Abstract
We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain et al. (Another look at Sobolev spaces. In: Menaldi, Rofman, Sulem (eds) Optimal control and partial differential equations-innovations and applications: in honor of professor Alain Bensoussan’s 60th anniversary, IOS Press, Amsterdam, pp 439–455, 2001). For the proposed regularization functionals, we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.
Highlights
Functions with values in a subset of a vector space appear in several applications of imaging and in inverse problems, e.g., Interferometric Synthetic Aperture Radar (InSAR) is a technique used in remote sensing and geodesy to generate, for example, digital elevation maps of the earth’s surface
The main objective of this paper is to introduce a general class of regularization functionals for functions with values in a set of vectors
In this paper we developed a functional for regularization of functions with values in a set of vectors
Summary
Functions with values in a (nonlinear) subset of a vector space appear in several applications of imaging and in inverse problems, e.g., Interferometric Synthetic Aperture Radar (InSAR) is a technique used in remote sensing and geodesy to generate, for example, digital elevation maps of the earth’s surface. The imaging function is from ⊂ R2, a part of the Earth’s surface, into S2 ⊆ R3, representing foliage angle orientation. Estimation of functions with values in S O(3) ⊆ R3×3
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