Sonar Equation: The Wave Equation - University of Washington
Image restoration - Washington State University
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Image restoration Adaptive local noise reduction filter • Filter operation is not uniform at all pixel locations but depends on the local characteristics (local mean, local variance) of the observed image. • Consider an observed image g(m,n) and an b a × window ab S . Let 2 η σ be the noise variance and ) , ( n m m L , ) , ( 2 n m L σ be the local mean and variance of g(m,n) over an b a × window around (m,n). • The adaptive filter is given by: ( 29 ) , ( ) , ( ) , ( ) , ( ) , ( ˆ 2 2 n m m n m g n m n m g n m f L L - σ σ - = η • Usually, we need to be careful about the possibility of 2 2 ) , ( η σ < σ n m L , in which case, we could potentially get a negative output gray value. • This filter does the following: o If 0 2 = σ η (or is small), the filter simply returns the value of g(m,n). o If the local variance ) , ( 2 n m L σ is high relative to the noise variance 2 η σ , the filter returns a value close to g(m,n). This usually corresponds to a location associated with edges in the image. o If the two variances are roughly equal, the filter does a simple averaging over window ab S .
Transcript of Image restoration - Washington State University
Image restoration
Adaptive local noise reduction filter
• Filter operation is not uniform at all pixel locations but depends on
the local characteristics (local mean, local variance) of the observed image.
• Consider an observed image g(m,n) and an ba × window abS . Let 2ησ
be the noise variance and ),( nmmL , ),(2 nmLσ be the local mean and variance of g(m,n) over an ba × window around (m,n).
• The adaptive filter is given by:
( )),(),(),(
),(),(ˆ2
2
nmmnmgnm
nmgnmf LL
−σ
σ−= η
• Usually, we need to be careful about the possibility of 22 ),( ησ<σ nmL , in which case, we could potentially get a negative output gray value.
• This filter does the following:
o If 02 =ση (or is small), the filter simply returns the value of g(m,n).
o If the local variance ),(2 nmLσ is high relative to the noise
variance 2ησ , the filter returns a value close to g(m,n). This
usually corresponds to a location associated with edges in the image.
o If the two variances are roughly equal, the filter does a simple averaging over window abS .
Example
Adaptive median filter ing • Read from text (page 241-243).
Per iodic Inter ference/Noise
• Periodic noise or interference occurs in images due to electrical or electromechanical interference during image acquisition.
• It is an example of spatially dependent noise. • This type of noise can be very effectively removed using frequency
domain filtering. Recall that the spectrum of a pure sinusoid would be a simple impulse at the appropriate frequency location.
Bandreject filters • Bandreject filters remove (or attenuate) a band of frequencies, around
some frequency, say 0D . • An ideal bandreject filter is given by:
�����
+>+≤≤−
−<=
20
2020
20
),(1
),( if0
),( if1
),(W
WW
W
DvuD
DvuDD
DvuD
vuH
where 22),( vuvuD += .
• W is usually referred to the width of the (stop) band and 0D as the center frequency.
• A Butterworth bandreject filter of order n is given by
n
DvuD
WvuDvuH 2
20
2 ),(
),(1
1),( �
����
−+
= .
• A Gaussian bandreject filter is given by
220
2
21
),(
),(
1),( �� �� −−
−= WvuD
DvuD
evuH
Example
• Bandreject filters are ideally suited for filtering out periodic interference.
• Recall that the Fourier transform of a pure sine or cosine function is just a pair of impulses.
• Therefore the interference is “ localized” in the spectral domain and one can easily identify this region and filter it out.
Bandpass filters • Bandpass filters are the exact opposite of bandreject filters. They pass
a band of frequencies, around some frequency, say 0D (rejecting the rest).
• One can write: ),(1),( vuHvuH brbp −=
• Bandpass filter is usually used to isolate components of an image that correspond to a band of frequencies.
• It can also be used to isolate noise interference, so that more detailed analysis of the interference can be performed, independent of the image.
Notch filter
• It is a kind of bandreject/bandpass filter that rejects/passes a very narrow set of frequencies, around a center frequency.
• Due to symmetry considerations, the notches must occur in symmetric pairs about the origin of the frequency plane.
• The transfer function of an ideal notch-reject filter of radius 0D with center frequency ( )00,vu is given by
���� ≤≤=
otherwise1
),(or ),( if0),( 0201 DvuDDvuD
vuH
where
( ) ( )[ ] 5.020
201 2/2/),( vNvuMuvuD −−+−−=
and
( ) ( )[ ] 5.020
202 2/2/),( vNvuMuvuD +−++−=
• The transfer function of a Butterworth notch reject filter of order n is
given by
n
vuDvuD
DvuH �
����+
=
),(),(1
1),(
21
20
.
