AI 中文总结
该研究分析带线性和非线性全变分的一维正则化问题的极小化子，明确输入是否为极小化子的条件，得到线性与非线性情形的不同结论并辅以数值示例。
AI 中文摘要
本研究针对具有线性或非线性全变分及保真项的一维Rudin-Osher-Fatemi型泛函的极小化子展开探讨，得出输入数据及泛函参数的相关条件，以判定输入本身是否为极小化子。在线性情形下，参数的判别条件互为补集，凸显了结果的尖锐性；非线性情形则存在显著差异：数据的非极小性可通过与线性情形类似的方式处理，而对于输入的极小性，仅得到部分答案。非线性情形的结果依赖于辅助的约束或惩罚极小化问题，用于研究给定高度下最优跃迁的行为，此外还辅以若干数值示例。
英文摘要
A study of the minimizers of one-dimensional Rudin-Osher-Fatemi-type functionals with linear or nonlinear total variation and fidelity term is undertaken. Conditions on the input datum and the parameters of the functional are found that force the input itself to be the minimizer or not. In the linear setting the discriminating conditions on the parameters are complementary highlighting the sharpness of the results. The nonlinear setting is substantially different: while the non-minimality of the datum is treated in analogy with the linear case, for the minimality of the input only partial answers are found. The results in the nonlinear setting hinge on auxiliary constrained or penalized minimization problems investigating the behavior of optimal transitions with prescribed height. Additionally, they are complemented by some numerical examples.
Comments48 pages, 7 figures
