Statistical analysis of noise in MRI

The problem of noise and signal estimation for medical imaging consid-is analyzed from a statconsid-istical signal processing perspective.. 4 Part I Noise Models and the Noise Analysis P

Gonzalo Vegas-Sánchez-Ferrero

Statistical

Analysis of Noise in MRIModeling, Filtering and Estimation

Santiago Aja-Fern ández

Statistical Analysis

of Noise in MRI

Modeling, Filtering and Estimation

123

ISBN 978-3-319-39933-1 ISBN 978-3-319-39934-8 (eBook)

DOI 10.1007/978-3-319-39934-8

Library of Congress Control Number: 2016941078

© Springer International Publishing Switzerland 2016

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“How do you peel a porcupine?”

Medical imaging and thefield 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-raycomputed tomography (CT), ultrasound, positron emission tomography (PET), andmagnetic resonance imaging (MRI)

This book, “Statistical Analysis of Noise in MRI,” presents a modern signalprocessing approach for medical imaging with a focus on noise modeling andestimation for MRI MRI scanners use strong magnetic fields, radio waves, andmagneticfield gradients to form images of the body MRI has seen a tremendousdevelopment during the past four decades and is now an indispensable part ofdiagnostic medicine MRI is unparalleled in the investigation of soft tissues due toits 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 ofMathematics in Imaging at Brigham and Women’s Hospital, Harvard MedicalSchool, 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 knowledgeabout statistical signal processing He had a very productive year in Boston andsubsequently published several, now well-cited, papers on noise estimation in MRI.During Santiago’s year-long visit in my laboratory we were investigating theboundaries of what it meant to separate signals from noise What do you need toknow about the data to do this well? The more complicated the image formationprocess is, the less the commonly assumed model that the noise is Gaussian isapplicable This book is about exploring these questions and providing guidelines

on how to proceed One important message in this book is that you have tounderstand your data acquisition in detail Santiago Aja-Fernández continued towork on these questions when he returned to the University of Valladolid with thesecond author of this book, Dr Gonzalo Vegas-Sánchez-Ferrero They and theirco-workers have made tremendous progress during the past decade and havebecome authorities on the topic of noise modeling in MRI

vii

I expect that the importance of accurate noise modeling and estimation in thefield of MRI will increase over the next several years due to the increasing com-plexity of the MRI scanners Many commercial scanners now have the possibility toconnect multiple RF detector coil sets to allow the simultaneous acquisition ofseveral signals in a phased array system These systems were originally developed

to reduce the scanning time and therefore to avoid some problems with movingstructures, 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 whenestimating derived parameters from this more complex measured data For example,robust estimation of the diffusion tensor in diffusion MRI requires in-depthknowledge of the imaging process used for creating the multi-channel diffusionMRI data With today’s complex parallel imaging acquisition schemes commonlyused in the clinic, it is important to be able to understand how to model the dataappropriately for any subsequent signal processing task

Carl-Fredrik Westin, Ph.D.Director Laboratory of Mathematics in Imaging,

Brigham and Women’s HospitalHarvard Medical SchoolBoston, MA, USA

This work is the result of more than 10 years of research in the area of MRI from asignal and noise perspective Our interest has always been to properly model thenoise 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 comingalong with our own research, offering alternative points of view We realized thatmost of the works dealing with noise in MRI can be seen as complementary effortsrather than competitive It was necessary, thus, to systematize all that knowledgethat had arisen, in order to understand the problem as a whole It is precisely in therelations between distinct methods and philosophies where the real nature of thisquestion can be better understood In this work we gather different approaches tonoise analysis in MRI, systematizing and classifying the different methods, trying tobring them together to common ground So, instead of being seen as independentefforts, they can be considered as consecutive paces along the same way

This book is intended to serve as a reference manual for researchers dealing withsignal 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 basicprinciples of linear systems and signal processing, as well as being familiar withrandom variables, image processing, and calculus fundaments It could also serve as

a textbook for postgraduate students in engineering with an interest in medicalimage processing

We provide a complete framework to model and analyze noise in MRI, ering different modalities and acquisition techniques, focusing on three issues: noisemodeling, noise estimation, and noisefiltering To that end, the book is divided intothree parts Thefirst part analyzes the problem of noise in MRI, the modeling of theacquisition, and the definition of the most common statistical distributions used todescribe the noise The problem of noise and signal estimation for medical imaging

consid-is analyzed from a statconsid-istical signal processing perspective The second part of thebook is devoted to analyzing and reviewing the different techniques to estimate noiseout of a single MRI slice in single- and multiple-coil systems for fully sampledacquisitions The third part deals with the problem of noise estimation when

ix

accelerated acquisitions are considered and parallel imaging methods are used toreconstruct the signal The book is complemented with three appendices.

Our intention is to make the book comprehensive, thus many definitions andmethods have been included, and some ideas are repeated in different chapters fromdifferent perspectives That way, most of the chapters can be understood inde-pendently of the others, although relations between them will always be present.Some theoretical topics about random variables, image processing, and MRIacquisition have been omitted for the sake of compactness We provide a completebibliography that can be used tofill the gaps

Finally, note that this is afield of constant expansion, with new methods beingpublished every year In addition, acquisition techniques are also rapidly evolving,producing new models of noise that are not analyzed here We consider this book asthe framework that could serve as the basis for the analysis of all those noveltiesthat will surely arise in the next years

March 2016

The work presented in this book started at LMI (Harvard Medical School, Boston)almost 10 years ago, funded by a Fulbright Scholarship Many different researchershave contributed to the development of the main corpus on noise modeling andestimation that isfinally gathered here In particular, I want to thank Dr Tristán-Vegafor all the shared work in this field and to my coauthor, Gonzalo Vegas-

Sánchez-Ferrero, for his help and support in the elaboration of this book Let ushope we can work in new topics in the future The other researchers that have activelycontributed 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 thatcould not have been done without external support from other researchers

