Digital Image Processing - UGentsanja/ImageProcessingCourse/04a... · Digital Image Processing ......
Transcript of Digital Image Processing - UGentsanja/ImageProcessingCourse/04a... · Digital Image Processing ......
Digital Image Processing
Dr. ir. Aleksandra PizuricaProf. Dr. Ir. Wilfried Philips
7 December 2006
Aleksandra.Pizurica @telin.UGent.be Tel: 09/264.3415
Telecommunicatie en Informatieverwerking
UNIVERSITEIT GENT
Telecommunicatie en Informatieverwerking
UNIVERSITEIT GENT
Image Restoration
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Objectives of Image Restoration
• Image restoration likewise image enhancement attemts at improving the image quality
• Some overlap exists between image enhancement and restoration
• Important differences: image enhancement is largely subjective, while image restoration is mainly objective process
• Restoration attempts to recover an image that has been degraded by using a priori knowledge about degradation process
• Restoration techniques involve modelling of degradation and applying the inverse process in order to recover the image
• The restoration approach usually involves a criterion of goodness (e.g., mean squared error, smoothness, minimal desription length,…) that will yield an optimal estimate of the desired result
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Overview of restoration techniques
A categorization according to the degradation model (noise, blur or both)
Another possible categorization:• Spatial domain techniques• Frequency domain techniques• Other transform domain (e.g., wavelet) techniques
Model based approaches:• Bayesian techniques – make use of a priori knowledge about the
unknown, undegraded image statistical image modeling• Total variation – involves regularization – penalization of not-desired
local image structures
Statistical image modeling• Modeling marginal statistics (image histograms)• Context models modeling interactions among pixel intensities
–Powerful contextual models: Markov Random Field (MRF) models
Degradation model
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Reminder: camera model…
weighting function, e.g. w(x,y)=1for |x|<∆ and |y|<∆ and 0 otherwise
∫∫ −−== '')','()','(),( dydxyxwyyxxfyxff lkolkkl
A pixel sensor measures the image intensity in the neighborhood of (xk,yl)
),)(( lko yxwf ∗=
Remark: ),)('(),)((),(),( yxhfyxwhfyxyxff ilkkl f ∗=∗∗== where
⇒ Mathematical model: linear filter followed by ideal “sampling”
Camera: CCD (Charged-coupled device) pixel matrix
x, k
y, l
Optical system
fi(x,y)
h(x,y)f0(x,y)
kl
lk
fyxf
=),(
© W. Philips, Universiteit Gent, 1999-2006
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fkl=f(xk,yl)
…Reminder: camera model
The linear filter here is low-pass (attenuates high frequencies)⇒The image becomes blurred, fine details are lost
The sampling keeps only the values of f(x,y) at discrete positions (xk, yl)⇒ Aliasing appears if sampling frequency is not high enough
Remarks•Uniform sampling: xk=k∆, yl=l∆
linear filter
lensaveragingover pixels
idealsamplingfi(x,y) f(x,y)
=
lk
Vyx
l
k (sampling matrix V)•More general (sampling “lattice”)
© W. Philips, Universiteit Gent, 1999-2006
! For compactness we shall write f(x,y) instead of f(xk,yl) !
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Degradation model…
If there is no aliasing we can model the analogue PSF by a digital filter
Scene Not-idealanti-alias filter
fx
H( fx, fy)
lklklk
lk
NFHG
,,,
,
+
=
noise
+ lkG , (DFT-coefficients)Ideal sampling
½ sampling frequency
Scene Lens with idealfrequency
characteristicIdeal
samplingEquivalent
(digital) PSF (linear filter)
+
noise
fx
Hideaal( fx, fy)
kHk,l
lkB ,'
Equivalent model
lkB ,
© W. Philips, Universiteit Gent, 1999-2006
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Equivalent (digital) PSF +
noise
lkG ,Scene Lens with ideal
frequency characteristic
Idealsampling lkF ,
Scene Lens with ideal frequency
characteristicIdeal
sampling
…Degradation model
k
Hk,l
Digital filter: models imperfections in the lens, form of the pixels, … (if aliasing appears equivalent with analogue PSF may not hold)
lkN ,lklklklk NFHG ,,,, +=
Degradation function h(x,y) +
noisen(x,y)
),( yxg),( yxf
equivalent),(),(),(),( lklklkkk yxnyxfyxhyxg +∗=
equivalent
For compactness we write x,y instead of xk,yl
Noise reduction
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Why is denoising important
Not only visual enhancement, but also: automatic processing is facilitated!
