Contents I. Introduction
THE well-known benefit of time-of-flight (TOF) positron emission tomography (PET) over the conventional non-TOF PET is its ability to improve image statistics. The conventional rule states that the gain in the signal-to-noise ratio (SNR) in reconstructing a uniform disc is proportional to , where D is the disk diameter and is the expression in spatial unit of the system timing resolution. This benefit is a result of the redundancy found with the TOF-PET data. This redundancy can also be utilized for other advantages, such as reducing angular sampling and enabling limited-view tomography [1]. In a recent work [2], we derived a 2D analytic method that exploits this redundancy for reconstructing windowed image functions as summarily reviewed below. Let denote the image function and its TOF-PET projection data function given by p_{\phi}(\xi, \eta)=\int dtf(\vec{x})h(t)=\int dtf(\xi, \eta+t)h(t), \eqno{\hbox{(1)}} where is the projection angle, the radial coordinate, the TOF coordinate and the TOF blurring function. Given an arbitrary weight function , it induces a non- TOF data function from the TOF-PET data function by q_{\phi}^{(w)}(\xi)=\int d\eta\lambda_{\phi}^{\langle w)}(\xi, \eta)p_{\phi}(\xi, \eta), \eqno{\hbox{(2)}} where \lambda_{\phi}^{(w)}(\xi, \eta)=\int d\xi^{\prime}h(\xi-\xi^{\prime})w(\xi^{\prime}, \eta).\eqno{\hbox{(5)}}
