Data model for extracting line networks

 

Data term for the Candy model

Hypothesis tests :  

H₃ : we have three different regions  

  

Figure 5: Three region mask

Using a Gaussian assumption, the log of the likelihood function is :

$$\log L(H_{3}) - \frac{n_{D^{s}_{3}}+n_{D^{sl}_{3}}+n_{D^{sr}_{3}}}{2} - \frac{n_{D^{s}_{3}}}{2} \log (2\pi \sigma_{D^{s}_{3}}^ 2)$$ =

$$\frac{n_ {D^{sl}_{3}} }{2} \log (2\pi \sigma_{D^{sl}_{3}}^ 2) - \frac{n_ {D^{sr}_{3}} }{2} \log (2\pi \sigma_{D^{sr}_{3}}^ 2)$$ - (14)

H₂ : we have two different regions  

  

Figure 6: Two region mask

The log of the likelihood function :

$$\log L(H_{2}) - \frac{n_{D^{el}_{2}}+n_{D^{er}_{2}}}{2} - \frac{n_ {D^{el}_{2}} }{2} \log (2\pi\sigma_{D^{el}_{2}}^2)$$ =

$$\frac{n_ {D^{er}_{2}} }{2} \log (2\pi\sigma_{D^{er}_{2}}^2)$$ - (15)

H₁ : the segment is in the middle of an homogeneous region  

  

Figure 7: One region mask

The log of the likelihood function :

$$\log L(H_{1}) - \frac{n_{D^{h}_{1}}}{2} - \frac{n_ {D^{h}_{1}} }{2} \log (2\pi\sigma_{D^{h}_{1}}^2)$$ = (16)

  

 

The Total Energy for the Candy model

The conditional energy for a segment :

$$\min\{ \log L(H_{3}) - \log L(H_{2}),$$ U_D(s) =

$$\log L(H_{3})-\log L(H_{1}) \} + U_{ridge,valey}(s)$$ (17)

 
Depending on the type of image, we may add to the conditional term :

$$U_{ridge}(s)=\min\{(\mu_{D^{s}_{3}}-\mu_{D^{sl}_{3}}),(\mu_{D^{s}_{3}}-\mu_{D^{sr}_{3}})\}$$ (18)

or :

$$U_{valey}(s)=min\{(\mu_{D^{sl}_{3}}-\mu_{D^{s}_{3}})(\mu_{D^{sr}_{3}}-\mu_{D^{s}_{3}})\}$$ (19)

The total energy becomes :

U(S/D)=U_D(S)+U(S) (20)

with :

$$U(S/D)=-\sum_{s_{i} \in S} U(D/s_{i}) + \log g(s_{i}) + \sum_{s_{i} \diamond s_{j}, i < j} \log h(s_{i},s_{j})$$ (21)