While dealing with 2D images, the advection factor Fadv is not considered as part of the front motion. Hence, the two main speed terms, i.e., Fprop and Fcurv, are employed to deal with SAR images, where Fprop is derived from the image intensity gradient, and Fcurv from the curvature flow.2.2. Fast level set methodsAs the level set method is formulated from numerical equations for interface propagation, the iteration periods of the standard algorithm for boundary expansion are invariably longer. Taking into consideration a single pixel and its neighboring pixels, one solution is obtained by updating the value of each pixel till the final boundary is reached. For such a solution, O(N2) operations per time step are needed. Assuming the total number of iterations to be N, no less than O(N3) iterations will be needed.

To overcome the problem of longer time requirement, fast level set methods such as narrow-band level set and fast marching method have been introduced.(1) Narrow-band level setLevel set computations are usually carried out using the narrow-band algorithm as described by Malladi et al. [8]. The narrow-band algorithm, however, limits the propagation front’s requirements to update the properties of the neighboring pixels around the zero level set. As shown in Figure 2, the entire two-dimensional grid of data is stored in a square array.Figure 2.Narrow-band of level set.A one-dimensional array is employed to keep track of the points in the narrow band. Assuming the number of points in the front to be k, the band width to be m, the number of iterations to be N, the operation count drops down to O(kmN).

In the worst possible situation, the narrow band method will at most reduce the total operation count to (N3). Even though this indicates a significant progress over the brute-force approach, it is still considered slow for (near) real-time image processing applications Cilengitide [9].(2) Fast marching level setIn a situation wherein the speed function depends only on the interface position, the speed function F (Equation 7) will be reduced to F = Fcurv. Furthermore, if Fcurv > 0, it would be sufficient to solve the stationary perspective boundary problem |T|F = 1, given that x:T(x) = 0 (where T is the time of arrival of
There is an increasing demand for cost-effective and long-term stable measuring systems for gas monitoring in the environment [1, 2]. Beside traditional monitoring tasks (e.g., in research, emission analysis and safety) carbon capture and storage (CCS) develops to an important new application field for subsurface gas monitoring [3-6].