I specially want to thank Tomasz Pieziak, from AGH University of Science andTechnology, Krakow (Poland), whose work about VST is directly used in thisbook 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 forcomparison The filtering chapter takes many references from Dr VeroniqueBrion’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 dataused 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 Tristán-Vega’s thesis that was generated using HARDI data kindly provided by theAustralian eHealth Research Centre-CSIRO ICT Centre, Brisbane (Australia).The authors acknowledge Ministerio de Ciencia e Innovación for funding (grantTEC2013-44194-P) Gonzalo Vegas-Sánchez-Ferrero acknowledges Consejera deEducación, Juventud y Deporte de la Comunidad de Madrid and the People

xi

Programme (Marie Curie Actions) of the European Union’s Seventh FrameworkProgramme (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 notforget you when I become rich and famous

1 The Problem of Noise in MRI 1

1.1 Thermal Noise in Magnetic Resonance Imaging 1

1.2 Organization of the Book 4

Part I Noise Models and the Noise Analysis Problem 2 Acquisition and Reconstruction of Magnetic Resonance Imaging 9

2.1 Physics of Magnetic Resonance Imaging 10

2.2 The k-Space and the x-Space 12

2.3 Single-Coil Acquisition Process 14

2.4 Multiple-Coil Acquisition Process 15

2.5 Accelerated Acquisitions: Parallel Imaging 19

2.5.1 The Problem of Acceleration: Subsampling 19

2.5.2 Sensitivity Encoding (SENSE) 23

2.5.3 Generalized Autocalibrating Partially Parallel Acquisition (GRAPPA) 25

2.5.4 Other pMRI Methods 27

2.6 Final Remarks 29

3 Statistical Noise Models for MRI 31

3.1 Complex Single- and Multiple-Coil MR Signals 31

3.2 Single-Coil MRI Data 33

3.3 Fully Sampled Multiple-Coil Acquisition 35

3.3.1 Uncorrelated Multiple-Coil with SoS 35

3.3.2 Correlated Multiple-Coil with SoS 37

3.3.3 Multiple-Coil with SMF Reconstruction 40

3.4 Statistical Models for pMRI Acquisitions 42

3.4.1 General Noise Models in pMRI 42

3.4.2 Statistical Model in SENSE Reconstructed Images 46

xiii

3.4.3 Statistical Model in GRAPPA Reconstructed

Images 48

3.5 Some Practical Examples 54

3.5.1 Single-Coil Acquisitions 54

3.5.2 Multiple-Coil Acquisitions 54

3.5.3 pMRI Acquisitions 60

3.6 Final Remarks 69

4 Noise Analysis in MRI: Overview 73

4.1 The Problem of Noise Estimation: An Introductory Example 74

4.1.1 A Practical Problem 74

4.1.2 Analysis of the Data 74

4.1.3 Estimation Procedure 75

4.1.4 Other Estimation Issues 77

4.2 Main Issues About Noise Analysis in MRI 78

4.2.1 The Noise Model of the Data 78

4.2.2 The Stationarity of the Noise 79

4.2.3 The Background 81

4.2.4 Quantification of Data 82

4.2.5 Single Versus Multiple Sample Estimation 83

4.2.6 Practical Implementation 83

4.3 Noise Analysis Practical Methodology 85

5 Noise Filtering in MRI 89

5.1 Noise Filtering and Signal Estimation in MRI 89

5.2 The Importance of Noise Filtering 92

5.3 Noise Suppression/Reduction Methods 96

5.3.1 Noise Correction During the Acquisition 96

5.3.2 Generic Filtering Algorithms 98

5.3.3 Transform Domain Filters 103

5.3.4 Statistical Methods 105

5.3.5 Some Examples 109

5.4 Case Study: The LMMSE Signal Estimator 111

5.4.1 Original Formulation: Signal Estimation for the General Rician Model 111

5.4.2 Extension to Multiple Samples 113

5.4.3 Recursive LMMSE Filter 114

5.4.4 Extension to nc-´ Data 114

5.4.5 Extension for an Specific Application: DWI Filtering 115

5.5 Some Final Remarks 119

Part II Noise Analysis in Nonaccelerated Acquisitions

6 Noise Estimation in the Complex Domain 123

6.1 Single-Coil Estimation 124

6.2 Multiple-Coil Estimation 131

6.2.1 Variance in Each Coil 131

6.2.2 Covariance Matrix and Correlation Coefficient 131

6.2.3 Reconstruction Process 133

6.3 Non-stationary Noise Analysis 134

6.4 Examples and Performance Evaluation 134

7 Noise Estimation in Single-Coil MR Data 141

7.1 Noise Estimators for Rayleigh/Rician Data 142

7.1.1 Estimators Based on a Rayleigh Background 142

7.1.2 Estimators Based on the Signal Area 147

7.2 Estimators Based on Local Moments: A Detailed Study 153

7.3 Performance of the Estimators 160

7.3.1 Performance Evaluation with Synthetic Data 160

7.3.2 Performance Evaluation Over Real Data 164

7.4 Final Remarks 170

8 Noise Estimation in Multiple–Coil MR Data 173

8.1 Uncorrelated Data and SMF Reconstruction 174

8.2 Noise Estimation Assuming a nc-´ Distribution 174

8.2.1 Estimators Based on a c-´ Background 175

8.2.2 Estimators Based on the Signal Area 177

8.3 Performance of the Estimators 180

8.3.1 Performance Evaluation with Synthetic Data 180

8.3.2 Performance Evaluation Over Real Data 183

8.4 Final Remarks About the Estimators 185

9 Parametric Noise Analysis from Correlated Multiple-Coil MR Data 187

9.1 Parametric Noise Estimation for Correlated Multiple-Coil with SMF 188

9.1.1 Background-Based Estimation 189

9.1.2 Estimation Based on Signal Area 190

9.2 Noise Estimation for Correlated SoS 191

9.2.1 Estimation of 2 L 193

9.2.2 Estimation of Effective Values 194

9.2.3 Simplified Estimation 196

9.3 Performance of the Estimators 197

9.3.1 Correlated Coils with SMF 197

9.3.2 Correlated Coils with SoS 200

9.3.3 In Vivo Data 202

9.4 Final Remarks 206

Part III Noise Estimators in pMRI

10 Parametric Noise Analysis in Parallel MRI 211

10.1 Noise Estimation in SENSE 212

10.2 Noise Estimation in GRAPPA with SMF Reconstruction 215

10.3 Noise Estimation in GRAPPA with SoS Reconstruction 215

10.3.1 Practical Simplifications over the GRAPPA Model 216

