
Version traduite de la page Hyperspectral.txt
\ Chapter} {Hyperspectral

% TODO translate
A hyperspectral image acquired from the Earth
air carrier or a space contains a collection of pixels
spectral or, equivalently, a collection of tapes
spectral.

\ Begin {figure} [h]
  \ Centering
  \ Includegraphics [width = 0.7 \ textwidth] {} Cube_HPX.eps
  \ Itkcaption [Hyperspectral cube] {Illustration of a hyperspectral cube, a pixel spectral
a spectral band.}
  \ Label {fig: cube}
\ End {figure}

The hyperspectral cube \ ref {fig:} cube is an image acquired
radiance, each pixel contains spectral spectral information
fine that depends:

\ Begin {itemize}
\ Item {spectrum of the light source (in practice, the
sun) and atmospheric disturbances.}
\ Item {spectral responses
of different materials present in the overlap zone and the nature of the mixture.}
\ End {itemize}

Preliminary treatment of
atmospheric correction for estimating a reflectance cube
spectral subtraction of information extrinsic to the
scene.

\ Section {} Unmixing

\ Subsection {Linear mixing model}

The information depends only on reflectance spectra
reflectance of materials in the scene. When the mixture between
materials is macroscopic, the linear mixing model of spectra
is generally admitted. The imaged scene typically has the look
pictured here after:

\ Begin {figure} [h]
  \ Centering
  \ Includegraphics [width = 0.7 \ textwidth] {} Linear_Unmixing_HPX.eps
  \ Itkcaption [linear mixing model] {Schema zone MML verify the model.}
  \ Label {fig:} linear_unmixing
\ End {figure}

We note \ ref {fig:} linear_unmixing the
presence of pure pixels, and pixel-blending. The MML acknowledges that
reflectance spectrum associated with each pixel is a combination
Linear spectra of pure materials making up the area
recovery, commonly known as ``''endmembers. This can be
example \ ref {fig: decomp_mml}

\ Begin {figure} [h]
  \ Centering
  \ Includegraphics [width = 0.7 \ textwidth] {} Decomposition_MML_HPX.eps
  \ Itkcaption [Decomposition linear mixing model] {Decomposition of a hyperspectral cube according to the MML.}
  \ Label {fig:} decomp_mml
\ End {figure}

The term `` left''represents the different spectral bands of
data cube. The term `` right''represents a `` product''
between the reflectance spectra of endmembers and their cards
respective abundance. The map of an abundance of endmembers is
image grayscale between $ 0 and $ 1 USD. The pixel i of the map
abundance of endmember set to ja $ s_ {ji} $. This value is
abundance of endmember j in the pixel i. Under certain conditions
[Huck2009], this value can be interpreted as the ratio
surface of the material in the overlap zone (\ ref {fig: linear_unmixing}). In
practice, one can reasonably expect:

\ Begin {itemize}
\ Item {a is a number
limit of pure materials make up the scene.}
\ Item {a scene that the
contain pure pixels if the spatial resolution is sufficient, and
do not necessarily contain otherwise.}
\ End {itemize}
 
Many techniques are based hyperspectral image analysis
a geometric approach where each pixel is seen as a spectral
vector of L (number of spectral bands). The
spectral bands can then be written as vectors.

\ Begin {figure} [h]
\ Begin {tikzpicture}
\ Pgfmathsetmacro {\ Cubex} {2}
\ Pgfmathsetmacro {\ Cubey} {2}
\ Pgfmathsetmacro {\ Cubez} {2}
\ Draw [black, fill = white] (0,0,0) - + + (- \ Cubex, 0.0) - + + (0, - \ Cubey, 0) - + + (\ Cubex, 0,0) - cycle;
\ Draw [black, fill = white] (0,0,0) - + + (0.0 - \ Cubez) - + + (0 - \ Cubey, 0) - + + (0.0 \ Cubez) - cycle;
\ Draw [black, fill = white] (0,0,0) - + + (- \ Cubex, 0.0) - + + (0.0 - \ Cubez) - + + (\ Cubex, 0,0) - cycle;
\ Node [draw] (R) at (2,0.5) {R};
\ Draw [->] (2,0.5) - (R.west);
\ End {tikzpicture}
\ Itkcaption [Vectorization of the cube] {`` Trace''the hyperspectral cube. The spectral pixels
are stored in the columns of the matrix R and, in
equivalently, the spectral bands are assigned to the lines of R.}
\ Label {fig: mml}
\ End {figure}
%% Figure 5: "Trace" hyperspectral cube. The spectral pixels
%% Are placed in the columns of the matrix R and, in
%% Equivalent, the spectral bands are assigned to the lines of R.

