Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 82

4.4.3.2 Region of Constant Amplitude and Unknown
Noise Variance..............................................................113
4.4.4 Discussion....................................................................................114
4.4.4.1 CRLB ...........................................................................115
4.4.4.2 MSE..............................................................................116
4.4.5 Signal Amplitude Estimation from PCMR Data ........................118
4.4.5.1 Region of Constant Amplitude
and Known Noise Variance..........................................118
4.4.5.2 Experiments and Discussion ........................................122
4.5 Noise Variance Estimation ......................................................................123
4.5.1 Introduction..................................................................................123
4.5.2 Noise Variance Estimation from Complex Data.........................124
4.5.2.1 Region of Constant Amplitude and Phase...................124
4.5.2.2 Region of Constant Amplitude
and Different Phases ....................................................125
4.5.2.3 Background Region......................................................127
4.5.3 Noise Variance Estimation from Magnitude Data ......................128
4.5.3.1 Background Region......................................................128
4.5.3.2 Region of Constant Amplitude ....................................131
4.5.3.3 Double-Acquisition Method ........................................131
4.5.4 Discussion....................................................................................133
4.5.4.1 CRLB ...........................................................................133
4.5.4.2 MSE..............................................................................133
4.6 Conclusions..............................................................................................136
4.7 Appendix..................................................................................................136
4.7.1 Transformations of PDFs ............................................................136
4.7.1.1 Theorem........................................................................136
4.7.1.2 Example........................................................................137
4.7.2 General Theorem .........................................................................137
4.7.3 Approximation of the Mean of a Random Variable ...................138
4.7.3.1 Theorem........................................................................138
4.7.3.2 Example........................................................................138
Abbreviations ....................................................................................................139
Symbols .............................................................................................................139
References .........................................................................................................140

4.1 INTRODUCTION

Magnetic resonance imaging (MRI) is a magnificent imaging technique in bio-
medicine that is able to produce high-quality images containing an abundance of
physiological, anatomical, and functional information. Such information is often
not extracted by visual interpretation of the images alone; digital data from the
scanners are generally processed and analyzed in a quantitative way using
advanced digital-data-processing tools.