10.3.2 Noise Estimator 217

10.3.3 Estimation of Effective Values in GRAPPA 218

10.3.4 Gaussian Simplification 219

10.4 Examples and Performance of the Estimators 220

10.4.1 Noise Estimation in SENSE 220

10.4.2 Noise Estimation in GRAPPA 223

10.5 Final Remarks 227

11 Blind Estimation of Non-stationary Noise in MRI 229

11.1 Non-stationary Noise Estimation in MRI 230

11.1.1 Non-stationary Gaussian Noise Estimators 231

11.1.2 Rician Estimators 236

11.1.3 Noncentral´ Estimation 245

11.1.4 Estimation Along Multiple MR Scans 247

11.2 A Homomorphic Approach to Non-stationary Noise Estimation 249

11.2.1 The Gaussian Case 249

11.2.2 The Rayleigh Case 251

11.2.3 The Rician Case 253

11.3 Performance of the Estimators 256

11.3.1 Non-stationary Rician Noise 256

11.3.2 Non-stationary Nc-´ Noise 269

11.4 Final Remarks 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

ACS Auto Calibration Signal

ADC Apparent Diffusion Coefficient

ARC Autocalibrating Reconstruction of Cartesian imaging

ASL Arterial Spin Labeling

ASSET Array coil Spatial Sensitivity Encoding

AWGN Additive White Gaussian Noise

BOLD Blood Oxygen Level Dependent

c-´ Central chi

CA Conventional Approach

CAIPIRINHA Controlled Aliasing in Parallel Imaging Results

in Higher AccelerationCHARMED Composite Hindered and Restricted Model

of DiffusionCMS Composite Magnitude Signal

CURE Chi-square Unbiased Risk Estimator

CV Coefficient of Variation

DCT Discrete Cosine Transform

DFT Discrete Fourier Transform

DKI Diffusion Kurtosis Imaging

DoF Degrees of Freedom

DOT Diffusion Orientation Transform

DT Diffusion Tensor

DTI Diffusion Tensor Imaging

DWI Diffusion Weighted Imaging

DWT Discrete Wavelet Transform

EM Expectation Maximization

EPI Echo Planar Imaging

FFE Fast Field Echo

xvii

fMRI Functional Magnetic Resonance Imaging

GRAPPA Generalized Autocalibrating Partially Parallel AcquisitionHARDI High Angular Resolution Diffusion Image

HMF Homomorphic Filter

iDFT Inverse Discrete Fourier Transform

IID Independent and Identically Distributed

KDE Kernel Density Estimator

LLS Linear Least Squares

LMMSE Linear Minimum Mean Square Error

LPF Low-Pass Filter

MAD Median Absolute Deviation

MAP Maximum a Posteriori

MGF Moment generating function

MMSE Minimum Mean Square Error

MR Magnetic Resonance

MRI Magnetic Resonance Imaging

MRV Markov Random Field

nc-´ Non-central chi

NEX Number of Excitations

NLM Non-local Means

NLS Nonlinear Least Squares

NMR Nuclear Magnetic Resonance

ODF Orientation Density Function

OPDF Orientation Probability Density Function

OSRAD Oriented Rician Noise-Reducing Anisotropic Diffusion

PCA Principal Component Analysis

PDE Partial Differential Equation

PDF Probability Density Function

pMRI Parallel MRI

RMMSE Recursive Linear Minimum Mean Square Error

ROI Region of interest

SENSE Sensitivity Encoding for Fast MRI

SLV Sample Local Variance

SMASH Simultaneous Acquisition of Spatial Harmonics

SMF Spatial Match Filter

SNR Signal-to-Noise Ratio

SRRAD Scalar Rician Noise-Reducing Anisotropic Diffusion

STD Standard Deviation

SVD Singular Value Decomposition

SWT Stationary Wavelet Transform

TSE Turbo Spin Echo

UNLM Unbiased Non-local Means

VST Variance Stabilization Transform

WLS Weighted Least Squares

Notation

Probability, Estimation and Moments

pXð Þx Probability density function of X

E Xf g Expectation of random variable X

E Xf gp Order p moment of random variable X

Var Xf g Variance of random variable X

std Xf g Standard deviation of random variable X

CV Xf g Coefficient of variation of random variable X

hM xð Þix Local sample mean of image M xð Þ

hM xð Þix¼ 1

· x ð Þ

j j

Pp2· x ð ÞM pð Þwith· xð Þ a neighborhood centered in x

hM xð Þik Local sample mean of image M xð Þ calculated along N samples

hM xð Þik¼1

N

PN k¼1

mode I xf ð Þg Mode of the distribution of I xð Þ

ba Estimator of parameter a

Regions and Topology

M xð ÞB Background area of image M xð Þ

F1fs kð Þg Fourier inverse transform of s kð Þ

LPF S xð ð ÞÞ Low-passfilter of signal S xð Þ

MAD Median absolute deviation

C

k k1 L1 normk kC 1¼P

i

Pj

with ci;jthe elements of matrix C

CH Conjugate transpose of matrixC

C1 Inverse of matrixC

Functions

Inð Þx Modified Bessel Function of the first kind of order n

Lnð Þx Laguerre polymonial of order n

Lnð Þ ¼x 1F1ðn; 1; xÞ

1F1ða; b; xÞ Confluent hypergeometric function of the first kind

Γ nð Þ Gamma function

u xð Þ Heaviside step function

erf xð Þ Error function

erf xð Þ ¼p 2ffiffi…Rx

0et2dt

Chapter 1

The Problem of Noise in MRI

Magnetic Resonance (MR) data is known to be affected by several sources of qualitydeterioration due to limitations in the hardware, scanning time, or movement ofpatients 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 insome MR applications and acquisitions schemes or even acoustic sources (the soundproduced by the pulse sequences in the magnet) In this book, the term noise isstrictly 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 byeddy current losses in the patient, which are inductively coupled to the RF coil Thepresence 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 parametersrelated to diffusion, perfusion, or relaxometry Moreover, noisy data might seriouslyaffect to the diagnostic performance of the image-derived metrics like signal-to-noiseratio (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 ofsignal and noise statistics usually underlies the tools for processing and interpretationwithin 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

