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TU Graz - Advanced Signal Processing 1
Divide-and-Conquer Strategies for HyperspectralImage Processing [1]
Bernd Bachofner, Gernot Riegler
Advanced Signal Processing 1
14.01.2013
Bernd Bachofner, Gernot Riegler 14.01.2013 page 1/59
TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 2/59
TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 3/59
TU Graz - Advanced Signal Processing 1
Overview
I Human eye sees visible light in three bands (red, green andblue)
I Spectral imaging divides the spectrum into many more bands
I Can be extended beyond the visible
I Sensors look at objects using a vast portion of theelectromagnetic spectrum
I Certain objects leave unique fingerprints across theelectromagnetic spectrum
I Enables identification of materials (spectral signature of oil)
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TU Graz - Advanced Signal Processing 1
General
I Sensors collect information as a set of images
I Each image represents a range of the electromagneticspectrum (spectral band)
I Images are combined to form a 3-D data cube for processing
I Generated from airborne sensors or satellites
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Automatic Target Detection
I A hyperspectral remot sensig system has four partsI Radiation (or illuminating) sourceI Atmospheric pathI The imaged surfaceI The sensor
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TU Graz - Advanced Signal Processing 1
Quality of the image
I Spectral resolution(precision of the sensor) is equivalent to thewidth of each band of the spectrum
I Dependent from the spatial resolutionI Identify objects even if they are only captured with a few pixels
(spatial resolution ok)I If pixels are too large, objects cannot easily be identified
(spatial resolution to low)I If pixels are too small, the energy captured by the sensor is low
(spatial resolution to high)I Decrease of the signal to noise ratio
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TU Graz - Advanced Signal Processing 1
Consequence
I We need ’good’ data for further processing
I Airborne sensors or satellites have limited resources(calculation power, memory, transmission band width)
I We need efficient algorithms to reduce redundancy in data
I Divide and Conquer methods of the KLT(Karhunen-Loevetransform) are reasonable to use
Bernd Bachofner, Gernot Riegler 14.01.2013 page 8/59
TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 9/59
TU Graz - Advanced Signal Processing 1
General
I Decorrelation is used to reduce autocorrelation within a signalI Autocorrelation is the cross-correlation of a signal with itself
I It is a tool for finding repeating patterns
I Cross-correlation is a measure of similarity of two waveforms
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TU Graz - Advanced Signal Processing 1
Karhunen Loeve Transform (KLT)
I Is a powerful decorrelation transform
I Minimizes the total mean square error ⇒ optimally compactsthe energy
I Maximizes the variance and minimizes the reconstruction error
I No correlation remains among its outputs
I Very high computational cost
I Very high memory requirements
I Lack of component scalability (e.g. different results would beobtained if one uses Fahrenheit rather than Celsius)
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TU Graz - Advanced Signal Processing 1
Principal component analysis (PCA)I The discrete version (coefficients computed from samples) of
the KLTI Orthogonal transformation to convert a set of possibly
correlated variables into a set of values of linearly uncorrelatedvariables ⇒ principal components
I Number of principal components is ≤ the number of originalvariables
I This transformation is defined such that the first-pc has thelargest possible variance
I PCs are independent only if the date set is jointly normallydistributed
I PCA is sensitive to the relative scaling of the original variablesI Can be done by eigenvalue decomposition of the covariance
matrix or by Singular Value Decompostion (SVD) of the datamatrix
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TU Graz - Advanced Signal Processing 1
PCA continued
I Can be used for dimensional reduction of high dimensionaldata
I DetailsI XT ... data matrix with zero meanI The SVD of X is X = WΣV T
I W is the matrix of eigenvectors of the covariance matrix XXT
I Σ matrix is a rectangular diagonal matrix with non-negativereal numbers on the diagonal
I V is the matrix of eigenvectors of XTX
I PCA transform is then Y T = V ΣT
I Y T is a linear transformation of the corresponding row of XT
I First column of Y T contains the first principal components
I The second column of Y T contains the second principalcomponents and so on
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TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 14/59
TU Graz - Advanced Signal Processing 1
General
I Allows to approximate the KLT at a fraction of thecomputational cost
I Reduces the memory requirements
I Provides us with some component of scalabilityI The KLT is a transform that adapts to the statistics of its
inputI yi = KLTΣx = QT (xi − x)I QT matrix of the eigenvalue-decomposition of the covariance
matrix Σx of the input dataI X = {xi} ∀iI The term x is the input vector average, used to guarantee
zero-mean data
I The computational cost of the KLT is dominated by thequadratic cost of the matrix/vector multiplication, and by thecovariance matrix calculation
