Fast Level-Set Optimization Using Threshold Dynamics

In the method proposed by Esedoglu and Tsai, the embeddings, one for each phase, are maintained as piecewise-constant binary functions. This method, essentially, moves the level set by first updating the embeddings based on a gradient descent on the optimization metric, and then regularizing the region boundaries by Gaussian smoothing the embedding followed by thresholding. This approach does not keep track of points near interfaces or maintain distance transforms for embeddings. It allows new components of a region to crop up at remote locations--we have found that this property allows for very rapid level-set evolution when the level-set location is far from the optimum.

Let $R_k: T \rightarrow \{0,1\}$ denote the indicator function for region $\mathcal{T}_k$, i.e., $R_k(t) = 1$ for all $t \in \mathcal{T}_k$ and $R_k(t) = 0$ otherwise. The optimal segmentation, after incorporating this penalty using a Lagrange multiplier, is

$$\mathop{\mbox{argmin }}_{ \{ R_k \}_{k=1}^K } \Bigg( - ... ...in \mathcal{T}} \parallel \nabla_t R_k(t) \parallel_2 \Bigg),$$ (156)

where $\alpha \ge 0$ is the regularization parameter and $\nabla_t$ denotes a discrete spatial-gradient operator.

We now let $\{R_k\}_{k = 1}^K$ be a set of level-set functions. The segment for texture $k$ is then defined as $\mathcal{T}_k = \{t \in \mathcal{T} \vert R_k(t) > R_j(t), \forall j \neq k \}$. Coupling (7.5) and (2.53) creates nested region integrals that complicate the analytical expressions for the gradient flow associated with the level-set evolution [87,144,17]. Besson et al. [17] give the level-set speed term for minimizing the energy defined in (7.5) using a gradient-descent optimization scheme as

$$\frac {\partial R_k(t)} {\partial \tau} = \log {P_k ({\bf ... ...R_k(t) } { \parallel \nabla_t R_k(t) \parallel_2 } \Bigg),$$ (157)

where $\tau$ denotes the time-evolution variable [87,144].