The most common use of noise modeling in MR data is signal estimation via noiseremoval Noise filtering techniques in different fields are based on a well-definedprior statistical model of data, usually a Gaussian one Noise models in MRI haveallowed the natural extension of many well-known techniques to cope with featuresspecific to MRI However, an accurate noise modeling may be useful in MRI not onlyfor 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

(nc-χ) distribution [229] Other approaches for the estimation of the DT also assume

an underlying Rician model of the data: Maximum Likelihood and Maximum aPosteriori (MAP) estimation [19, 124], or sequential techniques for online estimation[46, 185] have been proposed The use of an appropriated noise model is crucial inall these methods to attain a statistically correct characterization of the underlyingsignals Other methods in MRI processing that benefit from relying on a precise noisedistribution model include automatic segmentation of regions [197, 250], compressedsensing for signal reconstruction [71, 161], and fMRI activation and simulation [165,

166, 246] Many examples in literature have shown the advantage of statisticallymodeling 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 analysiswill 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 Forpractical purposes, it has been usually assumed that the noise in the image domain

is a zero-mean, spatially uncorrelated Gaussian process, with equal variance inboth the real and imaginary parts In case the data is acquired by several receiv-ing coils, the exact same distribution is assumed for all of them As a result, insingle-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 literaturewhenever a noise model is needed, in modern acquisition systems they may nolonger hold as reliable distributions MRI systems often collect subsampled ver-

sions 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 struction methods are to be used, the so-called Parallel MRI (pMRI) techniques.This reconstruction will drastically change the features of noise As a conse-quence, some models adapted to the processed data must be considered

recon-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 individualsource; 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 mon in the literature, and otherwise necessary to achieve a reasonable trade-offbetween the accuracy of the model and its generalization capabilities

com-As a consequence of these assumptions, some important issues could be leftaside: interpolations due to nonCartesian sampling, ghost-correction postprocess-ing for acquisitions schemes such as EPI [63, 220], fat-suppression algorithms,manufacturer-specific systems for noise and artifacts reduction, or coil uniformitycorrection techniques will dramatically alter spatial noise characteristics, makingthe data differ from the models These specific cases are usually manufacturer-dependent, devise-dependent, or they may even depend on the particular imagingsequence 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 derivedfrom 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 erally, the parameter to estimate is the variance of noiseσ2 The way to estimatethis parameter changes if the complex data is available or if the estimation has

Gen-to be made over the magnitude signal There will be also variations when coil or multiple-coil are considered Finally, in modern acquisition systems, due

single-to different processing, noise becomes non-stationary andσ 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: approachesthat use a single magnitude image and approaches using multiple repetitions ofthe same slice Although both will be reviewed along the book, we will mainlyfocus 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 theliterature under very different names: noise filtering, noise removal, denoising, ornoise reduction, but they all denote the same operation, the reduction of the noisepattern present in the image This application is the counterpart to the previousone, 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 sameconfiguration 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 notfiltering 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

filtering scheme must choose the most likely or possible original signal based onthe 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 tocover the different aspects that concern noise analysis in MRI The disposition of thechapters is incremental, the basic concepts set on the first ones is lately used alongthe 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 commonstatistical distributions used to describe the noise The problem of noise and sig-nal estimation for medical imaging is analyzed from a statistical signal processingperspective

In Chap.2, we review some basic concepts about MRI acquisition that are essary to understand the signal/noise assumptions used along the book We will

nec-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 fromsingle- and multiple-coil are reviewed The chapter concludes with the analysis ofsome 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 signalacquired in each coil From there, the different processing and reconstruction schemesthat happen in the scanner are analyzed to generate the models of noise on the finalcomposite magnitude signals Gaussian, Rician, and noncentralχ distributions will

be considered, as well as stationary and non-stationary models

Chapter4makes 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 thistask 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 datacan alter the estimation; and the use of multiple samples Additionally, a practicalscheme to effectively estimate noise out of MRI is proposed

Chapter5is complementary to Chap.4 In it, we analyze the problem of noise tering from a signal estimation perspective First, we establish the basic requirements

fil-to use a filtering scheme in medical imaging in general and in MRI in particular Wereview the different uses that filtering can have and we show some examples of theadvantage of carrying out a noise reduction procedure on MRI Later, we analyzethe 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 awell-known filter (LMMSE for Rician noise) in order to better cope with differentmodalities of imaging

The second part of the book (Chaps.6 9) is devoted to analyze and review ent techniques to estimate noise out of a single MRI slice in single- and multiple-coilsystems for fully sampled acquisitions The scheme of the chapters will be very simi-lar: first the main estimators in the literature are described and then some performanceanalysis is carried out To that end, synthetic and real data are considered

differ-Chapter6is the first chapter that deals with the problem of noise estimation inMRI In this chapter, we focus on the case of stationary additive Gaussian noise Thederivations can be used for the complex signal before the magnitude is calculated orfor high SNR simplifications We review some methods to estimate the variance ofnoiseσ2and the covariance between coilsσ lm under the Gaussian assumption

In Chap.7, we review and classify the different approaches to estimateσ2out ofRician magnitude MR images In this chapter, we gather the most popular approachesfound in the literature The advantages and drawbacks of the different methods areanalyzed through synthetic and real data controlled experiments A special kind ofestimators, those based on the calculation of the mode, is deeply studied

The estimators for Rician noise of Chap.7are the basis for many of the estimatorsproposed in the following chapters, which can be seen as extensions of the Ricianestimators The next two chapters deal with MRI data from multiple-coil acquisitions.However, in Chap.8 the correlations between coils are not considered, producingsimpler statistical models, while in Chap.9 the correlations are included into theanalysis

