Transcript Lecture06_HI_Denoisi..

HYPERSPECTRAL IMAGE
DENOISING
Wissam
Mutlak
4/9/2015
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Introduction
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Image noise is generated unavoidably
during the hyperspectral image
acquision process.
Has a negative effect on subsequent
image processing and analysis
(classification, target detection,
unmixing …).
image denoising for hyperspectral
images is necessary in order to support
and improve image analysis capabilities.
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Denoising Algorithms cont.
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Hyperspectral Image Denoising with Cubic
Total Variation.
http://www.rle.mit.edu/stir/documents/ZelinskiG_IGARSS
2006.pdf
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Denoising Hyperspectral Imagery and
Recovering Junk Bands using Wavelets and
Sparse Approximation.
http://www.isprs-ann-photogramm-remote-sens-spatialinf-sci.net/I-7/95/2012/isprsannals-I-7-95-2012.pdf
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The Cubic Total Variation Model
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CTV Denoising method
treats the
hyperspectal image as
a whole 3-D integrity
from both the spatial
and spectral
dimension instead of
dealing with
hyperspectral image
band-by-band.
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CTV Model
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The CTV model is a combination of
the 2-D total variation model for
spatial domain with the 1-D total
variation model for spectral domain.
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Total Variation (TV) model
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Firstly proposed by Rudin in 1992.
Widely used in Image denoising.
Excellent denoising performance and edgepreserving property.
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TV model cont.
represents the gradient operator of
- the horizontal direction
represents the gradient operator of
- the vertical direction
- is the total number
of pixels in gray-level image v
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TV model for Hyperspectral
images – BBB
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Let us consider the TV model for
hyperspectral image from the
perspective of the 2-D spatial domain
band by band.
each band is a 2-D gray-level image.
add the standard TV model of each
band.
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TV model for Hyperspectral
images – BBB cont.
- the bth band of hyperspectral image u
- Number of bands
- the total number of pixels in band b
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TV model for Hyperspectral
images - PBP
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Now let us take a look from the
perspective of the 1-D spectral
domain pixel by pixel.
each pixel is a 1-D spectral signal.
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TV model for Hyperspectral
images - PBP cont.
the 1-D spectral signal
- of the mth pixel of the
hyperspectral image
represent the
gradient operator
- of the spectral
domain.
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CTV Model
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The hyperspectral image exhibits a
3-D concept.
Now we can combine the 2-D total
variation model for spatial domain
with the 1-D total variation model for
spectral domain.
propose the termed cubic total
variation model for the hyperspectral
image.
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CTV Model
Where
- represents the weight of spectral
dimension relative spatial domain.
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CTV based Hyperspectral Image
Denoising
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Based on the MAP (maximum a
posteriori) estimation theory and CTV
the denoising model for a
hyperspectral image can be
represented as a constrained least
squares problem:
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CTV based Hyperspectral Image
Denoising cont.
- Represents the noisy Hyperspectral image
- Represents the similarity between the
noisy HI g and clean HI .
u
- Gives us a prior model of the original
clear and smooth hyperspectral image.
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Is a parameter which controls the relative
contribution of
and
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CTV based Hyperspectral Image
Denoising cont.
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Apply the CTV model to the MAP
based denoising function:
Is The desired “clean” hyperspectral
image, witch can be solved by
optimizing the above cost function.
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Optimization Method
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The augmented Lagrangian method
is utilized to optimize the CTV
denoising model.
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Improves the speed of solution of the
desired hyperspectral image.
http://www.ccom.ucsd.edu/~peg/papers/A
Lvideopaper.pdf
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Experiments …
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Simulation Results
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In this next experiment we are going to
take a look at band 1 and 21 of a
Hyperspectral Image took from Washington
DC Mall.
Zero-Mean-Gaussian noise and Salt-AndPepper noise was added to the
hyperspectral image.
Parameters of the proposed method are set
as
and
.
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Simulation Results cont.
Band 1
Original Image
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Noisy Image
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Simulation Results cont.
Band 1
Denoising with
band-by-band
TV model
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Denoising with
CTV Model
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Simulation Results cont.
Band 21
Original Image
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Noisy Image
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Simulation Results cont.
Band 21
Denoising with
band-by-band
TV model
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Denoising with
CTV Model
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Simulation Results cont.
The hyperspectral image denoising method
with the CTV Model clearly outperforms the the
hyperspectral image denoising method with the
traditional band-by-band TV model.
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Real Results
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Now we will take a look at a real
noisy Hyperspectral Image of Indian
Pine.
Certain band were taken from the HI
and applied denoising with CTV Model
and with band-by-band TV Model for
comparison.
The parameters of the method are
set as
and
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Real Results cont.
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Real Results cont.
