Finite rate of innovation (FRI) is a powerful reconstruction framework enabling the recovery of sparse Dirac streams from uniform low-pass filtered samples. An extension of this framework, called generalised FRI (genFRI), has been recently proposed for handling cases with arbitrary linear measurement models. In this context, signal reconstruction amounts to solving a joint constrained optimisation problem, yielding estimates of both the Fourier series coefficients of the Dirac stream and its so-called annihilating filter, involved in the regularisation term. This optimisation problem is however highly non convex and non linear in the data. Moreover, the proposed numerical solver is computationally intensive and without convergence guarantee. In this work, we propose an implicit formulation of the genFRI problem. To this end, we leverage a novel regularisation term which does not depend explicitly on the unknown annihilating filter yet enforces sufficient structure in the solution for stable recovery. The resulting optimisation problem is still non convex, but simpler since linear in the data and with less unknowns. We solve it by means of a provably convergent proximal gradient descent (PGD) method. Since the proximal step does not admit a simple closed-form expression, we propose an inexact PGD method, coined as Cadzow plug-and-play gradient descent (CPGD). The latter approximates the proximal steps by means of Cadzow denoising, a well-known denoising algorithm in FRI. We provide local fixed-point convergence guarantees for CPGD. Through extensive numerical simulations, we demonstrate the superiority of CPGD against the state-of-the-art in the case of non uniform time samples.

中文翻译：

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CPGD：用于广义 FRI 的 Cadzow 即插即用梯度下降

有限创新率 (FRI) 是一个强大的重建框架，能够从均匀的低通滤波样本中恢复稀疏的狄拉克流。该框架的扩展，称为广义 FRI (genFRI)，最近被提出用于处理具有任意线性测量模型的情况。在这种情况下，信号重建相当于解决一个联合约束优化问题，产生狄拉克流的傅立叶级数系数​​及其所谓的湮没滤波器的估计，涉及正则化项。然而，这个优化问题在数据中是高度非凸的和非线性的。此外，所提出的数值求解器是计算密集型的，并且没有收敛保证。在这项工作中，我们提出了 genFRI 问题的隐式公式。为此，我们利用了一个新的正则化项，它不明确依赖于未知的湮灭滤波器，但在解决方案中强制执行足够的结构以实现稳定恢复。由此产生的优化问题仍然是非凸的，但更简单，因为在数据中是线性的，并且未知数更少。我们通过可证明收敛的近端梯度下降（PGD）方法来解决它。由于近端步骤不允许简单的闭式表达式，我们提出了一种不精确的 PGD 方法，称为 Cadzow 即插即用梯度下降 (CPGD)。后者通过 Cadzow 去噪（FRI 中众所周知的去噪算法）来逼近近端步骤。我们为 CPGD 提供局部定点收敛保证。通过大量的数值模拟，