Experimental results

Experimental results (more experiments were conducted in the research report)

• non noisy synthetic data
  
• noisy synthetic data
  
• first real data
  
• second real data

About initialization

We first need to initialize each level set function $\Phi_i$ defined in (4). We have chosen to initialize the $\Phi_i$'s with circular signed distance function. We use 2 methods for selecting the initial $\Phi_i$'s. The first one is completely manually handled : we manually select initial circular level sets.
The second one is automatic and we called it "seed initialization". This method consists of cutting the data image of u₀ into N windows $W_{l,\, l=1..N}$ of predefined size. We compute the average m_l of u₀ on each window W_l. We then select the index k such that $k=arg\min_j(m_l-\mu_j)^2$. And we initialize the corresponding circular signed distance function $\Phi_k$ on each W_l. Windows are not overlapping and each of them is supporting one and only one function $\Phi_k$, therefore we avoid overlapping of initial $\Phi_k$'s. The size of the windows is related to the smallest details we expect to detect. The major avantages of this simple initialization method are : it is automatic (only the size of the windows has to be fixed), it accelerates the speed of convergence (the smaller the windows, the faster the convergence), and it is less sensitive to noise (in the sense that we compute the average m_l of u₀over each window before selecting the function $\Phi_k$ whose mean $\mu_k$is the closest one to m_l).