Statistical analysis of noise in MRI

Santiago Aja-Fernández
Gonzalo Vegas-Sánchez-Ferrero

Statistical
Analysis of
Noise in MRI
Modeling, Filtering and Estimation


Statistical Analysis of Noise in MRI


Santiago Aja-Fernández
Gonzalo Vegas-Sánchez-Ferrero

Statistical Analysis
of Noise in MRI
Modeling, Filtering and Estimation

123


Gonzalo Vegas-Sánchez-Ferrero
Harvard Medical School
Brigham and Women’s Hospital
Boston, MA
USA

Santiago Aja-Fernández
ETSI Telecomunicación
Universidad de Valladolid

Valladolid
Spain

ISBN 978-3-319-39933-1
DOI 10.1007/978-3-319-39934-8

ISBN 978-3-319-39934-8

(eBook)

Library of Congress Control Number: 2016941078
© Springer International Publishing Switzerland 2016
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Printed on acid-free paper
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The registered company is Springer International Publishing AG Switzerland


“How do you peel a porcupine?”

Foreword

Medical imaging and the field of radiology have come a long way since Wilhelm
Röntgen’s discovery of the X-ray in 1895. Medical imaging is today an integral part
of modern medicine and includes a large number of modalities such as X-ray
computed tomography (CT), ultrasound, positron emission tomography (PET), and
magnetic resonance imaging (MRI).
This book, “Statistical Analysis of Noise in MRI,” presents a modern signal
processing approach for medical imaging with a focus on noise modeling and
estimation for MRI. MRI scanners use strong magnetic fields, radio waves, and
magnetic field gradients to form images of the body. MRI has seen a tremendous
development during the past four decades and is now an indispensable part of
diagnostic medicine. MRI is unparalleled in the investigation of soft tissues due to
its superior contrast sensitivity and tissue discrimination.
I met the lead author of this book Dr. Santiago Aja-Fernández for the first time in
2006 when he was a visiting Fulbright scholar in my laboratory, Laboratory of
Mathematics in Imaging at Brigham and Women’s Hospital, Harvard Medical
School, Boston. His goal was clear from the beginning: to learn more about MRI.
His plan was to combine this knowledge with his then already vast knowledge
about statistical signal processing. He had a very productive year in Boston and
subsequently published several, now well-cited, papers on noise estimation in MRI.
During Santiago’s year-long visit in my laboratory we were investigating the
boundaries of what it meant to separate signals from noise. What do you need to
know about the data to do this well? The more complicated the image formation
process is, the less the commonly assumed model that the noise is Gaussian is
applicable. This book is about exploring these questions and providing guidelines
on how to proceed. One important message in this book is that you have to
understand your data acquisition in detail. Santiago Aja-Fernández continued to

work on these questions when he returned to the University of Valladolid with the
second author of this book, Dr. Gonzalo Vegas-Sánchez-Ferrero. They and their
co-workers have made tremendous progress during the past decade and have
become authorities on the topic of noise modeling in MRI.

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viii

Foreword

I expect that the importance of accurate noise modeling and estimation in the
field of MRI will increase over the next several years due to the increasing complexity of the MRI scanners. Many commercial scanners now have the possibility to
connect multiple RF detector coil sets to allow the simultaneous acquisition of
several signals in a phased array system. These systems were originally developed
to reduce the scanning time and therefore to avoid some problems with moving
structures, as well as to enhance the signal-to-noise ratio of the magnitude image.
Noise modeling is important in noise removal, but perhaps even more so when
estimating derived parameters from this more complex measured data. For example,
robust estimation of the diffusion tensor in diffusion MRI requires in-depth
knowledge of the imaging process used for creating the multi-channel diffusion
MRI data. With today’s complex parallel imaging acquisition schemes commonly
used in the clinic, it is important to be able to understand how to model the data
appropriately for any subsequent signal processing task.
Carl-Fredrik Westin, Ph.D.
Director Laboratory of Mathematics in Imaging,
Brigham and Women’s Hospital
Harvard Medical School
Boston, MA, USA