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TU Graz - Advanced Signal Processing 1
General continued II Divide and conquer methods overcome this problem by
dividing the KLT into a collection of smaller transforms with alesser overall cost
I Smaller transforms have to be arranged in a way that they areapplied only where they are more effective
I It is worth noting that transforms provide little overall benefitsin portions of data with low amounts of information
I Example: A first level of KLT transforms is applied to providelocal decorrelation, with the most significant half of theoutputs of each transform forwarded to a next level
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TU Graz - Advanced Signal Processing 1
General continued III In the above example one large transformation is replaced by
seven smaller transforms each of1
4of the original size
I The transform cost is mainly quadratic, each smaller
transform has1
16of the original cost
I Yielding a cost for the whole approach of7
16≈ 45% of the
original costI The approach improves component scalability, which allows
random access to specific components in a compressedcodestream
I Reduces cost for inverse transform operations (In the aboveexample only 8 outputs required to perform the operation)
I Allows decoding of portions of a compressed image withouthaving to process or download the full compressed data
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TU Graz - Advanced Signal Processing 1
Problems with Divide and Conquer methods
I With D&C methods we have the problem of combinatorialexplosion in the number of possible D&C schemes
I With no constraints we have 8.77 ∗ 1026 possible D&Cschemes for a 16-input KLT
I Not all of the D&C schemes have equal decorrelationperformance
I Data do not always follow the Gaussian model on which thetheory is based
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TU Graz - Advanced Signal Processing 1
Divide and Conquer Strategies
I Can be classified in four families:I Recursive,I Single-Level,I Two-Level andI Multilevel strategies
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TU Graz - Advanced Signal Processing 1
RecursiveI Based on successive subdivision of a KLT into three half-sized
KLTsI Two half-sized KLTs provide a first level of local decorrelation,
while the third one provides partial global decorrelation fromthe outputs of the other two
I The use of recursion proves a computational complexity belowthat of the KLT
I Performance very close to the KLT
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TU Graz - Advanced Signal Processing 1
Single LevelI Based on a single level of small transforms that provide only
local decorrelationI Decorrelation properties are limited, since it produces low
amounts of side informationI May work well in situations where the size of side information
is a significant portion of the bit rateI Example: Very low bit rates
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TU Graz - Advanced Signal Processing 1
Static Two LevelI Works without any recursionI Achieve decorrelation locally on a first level and globally on a
second levelI Segments the first level of decorrelation in a large number of
small KLTsI In the second level the important outputs of the first level
KLT are decorrelated together with the equivalent output ofthe other first-level KLTs
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TU Graz - Advanced Signal Processing 1
Dynamic Two LevelI Works without any recursionI Achieve decorrelation locally on a first level and globally on a
second levelI Segments the first level of decorrelation in a large number of
small KLTsI In the second level the important outputs of the first level
KLT are decorrelated together with the equivalent output ofthe other first-level KLTs
I Pruning is performed after the transform is trained to removeless contributing inputs of second-level KLTs
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TU Graz - Advanced Signal Processing 1
Multilevel Strategies
I At each level, components are sliced into clusters of KLTs,and for each cluster some of the outputs are forwarded to anext level
I Until one last level decorrelates together all the remainingcomponents
I No permutation of components between each level, howeverthey provide good performance
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TU Graz - Advanced Signal Processing 1
Regular MultilevelI The most simplest and naive member of the family of
multilevel strategiesI Includes strong regularity constraints to avoid
explosion on multilevel structures
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TU Graz - Advanced Signal Processing 1
Static MultilevelI Uses eigenthresholding methods as constraintsI Eigenthresholding are analytically methods used to quantify
the relevant outputs of each KLTI On the static variant, the possible structures are reduced from
millions to a few hundredI Clusters are all of the same size, and the same number of
components are forwarded to the next levelI Best structures are empirically selected for and from a training
data set
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TU Graz - Advanced Signal Processing 1
Dynamic MultilevelI Produces one structure of equal cluster size in all levelsI Different number of important outputs for each small KLT
may be selected as the transform is applied
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TU Graz - Advanced Signal Processing 1
Pairwise Orthogonal Transform (POT)I Characterized by its minimal structure of two component
