基于TV的傅里叶测量样条重构：网格法的唯一性与收敛性
TV-based Spline Reconstruction with Fourier Measurements: Uniqueness and Convergence of Grid-Based Methods
摘要：
本文研究了从分段多项式周期函数的低频信息中恢复分段多项式周期函数的问题。这意味着我们只能访问最大截止频率$K_c$以下的地面实况的傅立叶样本的可能损坏版本。重建任务被指定为具有涉及$M$阶导数正则化算子$\mathrm{L} = \mathrm{D}^M$的全变差（TV）正则化（在测量的意义上）的最优化问题。阶数$M \geq 1$决定了重建的分段多项式样条的阶数，而TV正则化范数（已知可提高稀疏性）保证了少量的片段。我们表明，我们的最优化问题的解决方案总是唯一的，据我们所知，这是第一个基于电视的问题。此外，我们表明，这个解决方案是一个周期样条匹配的正则化算子$\mathrm{L}$的节点数是上限由$2 K_c$。然后，我们考虑基于网格的离散化我们的最优化问题的空间中的均匀$\mathrm{L}$样条。在理论方面，我们表明，任何序列的离散化问题的解决方案一致收敛到唯一的解决方案的网格大小为零的无网格问题。最后，在算法方面，我们提出了一个基于B样条的算法来解决基于网格的问题，并通过实验证明了其数值可行性。在这两个方面，我们利用了原始问题解的唯一性。
英文摘要：
We study the problem of recovering piecewise-polynomial periodic functions from their low-frequency information. This means that we only have access to possibly corrupted versions of the Fourier samples of the ground truth up to a maximum cutoff frequency $K_c$. The reconstruction task is specified as an optimization problem with total-variation (TV) regularization (in the sense of measures) involving the $M$-th order derivative regularization operator $\mathrm{L} = \mathrm{D}^M$. The order $M \geq 1$ determines the degree of the reconstructed piecewise polynomial spline, whereas the TV regularization norm, which is known to promote sparsity, guarantees a small number of pieces. We show that the solution of our optimization problem is always unique, which, to the best of our knowledge, is a first for TV-based problems. Moreover, we show that this solution is a periodic spline matched to the regularization operator $\mathrm{L}$ whose number of knots is upper-bounded by $2 K_c$. We then consider the grid-based discretization of our optimization problem in the space of uniform $\mathrm{L}$-splines. On the theoretical side, we show that any sequence of solutions of the discretized problem converges uniformly to the unique solution of the gridless problem as the grid size vanishes. Finally, on the algorithmic side, we propose a B-spline-based algorithm to solve the grid-based problem, and we demonstrate its numerical feasibility experimentally. On both of these aspects, we leverage the uniqueness of the solution of the original problem.
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