Preface

This work is the result of more than 10 years of research in the area of MRI from a
signal and noise perspective. Our interest has always been to properly model the
noise that affects our signals, in order to design the best possible algorithms based
on that knowledge. All this time we have found many great works that were coming
along with our own research, offering alternative points of view. We realized that
most of the works dealing with noise in MRI can be seen as complementary efforts
rather than competitive. It was necessary, thus, to systematize all that knowledge
that had arisen, in order to understand the problem as a whole. It is precisely in the
relations between distinct methods and philosophies where the real nature of this
question can be better understood. In this work we gather different approaches to
noise analysis in MRI, systematizing and classifying the different methods, trying to
bring them together to common ground. So, instead of being seen as independent
efforts, they can be considered as consecutive paces along the same way.
This book is intended to serve as a reference manual for researchers dealing with
signal processing in MRI acquisitions. It is written from a signal theory perspective,
using probabilistic modeling as a basic tool. Readers are assumed to know the basic
principles of linear systems and signal processing, as well as being familiar with
random variables, image processing, and calculus fundaments. It could also serve as
a textbook for postgraduate students in engineering with an interest in medical
image processing.
We provide a complete framework to model and analyze noise in MRI, considering different modalities and acquisition techniques, focusing on three issues: noise
modeling, noise estimation, and noise filtering. To that end, the book is divided into
three parts. The first part analyzes the problem of noise in MRI, the modeling of the
acquisition, and the definition of the most common statistical distributions used to
describe the noise. The problem of noise and signal estimation for medical imaging
is analyzed from a statistical signal processing perspective. The second part of the

book is devoted to analyzing and reviewing the different techniques to estimate noise
out of a single MRI slice in single- and multiple-coil systems for fully sampled
acquisitions. The third part deals with the problem of noise estimation when

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x

Preface

accelerated acquisitions are considered and parallel imaging methods are used to
reconstruct the signal. The book is complemented with three appendices.
Our intention is to make the book comprehensive, thus many definitions and
methods have been included, and some ideas are repeated in different chapters from
different perspectives. That way, most of the chapters can be understood independently of the others, although relations between them will always be present.
Some theoretical topics about random variables, image processing, and MRI
acquisition have been omitted for the sake of compactness. We provide a complete
bibliography that can be used to fill the gaps.
Finally, note that this is a field of constant expansion, with new methods being
published every year. In addition, acquisition techniques are also rapidly evolving,
producing new models of noise that are not analyzed here. We consider this book as
the framework that could serve as the basis for the analysis of all those novelties
that will surely arise in the next years.
Valladolid, Spain
March 2016

Santiago Aja-Fernández

Acknowledgments

The work presented in this book started at LMI (Harvard Medical School, Boston)
almost 10 years ago, funded by a Fulbright Scholarship. Many different researchers
have contributed to the development of the main corpus on noise modeling and
estimation that is finally gathered here. In particular, I want to thank Dr. Tristán-Vega
for all the shared work in this field and to my coauthor, Gonzalo VegasSánchez-Ferrero, for his help and support in the elaboration of this book. Let us
hope we can work in new topics in the future. The other researchers that have actively
contributed with their knowledge are Prof. C.F. Westin, Prof. Alberola-López,
Dr. K. Krissian, Dr. M. Niethammer, Dr. V. Brion, and Dr. W.S. Hoge.
Our intent to make a comprehensive book implies a great amount of work that
could not have been done without external support from other researchers.
I specially want to thank Tomasz Pieziak, from AGH University of Science and
Technology, Krakow (Poland), whose work about VST is directly used in this
book. We use some parts of his Ph.D. thesis for the chapter about blind estimation,
and he was also a great help in the implementation of some of the methods for
comparison. The filtering chapter takes many references from Dr. Veronique
Brion’s Ph.D. thesis, to whom I must be very grateful for saving me a great amount
of time.
The data used in this book come from different sources, but I want to thank
Dr. W. Scott Hoge and Dr. Diego Hernando for providing the valuable raw data
used along the book for validation. Additional scanning was done in Q-Diagnóstico
(Valladolid) and the 3T- scanner of Instituto de Técnicas Intrumentales
(Universidad de Valladolid). We also use an ilustration taken from Dr. TristánVega’s thesis that was generated using HARDI data kindly provided by the
Australian eHealth Research Centre-CSIRO ICT Centre, Brisbane (Australia).
The authors acknowledge Ministerio de Ciencia e Innovación for funding (grant
TEC2013-44194-P). Gonzalo Vegas-Sánchez-Ferrero acknowledges Consejera de
Educación, Juventud y Deporte de la Comunidad de Madrid and the People

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xii

Acknowledgments

Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme (FP7/2007−2013) for REA grant agreement n. 291820.
Last but not least, I am in great debt with my wife Isabel and my child Juan,
from whom I steal the many hours I dedicated to the writing of this book. I will not
forget you when I become rich and famous.
March 2016

Santiago Aja-Fernández


Contents

1

The Problem of Noise in MRI . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Thermal Noise in Magnetic Resonance Imaging. . . . . . . . . . . .
1.2 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I
2

