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.

中文翻译：

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具有一组向量中的值的函数的度量双积分正则化。

我们提出了一种对逆和成像问题进行变分正则化的方法，用于恢复具有一组向量中的值的函数。我们引入了正则化泛函，它们是此类函数的无导数双积分。这些正则化函数是由双积分激发的，它近似于强度函数的 Sobolev 半范数。这些是在 Bourgain 等人中介绍的。（再看 Sobolev 空间。在：Menaldi、Rofman、Sulem (eds) 最优控制和偏微分方程——创新和应用：纪念 Alain Bensoussan 教授 60 周年，IOS 出版社，阿姆斯特丹，第 439-455 页，2001 年）。对于提出的正则化泛函，我们证明了极小值的存在以及具有一组向量值的函数的稳定性和收敛性结果。