KLTsI Provides the possibility of operation under strong memory
constraintsI Eliminates the numerically cumbersome eigendecomposition
procedure which is required in other structures
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TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
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TU Graz - Advanced Signal Processing 1
Motivation
I Provide moderate spectral decorrelation [2]
I Low computational cost
I Presence in the hyperspectral image coding literature [3]
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TU Graz - Advanced Signal Processing 1
Short Term Fourier Transform
STFT
X(f, τ) =
∫x(t)w(t− τ)e−j2πftdt
I Windowed FT
I Gabor-Transformation: w(t) = e− t2
(∆t)2
I narrow window: good time resolution, bad frequencyresolution
I broad window: bad time resolution, good frequency resolution
I STFT: local analysis with fixed frequency resolution
I Goal: local analysis with variable frequency resolution
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CWT
I Idea: Window function contains frequency information
I Adjust window function during analysis
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TU Graz - Advanced Signal Processing 1
Wavelets
I Basic operationsI TranslationI Dilation
Example: Haar-Wavelet
Ψ(t) =
1 0 ≤ t < 0.5
−1 0.5 ≤ t < 1
0 else
Ψab =1√a
Ψ(t− ba
)
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Wavelet
I Ψ(t) is an oscillating function
I Ψ(t) is localized in a finite interval
I If Ψ(t) fulfills: CΨ = 2π∫∞−∞
| ˆΨ(ω)|2|ω| dω <∞
I Then Ψab(t) = 1√aΨ( t−ba ) is a wavelet-basis
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TU Graz - Advanced Signal Processing 1
CWT
I WΨ(a, b) = 1√a
∫∞∞ x(t)Ψ( t−ba )dt
I x(t) is projected onto Ψab(t)
I The coefficient WΨ(a, b) measures how well x(t) fits Ψab(t)
I Small a⇒ small frequencies and vice versa
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DWT
I Introduced by Mallat [4]
I Successive lowpass and highpass filtering
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TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 37/59
TU Graz - Advanced Signal Processing 1
Recursive KLT
I Origninally not proposed for hyperspectral image coding [5],[6]
I Successive subdivision of a KLT
I Local decorrelation
I Global decorrelation
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Recursive KLT
I The N -dimensional signal is equally divided into two groups
I Each group is decorrelated by the KLT
I Outputs are sorted according to the variance
I Outputs with high variance are transformed again
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Airborne Visible Infrared Imaging Spectrometer
I AVIRIS is an airborne hyperspectral sensor from the NASA
I 224 contiguous spectral bands each .01µm wide
I Ranging from 0.4µm to 2.5µm
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Recursive Strategy
I 614× 2206× 224 pixel Image
I ⇒ 614× 2206 input vectors ∈ R224
I 81 small KLTs, each of 14 inputs
I For non-zero mean inputs the mean is subtractedI And the Q matrix is computed
I QR algorithm on covariance matrix
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Structure of the recursive divide-and-conquer strategyapplied to 224 components, recursion depth of four
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Comparison
I Recursive KLT and JPEG2000 yields a SNR of 54.12 dB
I 1.2 min on a 1 Gigaflop/s CPU
I KLT and JPEG2000 yields a SNR of 54.13 dB
I 4 min on a 1 Gigaflop/s CPU
I DWT CDF 9/7 and JPEG2000 yields a SNR 50.67 dB
I 9 s on a 1 Gigaflop/s CPU
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TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 45/59
TU Graz - Advanced Signal Processing 1
Pairwise Orthogonal Transform - Overview
I The POT [7] is a special case of a multilevel strategy
I Minimal structure of two-component KLTs
I Low memory consumption
I Numerically stable
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TU Graz - Advanced Signal Processing 1
Transform
I In the first level: Two-component KLT for every pairI The first component will be further decorrelated in the next
levelI The most energy is in the first component
I The same approach is repeated in the next levelI Direct forwarding in an odd case
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TU Graz - Advanced Signal Processing 1
Transform (Math)
I Recall KLT: Y = QTX X ∈ RM×N
I The ED procedure in 2D is simple
2D Eigenvalue Decomposition
Σ = QΛQ−1
Σ =
[a bb d
]Q =
[p qt u
]Λ =
[λ1 00 λ2
]
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TU Graz - Advanced Signal Processing 1
Transform (Math)
2D Eigenvalue Decomposition
t = −q =b
|b|
√1
2−a− d
2s
p = u =
√1
2+a− d
2s=
√1− t2
λ1 =a+ d+ s
2λ2 =
a+ d− s2
s =√
(a− d)2 + 4b2
I Assuming b|b| ∈ {−1, 1} is a sufficient solution
I s ' 0⇒ Inputs are similar and share almost no energyI Division by zeroI Remedy: Assume identity matrix as Q
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Side Information
I The KLT needs for the inverse transform side information
I For the KLT the full transformation matrix (and mean vector)
I The POT is more efficient
I For every KLT the parameter t is sufficient
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TU Graz - Advanced Signal Processing 1
Performance
Bernd Bachofner, Gernot Riegler 14.01.2013 page 51/59
TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 52/59
TU Graz - Advanced Signal Processing 1
Definitions
Coding performance
Tradeoff between quality (SNR) and bit rate (bpppb).