In Chap.8, we extend those results to the particular case of a multiple-coil sition in which the magnitude signal is reconstructed using Sum of Squares (SoS) or

acqui-a Spacqui-atiacqui-al macqui-atched filter (SMF), no correlacqui-ations acqui-are acqui-assumed between coils, acqui-and acqui-all

of them show the same variance of noise As a consequence, the magnitude signalfollows 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 takinginto account the correlations between the acquisition coils As a consequence, thedistributions become non-stationary and the estimation of single values carried out

in Chaps.6 8is no longer valid The parameters of noise in the magnitude imagebecomes position dependent and, therefore, a noise map σ2(x) must be estimated

instead We consider two cases, for the magnitude signal being constructed usingeither a SMF or a SoS approach In the first case, a non-stationary Rician distributionarises In the second, a nc-χ approximation of the data is considered, using effective

values forσ2and the number of coils

There are two main ways to approach the non-stationary noise estimation: a metric 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

para-of noise Since not all the parameters needed may be available, some simplificationsare done We propose some estimation guidelines for this specific problem under therestrictions posed by the model The results here presented may be extended to morecomplex estimators.

The third part of the book deals with the problem of noise estimation when

accelerated acquisitions are considered and parallel imaging methods are used toreconstruct the signal Chapters8and9studied noise estimators for nonacceleratedacquisitions However, the common trend in acquisition, due to time restrictions, isprecisely to use parallel reconstruction techniques for subsampled acquisitions InChaps.10and11, noise estimation techniques for parallel signals are considered.Two different approaches will be used: parametric estimation and blind estimation.Chapter10proposes parametric methods to estimate noise for two specific parallelimaging methods, SENSE and GRAPPA The details of each of the reconstructionalgorithms are taken into account in order to estimate the noise In each case, someparameters from the reconstruction process may be needed

In Chap.11, we revise some methods to carry out a blind estimation of the meters of noise for non-stationary models The only requirement for these methods isthat a statistical model has to be adopted for the acquisition noise The main difficulty

para-of this kind para-of analysis is that a single value para-ofσ no longer characterizes the whole

image, on the contrary, a value for each position x must be calculated The

differ-ent proposals in the literature for blind non-stationary noise estimation are reviewedand validated, with a deeper insight in one specific methodology, the homomorphicapproach to noise estimation

Three appendices complements the book The first one provides information

about the probability density functions used along the book, together with their ments and relevant features We also include some combinations of random variablesthat are used to derive the estimators The second appendix reviews a very power-ful technique used for parameter estimation, the variance stabilizing transformation(VST) The VST inspires the stabilization process performed to develop estimatorsfor non-stationary Rician and nc-χ by means of an alternative parametric formula-

mo-tion In the last appendix, we collect the different MRI data sets used along the bookfor illustration and evaluation

Part I Noise Models and the Noise Analysis

Problem

Acquisition and Reconstruction

of Magnetic Resonance Imaging

Magnetic Resonance Imaging (MRI) is based on a phenomenon known as NuclearMagnetic Resonance (NMR), first described by Bloch [31] and Purcell [187] in 1946.Under the effect of a magnetic field strong enough, atomic nuclei with unpaired

protons rotate with a frequency depending on the strength of the magnetic field (and

the nature of the atom) This is in fact the resonance frequency of the nuclei for theparticular magnetic field strength applied, and the atoms are able to absorb energy atthis radio frequency (RF) In other words, a RF pulse can be used to excite the nuclei,which, once the pulse is removed, emit this electromagnetic energy at the resonancefrequency

The use of NMR to image a given tissue requires the localization of the source ofthe electromagnetic energy emitted in order to infer the spatial position of a givenspin density (the concept of spin will be reviewed later on) and, therefore, the localproperties of a given tissue Since the resonance frequency depends on the strength

of the magnetic field applied, the spatial resolution is based on the design of aspatial gradient of the magnetic field: different locations are associated to different

magnetic field strengths, and thus to different resonance frequencies: listening to

different frequencies is the same as studying different locations This principle wasused for the first time by Lauterbur in 1973 [128] to obtain a two-dimensional image.This discovery, together with the Fourier relationship between spin densities andNMR signals, proved by Mansfield and Grannell that same year [153], constitutesthe basis for modern MRI scanners

MRI has been used for medical purposes since 1980 This imaging modalityprovides an excellent contrast between tissues, it is non-invasive, and it does notrequire the use of ionizing radiations, which avoids any secondary effects (as far as it

is known) These features make MRI very attractive for the clinical practice, with theonly drawbacks of its higher cost compared to other modalities (such as ultrasoundimaging) or its relatively high acquisition times Besides, NMR-derived effects may

be used in other MRI modalities: apart from anatomical MRI, functional MRI ordiffusion MRI provide complimentary information and are the focus of importantresearch efforts and interest The information on the first sections of this chapter hasbeen retrieved mainly from [25, 111, 127, 132]

© 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_2

9

10 2 Acquisition and Reconstruction of Magnetic Resonance Imaging

Fig 2.1 General process of a MRI acquisition, from patient to final image

For the whole book a simple pipeline like the one depicted in Fig.2.1will beconsidered: the data is acquired in the scanner using one or multiple-coil, the data

is processed (for different purposes) and a final image is achieved Although in thefollowing sections we will slightly review the physics involved in MRI and someacquisition basics, these concepts are not necessary for the understanding of thegeneral analysis carried out in this book For the sake of simplicity and compactness,

we take a higher level point of view, in which we consider the data already acquired

in the so-called k-space From there, basic transformations will be used in order to

obtain a single magnitude signal

2.1 Physics of Magnetic Resonance Imaging

Protons, neutrons, and electrons show an angular moment known as spin, whichmay have the values±1

2, ±1, ±3

2, ±2, ±5

2 When these particles are paired, their

spins are paired as well, so they cancel each other This is the reason why NMR isonly feasible with unpaired protons In MRI, the particles considered are hydrogennucleus associated to the concentration of water molecules In this case, the spin isreduced to values±1

2 The spin is a property of elemental particles, so it has to beanalyzed in the scope of quantum mechanics However, in MRI, spin systems andnot individual spins are analyzed, so their macroscopic behavior may be accuratelydescribed with classic magnetic field theory In this sense, the spin may be seen as amicroscopic magnetization vector originated by the movement of electrons aroundthe nuclei, much like the magnetic vector induced by a round wire conducting anelectric current In the absence of an external stimulus, spins are randomly distributed,

so the macroscopic magnetization is M = 0 When an external magnetic field B0isapplied, the spins have a slight tendency to point along the field’s direction (by

convention, it is assumed to be the z axis) so an overall magnetization M appears

aligned with B0 At the same time, the magnetization vector of individual spins is

subject to a precession movement around M Its frequencyω0is commonly known

as the Larmor frequency or the resonance frequency, and may be written as

ω0= γB0 = γ B0, (2.1)withγ the gyromagnetic ratio As previously stated, the Larmor frequency depends

on the strength of B0, and on the properties of the tissue being imaged through γ.

