Department of Mathematics, Claremont McKenna College, Claremont, CA 91711, USA.
IEEE Trans Image Process. 2013 Oct;22(10):3941-9. doi: 10.1109/TIP.2013.2264681. Epub 2013 May 22.
Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for two-dimensional images [Formula: see text] to signals [Formula: see text] of arbitrary dimension d ≥ 2 and to isotropic total variation problems. In this paper, we show that a multidimensional signal [Formula: see text] can be reconstructed from O(s dlog(N(d))) linear measurements [Formula: see text] using total variation minimization to a factor of the best s -term approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension d.
考虑从欠定测量集中重建多维信号的问题，如在压缩感知的设置中。在没有任何其他假设的情况下，这个问题是不适定的。然而，对于自然图像或电影等信号，与测量值一致的最小全变差估计通常可以很好地逼近基础信号，即使测量值的数量远远小于环境维度。本文将二维图像[Formula: see text]的最新重建保证扩展到任意维度 d≥2 的信号[Formula: see text]和各向同性全变差问题。在本文中，我们表明，可以从 O(s dlog(N(d)))个线性测量值[Formula: see text]使用全变差最小化重建多维信号[Formula: see text]，将其梯度的最佳 s 项逼近的因子减少到一个因子。我们提供的重建保证在空间维度 d 的多项式因子内是必要最优的。
