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Department of Computer Science and Engineering, Hanyang University
Restoration vs. EnhancementRestoration vs. Enhancement
Restoration
Objective process
A priori knowledge on degradation model
Modeling the degradation and applying the inverse process to recover the original
To improve an image in some predefined sense
Enhancement
Subjective process
Department of Computer Science and Engineering, Hanyang University
Restoration processRestoration process
Department of Computer Science and Engineering, Hanyang University
Noise modelsNoise models
Assume noise is independent of spatial coordinates and it isuncorrelated w.r.t. the image.
• Gaussian: electronic circuit noise, sensor noise• Rayleigh: range images• Exponential and gamma: laser images• impulse(salt-and-pepper): faulty switching
Department of Computer Science and Engineering, Hanyang University
Eg. Sample noisy imagesEg. Sample noisy images
Department of Computer Science and Engineering, Hanyang University
Eg. Sample noisy images(cont.)Eg. Sample noisy images(cont.)
Department of Computer Science and Engineering, Hanyang University
Periodic noisePeriodic noise
Spatially dependent noise Periodic noise can be
reduced significantly via frequency domain filtering
Department of Computer Science and Engineering, Hanyang University
Estimation of noise parametersEstimation of noise parameters
PDF from small patches
Department of Computer Science and Engineering, Hanyang University
When the only degradation is noiseWhen the only degradation is noise
Periodic noise subtraction gives a good result Random noise mean filter, order-statistics filter,…
),(),(),(
and
),(),(),(
vuNvuFvuG
yxyxfyxg
Department of Computer Science and Engineering, Hanyang University
Mean filtersMean filters
Arithmetic mean filters For Gaussian or uniform noise
Geometric mean filters For Gaussian or uniform noise
Harmonic mean filters Work well for salt noise but fail for pepper noise
Contraharmonic mean filters Suited for impulse noise but require identification(salt
or pepper)
Department of Computer Science and Engineering, Hanyang University
Arithmetic & Geometric mean filterArithmetic & Geometric mean filter
Department of Computer Science and Engineering, Hanyang University
Contraharmonic filtersContraharmonic filters
Q<0 : eliminates salt noiseQ=-1 harmonic mean filter
Q=0 : arithmetic mean filter Q>0: eliminates pepper noise
xy
xy
Sts
Q
Sts
Q
tsg
tsg
yxf
),(
),(
1
),(
),(
),(ˆ
Department of Computer Science and Engineering, Hanyang University
Eg. Contraharmonic filtersEg. Contraharmonic filters
Department of Computer Science and Engineering, Hanyang University
Wrong sign in contraharmonic filtersWrong sign in contraharmonic filters
Disaster!
Department of Computer Science and Engineering, Hanyang University
Order-Statistics filtersOrder-Statistics filters
Median filter Max filter Min filter Midpoint filter Alpha-trimmed mean filter
Department of Computer Science and Engineering, Hanyang University
Median filtersMedian filters
33x3x3medianmedian
33x3x3medianmedian
33x3x3medianmedian
blurred
Department of Computer Science and Engineering, Hanyang University
Max and Min filterMax and Min filter
• Max filter • Min filterRemoves pepper noiseRemoves dark pixels
Removes salt noiseRemoves light pixelsMakes dark objects larger
Department of Computer Science and Engineering, Hanyang University
Eg. ComparisonEg. Comparison
(a) Additive uniformnoise
(b) (a)+additive S&P
5x5 arithmetic mean 5x5 geometric mean
5x5 median 5x5 alpha-trimmedMean(d=5)
Department of Computer Science and Engineering, Hanyang University
Adaptive filtersAdaptive filters
Behavior changes locally based on statistical characteristics of local support
Simple adaptive filter based on mean and variance1. If global_var is zero, then f(x,y)=g(x,y)
2. If local_var>global_var, then f(x,y)=g(x,y) (high local var edge should be preserved)
3. If local_var==global_var, then arithmetic mean filtering
Department of Computer Science and Engineering, Hanyang University
Eg. Adaptive filterEg. Adaptive filter
Department of Computer Science and Engineering, Hanyang University
