Smoothing Smoothing frequency domain frequency domain

filtersfilters

Image enhancementImage enhancement

Spatial domain methodsSpatial domain methods

- Direct manipulation on pixels of- Direct manipulation on pixels of

image image Frequency domain methodsFrequency domain methods

- Modify the fourier transform of an- Modify the fourier transform of an

imageimage

Smoothing filtersSmoothing filters

Used for blurring and noise rednUsed for blurring and noise redn Used in preprocessing steps to remove Used in preprocessing steps to remove

small details small details Edges and noise in high frequency of Edges and noise in high frequency of

image F.Timage F.T Smoothing is achieved usingSmoothing is achieved using low pass low pass

filteringfiltering

Smoothing Frequency-Domain Smoothing Frequency-Domain FiltersFilters

IdealIdeal

Very sharp filter functionVery sharp filter function ButterworthButterworth

Transition between 2 extremes Transition between 2 extremes GaussianGaussian

Very smooth filter functionVery smooth filter function

Basics of filtering in the frequency Basics of filtering in the frequency domaindomain

)],([Image Filtered

),(),(),(1 vuG

vuFvuHvuG

Ideal lowpass filterIdeal lowpass filter

0

0

),( 0

),( 1 {),(

DvuDif

DvuDifvuH

2/122 ])2/()2/[(),( NvMuvuD

Ideal lowpass filterIdeal lowpass filter

Cut off frequency-pt of transition from Cut off frequency-pt of transition from H(u,v)=1 and H(u,v)=0H(u,v)=1 and H(u,v)=0

LPF are compared by studying their LPF are compared by studying their behaviour as a function of same cutoff behaviour as a function of same cutoff frequenciesfrequencies

Compute circles that enclose specified Compute circles that enclose specified amounts of total image poweramounts of total image power

Ideal lowpass filterIdeal lowpass filter

1

0

1

0

),(M

u

N

vT vuPP

u v

TPvuP ]/),([100

As the filter radius increases, As the filter radius increases, less and less power is less and less power is removed/filtered out, more and removed/filtered out, more and more details are preserved.more details are preserved.

Ringing effect is clear in most Ringing effect is clear in most cases except for the last one.cases except for the last one.

Ringing effect is the Ringing effect is the consequence of applying ideal consequence of applying ideal lowpass filterslowpass filters

Ringing EffectRinging Effect

Best explained in spatial domainBest explained in spatial domain

Convolution of a function with an impulse Convolution of a function with an impulse “copies” the value of that function at the “copies” the value of that function at the location of the impulselocation of the impulse

Center component Center component responsible for responsible for blurring,concentric blurring,concentric components components responsible for responsible for ringingringing

Narrower the filter in Narrower the filter in the freq domain, the freq domain, blurring and ringing blurring and ringing are more severeare more severe

Butterworth Lowpass FiltersButterworth Lowpass Filters

When D(u,v)= , H(u,v)=0.5When D(u,v)= , H(u,v)=0.50D

nDvuDvuH

20 ]/),([1

1),(

Butterworth Lowpass FiltersButterworth Lowpass Filters

Butterworth Lowpass FiltersButterworth Lowpass Filters

ILPF Vs BLPF….ILPF Vs BLPF….

Butterworth Lowpass FiltersButterworth Lowpass Filters

D0=80, n=1Original D0=80, n=2 D0=80, n=3 D0=80, n=5

D0=80, n=10 D0=80, n=20 D0=80, n=50

Disadvantage..Disadvantage..

Gaussian Lowpass filterGaussian Lowpass filter

LetLet

22 2/),(),( vuDevuH

20

2 2/),(),( DvuDevuH

0D

Gaussian Lowpass filterGaussian Lowpass filter

Gaussian Lowpass filterGaussian Lowpass filter

There is no ringing in GLPFThere is no ringing in GLPF

GLPF is used in Medical ImagingGLPF is used in Medical Imaging