Bayesian methods are commonly applied to solve image analysis problems such as noise-reduction, feature enhancement and object detection. A primary limitation of these approaches is the computational complexity due to the interdependence of neighboring pixels which limits the ability to perform full posterior sampling through Markov chain Monte Carlo (MCMC). To alleviate this problem, we develop a new posterior sampling method that is based on modeling the prior and likelihood in the space of the Fourier transform of the image. One advantage of Fourier-based methods is that many spatially correlated processes in image space can be represented via independent processes over Fourier space. A recent approach known as Bayesian Image Analysis in Fourier Space (or BIFS), has introduced parameter functions to describe prior expectations about image properties in Fourier space. To date BIFS has relied on Maximum a Posteriori (MAP) estimation for generating posterior estimates; providing just a single point estimate. The work presented here develops a posterior sampling approach for BIFS that can explore the full posterior distribution while continuing to take advantage of the independence modeling over Fourier space. As a result computational efficiency is improved over that for conventional Bayesian image analysis and mixing concerns that commonly have to be dealt with in high dimensional Markov chain Monte Carlo sampling problems are avoided. Implementation results and details are provided using simulated data.

翻译：贝叶斯方法广泛应用于解决图像分析问题，如降噪、特征增强和目标检测。这些方法的主要局限在于计算复杂度较高，原因是相邻像素间的相互依赖性限制了通过马尔可夫链蒙特卡洛（MCMC）对后验分布进行全采样。为解决此问题，我们提出了一种新的后验采样方法，该方法基于在图像的傅里叶变换空间中构建先验和似然模型。傅里叶方法的一个优势在于，图像空间中许多空间相关过程可通过傅里叶空间中的独立过程表示。近期提出的“傅里叶空间贝叶斯图像分析”（BIFS）方法，引入了参数函数来描述傅里叶空间中图像属性的先验期望。迄今为止，BIFS依赖最大后验（MAP）估计生成后验估计，仅提供单一的点估计。本文针对BIFS开发了一种后验采样方法，能够在利用傅里叶空间独立性建模优势的同时，探索完整的后验分布。因此，与传统贝叶斯图像分析相比，该方法提升了计算效率，并避免了高维马尔可夫链蒙特卡洛采样问题中常见的混合困难。文中通过模拟数据给出了实现结果与细节。