DOC:update MDMDNMF class documentation

Code/Hyperspectral/otbMDMDNMFImageFilter.h

@@ -37,79 +37,86 @@ namespace otb
  *  M. G. A. Huck and J. Blanc-Talon, IEEE TGRS, vol. 48, no. 6, pp. 2590-2602, 2010.
  *  A. Huck and M. Guillaume, in WHISPERS, 2010, Grenoble.
  *
- *  Let $\matR$ be the matrix of the hyperspectral data, whose $I$ columns are the
- *  spectral pixels and the $L$ rows are the vectorial spectral band
+ *  Let \f$ \matR \f$ be the matrix of the hyperspectral data, whose
+ *  \f$ I \f$ columns are the
+ *  spectral pixels and the \f$ L \f$ rows are the vectorial spectral band
  *  images.  The linear mixing model can be written as follow :
- *  \begin{equation}
+ *  \f$ \begin{equation}
  *     \matR=\matA \matS + \matN= \matX + \matN
- *  \end{equation}
- *  The $I$ columns of $\matR$ contain the spectral pixels
- *  and the $I$ columns of $\matS$ hold their respective sets of abundance
- *  fractions.  The $J$ rows of $\matS$ are the abundance maps
- *  corresponding to the respective end-members. The $J$ columns of
- *  $\matA$ are the end members spectra, and $\matX$ is the signal
- *  matrix. Both $\matA$ and $\matS$ are unknown.
+ *  \end{equation} \f$
+ *  The \f$ I \f$ columns of \f$ \matR \f$ contain the spectral pixels
+ *  and the \f $I \f$ columns of \f $\matS \f$ hold their respective sets of abundance
+ *  fractions.  The \f$ J \f$ rows of \f$ \matS \f$ are the abundance maps
+ *  corresponding to the respective end-members. The \f$ J \f$ columns of
+ *  \f$ \matA \f$ are the end members spectra, and \f$ \matX \f$ is the signal
+ *  matrix. Both \f$ \matA \f$ and \f$ \matS \f$ are unknown.
  *
- *  The basic NMF formulation is to find two matrices $\hat{\matA}$ and
- *  $\hat{ \matS}$ such as:
- *  \begin{equation}
+ *  The basic NMF formulation is to find two matrices \f$ \hat{\matA} \f$ and
+ *  \f$ \hat{ \matS} \f$ such as:
+ *  \f$ \begin{equation}
  *    \matX\simeq \hat{\matA} \hat{\matS}
- *  \end{equation}
+ *  \end{equation} \f$
  *  NMF based algorithms consider the
- *  properties of the dual spaces $span^+(\matA')$ and $span^+(\matS)$, in
- *  which $span^+(\mathbf m^1 ...\mathbf m^d)=\{\mathbf v=\sum_i \mathbf
- *  m^i\mathbf a_i|\mathbf a\in \matR _+^d\}$. The
+ *  properties of the dual spaces \f$ span^+(\matA')\f$ and
+ *  \f $span^+(\matS) \f$, in
+ *  which \f$ span^+(\mathbf m^1 ...\mathbf m^d)=\{\mathbf v=\sum_i \mathbf
+ *  m^i\mathbf a_i|\mathbf a\in \matR _+^d\} \f$. The
  *  positiveness is then a fundamental assumption and is exploited to
  *  restrict the admissible solutions set.
  *
  *  A common used solution is to minimize the reconstruction quadratic
- *  error $RQE({\matA}, {\matS})=\|\matR-{\matA} {\matS}\|^2_F$. In order to
+ *  error :
+ *  \f$ RQE({\matA}, {\matS})=\|\matR-{\matA} {\matS}\|^2_F \f$. In order to
  *  satisfy the sum-to-one constraint for hyperspectral data, a
- *  regularization term $STU(\matS)$ can be added to the objective
+ *  regularization term \f$ STU(\matS) \f$ can be added to the objective
  *  function.
  *
- *  A generic expression for the optimized function is $$
+ *  A generic expression for the optimized function is \f$
  *  f(\matA,\matS)=\|\matA \matS-\matR\|_{norm}+\sum_i \lambda_i
- *  D_i(\matA) + \sum_j \lambda_j D_j(\matS)$$ in which $\|\matA
- *  \matS-\matR\|_{norm}$ is the connection-to-the-data term, and
- *  $\lambda_{\{i, j\}}$ are regularization parameters for end members and
- *  abundances constraints $D_{\{i, j\}}$.
- *  In \cite{Huck2010a}, they
+ *  D_i(\matA) + \sum_j \lambda_j D_j(\matS) \f$ in which \f$ \|\matA
+ *  \matS-\matR\|_{norm} \f$ is the connection-to-the-data term, and
+ *  \f$ \lambda_{\{i, j\}} \f$ are regularization parameters for end members and
+ *  abundances constraints \f$ D_{\{i, j\}} \f$.
