Tensor分解字典学习提升能谱CT重建效果

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本文主要探讨了在低剂量情况下利用Tensor（多维数组）进行空间-能量（spatio-spectral）字典学习的新型方法，以优化spectral CT（能量分辨CT）图像重建。Spectral CT技术使用能量分辨的光子计数探测器，能够在单次扫描中同时获取不同能量通道的投影信息，这为提高图像质量和减少噪声提供了前所未有的机会。传统的基于向量（vector-based）的字典学习方法在常规CT中已显示出显著的降噪效果。然而，对于spectral CT这种复杂的数据结构，单纯将多通道数据视为一组常规CT数据并进行迭代重建可能不足以充分利用其能量分辨率的优势。因此，本研究旨在发展一种新的策略，即采用Tensor表示来捕捉和分析空间和能量维度之间的潜在关系。在实验部分，研究者将提出的Tensor-based字典学习方法与两种其他算法进行了比较，这些算法分别在小型动物模型上，通过Medipix MXR CdTe探测器在MARs微CT扫描仪上进行了实时成像。结果显示，新方法在图像质量、细节保留和噪声抑制方面表现出色，超越了现有竞争者。 Tensor-based字典学习方法的核心在于，它能够分解多通道数据为一组共享的基础元素（atoms），这些元素不仅考虑了空间信息，还捕捉到了能量特征的空间分布模式。这种方法通过联合优化空间和能量字典，使得在处理spectral CT数据时，可以更准确地提取和重构各个能量通道的特征，从而提升图像重建的精度和信噪比。此外，文章的关键词"Tensor", "Spectral CT", "dictionary learning", 和 "CPD (Canonical Polyadic Decomposition)"，揭示了研究的核心技术，即使用多模式分解（如CPD）来处理Tensor数据，这是实现高效能谱CT重建的关键步骤。总结来说，这项研究对于推动spectral CT技术的发展具有重要意义，通过引入Tensor-based字典学习方法，为处理复杂多维能量数据提供了一种创新且高效的解决方案，有望在实际临床应用中带来显著的图像重建性能提升。

Abstract Spectral CT with an energy-discriminating photon

counting detector collects each projection in different energy

channels simultaneously. How to fully utilize such a spectral

dataset for image reconstruction attracts an increasing attention.

Our previous work has shown that the conventional vector based

dictionary learning for low-dose CT can reduce image noise

remarkably. In this paper, we develop a tensor-based

spatio-spectral dictionary learning approach for spectral CT. The

proposed method is compared with two other algorithms using an

in vivo mouse scanned on the MARS micro-CT scanner with a

Medipix MXR CdTe detector. The results demonstrate that the

proposed method outperforms the competing algorithms.

Index Terms Spectral CT, dictionary learning, tensor, CPD

I. I

NTRODUCTION

PECTRALCT with a photon counting detector can

discriminate x-ray energy and collect multiple projections

from different energy channels simultaneously. How to fully

use the acquired information to reconstruct a multichannel

image is a hot topic. A direct way is to treat multichannel data

as a group of conventional CT data, and iteratively reconstruct

each channel image independently with regularization [1].

Because data in different channels are collected from the same

object, the corresponding images are highly correlated and low

rank priori can be applied to encourage correlation among

channels [2, 3]. Furthermore, the x-ray attenuation coefficients

for all channels can be reconstructed as a whole simultaneously.

For example, Semerci et al. treated x-ray attenuation

coefficients at different energy channels as a tensor and the

tensor nuclear norm regularization was designed for spectral

CT reconstruction [4].

In our previous work, vector based dictionary learning was

applied to conventional low dose CT, which demonstrated the

This work was supported in part by the National Natural Science Foundation

of China (NSFC) under Grant Nos. 61172163, 90920003, 61401349 and

61302136, in part by NIH/NIBIB U01 grant EB017140, in part by Natural

Science Basic Research Plan in Shaanxi Province of China (Program No.

2014JQ8317), in part by China Postdoctoral Science Foundation (Program No.

2012M521775).

Yanbo Zhang, Xuanqin Mou and Qiong Xu are with the Institute of Image

Processing and Pattern Recognition, Xi'an Jiaotong University, Xi'an, Shaanxi

710049, China. (e-mail:yanbozhang007@gmail.com,

xqmou@mail.xjtu.edu.cn, xuqiong@mail.xjtu.edu.cn).

Hengyong Yu is with Department of Electrical and Computer Engineering,

University of Massachusetts Lowell, Lowell, MA, 91854 (e-mail:

hengyong-yu@ieee.org).

Ge Wang is with Biomedical Imaging Center/Cluster, CBIS, Rensselaer

Polytechnic Institute, Troy, NY, 12180 (e-mail: wangg6@rpi.edu).

superior reconstruction performance [5]. Later on, it was

extended to spectral CT reconstruction [6, 7]. Very recently, we

proposed a tensor based spatio-temporal dictionary learning

method for dynamic CT, and our preliminary results

demonstrated that tensor based dictionary outperforms the

vector based dictionary [8]. Based on our previous work, in this

paper we will develop a tensor based spatio-spectral dictionary

learning approach for spectral CT reconstruction.

II. T

ENSOR

-B

ASED

ICTIONARY

EARNING

A tensor is a multidimensional array. An n

-order (n-way)

tensor is defined as , whose element in

is , , .

Specifically, for and , tensors are vectors and

matrices, respectively. For , is a

third-order tensor. By an element reordering transformation, a

tensor can be unfolded to a matrix. The mode-k unfolding of

is denoted by , , i.e. ,

and . A third-order tensor can

be factorized into a sum of weighted rank-one tensors using

CANDECOMP/PARAFAC decomposition (CPD) [9]

, (1)

where R is a positive integer,

a ,

b and

are normalized vectors for , and is the weight. The

Given a set of third-order training tensors ,

, the tensor based dictionary learning can be

formulated as the following optimization problem,

(2)

where L represents the sparsity level and

1 2 3

( )

{ }

I I I K

is the learned tensor dictionary, K is the

number of atoms, is the representation coefficients, the

-mode product of a tensor with a

vector resulting in a third-order tensor, represents L

-norm,

namely, the number of nonzero elements, and

represents

Frobenius norm. is the k

atom which is a

rank-1 tensor, so the atom can be rewritten as

Tensor-based Dictionary Learning for Spectral

CT Reconstruction

Yanbo Zhang, Xuanqin Mou, Qiong Xu, Hengyong Yu, and Ge Wang

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