3

1

1
4

Noise Models and the Noise Analysis Problem

Acquisition and Reconstruction of Magnetic Resonance
Imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Physics of Magnetic Resonance Imaging. . . . . . . . . .
2.2 The k-Space and the x-Space. . . . . . . . . . . . . . . . . .
2.3 Single-Coil Acquisition Process . . . . . . . . . . . . . . . .
2.4 Multiple-Coil Acquisition Process . . . . . . . . . . . . . .
2.5 Accelerated Acquisitions: Parallel Imaging . . . . . . . .
2.5.1 The Problem of Acceleration: Subsampling . .
2.5.2 Sensitivity Encoding (SENSE) . . . . . . . . . . .
2.5.3 Generalized Autocalibrating Partially Parallel
Acquisition (GRAPPA). . . . . . . . . . . . . . . .
2.5.4 Other pMRI Methods . . . . . . . . . . . . . . . . .
2.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical Noise Models for MRI . . . . . . . . . . . . . . . .
3.1 Complex Single- and Multiple-Coil MR Signals . .
3.2 Single-Coil MRI Data. . . . . . . . . . . . . . . . . . . . .
3.3 Fully Sampled Multiple-Coil Acquisition . . . . . . .
3.3.1 Uncorrelated Multiple-Coil with SoS . . . .
3.3.2 Correlated Multiple-Coil with SoS . . . . . .
3.3.3 Multiple-Coil with SMF Reconstruction . .
3.4 Statistical Models for pMRI Acquisitions . . . . . . .
3.4.1 General Noise Models in pMRI . . . . . . . .
3.4.2 Statistical Model in SENSE Reconstructed
Images . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

3.4.3

3.5

3.6
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5

Statistical Model in GRAPPA
Images . . . . . . . . . . . . . . . .
Some Practical Examples . . . . . . . . .
3.5.1 Single-Coil Acquisitions . . . .
3.5.2 Multiple-Coil Acquisitions . .
3.5.3 pMRI Acquisitions . . . . . . . .
Final Remarks . . . . . . . . . . . . . . . . .

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Noise Analysis in MRI: Overview . . . . . . . . . . . . . . . .
4.1 The Problem of Noise Estimation: An Introductory
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 A Practical Problem . . . . . . . . . . . . . . . .
4.1.2 Analysis of the Data. . . . . . . . . . . . . . . .
4.1.3 Estimation Procedure . . . . . . . . . . . . . . .
4.1.4 Other Estimation Issues . . . . . . . . . . . . .
4.2 Main Issues About Noise Analysis in MRI . . . . . .
4.2.1 The Noise Model of the Data . . . . . . . . .
4.2.2 The Stationarity of the Noise . . . . . . . . . .
4.2.3 The Background . . . . . . . . . . . . . . . . . .
4.2.4 Quantification of Data. . . . . . . . . . . . . . .
4.2.5 Single Versus Multiple Sample Estimation
4.2.6 Practical Implementation . . . . . . . . . . . . .
4.3 Noise Analysis Practical Methodology . . . . . . . . .
Noise
5.1
5.2
5.3

5.4

5.5

Filtering in MRI . . . . . . . . . . . . . . . . . . . . . . .
Noise Filtering and Signal Estimation in MRI . . .
The Importance of Noise Filtering . . . . . . . . . . .
Noise Suppression/Reduction Methods . . . . . . . .
5.3.1 Noise Correction During the Acquisition.
5.3.2 Generic Filtering Algorithms . . . . . . . . .
5.3.3 Transform Domain Filters . . . . . . . . . . .
5.3.4 Statistical Methods . . . . . . . . . . . . . . . .
5.3.5 Some Examples . . . . . . . . . . . . . . . . . .
Case Study: The LMMSE Signal Estimator . . . . .
5.4.1 Original Formulation: Signal Estimation
for the General Rician Model . . . . . . . .
5.4.2 Extension to Multiple Samples. . . . . . . .
5.4.3 Recursive LMMSE Filter . . . . . . . . . . .
5.4.4 Extension to nc-´ Data . . . . . . . . . . . . .
5.4.5 Extension for an Specific Application:
DWI Filtering . . . . . . . . . . . . . . . . . . .
Some Final Remarks . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . 115
. . . . . . . . . . 119


Contents

Part II

xv

Noise Analysis in Nonaccelerated Acquisitions

6

Noise Estimation in the Complex Domain . . . . . . . . . . . . .
6.1 Single-Coil Estimation . . . . . . . . . . . . . . . . . . . . . . .
6.2 Multiple-Coil Estimation . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Variance in Each Coil. . . . . . . . . . . . . . . . . .
6.2.2 Covariance Matrix and Correlation Coefficient .
6.2.3 Reconstruction Process . . . . . . . . . . . . . . . . .
6.3 Non-stationary Noise Analysis . . . . . . . . . . . . . . . . . .
6.4 Examples and Performance Evaluation . . . . . . . . . . . .