Adaptive median filterAdaptive median filter
Cope with impulse noise with large probability Preserve detail while smoothing non-impulse noise
Level A:A1=zmed-zmin
A2=zmed-zmax
If A1>0 AND A2<0, go to level BElse increase the window sizeIf window size<=Smax repeat level AElse output zxy
Level B:B1=zxy-zmin
B2=zxy-zmax
If B1>0 AND B2<0, output zxy
Else output zmed
Algorithm
Department of Computer Science and Engineering, Hanyang University
Eg. Adaptive median filterEg. Adaptive median filter
median adaptive median
Department of Computer Science and Engineering, Hanyang University
Periodic noise reductionPeriodic noise reduction
By frequency domain filtering Band reject filter
Department of Computer Science and Engineering, Hanyang University
Eg. Periodic noise reductionEg. Periodic noise reduction
Department of Computer Science and Engineering, Hanyang University
Noise extractionNoise extraction
By bandpass filter
Help understanding noise pattern
Department of Computer Science and Engineering, Hanyang University
Eg. Notch filteringEg. Notch filtering
Removing sensor scan-line patterns
Department of Computer Science and Engineering, Hanyang University
Optimum notch filteringOptimum notch filtering
First isolating the principal contributions of the interference pattern
Then subtracting weighted portion of the pattern from the corrupted image
Department of Computer Science and Engineering, Hanyang University
Eg. Periodic interference(1/3)Eg. Periodic interference(1/3)
Noisy image
Department of Computer Science and Engineering, Hanyang University
Eg. Periodic interference(2/3)Eg. Periodic interference(2/3)
Extraction of noise interference pattern
Department of Computer Science and Engineering, Hanyang University
Eg. Periodic interference(3/3)Eg. Periodic interference(3/3)
Restored image by subtracting weighted portion of periodic interference (Refer to the derivation of weights in pp.250-252)
Department of Computer Science and Engineering, Hanyang University
Linear, Position-Invariant DegradationLinear, Position-Invariant Degradation
),(),(),(),(
:DomainFrequency
),(),(),(),(
:Domain Spatial
vuNvuFvuHvuG
yxyxfyxhyxg
Department of Computer Science and Engineering, Hanyang University
Degradation knowledgeDegradation knowledge
Degradation knowledge about 1. A priori (known)
2. A posteriori (unknown) blind restoration or blind deconvolution
fg
H
Restoration:Restoration:determine the original image ,
given the observed image and
knowledge about the degradation (H).
Department of Computer Science and Engineering, Hanyang University
Fundamental issueFundamental issue
Restoration problem
restoration is to find , such that
but, 1. does not exist: singular
2. may exist, but not be unique: ill-conditioned
3. may exist and unique, but there exists ,
which can be made arbitrarily small, such that
which is not negligible
Image restoration is ill-conditioned at best and Image restoration is ill-conditioned at best and singular at worstsingular at worst
gfT }{1T fgT }{1
1T1T1T
,,}{1 fgT
Department of Computer Science and Engineering, Hanyang University
Estimation of degradation functionEstimation of degradation function
Approaches
Observation
Experimentation
Mathematical modeling
Department of Computer Science and Engineering, Hanyang University
Estimation by observationEstimation by observation
Looking at a small section of the image containing simple structures and then obtaining degradation function
Observed sub-image:
Estimate of original image:
),( yxg s),(ˆ yxf s
),(ˆ),(
),(vuF
vuGvuH
s
ss
Department of Computer Science and Engineering, Hanyang University
Estimation by experimentationEstimation by experimentation
Possible only if equipment similar to the equipment used to acquire the degraded images is available
Eg. Use an impulse
Department of Computer Science and Engineering, Hanyang University
Estimation by modelingEstimation by modeling Based on either physical characteristics or basic
principles
Eg.1. Physical characteristics: atmospheric turbulence
Eg.2. Math derivation: motion blur Starting from
After some manipulation(p.259)
Setting the motion model, we obtain the degradation func.
