总变分最小化在PET等医学图像去噪中的应用及其快速实现算法

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摘要

医学图像的去噪处理是对医学图像进行后续分析的基础。正电子发射计算机

断层显像(Positron Emission Computed Tomography，PET)可以从分子水平观察代谢

物或药物在人体内的活动，它在肿瘤学、临床医学影像和癌扩散方面上有着大量

的应用，有助于医生更早地发现疾病进而准确、客观地进行诊断和治疗。PET 在

成像过程中会受到放射性核素衰变、散射光子以及其它的影响使得图像受到噪声

污染，造成局部区域不清晰的现象，这给 PET 图像的诊断分析和利用带来许多的

困难。PET 的空间分辨率一般为 4-5mm，它主要反映的是组织的功能信息，并不

能清晰地反映解剖结构信息。因此，对 PET 图像进行去噪有着重要的临床意义和

应用价值。

本文首先介绍了各类医学图像噪声的模型以及相应的去噪方法。然后重点对

总变分最小化算法应用到 PET 图像去噪中进行了研究，分析了基于偏微分方程去

噪模型的推导过程，论述了偏微分方程去噪模型从线性到非线性、从各向同性到

各向异性、从低阶到高阶的发展历程，在偏微分方程去噪模型的基础上导出了总

变分最小化去噪模型，并结合 PET 图像中噪声的特点给出了针对 PET 图像去噪的

总变分最小化模型。最后在求解总变分最小化模型对应的能量泛函时，考虑到这

是一个非线性椭圆型的偏微分方程，文中给出了三种求解数值算法：固定点迭代

法、共轭梯度法、多网格方法。

文中进行了多次仿真实验来验证算法的有效性。PET 图像中的噪声主要为泊

松噪声，其次是高斯噪声。为了证明总变分最小化去噪算法模型对抑制泊松噪声

和高斯噪声有用，本文先将普通的非医学图像添加高斯噪声和泊松噪声，然后用

文中的算法对添加噪声后的图像分别去噪，去噪后的效果评价证明了总变分最小

化算法在去除泊松噪声和高斯噪声上的有效性。总变分最小化算法对于 PET 图像

的噪声去除，文中分了两种情况来说明，一种是去除 PET 图像中的泊松噪声，另

一种是去除 PET 图像中的高斯噪声。文中还用了高斯滤波和中值滤波来处理 PET

噪声图像，处理后的结果与本文算法处理的结果对比，用最小均方差、信噪比和

峰值信噪比来客观评价每种方法的处理结果。实验的结果说明了总变分最小化算

法可以有效地去除 PET 图像中的噪声。相对于其它经典的去噪方法，总变分最小

化算法还有一个优势就是在去噪的同时能够保留图像的边缘不被平滑。本文对去

噪前后的图像分别用 Prewitt 算子提取边缘，然后线性地叠加在去噪后的图像上，

通过比较这两幅叠加图像，来说明总变分最小化算法保留边缘的特点。

关键词：偏微分方程总变分最小化共轭梯度多网格方法

ABSTRACT

Medical image denoising processing is the basis for the subsequent analysis of

medical images. With the help of Positron Emission Computed Tomography(PET),

Metabolite or drug activity can be observed in the human body from the molecular level.

It has a large number of applications in oncology and clinic medical imaging and cancer

proliferation, which helps physicians detect disease earlier then accurately and

objectively diagnose and treat. In the PET imaging process, radionuclides decay and

scattered photons and many other reasons make the PET image affected by noise. This

brought many difficulties for PET image diagnostic and analysis. The spatial resolution

of PET is generally 4-5mm, it mainly reflects the function information of structure with

unclear anatomical structure information. therefore, the PET image denoising has

important clinical significance and application value.

First of all, this paper describes the noise model and the corresponding denoising

method of the medical image. Then the paper focus on the total variation minimization

algorithm applied to the PET image denoising. The derivation of denoising model based

on partial differential equations is studied. In the paper ,we also discusses the

development of denoising model based on partial differential equations which is from

linear to nonlinear, from isotropic to anisotropic, from low-end to high-end. On the

basis of partial differential equations denoising, the denoising model of minimizing the

total variation is derived. Combined with the characteristics of PET image noise, a

special total variation minimization model for PET image denoising is given. Finally,

solving the energy functional which corresponding to total variation minimization

model. Considering that it is a nonlinear elliptic partial differential equations, this paper

gives three numerical algorithm to solve the equations: a fixed point iteration method,

conjugate gradient method and multi-grid method.

In the paper, a number of simulations is conducted to verify the effectiveness of the

algorithm. The noise in PET image is mainly Poisson noise and a little Gaussian noise.

