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.

In this paper, we study existence of invariant curves of an iterative equation h[2](x)−(1+b)h(x)+bx−ϵf(h(x))+(b−1)α=0, which is from dissipative standard map. By constructing an invertible analytic solution g(x) of an auxiliary equation of the form g(λ2x)−(1+b)g(λx)+bg(x)+(b−1)α=ϵf(g(λx)), invertible analytic solutions of the form g(λg − 1(x)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |λ| < 1, we focus on those λ on the unit circle S1, that is, |λ| = 1. We discuss not only those λ at resonance, that is, at a root of the unity, but also those λ near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.