Ahn Sangtae, Leahy Richard M
Signal and Image Processing Institute, University ofSouthern California, Los Angeles, CA 90089, USA.
IEEE Trans Med Imaging. 2008 Mar;27(3):413-24. doi: 10.1109/TMI.2007.911549.
We present accurate and efficient methods for estimating the spatial resolution and noise properties of nonquadratically regularized image reconstruction for positron emission tomography (PET). It is well known that quadratic regularization tends to over-smooth sharp edges. Many types of edge-preserving nonquadratic penalties have been proposed to overcome this problem. However, there has been little research on the quantitative analysis of nonquadratic regularization due to its nonlinearity. In contrast, quadratically regularized estimators are approximately linear and are well understood in terms of resolution and variance properties. We derive new approximate expressions for the linearized local perturbation response (LLPR) and variance using the Taylor expansion with the remainder term. Although the expressions are implicit, we can use them to accurately predict resolution and variance for nonquadratic regularization where the conventional expressions based on the first-order Taylor truncation fail. They also motivate us to extend the use of a certainty-based modified penalty to nonquadratic regularization cases in order to achieve spatially uniform perturbation responses, analogous to uniform spatial resolution in quadratic regularization. Finally, we develop computationally efficient methods for predicting resolution and variance of nonquadratically regularized reconstruction and present simulations that illustrate the validity of these methods.
我们提出了精确且高效的方法，用于估计正电子发射断层扫描（PET）中非二次正则化图像重建的空间分辨率和噪声特性。众所周知，二次正则化往往会过度平滑锐利边缘。为克服此问题，人们提出了多种保边非二次惩罚项。然而，由于其非线性，关于非二次正则化的定量分析研究甚少。相比之下，二次正则化估计器近似线性，并且在分辨率和方差特性方面已得到充分理解。我们使用带有余项的泰勒展开式，推导出线性化局部扰动响应（LLPR）和方差的新近似表达式。尽管这些表达式是隐式的，但对于基于一阶泰勒截断的传统表达式失效的非二次正则化情况，我们可以用它们来准确预测分辨率和方差。它们还促使我们将基于确定性的修正惩罚项的应用扩展到非二次正则化情形，以实现空间均匀的扰动响应，类似于二次正则化中的均匀空间分辨率。最后，我们开发了计算高效的方法来预测非二次正则化重建的分辨率和方差，并给出了说明这些方法有效性的模拟结果。
