

H( u, v) =1/ ( 1+( sqrt ( 2)- 1)*( sqrt ( ( ( umax/ 2-( u- 1)) . 2+( vmax/ 2-( v-1)) . 2)) / cutof f ). ( 2*order) ) ;

end;

end;

G=H. *F;

G=i f f t 2(G) ;

g=sqr t ( r eal ( G) . 2+i mag(G). 2);


Example MATLAB output

Cutoff frequency=10 Cutoff frequency=100


Butterworth lowpass filter function


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Ringing in BLPF


Gaussian lowpass filter

The transfer function of a Gaussian lowpass filter (GLPF) isgiven by

Here, is a measure of spread about the center Let =D0, then

where D0 is the cutoff frequency

22

2/),(),(vuDevuH =

20

2 2/),(),( DvuDevuH =


Gaussian lowpass filter function


Ideal highpass filter (IHPF)

A transfer function for a 2-D ideal highpass filter (IHPF) isgiven as

where D0 is a stated nonnegative quantity (the cutofffrequency) and D(u,v) is the distance from the point (u,v) tothe center of the frequency plane

>

=

0

0

Dv)D(u,if1

Dv)D(u,if0),( vuH

22),( vuvuD +=

v

u

H(u,v)


Butterworth highpass filter

The transfer function of aButterworth highpass filter(BHPF) of order n with cutofffrequency D0 is given by

where D(u,v)=[u2+v2]1/2

For this smooth transition filter, acutoff frequency locus is chosensuch that D(u,v) is a certainpercentage of its maximum

Designed such that at D(u,v)=D0H(u,v)=0.50 (50% of its maximumvalue)

[ ] nvuDDvuH

2

0 ),(/1

1),(

+=


Butterworth highpass filter (continued)

Another transfer function ofa Butterworth highpassfilter (BHPF) of order n withcutoff frequency D0 is givenby

Designed such that atD(u,v)=D0

nvuDDvuH

20 )],(/][12[1

1),(

+=

2

1),( =vuH


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Gaussian highpass filter

The transfer function of a Gaussian highpass filter (GHPF) isgiven by

Here, is as in the Gaussian lowpass case Let =D0, then

where D0 is the cutoff frequency

22

2/),(1),(vuDevuH =

20

2 2/),(1),( DvuDevuH =


Highpass filter functions


Unsharp masking, highboost filtering, and high-

frequency-emphasis filtering

Let

where

Where HLP(u,v) is a lowpass filter andF(u,v) is theFourier transform of f(x,y)

Then

If k=1 this is an unsharp mask

If k>1 this is a highboost filter

),(),(),( yxfyxfyxg LPmask =

)],(),([),( 1 vuFvuHyxf LPLP=

),(*),(),( yxgkyxfyxg mask+=


Unsharp masking, highboost filtering, and high-

frequency-emphasis filtering

In frequency domain only terms

This is a high-frequency-emphasisfilter

A more general form is

Here k10 controls an offset from the origin and k20 controlscontributions of high frequencies

)},()],(*1{[),(

filterhighpassaofin termsor

)},()]],(1[*1{[),(

1

1

vuFvuHkyxg

vuFvuHkyxg

HP

LP

+=

+=

)},()],(*{[),( 211 vuFvuHkkyxg HP+=


Example use of a high-frequency-emphasis filter


Homomorphic filtering

Recall f(x,y) can be expressed as f(x,y)=i(x,y)r(x,y)

We cannot use this directly to operate on the frequencycomponents of i(x,y) and r(x,y) because

But if we define

then

)},({)},({)},({ yxryxiyxf

)],(ln[)],(ln[

)],(ln[),(

yxryxi

yxfyxz

+=

=

)]},({ln[)]},({ln[

)]},({ln[)},({

yxryxi

yxfyxz

+=

=


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Homomorphic filtering (continued)

Then,

where I(u,v) and R(u,v) are the Fourier transforms of ln[i(x,y)]and ln[r(x,y)] respectively