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7

Noise Estimation in Single-Coil MR Data . . . . . . . . . . . . .
7.1 Noise Estimators for Rayleigh/Rician Data . . . . . . . . .
7.1.1 Estimators Based on a Rayleigh Background . .
7.1.2 Estimators Based on the Signal Area . . . . . . .
7.2 Estimators Based on Local Moments: A Detailed Study
7.3 Performance of the Estimators . . . . . . . . . . . . . . . . . .
7.3.1 Performance Evaluation with Synthetic Data . .
7.3.2 Performance Evaluation Over Real Data . . . . .
7.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Noise Estimation in Multiple–Coil MR Data . . . . . . . . .
8.1 Uncorrelated Data and SMF Reconstruction . . . . . .
8.2 Noise Estimation Assuming a nc-´ Distribution. . . .
8.2.1 Estimators Based on a c-´ Background. . . .
8.2.2 Estimators Based on the Signal Area . . . . .

8.3 Performance of the Estimators . . . . . . . . . . . . . . . .
8.3.1 Performance Evaluation with Synthetic Data
8.3.2 Performance Evaluation Over Real Data . . .
8.4 Final Remarks About the Estimators. . . . . . . . . . . .

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9

Parametric Noise Analysis from Correlated Multiple-Coil
MR Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Parametric Noise Estimation for Correlated
Multiple-Coil with SMF . . . . . . . . . . . . . . . . . . . . .
9.1.1 Background-Based Estimation . . . . . . . . . . .
9.1.2 Estimation Based on Signal Area . . . . . . . . .
9.2 Noise Estimation for Correlated SoS . . . . . . . . . . . .
9.2.1 Estimation of ¾2L . . . . . . . . . . . . . . . . . . . .
9.2.2 Estimation of Effective Values . . . . . . . . . . .
9.2.3 Simplified Estimation . . . . . . . . . . . . . . . . .
9.3 Performance of the Estimators . . . . . . . . . . . . . . . . .
9.3.1 Correlated Coils with SMF . . . . . . . . . . . . .
9.3.2 Correlated Coils with SoS . . . . . . . . . . . . . .

9.3.3 In Vivo Data . . . . . . . . . . . . . . . . . . . . . . .
9.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xvi

Part III

Contents

Noise Estimators in pMRI

10 Parametric Noise Analysis in Parallel MRI . . . . . . . . . . . . . . .
10.1 Noise Estimation in SENSE . . . . . . . . . . . . . . . . . . . . . .
10.2 Noise Estimation in GRAPPA with SMF Reconstruction . .
10.3 Noise Estimation in GRAPPA with SoS Reconstruction . . .
10.3.1 Practical Simplifications over the GRAPPA Model .
10.3.2 Noise Estimator . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.3 Estimation of Effective Values in GRAPPA. . . . . .
10.3.4 Gaussian Simplification. . . . . . . . . . . . . . . . . . . .
10.4 Examples and Performance of the Estimators. . . . . . . . . . .
10.4.1 Noise Estimation in SENSE . . . . . . . . . . . . . . . .
10.4.2 Noise Estimation in GRAPPA . . . . . . . . . . . . . . .
10.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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211
212
215
215
216
217
218
219
220
220
223
227

11 Blind Estimation of Non-stationary Noise in MRI . . . . .
11.1 Non-stationary Noise Estimation in MRI. . . . . . . . .
11.1.1 Non-stationary Gaussian Noise Estimators. .
11.1.2 Rician Estimators . . . . . . . . . . . . . . . . . . .
11.1.3 Noncentral ´ Estimation . . . . . . . . . . . . . .
11.1.4 Estimation Along Multiple MR Scans. . . . .
11.2 A Homomorphic Approach to Non-stationary Noise
Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.2.1 The Gaussian Case. . . . . . . . . . . . . . . . . .
11.2.2 The Rayleigh Case . . . . . . . . . . . . . . . . . .
11.2.3 The Rician Case . . . . . . . . . . . . . . . . . . .
11.3 Performance of the Estimators . . . . . . . . . . . . . . . .
11.3.1 Non-stationary Rician Noise . . . . . . . . . . .
11.3.2 Non-stationary Nc-´ Noise . . . . . . . . . . . .
11.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

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229
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231
236
245
247

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249
249
251
253
256
256
269

273

Appendix A: Probability Distributions and Combination
of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Appendix B: Variance-Stabilizing Transformation . . . . . . . . . . . . . . . . 295
Appendix C: Data Sets Used in the Experiments . . . . . . . . . . . . . . . . . 305
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323


Acronyms and Notation

Acronyms
ACS
ADC
ARC
ASL
ASSET
AWGN
BOLD
c-´
CA
CAIPIRINHA
CHARMED
CMS
CURE
CV
DCT
DFT
DKI

DoF
DOT
DT
DTI
DWI
DWT
EM
EPI
FFE

Auto Calibration Signal
Apparent Diffusion Coefficient
Autocalibrating Reconstruction of Cartesian imaging
Arterial Spin Labeling
Array coil Spatial Sensitivity Encoding
Additive White Gaussian Noise
Blood Oxygen Level Dependent
Central chi
Conventional Approach
Controlled Aliasing in Parallel Imaging Results
in Higher Acceleration
Composite Hindered and Restricted Model
of Diffusion
Composite Magnitude Signal
Chi-square Unbiased Risk Estimator
Coefficient of Variation
Discrete Cosine Transform
Discrete Fourier Transform
Diffusion Kurtosis Imaging
Degrees of Freedom

