The removal of the background magnetic field is a critical step in generating phase images and quantitative susceptibility maps, which have recently been receiving increasing attention. Although it is known that the background field satisfies Laplace's equation, the boundary values of the background field for the region of interest have not been explicitly addressed in the existing methods, and they are not directly available from MRI measurements. In this paper, we assume simple boundary conditions and remove the background field by explicitly solving the boundary value problems of Laplace's or Poisson's equation. The proposed Laplacian boundary value (LBV) method for background field removal retains data near the boundary and is computationally efficient. Tests on a numerical phantom and an experimental phantom showed that LBV was more accurate than two existing methods.
One of the major problems in applying automatic verication tools to industrial-size systems is the excessive amount of memory required during the state-space exploration of a<br />model. In the setting of real-time, this problem of state-explosion requires extra attention as information must be kept not only on the discrete control structure but also on the values of continuous clock variables. In this paper, we present Clock Dierence Diagrams, CDD's, a BDD-like data-structure for<br />representing and eectively manipulating certain non-convex subsets of the Euclidean space, notably those encountered during verication of timed automata. A version of the real-time verication tool Uppaal using CDD's as a compact datastructure<br />for storing explored symbolic states has been implemented. Our experimental results demonstrate signicant space-savings: for 8 industrial examples, the savings are between 46%<br />and 99% with moderate increase in runtime. We further report on how the symbolic state-space exploration itself may be carried out using CDD's.
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