Segmentation of 3D echographies

Level Sets

• $S(t) = \{ \mbox{\boldmath$\textstyle\bf x$} \in I\!\!R^3 \, / \, \Psi( \mbox{\boldmath$\textstyle\bf x$},t)=0 \} \quad$

  
  
  
  

• Evolution of $\Psi$: $\quad \frac{\partial \Psi}{\partial t} + F\vert\nabla \Psi\vert = 0$

  
  
  
  

• Evolution function: $\quad F = h_I(\nu + \rho \kappa)$
  
  
  • $\nu$ propagation force (constant)
    
    
  • $\kappa$ regularisation term
    
    
  • $h_I$ speed adjustment (related to the gradients)

Adaptation to ultrasound data

• Gradient barely significant
  
  
• Signal model:
  Mixture : Gaussian + shifted Rayleigh law
  
  
  $${\cal R}( y) = (\frac{y-min}{\alpha^2})exp(-\frac{(y-min)^2}{2\alpha^2})$$
  
  

  
  
  
  $\Rightarrow$ estimation of a posteriori probability
  densities $p_e(I)$ and $p_i(I)$

Estimation of $p_e(I)$ and $p_i(I)$

Use of the initialization:

Estimation of $p_e(I)$ and $p_i(I)$

Evolution function $\quad F = h_I(\nu + \rho \kappa)$

• $\nu( \mbox{\boldmath$\textstyle\bf x$})$ function of belonging probabilities to one of both classes:
  
  $$\begin{array}{ll} \nu &= \mbox{Sign} \{ p(\lambda_i\vert I)... ...box{Sign} \{ \alpha_i p_i(I) - (1-\alpha_i)p_e(I)\} \end{array}$$
  
  

  
  
  
  

• $h_I( \mbox{\boldmath$\textstyle\bf x$})$ decreasing function of probability to have a transition on $\mbox{\boldmath$\textstyle\bf x$}$ :

  
  

  $$h_I({ \mbox{\boldmath$\textstyle\bf x$}}) = exp \{-k. p_T({ \mbox{\boldmath$\textstyle\bf x$}'}\vert I)\}$$
  
  

  
  
  with :
  

  $$\begin{array}{ll} p_T({ \mbox{\boldmath$\textstyle\bf x$}}\v... ...al N}_e))}p_T({ \mbox{\boldmath$\textstyle\bf x$}}) \end{array}$$
  
  

  
  
  $\cal N$ neighborhood of $\mbox{\boldmath$\textstyle\bf x$}$ split in 2 regions :
  ${\cal N}_i$ and ${\cal N}_e$

Results