The phase of the precession movement for each spin is random, so the macroscopic

effect is that the component of M in the transverse (x y) plane is null, while there

is a net longitudinal component in the z direction Once the spins are precessing at

frequency ω0, they are able to absorb energy from a radio frequency pulse B1(t).

This pulse may be thought of as a circularly polarized magnetic field rotating atfrequencyω0in the plane x y, so it may be coupled with the precession movement.

The first effect is the rotation of the spins around the B1(t) RF pulse, which induces a

rotating component in the transverse plane x y Besides, the particle is able to absorb

energy, and the overall effect is that an effective magnetic field Beffappears aligned

with one of the directions x or y, the precession of the spin follows this direction

and the net magnetization drifts from the direction of B0 a time-dependent angle

α Controlling the duration τ of the pulse B1(t), the final value of α may be fixed.

For real-world applications, durations producing anglesα = 90◦ or α = 180◦ areused, and are commonly known as 90◦or 180◦pulses The angle of M is changed

by exciting the spins with electromagnetic energy at the Larmor frequency When

the pulse B1is removed, the spins free the energy they have previously absorbed,

going back to their initial state so the net magnetization aligned with B0 recovers.This process is called relaxation and, during this, the energy is emitted in the form

of a RF signal which may be received by an antenna (in MRI, antennas are receivingcoils placed in the MRI scanner)

The relaxation of the spins is associated with two different physical processes.First, spins will rapidly dephase after the excitation occurs, pointing in all directions

perpendicular to the static B0field and as a result removing the transverse componentcreated by the RF pulse These effects cause spins to precess at different Larmorfrequencies depending on their position Second, when the RF pulse is removed, the

B0field is still present, thus the spins tend to point along the field’s direction, risingthe longitudinal component Both processes occur at the same time and are basicallyindependent, though the first one is generally much faster Therefore, there exist tworelaxation times

T1: is the time for the longitudinal component to return to its original state throughthe emission of electromagnetic energy at Larmor frequency This is the spin–lattice relaxation, corresponding to the exchange of energy between the spinsystem and its surroundings

T2: is the time for the transverse component to return to its original state, associated

to thermal equilibrium between spins This is the spin–spin relaxation

Both times, T1 and T2 refer to the time constant of the exponential laws ruling

the relaxation processes In general, T1 T2 Measuring relaxation times of the

longitudinal and transverse components of M, different properties of the tissues may

be inferred This is the principle of T1and T2 imaging modalities (see Fig.2.2) Insome example of this book, we will also work with a third kind of imaging modality:

12 2 Acquisition and Reconstruction of Magnetic Resonance Imaging

Fig 2.2 Examples of

anatomical MRI images of

different modalities: T1

(left), and T2 (right) In

these modalities, each pixel

represents the relaxation time

(longitudinal component for

T1, and transverse

component for T2) after the

application of the RF pulse

proton density (PD)-weighted imaging A PD image is obtained by minimizing theeffects of T1 and T2 with long TR (2000–5000 ms) and short TE (10–20), resulting

in an image mainly dependent on the density of protons in the imaging volume.Thus, the tissues with the higher concentration or protons (hydrogen atoms) producestronger signals and are those showing the higher intensity values

2.2 The k-Space and the x-Space

The Larmor frequency depends on the strength of the external magnetic field, B0.This property may be used to infer spatial information by the use of field gradients

A spatial gradient is applied to B0in the z direction while the radio frequency pulse

B1(t) is active This implies that the Larmor frequency varies for each plane z p, soonly one of the planes is excited by the pulse, being able to absorb electromagneticenergy This principle is used in MRI to select an image slice

ω0(z p ) = γ B0+ γG z z p , (2.2)

where G z is the modulus of the gradient applied in the slice direction The spatial

encoding for the x y plane is more complex A combination of gradients in the x and

y directions simultaneously may be considered With this strategy, for each selected

slice z p , there is a plane defined by the two gradients G x and G y

ω0(x, y, z p ) = γ B0+ γG x x + γG y y + γG z z p , (2.3)which defines lines in the angle tan−1(G x /G y ) with the same Larmor frequency.

The collected signals will be the superposition of the spins along these lines, and

a projection image can be obtained by varying the ratio G x /G y similarly as incomputerized axial tomographies The main drawback is the need to infer the spatialinformation from projections, as it is the case with tomographies

Fig 2.3 Magnetic field gradients for the encoding of spatial information in MRI The pulse G z

used for the encoding of the slice z pis applied at the same time as the radio frequency pulse B1(t),

with a strength greater than G x and G y Therefore, only the spins in the slice z pare able to absorb

the radio frequency energy provided by B1(t) G yalters the frequency of the precession of spins

for different y positions Before G xis applied, all spins return to the Larmor frequencyω0 , with

different phases depending on y When G x is applied, the resonance frequency changes in the

direction x, so each pair x , y is identified by a unique frequency and phase in the composite RF

signal detected by the coils

A phase/frequency encoding is used in practice: once the plane z phas been chosen

with G z , a pulsed gradient G yis applied, so the Larmor frequency is different for

each point along the y axis If the duration and amplitude of G yare properly chosen,

when G y is removed the points along y have linearly spaced phases, so that their

Larmor frequencies return to their original value but their spins have different phases

Then, a pulsed gradient G x is applied, varying the Larmor frequency along x The

RF signal is measured while G xis being applied (see Fig.2.3)

The x direction is encoded in the frequency of the emitted signal, while the y

direction is encoded in its phase Unfortunately, this scheme is prone to an ambiguity

in the phase encoding: the superposition of several signals with different phases has

a phase which is a function not only of the phases of the original signals, but itdepends on their amplitudes as well In practice, this means that the acquisition has

to be repeated several times for slightly different values of G y The resolution in the

y axis is given by the number of repetitions used in the acquisition process, while

the resolution along x depends on the number of samples taken at each line.