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Wavelets and Sparse
Approximation techniques
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Two algorithms are proposed:
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The first denoises each band in a data
cube.
The second is for noisy “junk band”
recovery.
Both of these algorithms utilize a
combination of wavelet and sparse
approximation techniques
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Wavelets
Freq.
Freq.
Fourier basis
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Wavelet basis
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Wavelets cont.
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Sparsity of wavelet representations
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Applying a 2-D DWT to an image
results in a sparse representation.
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only a few of the wavelet coefficients are
large (those containing signal + noise)
and the majority are small (noise only).
Denoising - zero out the noise
coefficients while retaining signal
coefficients.
Threshold each coefficient
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Sparsity of wavelet representations
cont.
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= 2-D DWT operator.
I is a noisy image
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vector.
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Note that
vector.
is sparse.
= ordering
into a
is a sparse
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Sparse approximation
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The goal is to find a vector of
unknowns with a small number of
nonzero elements such that a system
of equations (approximately) holds.
enforcing sparsity on by requiring
it to have few nonzero coefficients
 requiring that
be small.
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Sparse approximation cont.
given
given
Need to find
The problem above is computationally infeasible,
thus we minimize the
norm of instead.
This we can solve with linear programming
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Sparse approximation cont.
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Unlikely that a sparse
that
exactly.
exists such
Find that approximately solves the
noisy system.
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Sparse approximation cont.
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Let
vectors,
be a set of noisy
,
Generalize the previous problem to
find
.
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Denoising Algorithms
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First algorithm takes a noisy data cube
as input and outputs a denoised cube.
Consider a set of noisy band images
The clean (non-noisy) version of these
images must have similar sparse
representation in the wavelet domain.
Use sparse approximation and formulate
an optimization problem to enforce our
sparsity assumption across the
.
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Algorithm I
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Generate
where
.cccc
.
Find
so that for each
is close to
in a
norm sense,
while at the same time making sure
that the are simultaneously
sparse. This is where we use the
Sparse approximation technique.
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Algorithm I cont.
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This is done by optimizing the
following problem:
This is a special case of the equation
we mentioned earlier.
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…..
Wavelet coef. of
…..
Wavelet coef. of
Wavelet coef. of
Algorithm I cont.
‘s columns are the Wavelet coefficients vectors of the bands
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…..
…..
Wavelet coef. of
…..
Wavelet coef. of
Algorithm I cont.
…..
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Algorithm I cont.
- is the
norm of the
the rows of .
Result of
norms of
norm
Coefficients
coef. 1 of all bands
coef. i of all bands
norm
of the
rows
norm of
the result
Bands
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Algorithm I (final stage)
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Each column of
contains the
denoised detail coefficients of
image .
Extract the columns of to yield
Reform a set of 2-D wavelet
coefficients from each
.
Perform
on this data and
finally, we get - a denoised
version of .
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Junk band recovery algorithm
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Let
a set of user designated
good bands, those with high SNR.
Let
be the rest of the bands,
“Junk bands”, those with low SNR.
Compute
, so that
is
based on clean data with high SNR.
Now we take
and compute :
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Junk band recovery algorithm cont.
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For each Junk band we also
compute:
Let
, and
The goal is to generate
so that it has two properties:
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.
Its columns should be close to those of
each column of
should have a sparsity
profile similar to
‘s.
Sparsity Approximation…
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Junk band recovery algorithm cont.
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By optimizing this problem:
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Find
and extract
.
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Junk band recovery algorithm (final
stage)
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Reform a set of 2-D wavelet
coefficients from each
from
Finally we can recover a denoised
version of every junk band by
performing
on each and every
column of .
Junk bands are now clean…
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Experiments
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First Experiment :
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data cube – 220 band images.
perturb first ten bands with zero-mean AWGN
with a known standard deviation σ.
run our first algorithm
denoise the images using wavelet thresholding
techniques.
Compare the results
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Experiments
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Experiments
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Second Experiment:
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bands 104-113 as junk bands
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114-135 are good bands
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Run Junk band recovery Algorithm.
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Compare the denoised bands with the
original noisy bands.
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Experiments
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Hyperspectral Image Denoising
References
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Hyperspectral Image Denoising with Cubic Total
Variation.
http://www.rle.mit.edu/stir/documents/ZelinskiG_IG
ARSS2006.pdf
Denoising Hyperspectral Imagery and Recovering
Junk Bands using Wavelets and Sparse
Approximation.
http://www.isprs-ann-photogramm-remote-sensspatial-inf-sci.net/I-7/95/2012/isprsannals-I-7-952012.pdf
The augmented Lagrangian method
http://www.ccom.ucsd.edu/~peg/papers/ALvideopa
per.pdf
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