Z(u,v) can be processed by a filter function

where S(u,v) is the Fourier transform of the result

In the spatial domain,

),(),(),( vuRvuIvuZ +=

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

),(),(),(

vuRvuHvuIvuH

vuZvuHvuS

+=

=

)},(),({)},(),({

)},({),(11

1

vuRvuHvuIvuH

vuSyxs

+=

=



If,

then

Taking the exponential yields the final result

)},(),({),('

and)},(),({),('1

1

vuRvuHyxr

vuIvuHyxi

=

=

),('),('),( yxryxiyxs +=

),(),(

)],('exp[*)],('exp[

)],(exp[),(

00 yxryxi

yxryxi

yxsyxg

=

=

=



The process can be viewed graphically as above

The illumination of an image is generallycharacterized byslow spatial variations (associated with the low frequenciesof the Fourier transform of the logarithm)

The reflectance of an image tends to vary abruptly,especially at the junctions of dissimilar objects (associatedwith the high frequencies of the Fourier transform of thelogarithm)

ln

f(x,y)

FFT H(u,v) FFT-1 exp

g(x,y)



The filter function H(u,v) should/will affect the low-and high-frequencycomponents in different ways and can be approximated by

If the filter function chosen is such thatL1 then the lowfrequencies tend to be decreased and the high frequencies are amplified

functiontheofslopethecontrolswhere

]1)[(),( ]/),([20

2

c

evuH LDvuDc

LH +=


Homomorphic filtering (example)


Homomorphic filtering (example)


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Selective fil tering

Previously discussed filters operate over the entirefrequency rectangle (i.e. complete representation in

the frequency domain) Occasionally it is useful to operate on specific

frequency bands or small regions of the frequencyrectangle

Bandrejector Bandpass filters operate on specificfrequency bands

Notch filters operate on small regions of thefrequency rectangle


Bandreject filters

Bandreject filters can, in general, be easilyconstructed using the same concepts as described

for other filters Assume the following:

D(u,v) is the distance from the center of the frequencyrectangle

D0 is the radial center of the band of interest

Wis the width of the band of interest


Bandreject filters

Ideal bandreject filter

Butterworth bandreject filter

Gaussian bandreject filter

+=

otherwise122

if0),( 00

WDD

WD

vuH

n

DD

DWvuH

2

20

21

1),(

+

=

220

2

1),(

= DWDD

evuH


Bandpass filters

Bandpass filters can be derived from any of thebandrejectexpressions as

HBP(u,v)=1- HBR(u,v)

HBR(u,v) is the corresponding bandrejectfilter

This formulation is exactly as in thehighpass/lowpass case


Gaussian filter example


Notch filters

A notch filter rejects (or passes depending on itsconstruction) frequencies in a pre-defined area(neighborhood) about the center of the frequencyrectangle

We desire that the filters be zero-phase-shift Must be symmetric about the origin

A notch with center at (u0,v0) must have a correspondingnotch at (-u0,-v0)


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Notch reject filters

Notch reject filters are easily constructed asproducts of highpass filters whose centers have

been translated to the center of the notchesThe general form is:

Where Hk(u,v) and H-k(u,v) are highpass filterswhose centers are at (uk,vk) and (-uk,-vk)

Q is the number of notches

=

=Q

kkkNR vuHvuHvuH

1

),(),(),(


Notch reject filters (continued)

The centers at (uk,vk) and (-uk,-vk) are specified withrespect to the center of the frequency rectangle,

(M/2,N/2) Distances can be calculated as:

22

22

)2/()2/(),(

)2/()2/(),(

kkk

kkk

vNvuMuvuD

and

vNvuMuvuD

+++=

+=


Notch reject filters (continued)

A general form for a Butterworth notch reject filterof order n and containing three notch pairs is:

The constant D0kis the same for each pair ofnotches, but can be different for different pairs

A notch pass filtercan be expressed as

+

+=

= n

kkkn

kk

NRvuDDvuDD

vuH2

0

3

12

0 )],(/[1

1

)],(/[1

1),(

),(1),( vuHvuHNRNP

=

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Low Pass and High Pass Filters