The advantage of this encoding scheme is that it can be proved that the quency/phase plane is in fact the two-dimensional inverse Fourier transform of thespatial information [116] Without entering into unnecessary details, note that foreach phase encoding the radio frequency signal is the superposition of all the harmon-icsω0(x, y j , z p ) ≡ ω(x) (with y jthe location corresponding to this phase encoding),

weighted by the actual value of the energy emitted at location x with Larmor

fre-quencyω(x) The relation with the Fourier transform in the direction of the y axis is

not so trivial, but in general the received radio frequency signal s may be modeled as

14 2 Acquisition and Reconstruction of Magnetic Resonance Imaging

s(k) =



V

C(r)ρ(r)e j 2πr·x dr, (2.4)

whereρ(r) is the spin density at spatial location r within the Field Of View (FOV) of

the scanner, V , which is the whole spatial domain for which the tissues are imaged.

C (r) accounts for the possibility that the sensitivity of the receiving coil is different

for each location Equation (2.4) is obviously the (weighted) inverse 2D Fouriertransform ofρ(r) in the dual variable k for each slice z p Following this traditional

notation, the signal acquired by the receiving coil is said to be in the k-space, while the

signal of interest, i.e., the spin density, is defined on the image domain, which in this

dissertation will be referred to as the x-space Note that there is a direct equivalence

betweenρ(r) and ρ(x), as we will see in the following section.

MRI scanners use the protocol described in Fig.2.3to acquire the k-space line

by line: for each repetition of the phase encoding, a pulsed G x is applied Thefrequency encoded radio frequency signal is sampled to achieve a whole line of thetwo-dimensional inverse Fourier transform ofρ(r) Then, a two-dimensional discrete

Fourier transform (DFT) is used to recover the x-space from the sampled k-space

for each slice z p The entire acquisition process is often repeated several times, so

that multiple samples of each point in the k-space are available The average of all

these measurements serves to improve the signal-to-noise ratio of the data set Thenumber of measurements is commonly referred to as the Number of Excitations(NEX) This process will be deeply studied in the following sections for differentcoil configurations

2.3 Single-Coil Acquisition Process

The general acquisition scheme in Eq (2.4) is valid for single- and multiple-coilsystems Let us take a signal-oriented approach to the MRI acquisition process pre-viously described, simplified for a single-coil acquisition The basic block diagram

is surveyed in Fig.2.4 The signal acquired by the scanner coil in a single-coil systemcan be modeled by the following equation

s (k) =



V

where S (x) is the 2D slice in the image-space or x-space Note that both s(k) and

S (x) are complex signals, that can be seen as

Fig 2.4 Single-coil acquisition process of MR data

s (k) = s r (k) + j · s i (k) = |s(k)| · exp{ j · ∠s(k)}

In single-coil systems, one complex 2D signal is generated in that space, i.e., s (k)

for each slice of the whole MRI volume As we will show in the next chapters, thissignal is already corrupted by acquisition noise, and the way that noise is propagatedalong the reconstruction pipeline will define the nature and features of the noise inthe final image

This representation of the acquired signal is typically discretized and the imagereconstruction is performed computationally to form an estimate of the spin distrib-ution from the sampled data The complex image domain is obtained as the inverse

2D discrete Fourier transform (iDFT) of s (k) for each slice.

S (x) = F−1{s(k)} (2.7)The signal obtained by the iDFT is a complex signal defined over the image domain

In order to generate real data, the phase information is discarded

M (x) = |S(x)| =S2

r (x) + S2

where M (x) is the magnitude image, i.e., the final image given by the scanner.

2.4 Multiple-Coil Acquisition Process

Many commercial scanners nowadays have the possibility to connect multiple RFdetector coil sets that allows the simultaneous acquisition of several signals in aphased array system These systems were originally developed to reduce the scanning

16 2 Acquisition and Reconstruction of Magnetic Resonance Imaging

Fig 2.5 Head coil for MRI acquisition

Fig 2.6 Multiple-coil acquisition process of MR data An eight-coil system is considered

time and therefore to avoid some problems with moving structures [254], as well as

to enhance the SNR of the magnitude image while maintaining a large Field ofView [58]

Basically a coil is a hardware item of the MR system that acts as an antenna

In multiple-coil systems, several coils are gathered together around the object to bescanned, conforming a coil array In Fig.2.5we show a multiple-coil array used ofhead imaging acquisition The presence of various signals at the same time makesthe global pipeline slightly different to the single-coil one The basic block diagram

is surveyed in Fig.2.6

Let us assume a system with L RF-coils The acquired signal in coil l=

1, 2, , L, can be modeled by the following equation

Fig 2.7 Distribution of a

8-coil system around an

object in a MRI system.

Spatial sensitivity of a

single-coil (Right)

S l (x) is the complex signal at the lth coil in the x-space, which corresponds with the

inverse Fourier transform of s l (k):

S l (x) = F−1{s l (k)}. (2.11)Note that in Eqs (2.9) and (2.10) a new term when compared to the single-coil sys-

tem: the spatial sensitivity of coil l, C l (x) Each of the RF coils that conforms the

acquisition array presents nonuniform spatial sensitivity, which leads to geneous intensities across the image acquired by that coil In Fig.2.7, the effect ofthe sensitivity terms is shown in a muti-coil system where the coils are distributedaround the object that will be scanned Each coil is more sensitive to those areas ofthe object closer to it

nonhomo-Typically, this behavior is mathematically modeled as a point to point productbetween the underlying image and a sensitivity map

S l (x) = C l (x) · S(x), (2.12)

where S (x) is the original image, i.e., the image assuming uniform sensitivity It

corresponds to the excited spin density functionρ(r) An illustration of this effect

is in Fig.2.8 In Fig.2.9, a real acquisition of brain imaging for a 8-coil system isdepicted, together with the sensitivity map estimated for each coil

