title: The Asymptotic Noise Distribution in Karhunen-Loeve Transform Eigenmodes.
abstract: 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.
authors.affiliation: 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.
authors.forename: Yu
authors.initials: Y
authors.lastname: Ding
authors.affiliation: Siemens Corporate Research, Princeton, USA.
authors.forename: Hui
authors.initials: H
authors.lastname: Xue
authors.affiliation: Siemens Medical Solutions, Inc., Columbus, USA.
authors.forename: Ning
authors.initials: N
authors.lastname: Jin
authors.affiliation: Shenzhen Institute of Advanced Technology of Chinese Academy of Science, Shenzhen, Guangdong, China.
authors.forename: Yiu-Cho
authors.initials: YC
authors.lastname: Chung
authors.affiliation: Shenzhen Institute of Advanced Technology of Chinese Academy of Science, Shenzhen, Guangdong, China.
authors.forename: Xin
authors.initials: X
authors.lastname: Liu
authors.affiliation: Shenzhen Institute of Advanced Technology of Chinese Academy of Science, Shenzhen, Guangdong, China.
authors.forename: Yongqin
authors.initials: Y
authors.lastname: Zhang
authors.affiliation: Davis Heart and Lung Research Institute, The Ohio State University, Columbus, USA ; Department of Internal Medicine, The Ohio State University, Columbus, USA ; Department of Biomedical Engineering, The Ohio State University, Columbus, USA ; Department of Radiology, The Ohio State University, Columbus, USA.
authors.forename: Orlando P
authors.initials: OP
authors.lastname: Simonetti
keywords: Independently and identically-distributed noise; Karhunen-Loeve transform; Random matrix theory