Ajout de metrique dans AT et approche multiresolution. Noemie Tasca, …

doc/imageprocessing.dox

@@ -43,5 +43,20 @@
 </table>
 </center>
 
+ - @ref DocATu2v0-m : Computes a piecewise smooth approximation of an image, by optimizing the Ambrosio-Tortorelli functional (with u a 2-form and v a 0-form) including metric.
+
+<center>
+<table>
+  <tr>
+        <td> @image html images/LenaBruit256.png height=200px </td>
+        <td> @image html images/LenaBruit256Restaure.png height=200px </td>
+        <td> @image html images/LenaBruit256Contours.png height=200px </td>
+</tr>
+<tr>
+        <td colspan=3 align = center>@ref DocATu2v0-m </td>
+</tr>
+</table>
+</center>
+
 
 */

doc/moduleAT.dox

@@ -18,7 +18,7 @@
   Part of tool \ref imageProcessing.
   [TOC]
 
-- related tools: \ref DocATu2v0, \ref DocATu0v1
+- related tools: \ref DocATu2v0, \ref DocATu0v1, \ref DocATu2v0-m
 - related classes: \ref DGtal::ATu0v1, \ref DGtal::ATu2v0, \ref DGtal::DECImage2D
 - related namespace: \ref DGtal::functions::dec
 - related functions: \ref DGtal::functions::dec::diagonal, \ref DGtal::functions::dec::squares, \ref DGtal::functions::dec::squaredDiagonal, \ref DGtal::functions::dec::writePixel, \ref DGtal::functions::dec::writePrimalLinel, \ref DGtal::functions::dec::writeDualLinel, \ref DGtal::functions::dec::form2ToImage, \ref DGtal::functions::dec::form1ToImage, \ref DGtal::functions::dec::dualForm1ToImage, \ref DGtal::functions::dec::primalForm1ToImage, \ref DGtal::functions::dec::threeForms2ToImage, \ref DGtal::functions::dec::form2ToGreyLevelImage, \ref DGtal::functions::dec::primalForm1ToGreyLevelImage, \ref DGtal::functions::dec::dualForm1ToGreyLevelImage, \ref DGtal::functions::dec::primalForm1ToRGBColorImage, \ref DGtal::functions::dec::dualForm1ToRGBColorImage, \ref DGtal::functions::dec::threeForms2ToRGBColorImage.
@@ -426,6 +426,102 @@ triple points in the L2-metric sense.
 
 
 
+<!-- Partie de Noemie ---------------------------------------------------------------------------------------------- -->
+
+@section AT2m Discretization of AT with discrete calculus inluding euclidean metric
+
+
+@subsection AT2m_1 Formulations AT20-m with metric
+
+Let \f$ N \f$ be the size of image.
+Let \f$ h \f$ be the step associated to the image resolution and defined as
+\f[
+    \displaystyle
+    h = \frac{1}{N}
+\f]
+
+The euclidean metric appears in the scalar product
+\f[
+  \displaystyle
+  \langle
+  a , b
+  \rangle
+  =
+  a^T \, G_k \, b
+\f]
+where \f$ G_k \f$ are diagonal matrices, called metric tensors of k-forms (for \f$ k \in 0,1,2 \f$) and defined as
+\f[
+    \displaystyle
+    G_0 =
+    \begin{bmatrix}
+    	h^2 & & \\
+    	& \ddots & \\
+    	& & h^2
+    \end{bmatrix}
+    \quad \quad
+    G_1 =
+    \begin{bmatrix}
+    \mathbf{1} & & \\
+    & \ddots & \\
+    & & \mathbf{1}
+    \end{bmatrix}
+    \quad \quad
+    G_2 =
+    \begin{bmatrix}
+    \dfrac{1}{h^2} & & \\
+    & \ddots & \\
+    & & \dfrac{1}{h^2}
+    \end{bmatrix}
+\f]
+
+Hence, the formulation of AT20-m including metric become
+\f[
+    \displaystyle
+    \begin{array}{rcl}
+        AT_\varepsilon^{2,0} (u,v)
+        &=&
+        \alpha \, \sum_{i=1}^n \left( u_i - g_i \right)^T \, G_2 \, \left( u_i - g_i \right) \\[0.2cm]
+        &+&
+        \sum_{i=1}^n  u_i^T \, G_2^T \, B'^T \, diag\left( M_{01} \, v \right)^2 \, B' \, G_2 \, u_i \\[0.2cm]
+        &+&
+        \lambda \, \varepsilon \, v^T \, A^T \, G_1 \, A \, v + \dfrac{\lambda}{4 \, \varepsilon} \, \left( 1 - v \right)^T \, G_0 \, \left( 1 - v \right)
+    \end{array}
+\f]
+
+
+@subsection AT2m_2 Solutions to AT20-m
+
+Functional AT20-m is sum of quadratic terms, independently,
+but not simultaneously, convex in \f$ u \f$ and \f$ v \f$.
+It must has null derivative at optimum.
+We thus propose to alternatively solve for \f$ u \f$, then \f$ v \f$.
+The derivatives can be given explicitly as linear systems.
+To simplify notations, let \f$ \mathbf{A} := \mathbf{d_0} \f$,
+\f$  \mathbf{B} := \mathbf{d_1} \f$,
+\f$ \mathbf{A'} := \bar{\mathbf{\star}} \bar{\mathbf{d_1}} \mathbf{\star} \f$
+and \f$ \mathbf{B'} := \bar{\mathbf{\star}} \bar{\mathbf{d_0}} \mathbf{\star} \f$.
+We get at optimum:
+
+\f[
+    \displaystyle
+    \left\{
+        \begin{array}{rcl}
+            \left[ \alpha \, G_2 \, + \, G_2^T \, B'^T \, diag\left( M_{01} v \right)^2 \, B' \, G_2 \right] u_i
+                & = &
+                \alpha \, G_2 \, g_i \\
+            \left[ \dfrac{\lambda}{4 \, \varepsilon} \, G_0 \, + \, \lambda \, \varepsilon \, A^T \, G_1 \, A \, + \, M_{01} \, \left( diag\left( B' \, G_2 \, u_i \right)^2 \right)^T \, M_{01} \right] v
+                & = &
+                \dfrac{\lambda}{4 \, \varepsilon} \, G_0 \, \mathbf{1}
+        \end{array}
+    \right.
+\f]
+
+Since all matrices are symmetric, definite and positive, we use a Cholesky factorization to
+solve alternatively the two equations of each system.
+
+<!-- Partie de Noemie ---------------------------------------------------------------------------------------------- -->
+
+
 
 @section References