6/5)( 22
),( vukevuH
dtevuHT tvytuxj 0
)]()([2 00),(
T
dttyytxxfyxg0 00 )](),([),(
Department of Computer Science and Engineering, Hanyang University
Eg.1. Physical modelEg.1. Physical model
Atmospheric turbulence
Department of Computer Science and Engineering, Hanyang University
Eg.2. Math modelingEg.2. Math modeling
Motion blur
Department of Computer Science and Engineering, Hanyang University
Restoration methodsRestoration methods
Inverse filtering Wiener filtering Constrained least square filtering Geometric mean filtering Etc..
Department of Computer Science and Engineering, Hanyang University
Inverse filteringInverse filtering
Poor performance! Very sensitive to noise
),(),(),(),( vuNvuFvuHvuG
),(
),(),(),(ˆ
vuH
vuNvuFvuF
),(
),(),(ˆ
vuH
vuGvuF
Noise amplificationwhen H(u,v) is small
Department of Computer Science and Engineering, Hanyang University
Eg. Inverse filteringEg. Inverse filtering
Department of Computer Science and Engineering, Hanyang University
Minimum mean-square error filterMinimum mean-square error filter
Necessary to handle noise explicitly Statistical characteristics of noise should be
incorporated into the restoration process MMSE filter
To find an estimate of the uncorrupted image such that the mean square error between them is minimized:
Assume: the noise and the image are uncorrelated The one or the other has zero mean The gray levels in the estimate are a linear function of
the levels in the degraded image Derivation: Homework
})ˆ{( 22 ffEe
f̂ f
Department of Computer Science and Engineering, Hanyang University
MMSE filter (cont.)MMSE filter (cont.)
Frequency domain expression:
Approximation of the Wiener filter
),(),(/),(|),(|
),(*
),(),(|),(|),(
),(),(*),(ˆ
2
2
vuGvuSvuSvuH
vuH
vuGvuSvuHvuS
vuSvuHvuF
f
f
f
Wiener filter
),(|),(|
|),(|
),(
1),(ˆ
2
2
vuGKvuH
vuH
vuHvuF
PS of noise PS of image f
Department of Computer Science and Engineering, Hanyang University
Eg. Wiener filteringEg. Wiener filtering
Using the approximation K is chosen interactively
Department of Computer Science and Engineering, Hanyang University
Eg. Restoration by Wiener filterEg. Restoration by Wiener filter motion blurmotion blur
Severe noise
Moderate noise
Negligible noise
Department of Computer Science and Engineering, Hanyang University
Constrained Least Square FilteringConstrained Least Square Filtering
Difficulty in Wiener filter The power spectra of the undegraded image and noise
must be known Minimization in a statistical sense
The constrained LS filtering requires knowledge of
Mean of the noise Variance of the noise
Optimal result for each image
Department of Computer Science and Engineering, Hanyang University
Vector-matrix form of convolutionVector-matrix form of convolution
g: MN-vector (lexicographical order of an image) f: MN-vector H: MNxMN matrix
ηHfg
Department of Computer Science and Engineering, Hanyang University
Formulation: Constrained LS filterFormulation: Constrained LS filter
To find the minimum of a criterion function C defined as
subject to the constraint
where is the Euclidean vector norm
1
0
1
0
22 ),(M
x
N
y
yxfC
22 ||||||ˆ|| ηfHg
www T2||||
Department of Computer Science and Engineering, Hanyang University
Freq. Domain Sol.Freq. Domain Sol.
: adjustable parameter
: Fourier transform of the Laplacian operator
),(|),(||),(|
),(* ),(ˆ
22vuG
vuPvuH
vuHvuF
),( vuP
Department of Computer Science and Engineering, Hanyang University
Eg. Constrained LS filterEg. Constrained LS filter
Significant improvement over Wiener filter
Department of Computer Science and Engineering, Hanyang University
Procedure for computing Procedure for computing
Define a residual vector
Adjust so that
Calculation
a 22 ||||||||)( ηr
fHgr ˆ
][|||| 222 mMN η
In general, automatically determined restoration filteryields inferior results to manual adjustment of filter parameters
iteration