By deduction of \ ref {fig: decomp_mml} and \ ref {fig: mml}, the MML is to decompose R
in the following way:
\ Begin {center}
$ R = N + A.Ş. = X + N $
\ End {center}

J is the number of endmembers and the number of pixels I showtime:
\ Begin {itemize}
\ Item {The J columns of the matrix A contain the spectra of endmembers.}
\ Item {The J rows of the matrix S contain the abundance maps
vectorized, we call the columns vectors of abundances of
matrix S.}
\ Item {The matrix N, of dimensions $ LXI $ is a matrix of noise
additive.}
\ End {itemize}

   

The problem of unmixing is to estimate the matrices A and S
from R or possibly of \ "X, an estimate of the matrix signal
wreckage.

Several physical constraints can be taken into
account to allow the solution to this problem:
\ Begin {itemize}
\ Item {C1: positivity
reflectance spectra (non-negative matrix A).}
\ Item {C2: positivity
abundances (non-negative matrix S).}
\ Item {C3: additivity
abundance (the sum of the coefficients of each column of the matrix
S is 1).}
\ Item {The independence between the `` algebraic spectral vectors''
endmembers associated with linearity and mixtures of spectra, or
comes the property of the simplex described in paragraph below.}
\ End {itemize}

\ Subsection {} Simplex
Recent unmixing algorithms based on the `` property of
simplex.'' In a vector space of dimension $ 1 $ J-, we can
associate a vector J J algebraically independent point DEFINING
the vertices of a simplex-J \ ref {fig: simplex}.

\ Begin {figure} [h]
  \ Centering
  \ Includegraphics [width = 0.7 \ textwidth] {} Simplex_HPX.eps
  \ Itkcaption [Simplex] {Illustration of a 2-simplex, a 3-simplex and a
4-simplex.}
  \ Label {fig: simplex}
\ End {figure}

Hyperspectral cube of L bands, based on J endmembers,
may be contained in a subspace-dimensional affine $ J-$ 1.
A subspace relevant in the sense of signal-to-noise ratio is
generally obtained by Principal Component Analysis (PCA). In
practice, the J-$ $ 1 eigenvectors associated with higher values
own form the columns of a projection matrix V
dimension $ (W (J-1)) $. The reduced data Z, dimensions
$ ((J-1) xi) $ are obtained by the operation:
$ Z = V ^ {T} (\ tilde {R}) $

or each column of $ \ tilde {R} $ is to subtract the average spectrum,
generally estimated under maximum likelihood. In the
subspace carrying the column vectors of Z, the spectra of
endmembers are associated to the top of a simplex. If the noise is
negligible, the simplex circumscribed reduced data.

This property shows that the endmembers
research are the vertices of a simplex that circumscribes the
data. However, an infinity of different simplices can
identify a same data set. In fact, the problem of unmixing
generally does not fit all. This degeneration can
also be demonstrated by the formalism of the factorization
non-negative matrices (see Huck2010a), which will be addressed a little more
away. It is therefore necessary to choose the solution most physically
relevant. All unmixing techniques based on this
property of the simplex admit that the best solution is
defined by the allowable minimum volume simplex, or the notion of
volume is extended to all finite dimensions (possibly
different from 3).

