image restoration and reconstruction (noise removal)cnikou/courses/digital_image_processing/...2 c....
TRANSCRIPT
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University of Ioannina - Department of Computer Science
Image Restoration and Reconstruction(Noise Removal)
Christophoros [email protected]
Digital Image Processing
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C. Nikou – Digital Image Processing (E12)
Image Restoration and Reconstruction
Things which we see are not by themselves what we see…It remains completely unknown to us what the objects may be by themselves and apart from the receptivity of our senses. We know nothing but our manner of perceiving them.
Immanuel Kant
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Contents
In this lecture we will look at image restoration techniques used for noise removal
– What is image restoration?– Noise and images– Noise models– Noise removal using spatial domain filtering– Noise removal using frequency domain
filtering
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What is Image Restoration?
Image restoration attempts to restore images that have been degraded
– Identify the degradation process and attempt to reverse it
– Similar to image enhancement, but more objective
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Noise and Images
The sources of noise in digital images arise during image acquisition (digitization) and transmission
– Imaging sensors can be affected by ambient conditions
– Interference can be added to an image during transmission
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Noise Model
We can consider a noisy image to be modelled as follows:
where f(x, y) is the original image pixel, η(x, y) is the noise term and g(x, y) is the resulting noisy pixelIf we can estimate the noise model we can figure out how to restore the image
),( ),(),( yxyxfyxg η+=
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Noise Models (cont...)
Gaussian Rayleigh
Erlang Exponential
UniformImpulse
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There are many different models for the image noise term η(x, y):
– Gaussian• Most common model
– Rayleigh– Erlang (Gamma)– Exponential– Uniform– Impulse
• Salt and pepper noise
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Noise Example
The test pattern to the right is ideal for demonstrating the addition of noiseThe following slides will show the result of adding noise based on various models to this image
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Histogram to go here
Image
Histogram
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Noise Example (cont…)Im
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Gaussian Rayleigh Erlang
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Noise Example (cont…)Im
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Exponential Uniform Impulse
Histogram to go here
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Filtering to Remove Noise
We can use spatial filters of different kinds to remove different kinds of noiseThe arithmetic mean filter is a very simple one and is calculated as follows:
This is implemented as the simple smoothing filterIt blurs the image.
∑∈
=xySts
tsgmn
yxf),(
),(1),(ˆ
1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9
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Other Means
There are different kinds of mean filters all of which exhibit slightly different behaviour:
– Geometric Mean– Harmonic Mean– Contraharmonic Mean
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Other Means (cont…)
Geometric Mean:
Achieves similar smoothing to the arithmetic mean, but tends to lose less image detail.
mn
Sts xy
tsgyxf
1
),(
),(),(ˆ⎥⎥⎦
⎤
⎢⎢⎣
⎡= ∏
∈
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Other Means (cont…)
Harmonic Mean:
Works well for salt noise, but fails for pepper noise.Also does well for other kinds of noise such as Gaussian noise.
∑∈
=
xySts tsg
mnyxf
),( ),(1),(ˆ
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Other Means (cont…)
Contraharmonic Mean:
Q is the order of the filter.Positive values of Q eliminate pepper noise.Negative values of Q eliminate salt noise.It cannot eliminate both simultaneously.
∑
∑
∈
∈
+
=
xy
xy
Sts
QSts
Q
tsg
tsgyxf
),(
),(
1
),(
),(),(ˆ
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Noise Removal ExamplesIm
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Original imageImagecorrupted by Gaussian noise
3x3Geometric Mean Filter (less blurring than AMF, the image is sharper)
3x3 ArithmeticMean Filter
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Noise Removal Examples (cont…)
Image corrupted by pepper noise at 0.1
Filtering with a 3x3 Contraharmonic Filter with Q=1.5
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Noise Removal Examples (cont…)Im
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Image corrupted by salt noise at 0.1
Filtering with a 3x3 Contraharmonic Filter with Q=-1.5
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Contraharmonic Filter: Here Be Dragons
Choosing the wrong value for Q when using the contraharmonic filter can have drastic results
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Pepper noise filtered by a 3x3 CF with Q=-1.5
Salt noise filtered by a 3x3 CF with Q=1.5
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Order Statistics Filters
Spatial filters based on ordering the pixel values that make up the neighbourhood defined by the filter support.
Useful spatial filters include– Median filter– Max and min filter– Midpoint filter– Alpha trimmed mean filter
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Median Filter
Median Filter:
Excellent at noise removal, without the smoothing effects that can occur with other smoothing filters.
Particularly good when salt and pepper noise is present.
)},({),(ˆ),(
tsgmedianyxfxySts ∈
=
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Max and Min Filter
Max Filter:
Min Filter:
Max filter is good for pepper noise and Min filter is good for salt noise.
)},({max),(ˆ),(
tsgyxfxySts ∈
=
)},({min),(ˆ),(
tsgyxfxySts ∈
=
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Midpoint Filter
Midpoint Filter:
Good for random Gaussian and uniform noise.
