08 frequency domain filtering DIP

Frequency Domain Filtering : 1
Frequency DomainFrequency 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 D
H u v
D u v D
≤
= 
>

VisualizationVisualization

Effect of Different CutoffEffect of Different Cutoff
FrequenciesFrequencies

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 ( , ) /
n
H 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 DifferentButterworth Low-pass Filters of Different

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 2
0( , )/2
( , ) D u v D
H 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
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
1
sin(2 ) ( ) ( )
2
k x j k k k kπ δ δ⇔ + − −

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
W
D u v D
W W
H u v D D u v D
W
D 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 Filter
Transfer FunctionsTransfer Functions
Ideal High-pass filter
Butterworth High-pass filter
Gaussian High-pass filter
0
0
0, if ( , )
( , )
1, if ( , )
D u v D
H u v
D u v D
≤
= 
>
[ ]
2
0
1
( , )
1 / ( , )
n
H u v
D D u v
=
+
2 2
0( , )/2
( , ) 1 D u v D
H u v e−
= −

Sharpening Filter Transfer FunctionsSharpening Filter Transfer Functions

Spatial Representation ofSpatial 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
 = − + 144424443

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 yℑ ≠ ℑ ℑ

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 yℑ = ℑ + ℑ

FrameworkFramework

Homomorphic Filtering: ImageHomomorphic Filtering: Image
EnhancementEnhancement
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: ImageHomomorphic Filtering: Image
EnhancementEnhancement
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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