\ Subsection {Selection algorithms and state of the art}
The linear unmixing algorithms exploit LATEST
usually owned by the simplex. It is possible to classify
into several families:
\ Subsubsection {1} Family
A first family of algorithms
unmixing is based on research of the endmembers `` among''
data. This means that a pure pixel minimum must be associated with
each endmembers. If this hypothesis is not verified, it
produce an estimation error of the spectra of
endmembers. The historical advantage of these algorithms is their low
algorithmic complexity. The three best known are PPI (Pixel Purity
Index), and VCA NFINDER (Vertex Component Analysis)
[Nascimento2005]. In addition to its success recognized by the community and
very competitive algorithmic complexity, the estimation of spectra
of endmembers is unbiased in the absence of noise (and when there
pure pixels).

\ Paragraph {} VAC
Note: The VCA algorithm is systematically
used to initialize the various algorithms studied (except
VSS, based on a different initialization).

Important elements on the operation of VCA:
\ Begin {itemize}
\ Item {The algorithm VCA
is based on iterative projections of the data orthogonal to
subspace is already held by the endmembers.}
\ Item {Blaise when the
purity is less than 1.}

\ End {itemize}

\ Subsubsection {2} Family
A second family consists of looking for algorithms
simplex of minimum volume circumscribing the data. Phase
initialization is to determine an initial simplex any
circumscribing the data. Then, a numerical optimization scheme
minimizes a functional, increasing function of the volume
generalized, itself dependent endmembers estimated in the iteration
common. The optimization is constrained by the fact that the data
have remained on the simplex and possibly by
constraints C1, C2 and C3.

The existing algorithms are:
\ Begin {itemize}
\ Item {MVSA (Minimum Volume Simplex
Analysis) [Li2008].}
\ Item {VSS (Volume Minimum Encosing Simplex)
[Chan2009].}
\ Item {SISAL (Simplex Identification via Augmented Split
Lagrangian) [Dias2009].}
\ End {itemize}
  
The main differences between the algorithm considers
include:
\ Begin {itemize}
\ Item {The numerical optimization scheme.}
\ Item {The way
constraints are taken into account.}

\ End {itemize}

 
These issues impact the computational complexity and precision
estimation.
 
\ Paragraph {MVSA aLi2008]}
Important elements on the operation of MVSA:
\ Begin {itemize}
\ Item {Initialization by VCA.}
\ Item {All pixels included in the spectral estimates simplex
  (Approximately) VCA does not impact the stress
  registration data in the simplex search, they are
  simply delete the data used in the phase of
  simplex minimization to reduce the complexity
  algorithms.}
\ Item {The optimization technique is highly developed type
  sequential quadratic programming (Sequential Quadratic
  Programming - SQP) and more specifically to the category
  `` Quasi-Newton''under stress.}
\ End {itemize}

\ Paragraph {VSS [Chan2009]}
 Important elements on the functioning of VSS:
\ Begin {itemize}
\ Item {Initialization by
  iterative method (LP Linear Programming for Linear
  Programming) non-trivial and different from VCA.}
\ Item {Resolution of
  problem by LP.}
\ End {itemize}
 

\ Paragraph {SISAL [Dias2009]}
Important elements on the operation of SISAL:
\ Begin {itemize}
\ Item {Initialization
VCA.}
\ Item {Selection of a similar spectral pixels to reduce MVSA
computational complexity.}
\ Item {very advanced optimization technique
combining multiple features.}
\ Begin {itemize}
\ Item {Decomposition of the problem
non-convex convex set of problems.}
\ Item {Development of a
specific method of separation of variables for considering
Lagrangian increases with a good design properties.}
\ End {itemize}
\ End {itemize}

