From 8b75014e08be45295f1fbbed7c28841dc8c12421 Mon Sep 17 00:00:00 2001
From: Manuel Grizonnet <manuel.grizonnet@orfeo-toolbox.org>
Date: Mon, 17 Oct 2011 10:09:48 +0200
Subject: [PATCH] DOC:typo in MDMD doxygen class documentation
---
Code/Hyperspectral/otbMDMDNMFImageFilter.h | 70 +++++++++++-----------
1 file changed, 35 insertions(+), 35 deletions(-)
diff --git a/Code/Hyperspectral/otbMDMDNMFImageFilter.h b/Code/Hyperspectral/otbMDMDNMFImageFilter.h
index 56c3f57da3..c9375f7082 100644
--- a/Code/Hyperspectral/otbMDMDNMFImageFilter.h
+++ b/Code/Hyperspectral/otbMDMDNMFImageFilter.h
@@ -37,49 +37,49 @@ 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 \f$ \matR \f$ be the matrix of the hyperspectral data, whose
+ * Let \f$ \mathbf R \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 :
* \f$ \begin{equation}
- * \matR=\matA \matS + \matN= \matX + \matN
+ * \mathbf R=\mathbf A \mathbf S + \mathbf N= \mathbf X + \mathbf N
* \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
+ * The \f$ I \f$ columns of \f$ \mathbf R \f$ contain the spectral pixels
+ * and the \f$ I \f$ columns of \f$ \mathbf S \f$ hold their respective sets of abundance
+ * fractions. The \f$ J \f$ rows of \f$ \mathbf S \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.
+ * \f$ \mathbf A \f$ are the end members spectra, and \f$ \mathbf X \f$ is the signal
+ * matrix. Both \f$ \mathbf A \f$ and \f$ \mathbf S \f$ are unknown.
*
- * The basic NMF formulation is to find two matrices \f$ \hat{\matA} \f$ and
- * \f$ \hat{ \matS} \f$ such as:
+ * The basic NMF formulation is to find two matrices \f$ \hat{\mathbf A} \f$ and
+ * \f$ \hat{ \mathbf S} \f$ such as:
* \f$ \begin{equation}
- * \matX\simeq \hat{\matA} \hat{\matS}
+ * \mathbf X\simeq \hat{\mathbf A} \hat{\mathbf S}
* \end{equation} \f$
* NMF based algorithms consider 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$. The
+ * properties of the dual spaces \f$ span^+(\mathbf A')\f$ and
+ * \f$ span^+(\mathbf S) \f$, in
+ * which \f$ span^+(\mathbf hbf m^1 ...\mathbf hbf m^d)=\{\mathbf hbf v=\sum_i \mathbf hbf
+ * m^i\mathbf hbf a_i|\mathbf hbf a\in \mathbf R _+^d\} \f$. The
* positiveness is then a fundamental assumption and is exploited to
* restrict the admissible solutions set.
*
* A common used solution is to minimize the reconstruction quadratic
* error :
- * \f$ RQE({\matA}, {\matS})=\|\matR-{\matA} {\matS}\|^2_F \f$. In order to
+ * \f$ RQE({\mathbf A}, {\mathbf S})=\|\mathbf R-{\mathbf A} {\mathbf S}\|^2_F \f$. In order to
* satisfy the sum-to-one constraint for hyperspectral data, a
- * regularization term \f$ STU(\matS) \f$ can be added to the objective
+ * regularization term \f$ STU(\mathbf S) \f$ can be added to the objective
* function.
*
* 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) \f$ in which \f$ \|\matA
- * \matS-\matR\|_{norm} \f$ is the connection-to-the-data term, and
+ * f(\mathbf A,\mathbf S)=\|\mathbf A \mathbf S-\mathbf R\|_{norm}+\sum_i \lambda_i
+ * D_i(\mathbf A) + \sum_j \lambda_j D_j(\mathbf S) \f$ in which \f$ \|\mathbf A
+ * \mathbf S-\mathbf R\|_{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$.
* Huck
* propose an other regularization term,
- * \f$ D_A(\matA)=Tr(\matA^T\matA)-\frac{1}{L}Tr\left ( \matA^T \1_{LL}\matA
+ * \f$ D_A(\mathbf A)=Tr(\mathbf A^T\mathbf A)-\frac{1}{L}Tr\left ( \mathbf A^T \1_{LL}\mathbf A
* \right) \f$, which ensures low spectral dispersion on endmembers.
* The physical
* motivation is based on the assuption that in most situations, the
@@ -87,9 +87,9 @@ namespace otb
* selectively in multiple piece-wise convex sets. As a consequence, the mean value of the abundance,
* \f$ \frac{1}{J} \f$, is the least likely one. The maximum abundance
* dispersion condition is given by
- * \f$ 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$ D_s(\mathbf S)=-\|\mathbf S-\frac{1}{J}\1_{JI}\|^2_F \f$. MDMD-NMF algorithm
+ * minimizes the following function \f$ f(\mathbf A,\mathbf S) =RQE(\mathbf A,
+ * \mathbf S)+\delta.STU(\mathbf S)+\lambda_A D_A(\mathbf A)-\lambda_S D_S(\mathbf S)
* \f$,
* \f$ STU \f$ the sum-to-one constraint.
*
@@ -98,24 +98,24 @@ namespace otb
* iterations, or else on alternate least square methods. In MDMD-NMF, the update rules
* 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 ]
+ * \label{algomdmd} \mathbf S&\leftarrow &P\left [\mathbf S-\mu_S \left( \bar \mathbf A^T
+ * (\bar\mathbf A\mathbf S-\bar\mathbf R)-\lambda_S(\mathbf S-\frac{1}{J}\1_{JI})\right)\right
+ * ]\\ \nonumber \mathbf A &\leftarrow &P\left [\mathbf A-\mu_A \left(
+ * (\mathbf A\mathbf S-\mathbf R)\mathbf S^T +\lambda_A(\mathbf A-\frac{1}{L}\ \mathbf hbf
+ * 1_{LL}\mathbf A)\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 \\
+ * positivity constraint for \f$ \mathbf A \f$ and \f$ \mathbf S \f$, and \f$
+ * \bar\mathbf R \f$ and
+ * \f$ \bar\mathbf A \f$ include the sum-to-one constraint:
+ * \f$ \bar\mathbf R=\left[
+ * \begin{array}{c} \mathbf R \\
+ * \delta\cdot\1_{1I}\end{array}\right],\enspace \bar\mathbf A=\left[
+ * \begin{array}{c} \mathbf A \\
* \delta\cdot\1_{1J}\end{array}\right]\enspace \f$.
*
* \ingroup ImageFilters
--
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