+ *  Huck
  *  propose an other regularization term,
- *  $D_A(\matA)=Tr(\matA^T\matA)-\frac{1}{L}Tr\left ( \matA^T \1_{LL}\matA
- *  \right)$, which ensures low spectral dispersion on endmembers.
+ *  \f$ D_A(\matA)=Tr(\matA^T\matA)-\frac{1}{L}Tr\left ( \matA^T \1_{LL}\matA
+ *  \right) \f$, which ensures low spectral dispersion on endmembers.
  *  The physical
  *  motivation is based on the assuption that in most situations, the
  *  whole set of pure materials do not appear in each pixel, but
- *  selectively in multiple piece-wise convex sets., as pointed in
- *  \cite{Zare2010}. As a consequence, the mean value of the abundance,
- *  $\frac{1}{J}$, is the least likely one.  The maximum abundance
+ *  selectively in multiple piece-wise convex sets. As a consequence, the mean value of the abundance,
+ *  \f$ \frac{1}{J} \f$, is the least likely one.  The maximum abundance
  *  dispersion condition is given by
- *  $D_s(\matS)=-\|\matS-\frac{1}{J}\1_{JI}\|^2_F$. MDMD-NMF algorithm
- *  minimizes the following function $ f(\matA,\matS) =RQE(\matA,
- *  \matS)+\delta.STU(\matS)+\lambda_A D_A(\matA)-\lambda_S D_S(\matS)$,
- *  $STU$ the sum-to-one constraint.
+ *  \f$ D_s(\matS)=-\|\matS-\frac{1}{J}\1_{JI}\|^2_F \f$. MDMD-NMF algorithm
+ *  minimizes the following function \f$ f(\matA,\matS) =RQE(\matA,
+ *  \matS)+\delta.STU(\matS)+\lambda_A D_A(\matA)-\lambda_S D_S(\matS)
+ *  \f$,
+ *  \f$ STU \f$ the sum-to-one constraint.
  *
  *  In the literature, NMF based optimization algorithms are mainly based
  *  on multiplicative rules, or else alternate gradient descent
  *  iterations, or else on alternate least square methods. In MDMD-NMF, the update rules
- *  at each iteration become \begin{eqnarray}\label{algomdmd} \matS
- *  &\leftarrow &P\left [\matS-\mu_S \left( \bar \matA^T
+ *  at each iteration become :
+ *  \f$ \begin{eqnarray}
+ *  \label{algomdmd} \matS&\leftarrow &P\left [\matS-\mu_S \left( \bar \matA^T
  *  (\bar\matA\matS-\bar\matR)-\lambda_S(\matS-\frac{1}{J}\1_{JI})\right)\right
  *  ]\\ \nonumber \matA &\leftarrow &P\left [\matA-\mu_A \left(
  *  (\matA\matS-\matR)\matS^T +\lambda_A(\matA-\frac{1}{L}\ \mathbf
- *  1_{LL}\matA)\right)\right ] \end{eqnarray} where $\mu_A$ and $\mu_S$
- *  are the step sizes.  % and the projection.  % stands for
- *  non-negativity hard constraint : $P(M_{ij})=0$ if $M_{ij}<0$ for
- *  $\mathbf %M=\{\matA,\matS\}$.  Huck propose a
- *  multiscale method to determine the coefficients $\mu_A$ and
- *  $\mu_S$.  The projection operator $P$ at each step ensures the
- *  positivity constraint for $\matA$ and $\matS$, and $\bar\matR$ and
- *  $\bar\matA$ include the sum-to-one constraint: $ \bar\matR=\left[
+ *  1_{LL}\matA)\right)\right ] 
+ *  \end{eqnarray} \f$ 
+ *  where \f$ \mu_A\f$ and \f$\mu_S \f$
+ *  are the step sizes.
+ *  Huck propose a
+ *  multiscale method to determine the coefficients \f$ \mu_A \f$ and
+ *  \f$ \mu_S \f$.  The projection operator \f$ P \f$ at each step ensures the
+ *  positivity constraint for \f$ \matA \f$ and \f$ \matS \f$, and \f$
+ *  \bar\matR \f$ and
+ *  \f$ \bar\matA \f$ include the sum-to-one constraint: 
+ *  \f$ \bar\matR=\left[
  *  \begin{array}{c} \matR \\
  *  \delta\cdot\1_{1I}\end{array}\right],\enspace \bar\matA=\left[
  *  \begin{array}{c} \matA \\
- *  \delta\cdot\1_{1J}\end{array}\right]\enspace$.
+ *  \delta\cdot\1_{1J}\end{array}\right]\enspace \f$.
  *
  * \ingroup ImageFilters
  */