\ Subsubsection {3} family
This matrix factorization algorithms for non-negative
(NMF for Non-negative Matrix Factorization). The purpose of this
branch of applied mathematics is to factor a matrix
non-negative, X in our case, into a product of matrices
non-negative: AS by minimizing a distance between X and AS and
with a regularization adapted to lift the degeneration of
manner appropriate to the physical problems associated with unmixing.
\ Paragraph {MDMD-NMF [Huck2010b]}
Important elements on the operation of MDMD-NMF:
\ Begin {itemize}
\ Item {Initialization by VCA.}
\ Item {Minimizing the norm of standard
Frobenius with a spectral regularization and a regularization
`` Space''(the matrix of abundance).}
\ Begin {itemize}
\ Item {Minimum spectral
Dispersion: regularization encourages spectral variance
coefficients of a spectrum of endmembers to be low.}
\ Item {Maximum spatial dispersion: spatial regularization
  encourages vector abundance to occupy any part
  intake (more information in [Huck2009]). This action
  presents a certain analogy to the minimum volume constraint.}
\ End {itemize}
\ End {itemize}
 
 
  

\ Subsubsection {Further remarks}
The algorithms of families 1 and 2 estimate the spectra of
endmembers `` only''. The estimated abundance maps held
retrospectively and requires the application of an algorithm type Fully
Constrained Least Square (FCLS) [Heinz2001]. VSS includes an algorithm
specific estimation of the abundances, we have used in
our simulations (to VSS only).
The unmixing is a general non-convex,
explains the importance of the initialization of algorithms.
OVERVIEW taking into account the various constraints
Physical:


\ Begin {center}
   \ Begin {tabular} {l | l | l | l | l | l}
     \ Hline
     VCA MVSA & & & & VSS SISAL & MDMD \ \ \ hline
     C1 $ (A> 0) & $ dummy & & Mute Mute Mute & Lasts & \ \ \ hline
     C2 $ (S> 0) $ & Lasts & Lasts & dumb & Soft & Hard \ \ \ hline
     C3 (additivity) & Muette (via FCLS) & Lasts & Lasts & Lasts & Soft \ \ \ hline
     Simplex & Endmembers \ newline in the data & Circumscribed \ newline to data &
     Circumscribed \ newline DATA & Circumscribed \ newline to data &
     Indirectly by \ newline `` space''regularization \ \ \ hline
   \ End {tabular}
 \ End {center}

\ Section {} dimensionality reduction

\ Section {} Anomaly Detection
An anomaly in a scene is, by definition, an element that does
not expect to find. The unusual element is likely
different from its environment and its presence is in the minority
scene. Typically, a vehicle in the wild, a rock in
field, a wooden hut in a forest are sparse enough
many anomalies that can be desired to detect using a
hyperspectral imaging. This implies that the spectral response of
the anomaly can be discriminated in the spectral response of
`` Clutter-the-bottom''(background clutter). This also implies that
the pixels associated anomalies, the anomalous pixels are
sufficiently rare and / or one-time to be considered
such. These properties can be viewed as hypotheses spectral
and space that underpin the techniques of detection
anomalies in hyperspectral images.

We recall that the literature on hyperspectral imaging
generally distinguishes target detection and detection
anomalies:

\ Begin {itemize}
\ Item {There is talk of detection of targets when the response
spectral element of the research is used as input to
detection algorithm. This is a priori information that
can, in theory, to construct algorithms with very high
detection, such as the Adaptive Matched Filter (AMF) or Adaptive
Cosine / Coherence Estimator (ACE), to name but two. Nevertheless,
thin enough knowledge of the spectrum is a research
information difficult to hold in practice, leading many
often has the use of anomaly detection algorithms.}
\ Item {There is talk of detection of abnormalities when the spectrum
  The unusual element is not required by the algorithm
  detection. For this reason, we often associate the term
  `` Unsupervised''detection. Nevertheless, the algorithms depend
  generally of `` structural''parameters. For example, in the
  detection in case of a sliding window, selecting the right
  dimensions based on a priori knowledge of the size of
  abnormalities. This part of the study focuses on
  comparison of algorithms for anomaly detection.}
\ End {itemize}
    
\ Begin {figure} [h]
  \ Centering
  \ Includegraphics [width = 0.7 \ textwidth] {} Anomaly_HYP.eps
  \ Itkcaption [Concept of the detection] {Notions on detection: ground truth mask, card detection, face detection.}
  \ Label {fig:} anomaly_hyp
\ End {figure}
In \ ref {fig: anomaly_hyp}, we introduce some notions ALREADY
that will be useful later. Detection algorithms
abnormalities have an image as input and consider a map
serving as a detection tool for making a decision. A
adaptive thresholding provides a mask detection estimated that
we hope the most similar possible to the ground truth mask,
unknown in practice. This mask has estimated the vocation of a tool
automated decisions taken.