original denoised
Example: edge detection
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Noise models…Noise models can be categorized according to
• marginal statistics (first-order statistics, marginal probability density function):• Gaussian, Rayleigh, Poisson, impulsive,…
• higher-order statistics• white noise (uncorrelated)• colored (correlated)
• type of mixing with the signal• additive• multiplicative• other (more complex)
• dependence on the signal• statistically independent of the signal• statisticaly dependent of the signal
Many techniques assume additive white Gaussian noise (AWGN) model
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Noise models: marginal statistics
* Gaussian e.g., thermal noise and a variety of noise sources* Rayleigh e.g. amplitude of random complex numbers whose real and
imaginary components are normally and independently distributed. Examples: ultrasound imaging
* Rice e.g., MRI image magnitude (Gaussian and Rayleigh are special cases of this distribution)
* Poisson models photon noise in the sensor (an average number of photons within a given observation window)
* Bipolar impulsive (e.g., salt and pepper) noise…
Rayleigh Rice Impulsive
Some common probability densitu functions (pdf’s)of noise:
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Noise in MRI
- observed noisy image (complex-valued);
- ideal, noise-free data;],...,[ 1 Nff=f
],...,[ 1 Ndd=d
)sin()cos( Im,Re, lllll nfjnfd +++= θθ
low SNR)1,0( 2 == nf σ
m
high SNR
f1
)1,( 21 == nff σ
p(m)
2Im,
2Re, )sin()cos(|| llllll nfnfdm +++== θθ
The magnitude ml
is Rician distributed
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Noise in MRI
high SNR ( f =f1 )
f1
low SNR (f=0)
m
p(m)
noise-free f
noisy m
mag
nitu
de
cont
rast
SNR
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Original image Image with white noise
Difference with the original
Noise models: correlation properties
white ⇔ uncorrelated
colored ⇔ correlatednoise
Image with colored noise
Difference with the original
© B. Goossens, Universiteit Gent, 2006
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Some reasons behind noise correlation …
interpolatedcaptured image
Bayer pattern
© W. Philips, Universiteit Gent, 1999-2006
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RAW data – B channel Denoised RAW data - B ch.
Raw data in one color channel
noise is white
… Some reasons behind noise correlationInterpolated
noise is correlated
Resulting color imagewith correlated noise
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In many applications it is assumed that noise is additive and statistically independent of the signal
Types of mixing noise with signal
),(),(),( yxnyxfyxg =
),(),(),( yxnyxfyxg +=
In CMOS sensors there is a fixed-pattern noise and mixture ofadditive and multiplicative noise
This is a good model for example for thermal noise
Often, noise is signal-dependent. Examples: speckle, photon noise,…
Many noise sources can be modelled by a multiplicative model:
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2
4 3 2
0 9 1
1 2 3
Input image Output image
3 5
0 2
1 1
1 1 2 2 2
1 3 3 2 2
),( yxb ),( yxg
x
y
sorting 0 1 1 2 2 3 3 4 9
Order statistics filters: Median filter
Basic idea: remove “outliers”The median is a more robust statistical measure than mean
© W. Philips, Universiteit Gent, 1999-2006
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Reduction of impulse noise
median over 3x3
Median filter removes isolated noise peaks, without blurring the image
impulse noise
© W. Philips, Universiteit Gent, 1999-2006
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... Reduction of impulse noise …
median over 3x3Noise-free original
Median filter removes isolated noise peaks, without blurring the image
© W. Philips, Universiteit Gent, 1999-2006
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Median filter and reduction of white noise
median over 3x3
For not-isolated noise peaks (e.g., white Gaussian noise) median filter is not very efficient.
original
© W. Philips, Universiteit Gent, 1999-2006
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∑∈
=xySts
tsgmn
yxf),(
),(1),(ˆ
• Average within a local window Sxy• Aimed for Gaussian noise,
(but blurs edges)
Arithmetic mean
Some simple noise filters
Median
)},({median),(ˆ),(
tsgyxfxySts ∈
=
• Efficient for impulsive noise• Not efficient for Gaussian noise
MedianAlpha-trimmed mean
Discard d/2 lowest and d/2 largest values in Sxy
Denote by gr(s,t) the remaining mn-d pixels
∑∈−
=xyStsr tsg
dmnyxf
),(),(1),(ˆ
For d=0: mean filterFor d=mn-1: median filter