08 frequency domain filtering DIP

Frequency Domain Filtering : 1

Frequency Domain Frequency Domain FilteringFiltering


Blurring/Noise reductionBlurring/Noise reduction

Noise characterized by sharp transitions in image intensity

Such transitions contribute significantly to high frequency components of Fourier transform

Intuitively, attenuating certain high frequency components result in blurring and reduction of image noise


Ideal Low-pass FilterIdeal Low-pass Filter

Cuts off all high-frequency components at a distance greater than a certain distance from origin (cutoff frequency)

0

0

1, if ( , )( , )

0, if ( , )

D u v DH u v

D u v D


VisualizationVisualization


Effect of Different Cutoff FrequenciesEffect of Different Cutoff Frequencies





As cutoff frequency decreases

Image becomes more blurred

Noise becomes reduced

Analogous to larger spatial filter sizes

Noticeable ringing artifacts that increase as the amount of high frequency components removed is increased


Why is there ringing?Why is there ringing?

Ideal low-pass filter function is a rectangular function

The inverse Fourier transform of a rectangular function is a sinc function


RingingRinging


Butterworth Low-pass FilterButterworth Low-pass Filter

Transfer function does not have sharp discontinuity establishing cutoff between passed and filtered frequencies

Cutoff frequency D0 defines point at which H(u,v)=0.5

2

0

1( , )

1 ( , ) /nH u v

D u v D


Butterworth Low-pass FilterButterworth Low-pass Filter


Spatial RepresentationsSpatial Representations

Tradeoff between amount of smoothing and ringing


Butterworth Low-pass Filters of Different Butterworth Low-pass Filters of Different FrequenciesFrequencies


Gaussian Low-pass FilterGaussian Low-pass Filter

Transfer function is smooth, like Butterworth filter

Gaussian in frequency domain remains a Gaussian in spatial domain

Advantage: No ringing artifacts

2 20( , )/2( , ) D u v DH u v e






Low-pass Filtering: ExampleLow-pass Filtering: Example


Low-pass Filtering: ExampleLow-pass Filtering: Example


Periodic Noise ReductionPeriodic Noise Reduction

Typically occurs from electrical or electromechanical interference during image acquisition

Spatially dependent noise

Example: spatial sinusoidal noise


ExampleExample


ObservationsObservations

Symmetric pairs of bright spots appear in the Fourier spectra

Why?

Fourier transform of sine function is the sum of a pair of impulse functions

Intuitively, sinusoidal noise can be reduced by attenuating these bright spots

0 0 0

1sin(2 ) ( ) ( )

2k x j k k k k


Bandreject FiltersBandreject Filters

Removes or attenuates a band of frequencies about the origin of the Fourier transform

Sinusoidal noise may be reduced by filtering the band of frequencies upon which the bright spots associated with period noise appear


Example: Ideal Bandreject FiltersExample: Ideal Bandreject Filters

0

0 0

0

1, if ( , )2

( , ) 0, if ( , )2 2

1, if ( , )2

WD u v D

W WH u v D D u v D

WD u v D


ExampleExample


Notchreject FiltersNotchreject Filters

Idea:

Sinusoidal noise appears as bright spots in Fourier spectra

Reject frequencies in predefined neighborhoods about a center frequency

In this case, center notchreject filters around frequencies coinciding with the bright spots


Some Notchreject FiltersSome Notchreject Filters


ExampleExample


SharpeningSharpening

Edges and fine detail characterized by sharp transitions in image intensity

Such transitions contribute significantly to high frequency components of Fourier transform

Intuitively, attenuating certain low frequency components and preserving high frequency components result in sharpening


Sharpening Filter Transfer FunctionSharpening Filter Transfer Function

Intended goal is to do the reverse operation of low-pass filters

When low-pass filer attenuates frequencies, high-pass filter passes them

When high-pass filter attenuates frequencies, low-pass filter passes them

( , ) 1 ( , )hp lpH u v H u v


Some Sharpening FilterSome Sharpening FilterTransfer FunctionsTransfer Functions

Ideal High-pass filter

Butterworth High-pass filter

Gaussian High-pass filter

0

0

0, if ( , )( , )

1, if ( , )

D u v DH u v

D u v D

2

0

1( , )

1 / ( , )nH u v

D D u v

2 20( , )/2( , ) 1 D u v DH u v e


Sharpening Filter Transfer FunctionsSharpening Filter Transfer Functions


Spatial Representation of Spatial Representation of Highpass FiltersHighpass Filters


Filtered Results: IHPFFiltered Results: IHPF


Filtered Results: BHPFFiltered Results: BHPF


Filtered Results: GHPFFiltered Results: GHPF


ObservationsObservations

As with ideal low-pass filter, ideal high-pass filter shows significant ringing artifacts

Second-order Butterworth high-pass filter shows sharp edges with minor ringing artifacts

Gaussian high-pass filter shows good sharpness in edges with no ringing artifacts


High-boost filteringHigh-boost filtering

In frequency domain

( , ) ( , ) ( , )lpg x y Af x y f x y

( , ) ( 1) ( , ) ( , ) ( , )hpg x y A f x y f x y h x y

( , ) ( 1) ( , ) ( , ) ( , )lpg x y A f x y f x y f x y

( , ) ( 1) ( , ) ( , )hpg x y A f x y f x y

( , ) ( 1) ( , ) ( , ) ( , )G u v A F u v F u v H u v

( , ) ( 1) ( , ) ( , )hp

hb

G u v A H u v F u v

H


High frequency emphasisHigh frequency emphasis

Advantageous to accentuate enhancements made by high- frequency components of image in certain situations (e.g., image visualization)

Solution: multiply high-pass filter by a constant and add offset so zero frequency term not eliminated

Generalization of high-boost filtering

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


ResultsResults


Homomorphic FilteringHomomorphic Filtering

Image can be modeled as a product of illumination (i) and reflectance (r)

Can't operate on frequency components of illumination and reflectance separately

( , ) ( , ) ( , )f x y i x y y x y

( , ) ( , ) ( , )f x y i x y r x y


Homomorphic FilteringHomomorphic Filtering

Idea: What if we take the logarithm of the image?

Now the frequency components of i and r can be operated on separately

ln ( , ) ln ( , ) ln ( , )f x y i x y r x y

ln ( , ) ln ( , ) ln ( , )f x y i x y r x y


Homomorphic Filtering Homomorphic Filtering FrameworkFramework


Homomorphic Filtering: Image EnhancementHomomorphic Filtering: Image Enhancement

Simultaneous dynamic range compression (reduce illumination variation) and contrast enhancement (increase reflectance variation)

Illumination component characterized by slow spatial variations (low spatial frequencies)

Reflectance component characterized by abrupt spatial variations (high spatial frequencies)


Homomorphic Filtering: Image EnhancementHomomorphic Filtering: Image Enhancement

Can be accomplished using a high frequency emphasis filter in log space

DC gain of 0.5 (reduce illumination variations)

High frequency gain of 2 (increase reflectance variations)

Output of homomorphic filter

2( , ) ( , ) ( , )g x y i x y r x y


ExampleExample


Homomorphic Filtering: Noise ReductionHomomorphic Filtering: Noise Reduction

Multiplicative noise model

Transforming into log space turns multiplicative noise to additive noise

Low-pass filtering can now be applied to reduce noise

( , ) ( , ) ( , )f x y s x y n x y

ln ( , ) ln ( , ) ln ( , )f x y s x y n x y


ExampleExample

Date post:	10-May-2015
Category:	Education
Upload:	babak
View:	1,718 times
Download:	1 times

08 frequency domain filtering DIP

Education