Diffusion Orientation Transform
Diffusion Tensor
Diffusion Tensor Imaging
Diffusion Weighted Imaging
Discrete Wavelet Transform
Expectation Maximization
Echo Planar Imaging
Fast Field Echo

xvii


xviii

fMRI
FOV
GRAPPA
HARDI
HMF
iDFT
IID
KDE
LLS
LMMSE
LPF
LS
MAD
MAP
MGF
ML

MMSE
MR
MRI
MRV
nc-´
NEX
NLM
NLS
NMR
ODF
OPDF
OSRAD
PCA
PD
PDE
PDF
pMRI
RE
RF
RMMSE
ROI
RV
SENSE
SLV
SMASH
SMF
SNR
SoS
SRRAD

Acronyms and Notation

Functional Magnetic Resonance Imaging
Field of View
Generalized Autocalibrating Partially Parallel Acquisition
High Angular Resolution Diffusion Image
Homomorphic Filter
Inverse Discrete Fourier Transform
Independent and Identically Distributed
Kernel Density Estimator
Linear Least Squares
Linear Minimum Mean Square Error
Low-Pass Filter
Least Squares
Median Absolute Deviation
Maximum a Posteriori
Moment generating function
Maximum Likelihood
Minimum Mean Square Error
Magnetic Resonance
Magnetic Resonance Imaging
Markov Random Field
Non-central chi
Number of Excitations
Non-local Means
Nonlinear Least Squares
Nuclear Magnetic Resonance
Orientation Density Function
Orientation Probability Density Function
Oriented Rician Noise-Reducing Anisotropic Diffusion

Principal Component Analysis
Proton Density
Partial Differential Equation
Probability Density Function
Parallel MRI
Relative Error
Radio Frequency
Recursive Linear Minimum Mean Square Error
Region of interest
Random Variable
Sensitivity Encoding for Fast MRI
Sample Local Variance
Simultaneous Acquisition of Spatial Harmonics
Spatial Match Filter
Signal-to-Noise Ratio
Sum of Squares
Scalar Rician Noise-Reducing Anisotropic Diffusion


Acronyms and Notation

STD
SVD
SWT
TR
TSE
TV
UNLM
VST
WLS

xix

Standard Deviation
Singular Value Decomposition
Stationary Wavelet Transform
Repetition Time
Turbo Spin Echo
Total Variation
Unbiased Non-local Means
Variance Stabilization Transform
Weighted Least Squares

Notation
Probability, Estimation and Moments
pX ð x Þ
EfX g
E fX p g
VarfX g
stdfX g
CVfX g
hM ðxÞi
hM ðxÞix

Probability density function of X
Expectation of random variable X
Order p moment of random variable X
Variance of random variable X
Standard deviation of random variable X
Coefficient of variation of random variable X

stdfX g
CVf X g ¼
EfX g
(Global) Sample mean of image M ðxÞ
P
hM ðxÞi ¼ jΩ1 j
M ð xÞ
x2Ω

Local sample mean of image M ðxÞ
P
M ðpÞ
hM ðxÞix ¼ j·ð1xÞj
p2·ðxÞ

hM ðxÞik
V ðM ðxÞÞ
V x ðM ðxÞÞ
medianð X Þ
modefI ðxÞg
b
a

with ·ðxÞ a neighborhood centered in x
Local sample mean of image M ðxÞ calculated along N samples
N
P
M k ð xÞ
hM ðxÞik ¼ N1
k¼1

(Global) sample variance of M ðxÞ
V ðM ðxÞÞ ¼ hM 2 ðxÞi À hM ðxÞi2
Sample local variance of M ðxÞ
VðM ðxÞÞx ¼ hM 2 ðxÞix À hM ðxÞi2x
Median of random variable X
Mode of the distribution of I ðxÞ
Estimator of parameter a


xx

Acronyms and Notation

Regions and Topology
M ð xB Þ
M ð xR Þ

Background area of image M ðxÞ
xB ¼ xjAðxÞ ¼ 0
M ðxÞ in the region R
xR 2 R

Operators
F fSð xÞ g
F À1 fsðkÞg
LPFðSðxÞÞ
MAD

Fourier transform of SðxÞ

Fourier inverse transform of sðkÞ
Low-pass filter of signal SðxÞ
Median absolute deviation






