Stochastic Level Set Dynamics to Track Closed Curves Through Image Data

Abstract

We introduce a stochastic filtering technique for the tracking of closed curves from image sequence. For that purpose, we design a continuous-time dynamics that allows us to infer inter-frame deformations. The curve is defined by an implicit level-set representation and the stochastic dynamics is expressed on the level-set function. It takes the form of a stochastic partial differential equation with a Brownian motion of low dimension. The evolution model we propose combines local photometric information, deformations induced by the curve displacement and an uncertainty modeling of the dynamics. Specific choices of noise models and drift terms lead to an evolution law based on mean curvature as in classic level set methods, while other choices yield new evolution laws. The approach we propose is implemented through a particle filter, which includes color measurements characterizing the target and the background photometric probability densities respectively. The merit of this filter is demonstrated on various satellite image sequences depicting the evolution of complex geophysical flows.

Appendix

This section details the different terms involved in the expression of the evolution equation associated to the vectorial level set ψ.

1.1 6.1 Drift computation details

ψ is driven by the same velocity fields (5) as φ. For a fixed point y we have:

$$ d\psi^i_t(y)=b_i(y,t)dt + f_i(y, t) dB_{n,t}+g_i(y,t)dB_{\tau,t}. $$

(40)

In the same way as for φ, the differential of $\psi(\mathcal {X}_{t},t)$ at point $\mathcal{X}_{t} = x$ reads:

$$\begin{aligned} d\psi^i(\mathcal{X}_t,t)&=d\psi^i_t(x)+ \bigl(\nabla\psi^i_t\bigr)^T d\mathcal{X} \\ &\quad{} + \frac {1}{2}\sum_{i,j}{d\bigl \langle \mathcal{X}_t^i,\mathcal{X}_tj\bigr \rangle \frac {\partial ^2\psi^i_t}{\partial x_i\partial x_j}} \\ &\quad{} +\sum_{i}{d\biggl \langle \frac{\partial\psi^i_t}{\partial x_i}, \mathcal {X}^i\biggr \rangle } = 0. \end{aligned}$$

(41)

Developing the brackets with (40) and equating in (41) the deterministic and random terms we obtain

$$\begin{aligned} &f(x,t)=-\sigma_n \bigl(\nabla\psi^i_t \bigr)^T \frac{\nabla\varphi}{|\nabla \varphi|}, \end{aligned}$$

(42)

$$\begin{aligned} &g(x,t)=-\sigma_\tau\bigl(\nabla\psi^i_t \bigr)^T \frac{\nabla\varphi^\perp }{|\nabla\varphi|}. \end{aligned}$$

(43)

From these expressions the drift term reads then:

$$ b(x,t)=-\bigl(\nabla\psi^i_t\bigr)^{T} \frac{\nabla\varphi}{|\nabla \varphi |} w^*_n -\frac{1}{2}A_i + F_i + G_i, $$

(44)

with

$$\begin{aligned} \begin{aligned} A_i&=\sigma_n^2\nabla\varphi^T \nabla^2\psi_t^i\nabla\varphi + \sigma_\tau ^2\bigl(\nabla\varphi^\bot \bigr)^T\nabla^2\psi_t^i\nabla \varphi^\bot,\\ F_i&=\frac{\sigma_n}{|\nabla\varphi|}\nabla\varphi^T \biggl[\nabla ^2\psi ^i_t\nabla\varphi+ \nabla^2\varphi\nabla\psi_t^i\\ &\quad{} - \frac {1}{|\nabla \varphi|^2} \bigl(\bigl(\nabla\psi_t^i \bigr)^T\nabla\varphi\nabla^2\varphi \nabla \varphi \bigr) \biggr] ,\\ G_i&=\frac{\sigma_\tau}{|\nabla\varphi|} \bigl(\nabla\varphi^\bot \bigr)^T \biggl[\nabla^2\psi^i_t \nabla\varphi^\bot+\nabla^2\varphi \nabla\psi _t^i\\ &\quad{} -\frac{1}{|\nabla\varphi|^2} \bigl(\bigl(\nabla \psi_t^i\bigr)^T\nabla \varphi ^\bot\nabla^2\varphi\nabla\varphi \bigr) \biggr]. \end{aligned} \end{aligned}$$

(45)

The differential of ψ for a fixed point reads finally:

$$\begin{aligned} d\psi^i_t(x) &= - \bigl(\nabla\psi^i_t \bigr)^{T}\frac{\nabla\varphi }{|\nabla \varphi|} w^*_n dt \\ &\quad{} -\sigma_n \bigl(\nabla\psi^i_t\bigr)^T \frac{\nabla \varphi }{|\nabla\varphi|}dB_{n,t} - \sigma_\tau\bigl(\nabla \psi^i_t\bigr)^T \frac {\nabla\varphi^\bot}{|\nabla\varphi|}dB_{\tau,t} \\ &\quad{} - \frac {A_i dt}{2|\nabla\varphi|^2} + \frac{\sigma_n F_i dt}{|\nabla \varphi |} + \frac{\sigma_\tau G_i dt}{|\nabla\varphi|}. \end{aligned}$$

(46)