Estimating Precision in Functional Images

Abstract

Functional imaging of biologic parameters like in vivo tissue metabolism is made possible by Positron Emission Tomography (PET). Many techniques have been suggested for extracting such images from dynamic time-course sequences of reconstructed PET scans. Quantitating the precision of these estimates is important for drawing inferences on the biologic parameters. Analytic variance formulas are not immediate owing to the nonlinear methods used in extraction. The usual resampling approach is infeasible because each image reconstruction in PET is a computationally demanding solution to a high-dimensional linear inverse problem. We suggest an alternative simulation approach that approximates the distribution of reconstructed PET scans and performs a parametric bootstrap in the imaging domain. Results on a simplified model chosen to match the characteristics of PET reconstruction are very encouraging. Mixture analysis is used to estimate functional images; however, the suggested approach is general enough to extend to other techniques or imaging methods.     

有界平均振幅空间上的乘法算子

有界平均振幅空间的研究在算子理论及全纯空间的研究中具有重要的作用.主要研究了有界平均振幅空间上乘法算子的性质,并且得到了托普里兹算子有界性及紧性的条件.     

多重调和Bergman空间上径向Toeplitz算子的交换性

In this paper, we study the commutativity of Toeplitz operators with radial symbols on the pluriharmonic Bergman space. We obtain the necessary and sufficient conditions for the commutativity of bounded Toeplitz operator and Toeplitz operator with radial symbol on the pluriharmonic Bergman space.     

Commuting Toeplitz operators on the hardy space of the polydisk

In this paper, we show that two Toeplitz operators T_f and T_g on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [T_f, T_g] is n-harmonic.     

Schatten class Toeplitz operators on generalized Fock spaces

In this paper we characterize the Schatten p   class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range 0<p<∞$0<p<\infty$.     

Operators on the orthogonal complement of the Dirichlet space(Ⅱ)

In this paper we first prove that a dual Hankel operator R φ on the orthogonal complement of the Dirichlet space is compact for φ∈ W 1,∞(D),and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact.We also prove that a dual Hankel operator R φ with φ∈ W 1,∞(D) is of finite rank if and only if B φ is orthogonal to the Dirichlet space for some finite Blaschke product B,and give ...     

Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm

The structure preserving rank reduction problem arises in many important applications. The singular value decomposition (SVD), while giving the closest low rank approximation to a given matrix in matrix L ₂ norm and Frobenius norm, may not be appropriate for these applications since it does not preserve the given structure. We present a new method for structure preserving low rank approximation of a matrix, which is based on Structured Total Least Norm (STLN). The STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX B, preserving the given linear structure in the perturbation [E F] such that (A + E)X = B + F. The approximate solution can be obtained to minimize the perturbation [E F] in the L _p norm, where p = 1, 2, or . An algorithm is described for Hankel structure preserving low rank approximation using STLN with L _p norm. Computational results are presented, which show performances of the STLN based method for L ₁ and L ₂ norms for reduced rank approximation for Hankel matrices.     

Toeplitz预条件子的统一构造与性能分析

１．引言考虑线性方程组ＴＮｘ＝ｂ（１．１）其中ＴＮ＝（ｔｉ,ｊ）是ＮｘＮ对称正定（ＳＰＤ）Ｔｏｅｐｌｉｔｚ矩阵,即ｔｉ,ｊ＝ｔ｜ｉ－ｊ｜（ｉ,ｊ＝０,１,...,Ｎ－１）且ＴＮ的所有特征值均为正数,并表为ＴＮ：＝Ｔ（ｔ。,ｔｉ,...,ｔＮ－１）．如果我们用预条件子共轭梯度法（ＰＣＧ）求解方程组（１．１）,最关健的任务是构造出高效的预条件子．而预条件子最自然的选择似乎其逆矩阵易求且构成矩阵ＴＮ的某种最优逼近．由于循环矩阵ＣＮ的逆矩阵ＣＲ'仍为循环矩阵,因此ＣＮ和ＣＨ'与向量的乘积可通ｉｓ速Ｆｏｕｒｉｅｒ…     

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.