Ding Yu, Xue Hui, Jin Ning, Chung Yiu-Cho, Liu Xin, Zhang Yongqin, Simonetti Orlando P
Davis Heart and Lung Research Institute, The Ohio State University, Columbus, USA ; Shenzhen Institute of Advanced Technology of Chinese Academy of Science, Shenzhen, Guangdong, China.
Siemens Corporate Research, Princeton, USA.
J Health Med Inform. 2013 Jun;4(2):122. doi: 10.4172/2157-7420.1000122.
Karhunen-Loeve Transform (KLT) is widely used in signal processing. Yet the well-accepted result is that, the noise is uniformly distributed in all eigenmodes is not accurate. We apply a result of the random matrix theory to understand the asymptotic noise distribution in KLT eigenmodes. Noise variances in noise-only eigenmodes follow the Marcenko-Pastur distribution, while noise variances in signal-dominated eigenmodes still follow the uniform distribution. Both the mathematical expectation of noise level in each eigenmode and an analytical formula of KLT filter noise reduction effect with a hard threshold were derived. Numerical simulations agree with our theoretical analysis. The noise variance of an eigenmode may deviate more than 60% from the uniform distribution. These results can be modified slightly, and generalized to non-IID (independently and identically-distributed) noise scenario. Magnetic resonance imaging experiments show that the generalized result is applicable and accurate. These generic results can help us understand the noise behavior in the KLT and related topics.
卡尔胡宁 - 洛伊夫变换（KLT）在信号处理中被广泛应用。然而，普遍认可的结果是，噪声在所有本征模式中均匀分布这一说法并不准确。我们应用随机矩阵理论的一个结果来理解KLT本征模式中的渐近噪声分布。仅噪声本征模式中的噪声方差遵循马尔琴科 - 帕斯特分布，而信号主导本征模式中的噪声方差仍遵循均匀分布。推导了每个本征模式中噪声水平的数学期望以及具有硬阈值的KLT滤波器降噪效果的解析公式。数值模拟与我们的理论分析一致。一个本征模式的噪声方差可能与均匀分布偏差超过60%。这些结果可以稍作修改，并推广到非独立同分布（IID）噪声场景。磁共振成像实验表明，推广后的结果是适用且准确的。这些一般性结果有助于我们理解KLT中的噪声行为及相关主题。