Computational cost
Number of operations for a given decorrelation transform.
Component scalability
The ability to retrieve a single component (e.g. for false colorvisualization).
Memory requirements
The peak of computer memory capacity needed to apply thetransformation.
Bernd Bachofner, Gernot Riegler 14.01.2013 page 53/59
TU Graz - Advanced Signal Processing 1
Transform
Bernd Bachofner, Gernot Riegler 14.01.2013 page 54/59
TU Graz - Advanced Signal Processing 1
Transform
Bernd Bachofner, Gernot Riegler 14.01.2013 page 55/59
TU Graz - Advanced Signal Processing 1
OutlineIntroduction in Hyperspectral Image Processing
Image Acquisition
Decorrelation Methods
Devide and Conquer Methods
Wavelet TransformSTFTContinuous Wavelet Transform
Recursive KLTRecursive Strategy Example: AVIRIS Hyperspectral ImageCoding
POT
Comparison of Strategies
Another Example: Anomaly Detection
Bernd Bachofner, Gernot Riegler 14.01.2013 page 56/59
TU Graz - Advanced Signal Processing 1
Airborne Detection of Land Mines
I Demonstration of an RX anomaly detector based ondivide-and-conquer strategies
I State of the art methods: support vector methods [8], KernelRX [9]
I An RX anomaly detector [10] is based on the distance of apixel r to the overall background
I Using Mahalanobis distance: RX(r) = (r − µ)Σ−1(r − µ)
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TU Graz - Advanced Signal Processing 1
Airborne Detection of Land Mines
I Substitution Σ−1 = QΛ−1QT
I Yields RX(r) = (QT (r − µ))TΛ−1(QT (r − µ))
I QT (r − µ) is the KLT of (r − µ)
I Can be approximated by divide-and-conquer strategies
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TU Graz - Advanced Signal Processing 1
Visual Results
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TU Graz - Advanced Signal Processing 1
I. Blanes, J. Serra-sagrista, M. W. Marcellin, and J. Bartrina-rapesta,“Divide-andConquer Strategies for Hyperspectral Image Processing,” IEEESignal Processing Magazine, no. MAY, pp. 71–81, 2012.
P. Craigmile and D. Percival, “Asymptotic decorrelation of between-scale waveletcoefficients,” Information Theory, IEEE, 2005. [Online]. Available:http://ieeexplore.ieee.org/xpls/abs all.jsp?arnumber=1397939
J. E. Fowler and J. T. Rucker, “3-D wavelet-based compression of hyperspectralimages,” in Hyperspectral Data Exploitation: Theory and Applications, 2007, pp.379–407.
S. Mallat, “A theory for multiresolution signal decomposition: the waveletrepresentation,” Pattern Analysis and Machine Intelligence, vol. II, no. 7, 1989.[Online]. Available: http://ieeexplore.ieee.org/xpls/abs all.jsp?arnumber=192463
W. Yodchanan, “Lossless compression for 3-D MRI data using reversible KLT,”International Conference on Audio, Language and Image Processing, pp.1560–1564, Jul. 2008. [Online]. Available:http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4590113
Y. Wongsawat, S. Oraintara, and K. Rao, “Integer sub-optimal Karhunen-Loevetransform for multichannel lossless EEG compression,” Proc. European Signal,no. Figure 6, pp. 4–8, 2006. [Online]. Available: http://www.eurasip.org/Proceedings/Eusipco/Eusipco2006/papers/1568981760.pdf
Bernd Bachofner, Gernot Riegler 14.01.2013 page 59/59
TU Graz - Advanced Signal Processing 1
I. Blanes and J. Serra-Sagrista, “Pairwise orthogonal transform for spectralimage coding,” Geoscience and Remote Sensing, vol. 49, no. 3, pp. 961–972,2011. [Online]. Available:http://ieeexplore.ieee.org/xpls/abs all.jsp?arnumber=5599290
A. Banerjee, P. Burlina, and R. Meth, “Fast hyperspectral anomaly detection viaSVDD,” in Image Processing, 2007. ICIP 2007. IEEE International Conferenceon, vol. 4. IEEE, 2007, pp. IV—-101.
H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: a nonlinear anomalydetector for hyperspectral imagery,” Geoscience and Remote Sensing, IEEETransactions on, vol. 43, no. 2, pp. 388–397, 2005.
I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an opticalpattern with unknown spectral distribution,” Acoustics, Speech and SignalProcessing, IEEE Transactions on, vol. 38, no. 10, pp. 1760–1770, 1990.
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