In order to prove that it is useful for the denoising model of total variation minimization

algorithm to suppression Poisson noise and Gaussian noise, the paper firstly adds

Gaussian noise and Poisson noise to ordinary non-medical images, and then use the

total variation minimization algorithm to suppress noise respectively. The evaluation

proved that total variation minimization algorithm is effective to remove the Poisson

III

noise and Gaussian noise. Two cases is described for total variation minimization

algorithm for PET image noise removal, one is to remove the Poisson noise in the PET

image, and the other one is to remove the PET image Gaussian noise. The paper also

used the Gaussian filter and median filter to process the PET image noise, whose results

is compared with the result of total variation minimization algorithm. Mean squared

error, signal to noise ratio and peak signal to noise ratio is used to objectively evaluate

every noise removal result. Experiments show that the total variation minimization

algorithm can effectively remove noise in PET images. Compared to other classical

denoising methods, total variation minimization algorithm has another advantage that it

can preserve the image edges not to be smoothed while denoising. The Prewitt operator

is used to extract PET edge before and after denoising, then linearly superimpose these

edges on the denoised image. By comparing these two superimposed images to illustrate

that total variation minimization algorithm can retain image edge.

Key Word ：Partial differential equation, Total variation model,

Conjugate Gradient, Multi-Grid

IV
目 录
摘 要
ABSTRACT
第一章 绪 论................................................................................................................... 1
1 引言.................................................................................................................... 1
2 PET 图像............................................................................................................. 1
2.1 PET 成像的物理基础.............................................................................. 1
2.2 PET 图像去噪的意义.............................................................................. 2
2.3 PET 图像去噪现状研究.......................................................................... 2
3 本文主要研究内容和结构安排........................................................................ 3
4 本章小结............................................................................................................ 4
第二章 医学图像的噪声分析......................................................................................... 5
1 常见噪声简介.................................................................................................... 5
2 医学图像的噪声分析........................................................................................ 6
2.1 超声图像................................................................................................. 6
2.2 CT 图像.................................................................................................... 8
2.3 MRI 图像..................................................................................................9
2.4 PET 图像................................................................................................ 11
3 常见的去噪算法.............................................................................................. 13
3.1 均值滤波算法....................................................................................... 13
3.2 中值滤波算法....................................................................................... 14
3.3 维纳滤波算法....................................................................................... 14
4 本章小结.......................................................................................................... 15
第三章 总变分最小化算法........................................................................................... 16
1 基于偏微分方程的去噪问题.......................................................................... 16
1.1 基于偏微分方程去噪模型的导出....................................................... 16
1.2 基于偏微分方程去噪模型的改进....................................................... 17
2 图像变分去噪数学模型.................................................................................. 21
2.1 总变分最小化数学模型....................................................................... 21
2.2 PET 图像中的总变分最小化去噪模型................................................ 23
3 本章小结.......................................................................................................... 23
第四章 总变分最小化算法快速实现的数值方法....................................................... 25
1 固定点迭代法.................................................................................................. 25
2 共轭梯度法...................................................................................................... 26
3 多网格方法...................................................................................................... 29
4 本章小结.......................................................................................................... 30
第五章 去噪效果评价................................................................................................... 31
1 效果评价参数.................................................................................................. 31
2 仿真实验结果分析.......................................................................................... 32

5.2.1 普通图像去噪....................................................................................... 32

5.2.2 PET 图像去噪........................................................................................ 33

5.2.3 PET 图像边缘和细节的保护................................................................ 35

5.3 本章小结.......................................................................................................... 38

第六章全文总结与展望............................................................................................... 39

6.1 本文主要工作.................................................................................................. 39

6.2 研究展望.......................................................................................................... 39

参考文献......................................................................................................................... 41

在读期间公开发表的论文和承担科研项目及取得成果............................................. 45

致谢............................................................................................................................... 46

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摘要：

I摘要医学图像的去噪处理是对医学图像进行后续分析的基础。正电子发射计算机断层显像(PositronEmissionComputedTomography，PET)可以从分子水平观察代谢物或药物在人体内的活动，它在肿瘤学、临床医学影像和癌扩散方面上有着大量的应用，有助于医生更早地发现疾病进而准确、客观地进行诊断和治疗。PET在成像过程中会受到放射性核素衰变、散射光子以及其它的影响使得图像受到噪声污染，造成局部区域不清晰的现象，这给PET图像的诊断分析和利用带来许多的困难。PET的空间分辨率一般为4-5mm，它主要反映的是组织的功能信息，并不能清晰地反映解剖结构信息。因此，对PET图像进行去噪有着重...

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