Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA.
IEEE Trans Image Process. 2012 Mar;21(3):1084-96. doi: 10.1109/TIP.2011.2168410. Epub 2011 Sep 15.
Observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f*) from Poisson data (y) cannot be effectively accomplished by minimizing a conventional penalized least-squares objective function. The problem addressed in this paper is the estimation of f* from y in an inverse problem setting, where the number of unknowns may potentially be larger than the number of observations and f* admits sparse approximation. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods.
在许多应用中，观测结果由离散事件的计数组成，例如光子击中探测器，这些计数不能有效地使用加性有界或高斯噪声模型进行建模，而需要使用泊松噪声模型。因此，从泊松数据 (y) 中准确重建空间或时间分布现象 (f*) 不能通过最小化传统的惩罚最小二乘目标函数有效地完成。本文解决的问题是在反问题设置中从 y 估计 f*，其中未知数的数量可能大于观测数量，并且 f* 允许稀疏逼近。本文考虑的优化公式使用带非负约束的惩罚负泊松对数似然目标函数（因为泊松强度自然是非负的）。具体来说，所提出的方法在每次迭代时都使用可分离二次逼近目标函数的关键思想，并使用与系数向量的 l1 范数、总变差半范数和基于分区的多尺度估计方法相关的惩罚项。