MADðgi Þ ¼ median gi À medianðgk Þ

MADx

Local median absolute deviation





MADx ðSðxÞÞ ¼ medianSðpÞ À medianðSðqÞÞ

»ðµÞ

Koay correction factor

i

k

p2·ðxÞ

µ2

»ðµÞ ¼ 2 þ µ2 À …8 Á eÀ 2 Á
fstab ðÞ
~
div
r

q2·ðxÞ

hÀ

 2 i2
Á  2
2 þ µ2 I0 µ4 þ µ2 I1 µ4

Variance stabilization transform
(Circular) convolution
Divergence operator
Gradient operator

Matrix Operations
trðCÞ

Trace of P
matrix C
trðCÞ ¼ ci;i
i

kC k1

with ci;i the diagonal elements
of matrix C
P P  
L1 norm kCk1 ¼
ci;j

kC kF

with ci;j the elements of matrix C
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P P  2 pffiffiffiffiffiffiffiffiffiffi
Frobenius norm kCk ¼
ci;j ¼ CCH

CH
CÀ1

with ci;j the elements of matrix C
Conjugate transpose of matrix C
Inverse of matrix C

i

j

F

i

j

Functions
I n ð xÞ
Ln ð xÞ

Modified Bessel Function of the first kind of order n
Laguerre polymonial of order n
Ln ð xÞ ¼1F1 ðÀn; 1; xÞ


Acronyms and Notation
1 F1 ða; b; xÞ

Γ ðnÞ
uð x Þ
erfð xÞ

Confluent hypergeometric function of the first kind
Gamma function
Heaviside step function
Error function
Rx 2
erf ðxÞ ¼ p2ffiffi… 0 eÀt dt

xxi


Chapter 1

The Problem of Noise in MRI

Magnetic Resonance (MR) data is known to be affected by several sources of quality
deterioration due to limitations in the hardware, scanning time, or movement of
patients. One source of degradation that affects most of the acquisitions is noise. The
term noise in MR can have different meanings depending on the context. It has been
applied to degradation sources such as physiological and respiratory distortions in
some MR applications and acquisitions schemes or even acoustic sources (the sound
produced by the pulse sequences in the magnet). In this book, the term noise is
strictly limited to the thermal noise introduced during data acquisition, also known
as Johnson–Nyquist noise.

1.1 Thermal Noise in Magnetic Resonance Imaging
The principal source of thermal noise in most MR scans is the subject (object to
be imaged) itself, followed by electronic noise during the acquisition of the signal
in the receiver chain [76, 108, 121, 244]. It is produced by the stochastic motion
of free electrons in the radio frequency (RF) coil, which is a conductor, and by
eddy current losses in the patient, which are inductively coupled to the RF coil. The
presence of noise over the acquired MR signal not only affects the visual assessment
of the images, but it also may interfere with further processing techniques such
as segmentation, registration, fMRI analysis or numerical estimation of parameters
related to diffusion, perfusion, or relaxometry. Moreover, noisy data might seriously
affect to the diagnostic performance of the image-derived metrics like signal-to-noise
ratio (SNR) and contrast-to-noise ratio (CNR), or the evaluation of tumor tissues [69].
There are different ways to cope with thermal noise but, due to its random nature,
a probabilistic modeling is a proper and powerful solution. The accurate modeling of
signal and noise statistics usually underlies the tools for processing and interpretation
within magnetic resonance imaging (MRI).
© Springer International Publishing Switzerland 2016

S. Aja-Fern´andez and G. Vegas-S´anchez-Ferrero,
Statistical Analysis of Noise in MRI, DOI 10.1007/978-3-319-39934-8_1

1


2

1 The Problem of Noise in MRI

The most common use of noise modeling in MR data is signal estimation via noise
removal. Noise filtering techniques in different fields are based on a well-defined
prior statistical model of data, usually a Gaussian one. Noise models in MRI have
allowed the natural extension of many well-known techniques to cope with features
specific to MRI. However, an accurate noise modeling may be useful in MRI not only
for filtering purposes, but also for many other processing techniques. For instance,
weighted least squares methods to estimate the diffusion tensor (DT) have proved
to be nearly optimal when the data follows a Rician [200] or a noncentral Chi (ncχ) distribution [229]. Other approaches for the estimation of the DT also assume
an underlying Rician model of the data: Maximum Likelihood and Maximum a
Posteriori (MAP) estimation [19, 124], or sequential techniques for online estimation
[46, 185] have been proposed. The use of an appropriated noise model is crucial in
all these methods to attain a statistically correct characterization of the underlying
signals. Other methods in MRI processing that benefit from relying on a precise noise
distribution model include automatic segmentation of regions [197, 250], compressed
sensing for signal reconstruction [71, 161], and fMRI activation and simulation [165,
166, 246]. Many examples in literature have shown the advantage of statistically
modeling the specific features of noise for a specific type of data.
In the present book, we provide a complete framework to model and analyze noise
in MRI, considering different modalities and acquisition techniques. This analysis
will be focused on three main issues:

