08 frequency domain filtering DIP

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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)

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

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

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

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

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

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

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, if ( , )( , )

1, if ( , )

D u v DH u v

D u v D

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

G u v A H u v F u v

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

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

08 frequency domain filtering DIP

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