In single-coil systems, the final magnitude image is simply obtained by takingthe absolute value of the complex signal In the multiple-coil case, one compleximage is available per coil, so it is necessary to combine all that information into

Fig 2.8 The image acquired in the lth coil can be seen as the original image S (x) multiplied by

the sensitivity of that coil

18 2 Acquisition and Reconstruction of Magnetic Resonance Imaging

Fig 2.9 Actual brain imaging acquisition from a GE Signa 1.5 T scanner with 8 coils (top) and

estimated sensitivity maps for each coil (bottom)

one single real image That final image is the so-called Composite Magnitude Signal

(CMS) It will be denoted by M T (x), to distinguish it from the single-coil magnitude

image In [195], authors showed that, for optimal SNR and reduction of artifacts, thecombination must be done pointwise weighting the contribution of each coil by itssensitivity However, note that the coil sensitivity is not always available

Many different approaches have been proposed to reconstruct the CMS aftermultiple-coil acquisition, though the most frequently used are the spatial matchedfilter (SMF) and the Sum of Squares (SoS):

1 Spatial Matched filter (SMF) This method, also known as adaptive

reconstruc-tion, makes use of information of the coil sensitivities It calculates the mal reconstruction using the model in Eq (2.12), which in matrix form can beexpressed (for each pixel) as

opti-ST (x) = C(x) S(x), (2.13)where

In practical implementations, the sensitivities are not known and must be

esti-mated, so sensitivities C l (x) in Eq (2.14) must be replaced by their estimates



C l (x):

SSMF(x) =CH (x)C(x)−1CH (x)S T (x). (2.15)Many methods have been proposed to estimate the sensitivities, see for instance[45, 87, 236, 255] The correlation between coils can be incorporated to the filter,

in order to reduce the existing correlations

SSMF(x) =CH −1(x)C(x)−1CH (x)  S T (x). (2.16)More information about the covariance matrix will be given in the next chapter,where noise models are defined.

Finally, note that the estimated signal SSMF(x) is a complex signal in the x-domain.

In order to obtain a real value, the phase information is discarded, similar to theprocess done in single-coil systems

M T (x) = |SSMF(x)|. (2.17)

2 Sum of Squares (SoS) This alternative method does not require a prior estimation

of the coil sensitivity Instead, the CMS is directly constructed from the signal ineach coil

of both methods will produce different statistical models for the signal and the noise,

as we will see in the next chapters

2.5 Accelerated Acquisitions: Parallel Imaging

In the previous section, we have described the reconstruction process in multiple-coilsystems from acquisition to the final CMS Although the SNR may benefit from theuse of several receivers, the scanning time will be roughly similar to a single-coilacquisition for systems with similar features The high acquisition time is a probleminherent to the image formation process It may be computed for each slice as

20 2 Acquisition and Reconstruction of Magnetic Resonance Imaging

Fig 2.10 The scanning time

of a line in the k-space is

much smaller than the line

shift time

where T R is the repetition time, i.e., the time it takes for the selected plane z p to

return to its equilibrium after it has been excited by the pulses N is the number of steps used for phase encoding, directly related to the resolution in the y axis In many

acquisitions protocols, acquisition time may become an important limitation.Acquisition times may be reduced with modern MRI techniques, especially inthe case of multiple receiving coils scanners When a number of independent anten-

nas (receiving coils) work together, each of them acquiring a subset of the k-space,

Fourier domain information may be retrieved faster Note that there is a great amount

of redundancy in the acquired data: the same image is repeated for every coil, dered with different sensibilities This redundancy in the data can be exploited inorder to accelerate the acquisition using the so-called parallel MRI (pMRI) recon-struction techniques

pon-These pMRI protocols increase the acquisition rate by subsampling the k−space

data [104, 127], while reducing phase distortions when strong magnetic field dients are present The subsampling, assuming Cartesian coordinates, is done only

gra-over the lines of the k-space, since the time employed in scanning a whole line is very

small when compared with the time employed in shifting from one line to another(see Fig.2.10)

The immediate effect of the k−space subsampling is the appearance of aliased

replicas in the image domain retrieved at each coil An illustration of this effect isdepicted in Fig.2.11 In order to suppress or correct this aliasing, pMRI combines theredundant information from several coils to reconstruct a single non-aliased image.The way the information from each coil is combined to reconstruct the data willheavily impact the statistics of the noise, as we will see in the following chapter

An illustrative example may be found in Fig.2.12 In its simplest form, the use of

multiple-coil allows a subsampling of the k-space, whose most immediate effect

is the aliasing in the x-space due to the violation of Nyquist criterion The signals

received by each antenna has to be combined in some way to avoid this artifact Theway the signals are fused is the reconstruction scheme which defines each pMRIalgorithm

The main drawback of pMRI is the reduction of the SNR of the images due toreduced Fourier averaging For nonparallel schemes, the computation of the DFT

produces an averaging of the noise components over the samples of the k-space

Fig 2.11 Subsampling of one coil in the k-space by a factor 2: one out of two lines is not acquired The result on the x-space is the appearance of aliased replicas along the y axis

Fig 2.12 Effect of k-space subsampling From left to right, the original T2 image; the ideal k-space

for this image (computed as the DFT of the original image) in logarithmic units; the subsampled

k-space, in logarithmic units, resulting from the elimination of one of each two lines in the y direction;

the modulus of the image domain (x-space) reconstructed from the subsampled k-space The Fourier relation between the k-space and the x-space explains the aliasing in the image domain if the k-space

is subsampled violating Nyquist criterion As a result, two points (in this over-simplified scenario)

of the original image contribute to each image location of the reconstructed image Parallel imaging algorithms eliminate this artifact using the redundant information from several receiving coils

at each point of the x-space, notably improving the SNR With parallel tions, not all the samples in the k-space are acquired, so the increase in the SNR is

acquisi-minor Besides, reconstruction schemes introduce an amplification of the noise in the

x-space known as theg-factor (where ‘g’ stands for ‘geometric’).