Two approaches dominate the state of the art in anomaly detection
in hyperspectral images. These are the methods by Pursuit
Projection (PP) and methods based on a modeling
probabilistic class background and possibly the target class
with statistical hypothesis testing.
The PP is a linear project the pixels on spectral
vectors wi which optimizes a criterion sensitive to the presence of anomalies
(Eg kurtosis). This gives a series of maps or projections
anomalies contrast very strongly with the background. But the estimate
Automatic card detection presents difficulties
major, including:
\ Begin {itemize}
\ Item How many lights should I consider?
\ Item What (s) projector (s) to choose?
\ Item How to manage an inhomogeneous background?
\ Item detection performance varies with the spatial dimensions of the image (number of samples)
\ Item These algorithms usually have a little structure parallelizable
\ Item These algorithms are generally not a `` false alarm rate constant.''
\ End {itemize}

The algorithms selected are RX (presented in the first version
in [Reed1990] and GMRF [Schweizer2001] and specificity
respectively will be presented later. They are based on
concepts of probability models, statistics and hypothesis testing
sliding window. This type of approach is to meet the
question: `` The pixel (or set of neighboring pixels) tests
he looks like the pixels of the bottom?''by a process of test
statistics. More fully, this approach requires:
\ Begin {itemize}
\ Item A model for the class `` background''
\ Item The choice of a statistical test
\ Item Eventually, a model for the class `` anomaly''
\ Item estimators for the parameters a priori unknown
\ Item The hypothesis of homogeneity for a satisfactory compromise between:
\ Begin {itemize}
\ Item The number of samples compared to the number of parameters
  estimate
\ Item The homogeneity, including the bottom
\ Item algorithmic complexity (although the algorithms considered are
highly parallelizable)
\ End {itemize}
\ End {itemize}
The operating principle of RX and GLRT can be summarized by
Figure 27 and Figure 28.
(TODO do block diagram using tikz)

An optional first step (spontaneously regarded as part
this study) is to reduce the size of spectral data
while maintaining information related to anomalies. Then, the pixels
are tested spectral turn (parallelizable task). The pixel
test is given a sliding window (see Figure 28). This
window consists of two sub windows, centered on the pixel test,
respective dimensions L and LL, with $ L <$ LL.
\ Begin {itemize}
\ Item The pixels belonging to the crown thus formed belong
  the class `` local''background. The statistical parameters of the background,
  required by statistical tests, are estimated from these
  spectral pixels. One of the challenges is to find a
  compromise on the thickness of the crown (and therefore the number
  statistical sampling) where:
\ Begin {itemize}
\ Item If the crown is too thick, the bottom is no longer local
  sufficiently homogeneous
\ Item If the number of samples is too low, the precision of
  statistical estimation of model parameters of the background is too
  low
\ End {itemize}
\ Item The pixels of the central part of the sliding window
  belong to the class `` unknown''. If the test pixel (pixel
  central) pixel is abnormal, it is possible that its neighbors
  pixels are also abnormal, hence the interest not to choose
  a too small value of L if it is known that abnormalities
  sought can be spread over several pixels.
\ End {itemize}
Once all
estimated parameters, a statistical test is performed and assigns a
value $ \ Lambda (i) $ test pixel i. A map of detection is thus
formed.

Figure 28: Principle of the sliding window and definitions of
parameters L and LL of the sliding window. RX algorithms and
GMRF, taken herein in the following sections.
(TODO block diagram for using windowing tikz)

\ Subsection {Local} RX