⎥⎦⎤
⎢⎣⎡ +=
∈∈)},({min)},({max
21),(ˆ
),(),(tsgtsgyxf
xyxy StsSts
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Alpha-Trimmed Mean Filter
Alpha-Trimmed Mean Filter:
We can delete the d/2 lowest and d/2 highest grey levels.So gr(s, t) represents the remaining mn – dpixels.
∑∈−
=xySts
r tsgdmn
yxf),(
),(1),(ˆ
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Noise Removal ExamplesIm
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corruptedby Salt And
Pepper noiseat 0.2
Result of 1 pass with a 3x3 MedianFilter
Result of 2passes with
a 3x3 MedianFilter
Result of 3 passes witha 3*3 MedianFilter
Repeated passes remove the noise better but also blur the image
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Noise Removal Examples (cont…)Im
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Imagecorrupted
by Peppernoise
Imagecorruptedby Saltnoise
Filtering above with a 3x3 Min Filter
Filteringabove
with a 3x3Max Filter
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Noise Removal Examples (cont…)Im
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Imagecorrupted
by uniformnoise
Image furthercorruptedby Salt andPepper noise
Filtering by a5x5 Arithmetic
Mean Filter
Filtering by a5x5 Median
Filter
Filtering by a5x5 GeometricMean Filter
Filtering by a5x5 Alpha-TrimmedMean Filter (d=5)
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Adaptive Filters
The filters discussed so far are applied to an entire image without any regard for how image characteristics vary from one point to another.The behaviour of adaptive filters changes depending on the characteristics of the image inside the filter region.We will take a look at the adaptive median filter.
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Adaptive Median Filtering
The median filter performs relatively well on impulse noise as long as the spatial density of the impulse noise is not large.
The adaptive median filter can handle much more spatially dense impulse noise, and also performs some smoothing for non-impulse noise.
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Adaptive Median Filtering (cont…)
The key to understanding the algorithm is to remember that the adaptive median filter has three purposes:
– Remove impulse noise– Provide smoothing of other noise– Reduce distortion (excessive thinning or
thickenning of object boundaries).
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Adaptive Median Filtering (cont…)
In the adaptive median filter, the filter size changes depending on the characteristics of the image.Notation:
– Sxy = the support of the filter centerd at (x, y)– zmin = minimum grey level in Sxy
– zmax = maximum grey level in Sxy
– zmed = median of grey levels in Sxy
– zxy = grey level at coordinates (x, y)– Smax =maximum allowed size of Sxy
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Adaptive Median Filtering (cont…)
Stage A: A1 = zmed – zminA2 = zmed – zmaxIf A1 > 0 and A2 < 0, Go to stage BElse increase the window sizeIf window size ≤ Smax repeat stage AElse output zmed
Stage B: B1 = zxy – zminB2 = zxy – zmaxIf B1 > 0 and B2 < 0, output zxyElse output zmed
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Adaptive Median Filtering (cont…)Stage A: A1 = zmed – zmin
A2 = zmed – zmaxIf A1 > 0 and A2 < 0, Go to stage BElse increase the window sizeIf window size ≤ Smax repeat stage AElse output zmed
Stage A determines if the output of the median filter zmed is an impulse or not (black or white). If it is not an impulse, we go to stage B.If it is an impulse the window size is increased until it reaches Smax or zmed is not an impulse.Note that there is no guarantee that zmed will not be an impulse. The smaller the the density of the noise is, and, the larger the support Smax, we expect not to have an impulse.
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Adaptive Median Filtering (cont…)
Stage B: B1 = zxy – zminB2 = zxy – zmaxIf B1 > 0 and B2 < 0, output zxyElse output zmed
Stage B determines if the pixel value at (x, y), that is zxy, is an impulse or not (black or white). If it is not an impulse, the algorithm outputs the unchanged pixel value zxy .If it is an impulse the algorithm outputs the median zmed .
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Adaptive Filtering ExampleIm
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Image corrupted by salt and pepper noise with
probabilities Pa = Pb=0.25
Result of filtering with a 7x7 median filter
Result of adaptive median filtering with
Smax = 7
AMF preserves sharpness and details, e.g. the connector fingers.
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Periodic Noise
Typically arises due to electrical or electromagnetic interference.Gives rise to regular noise patterns in an image.Frequency domain techniques in the Fourier domain are most effective at removing periodic noise.
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Band Reject Filters
Removing periodic noise form an image involves removing a particular range of frequencies from that image.Band reject filters can be used for this purposeAn ideal band reject filter is given as follows:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+>
+≤≤−
−<
=
2),( 1
2),(
2 0
2),( 1
),(
0
00
0
WDvuDif
WDvuDWDif
WDvuDif
vuH
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Band Reject Filters (cont…)
The ideal band reject filter is shown below, along with Butterworth and Gaussian versions of the filter
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Ideal BandReject Filter
ButterworthBand Reject
Filter (of order 1)
GaussianBand Reject
Filter
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39
C. Nikou – Digital Image Processing (E12)
Band Reject Filter ExampleIm
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g (2
002) Image corrupted by
sinusoidal noiseFourier spectrum of
corrupted image
Butterworth band reject filter
Filtered image