• A Gaussian notch reject filter is given by
2
20
2121 ),(),(
1),( �� ��−
−= D
vuDvuD
evuH
• A notch pass filter can be obtained from a notch reject filter using: ),(1),( vuHvuH nrnp −=
I llustration of transfer function of notch filters
Example
• Image corrupted by periodic horizontal scan lines.
Optimum Notch Filter ing
• When interference patterns are more complicated, the preceding filters tend to reject more image information in an attempt to filter out the noise.
• In this case, we first filter out the noise interference using a notch
pass filter:
{ }),(),(
),(),(),(1 vuNnm
vuGvuHvuN−=η
=
F
• The image ),( nmη yields a rough estimate of the interference pattern. • We can then subtract off a weighted portion of ),( nmη from the
image g(m,n) to obtain our restored image: ),(),(),(),(ˆ nmnmwnmgnmf η−=
• It is possible to design the weighting function or modulation function
),( nmw in an optimal fashion. See section 5.4.4 (page 251,252) of text for details.
Linear , position-invar iant degradation
• We will now consider the general degradation equation (see page 254, 255 of text for a derivation of this equation):
),(),(),(),(
),(),(*),(),(
vuNvuFvuHvuG
nmnmfnmhnmg
+=η+=
• This consists of a “blurring” function h(m,n), in addition the random
noise component ),( nmη . • The blurring function h(m,n) is usually referred to as a point-spread
function (PSF) and represents the observed image corresponding to imaging an impulse or point source of light.
• In this case, we need to have a good knowledge of the PSF h(m,n), in
addition to knowledge of the noise statistics. This can be done in practice using one of the following methods:
Using Image observation
o Identify portions of the observed image (subimage) that are relatively noise-free and which corresponds to some simple structures.
o We can then obtain ),(ˆ),(
),(vuF
vuGvuH
s
ss = , where ),( vuGs is the
spectrum of the observed subimage, ),(ˆ vuFs is our estimate of the spectrum of the original image (based on the simple structure that the subimage represents).
o Based on the characteristic of the function ),( vuH s , once can rescale to obtain the overall PSF ),( vuH .
Exper imentation
• If feasible, image a known object, usually a point source of light, using the given imaging equipment and setup.
• If A is the intensity of light source and G(u,v) is the observed spectrum, we have
A
vuGvuH
),(),( = .
Modeling
• A physical model is often used to obtain the PSF. • Blurring due to atmospheric turbulence can be modeled by the
transfer function:
6/522 )(),( vukevuH +−= where k is a constant that depends on the nature of the turbulence. • Note that this is similar to a Gaussian lowpass filter.
• Gaussian lowpass filter is also often used to model mild uniform blurring.
• Precise mathematical modeling of the blurring process is sometime
used. For example, blurring due to uniform motion is modeled as:
[ ] )()(sin)(
),( vbuajevbuavbua
TvuH +π−+π
+π=
where T is the duration of exposure and a and b are the displacements in the x- and y-directions, respectively, during this time T.
Inverse Filter
• The simplest approach to restoration is direct inverse filtering. This is obtained as follows:
,1,,1,0, ),,(),(),(ˆ
),(),(
),(ˆ
−==�=
NvuvuGvuRvuF
vuH
vuGvuF
��
where��
),(
1),(
vuHvuR = .�
• We�can�rewrite�this�in�the�spatial�domain�as�follows:�
� ��
����==),(),(
IDFT),(*),(),(ˆvuH
vuGnmrnmgnmf .��
• In�practice,�we�actually�use�a�slightly�modified�filter:�
��
ε>=otherwise���,0
),(���,),(
1),(
vuHvuHvuR �
where� ε� is� a� small� value.� This� avoids� numerical� problems� when�),( vuH �is�small.��
• The� inverse� filter� works� fine� provided� there� is� no� noise.� This� is�illustrated�in�the�following�example.��
• Let� us� now� analyze� the� performance� of� the� inverse� filter� in� the�presence�of�noise.�Indeed,�in�this�case:��
),(),(),(),( vuNvuFvuHvuG += ,�
which�gives��
),(
),(),(
),(
),(),(ˆ
vuH
vuNvuF
vuH
vuGvuF +== �
• Hence�noise�actually�gets�amplified�at�frequencies�where� ),( vuH �is�zero� or� very� small.� In� fact,� the� contribution� from� the� noise� term�dominates�at�these�frequencies.��
• As�illustrated�by�an�example,� the� inverse�filter�fails�miserably� in�the�presence� of� noise.� It� is� therefore,� seldom� used� in� practice,� in� the�presence�of�noise.��
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02.0MSE = 008.0MSE =
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013.0MSE = 0016.0MSE =
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