1. Noise modeling: The adoption of a specific probability distribution to model the
behavior of noise is the basis of the different applications aforementioned. For
practical purposes, it has been usually assumed that the noise in the image domain
is a zero-mean, spatially uncorrelated Gaussian process, with equal variance in
both the real and imaginary parts. In case the data is acquired by several receiving coils, the exact same distribution is assumed for all of them. As a result, in
single-coil systems the magnitude data in the spatial domain are modeled using
a Rician distribution and as a noncentral-χ (nc-χ) for multiple-coil systems.
Although these two distributions have been extensively used in the MR literature
whenever a noise model is needed, in modern acquisition systems they may no
longer hold as reliable distributions. MRI systems often collect subsampled versions of the k-space to speed-up the acquisitions and palliate phase distortions. In
order to correct the aliasing artifacts produced by this subsampling, some reconstruction methods are to be used, the so-called Parallel MRI (pMRI) techniques.
This reconstruction will drastically change the features of noise. As a consequence, some models adapted to the processed data must be considered.
In this book, we review most of the models already presented for signal and noise
in MRI. We present them in the global context of MRI processing. Along the
whole book, we will consider that the data is obtained using a direct acquisition,
i.e., we will assume that: (1) data are acquired in the k-space using a regular
Cartesian sampling; (2) the different contributions of noise are all independent,
so that the total noise in the system is the noise contribution from each individual
source; and (3) postprocessing and correction schemes are not applied. Though


1.1 Thermal Noise in Magnetic Resonance Imaging

3

these assumptions may seem unrealistic for certain applications, they are common in the literature, and otherwise necessary to achieve a reasonable trade-off
between the accuracy of the model and its generalization capabilities.
As a consequence of these assumptions, some important issues could be left
aside: interpolations due to nonCartesian sampling, ghost-correction postprocessing for acquisitions schemes such as EPI [63, 220], fat-suppression algorithms,
manufacturer-specific systems for noise and artifacts reduction, or coil uniformity

correction techniques will dramatically alter spatial noise characteristics, making
the data differ from the models. These specific cases are usually manufacturerdependent, devise-dependent, or they may even depend on the particular imaging
sequence or imaged anatomy. Hence, they will need a more in-depth study, which
is far from the scope of this book, though in many cases such study can be derived
from the general models here described.
2. Noise estimation: once a statistical model is adopted for the signal and noise in a
MRI acquisition, the parameters of that model must be estimated from data. Generally, the parameter to estimate is the variance of noise σ 2 . The way to estimate
this parameter changes if the complex data is available or if the estimation has
to be made over the magnitude signal. There will be also variations when singlecoil or multiple-coil are considered. Finally, in modern acquisition systems, due
to different processing, noise becomes non-stationary and σ becomes dependent
with the position, i.e., σ(x). Thus, instead of estimating a single value for the
whole image, a value for each pixel must be considered instead.
Noise estimation methods may roughly be divided into two groups: approaches
that use a single magnitude image and approaches using multiple repetitions of
the same slice. Although both will be reviewed along the book, we will mainly
focus on the former.
3. Noise filtering: one of the most common applications of statistically modeling the
noise in MRI is precisely to remove or reduce it. Noise filtering can be found in the
literature under very different names: noise filtering, noise removal, denoising, or
noise reduction, but they all denote the same operation, the reduction of the noise
pattern present in the image. This application is the counterpart to the previous
one, since, from a statistical point of view, it can be seen as signal estimation in
a noisy environment.
Many methods have been reported in literature in order to remove noise out of
MRI data based on different approaches: signal estimation, anisotropic diffusion,
non-local means or wavelets. The goodness of a specific method must be related
to the purpose of the filtering. There is no all-purpose filter that, with the same
configuration parameters, could perform excellent in all situations. The only hard
requisite is that a good noise filtering method for MRI must not invent data or
clean an image. Instead, it must estimate the underlying signal out of noisy data

keeping all (and only) the information contained in the data.
There is always some controversy on the MRI community about filtering or not
filtering the data. We do not have a solution to that controversy here. However,
a statistical approach could help in understanding the procedure. Ideally, a good


4

1 The Problem of Noise in MRI

filtering scheme must choose the most likely or possible original signal based on
the available data.
Along this book, we deeply study these three important aspects of noise analysis
in MRI. Whenever it is possible, some examples are presented using synthetic data
(to provide quantitative results) and real data.

1.2 Organization of the Book
The book is organized in 3 parts, 11 chapters, and 3 appendices in an attempt to
cover the different aspects that concern noise analysis in MRI. The disposition of the
chapters is incremental, the basic concepts set on the first ones is lately used along
the book.
The first part of the book is committed to undertake the problem of noise in
MRI through the modeling of the acquisition and the definition of the most common
statistical distributions used to describe the noise. The problem of noise and signal estimation for medical imaging is analyzed from a statistical signal processing
perspective.
In Chap. 2, we review some basic concepts about MRI acquisition that are necessary to understand the signal/noise assumptions used along the book. We will
especially focus on modeling the k-space and x-space from a signal processing point
of view. Sequences and acquisition modalities will be left aside to confine ourselves
to an upper level modeling of the acquired signal. The formation processes from
single- and multiple-coil are reviewed. The chapter concludes with the analysis of

some parallel MRI methods.
In Chap. 3, the noise models for the different acquisitions reviewed in Chap. 2
are presented. The starting point will be the complex Gaussian model for the signal
acquired in each coil. From there, the different processing and reconstruction schemes
that happen in the scanner are analyzed to generate the models of noise on the final
composite magnitude signals. Gaussian, Rician, and noncentral χ distributions will
be considered, as well as stationary and non-stationary models.
Chapter 4 makes a profound analysis on how to estimate noise from MRI data.
The starting point will be an example that will raise the main issues concerning this
task. These issues will be deeply analyzed: the use of a noise model; the stationarity
of the data; the use of the background in estimation; how the quantification of the data
can alter the estimation; and the use of multiple samples. Additionally, a practical
scheme to effectively estimate noise out of MRI is proposed.
Chapter 5 is complementary to Chap. 4. In it, we analyze the problem of noise filtering from a signal estimation perspective. First, we establish the basic requirements
to use a filtering scheme in medical imaging in general and in MRI in particular. We
review the different uses that filtering can have and we show some examples of the
advantage of carrying out a noise reduction procedure on MRI. Later, we analyze
the different approaches and evaluate their performance for specific purposes. As a


1.2 Organization of the Book

5

case study, we review the different modifications provided in the literature over a
well-known filter (LMMSE for Rician noise) in order to better cope with different
modalities of imaging.
The second part of the book (Chaps. 6–9) is devoted to analyze and review different techniques to estimate noise out of a single MRI slice in single- and multiple-coil
systems for fully sampled acquisitions. The scheme of the chapters will be very similar: first the main estimators in the literature are described and then some performance
analysis is carried out. To that end, synthetic and real data are considered.

Chapter 6 is the first chapter that deals with the problem of noise estimation in
MRI. In this chapter, we focus on the case of stationary additive Gaussian noise. The
derivations can be used for the complex signal before the magnitude is calculated or
for high SNR simplifications. We review some methods to estimate the variance of
noise σ 2 and the covariance between coils σlm under the Gaussian assumption.
In Chap. 7, we review and classify the different approaches to estimate σ 2 out of
Rician magnitude MR images. In this chapter, we gather the most popular approaches
found in the literature. The advantages and drawbacks of the different methods are
analyzed through synthetic and real data controlled experiments. A special kind of
estimators, those based on the calculation of the mode, is deeply studied.
The estimators for Rician noise of Chap. 7 are the basis for many of the estimators
proposed in the following chapters, which can be seen as extensions of the Rician
estimators. The next two chapters deal with MRI data from multiple-coil acquisitions.
However, in Chap. 8 the correlations between coils are not considered, producing
simpler statistical models, while in Chap. 9 the correlations are included into the
analysis.
In Chap. 8, we extend those results to the particular case of a multiple-coil acquisition in which the magnitude signal is reconstructed using Sum of Squares (SoS) or
a Spatial matched filter (SMF), no correlations are assumed between coils, and all
of them show the same variance of noise. As a consequence, the magnitude signal
follows a stationary nc-χ distribution (if SoS is used) or a stationary Rician one (in
the case of SMF). We focus in the SoS case and the nc-χ distribution, since the
Rician case is studied in Chap. 7. The main noise estimators for the nc-χ are thus
reviewed and evaluated. Most of the methods proposed are basically extrapolations
of the Rician estimators to the nc-χ.
In Chap. 9, we also focus on nonaccelerated multiple-coil acquisitions, but taking
into account the correlations between the acquisition coils. As a consequence, the
distributions become non-stationary and the estimation of single values carried out
in Chaps. 6–8 is no longer valid. The parameters of noise in the magnitude image
becomes position dependent and, therefore, a noise map σ 2 (x) must be estimated
instead. We consider two cases, for the magnitude signal being constructed using

either a SMF or a SoS approach. In the first case, a non-stationary Rician distribution
arises. In the second, a nc-χ approximation of the data is considered, using effective
values for σ 2 and the number of coils.
There are two main ways to approach the non-stationary noise estimation: a parametric estimation and a blind estimation. In this chapter, we will focus on the former:
the estimation is done considering the process that has generated the specific model