-
Notifications
You must be signed in to change notification settings - Fork 0
/
preprocessing.html
838 lines (740 loc) · 33.3 KB
/
preprocessing.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta http-equiv="X-UA-Compatible" content="IE=EDGE" />
<title>Preprocessing</title>
<script src="site_libs/header-attrs-2.28/header-attrs.js"></script>
<script src="site_libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link href="site_libs/bootstrap-3.3.5/css/lumen.min.css" rel="stylesheet" />
<script src="site_libs/bootstrap-3.3.5/js/bootstrap.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/html5shiv.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/respond.min.js"></script>
<style>h1 {font-size: 34px;}
h1.title {font-size: 38px;}
h2 {font-size: 30px;}
h3 {font-size: 24px;}
h4 {font-size: 18px;}
h5 {font-size: 16px;}
h6 {font-size: 12px;}
code {color: inherit; background-color: rgba(0, 0, 0, 0.04);}
pre:not([class]) { background-color: white }</style>
<script src="site_libs/jqueryui-1.13.2/jquery-ui.min.js"></script>
<link href="site_libs/tocify-1.9.1/jquery.tocify.css" rel="stylesheet" />
<script src="site_libs/tocify-1.9.1/jquery.tocify.js"></script>
<script src="site_libs/navigation-1.1/tabsets.js"></script>
<script src="site_libs/navigation-1.1/codefolding.js"></script>
<script src="site_libs/navigation-1.1/sourceembed.js"></script>
<link href="site_libs/anchor-sections-1.1.0/anchor-sections.css" rel="stylesheet" />
<link href="site_libs/anchor-sections-1.1.0/anchor-sections-hash.css" rel="stylesheet" />
<script src="site_libs/anchor-sections-1.1.0/anchor-sections.js"></script>
<link href="site_libs/font-awesome-6.4.2/css/all.min.css" rel="stylesheet" />
<link href="site_libs/font-awesome-6.4.2/css/v4-shims.min.css" rel="stylesheet" />
<!DOCTYPE html>
<link rel="shortcut icon" href="images/logo.png" />
<style type="text/css">
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
</style>
<style type="text/css">
#rmd-source-code {
display: none;
}
</style>
<style type="text/css">
/* for pandoc --citeproc since 2.11 */
div.csl-bib-body { }
div.csl-entry {
clear: both;
}
.hanging div.csl-entry {
margin-left:2em;
text-indent:-2em;
}
div.csl-left-margin {
min-width:2em;
float:left;
}
div.csl-right-inline {
margin-left:2em;
padding-left:1em;
}
div.csl-indent {
margin-left: 2em;
}
</style>
<link rel="stylesheet" href="styles.css" type="text/css" />
<style type = "text/css">
.main-container {
max-width: 940px;
margin-left: auto;
margin-right: auto;
}
img {
max-width:100%;
}
.tabbed-pane {
padding-top: 12px;
}
.html-widget {
margin-bottom: 20px;
}
button.code-folding-btn:focus {
outline: none;
}
summary {
display: list-item;
}
details > summary > p:only-child {
display: inline;
}
pre code {
padding: 0;
}
</style>
<style type="text/css">
.dropdown-submenu {
position: relative;
}
.dropdown-submenu>.dropdown-menu {
top: 0;
left: 100%;
margin-top: -6px;
margin-left: -1px;
border-radius: 0 6px 6px 6px;
}
.dropdown-submenu:hover>.dropdown-menu {
display: block;
}
.dropdown-submenu>a:after {
display: block;
content: " ";
float: right;
width: 0;
height: 0;
border-color: transparent;
border-style: solid;
border-width: 5px 0 5px 5px;
border-left-color: #cccccc;
margin-top: 5px;
margin-right: -10px;
}
.dropdown-submenu:hover>a:after {
border-left-color: #adb5bd;
}
.dropdown-submenu.pull-left {
float: none;
}
.dropdown-submenu.pull-left>.dropdown-menu {
left: -100%;
margin-left: 10px;
border-radius: 6px 0 6px 6px;
}
</style>
<script type="text/javascript">
// manage active state of menu based on current page
$(document).ready(function () {
// active menu anchor
href = window.location.pathname
href = href.substr(href.lastIndexOf('/') + 1)
if (href === "")
href = "index.html";
var menuAnchor = $('a[href="' + href + '"]');
// mark the anchor link active (and if it's in a dropdown, also mark that active)
var dropdown = menuAnchor.closest('li.dropdown');
if (window.bootstrap) { // Bootstrap 4+
menuAnchor.addClass('active');
dropdown.find('> .dropdown-toggle').addClass('active');
} else { // Bootstrap 3
menuAnchor.parent().addClass('active');
dropdown.addClass('active');
}
// Navbar adjustments
var navHeight = $(".navbar").first().height() + 15;
var style = document.createElement('style');
var pt = "padding-top: " + navHeight + "px; ";
var mt = "margin-top: -" + navHeight + "px; ";
var css = "";
// offset scroll position for anchor links (for fixed navbar)
for (var i = 1; i <= 6; i++) {
css += ".section h" + i + "{ " + pt + mt + "}\n";
}
style.innerHTML = "body {" + pt + "padding-bottom: 40px; }\n" + css;
document.head.appendChild(style);
});
</script>
<!-- tabsets -->
<style type="text/css">
.tabset-dropdown > .nav-tabs {
display: inline-table;
max-height: 500px;
min-height: 44px;
overflow-y: auto;
border: 1px solid #ddd;
border-radius: 4px;
}
.tabset-dropdown > .nav-tabs > li.active:before, .tabset-dropdown > .nav-tabs.nav-tabs-open:before {
content: "\e259";
font-family: 'Glyphicons Halflings';
display: inline-block;
padding: 10px;
border-right: 1px solid #ddd;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open > li.active:before {
content: "\e258";
font-family: 'Glyphicons Halflings';
border: none;
}
.tabset-dropdown > .nav-tabs > li.active {
display: block;
}
.tabset-dropdown > .nav-tabs > li > a,
.tabset-dropdown > .nav-tabs > li > a:focus,
.tabset-dropdown > .nav-tabs > li > a:hover {
border: none;
display: inline-block;
border-radius: 4px;
background-color: transparent;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open > li {
display: block;
float: none;
}
.tabset-dropdown > .nav-tabs > li {
display: none;
}
</style>
<!-- code folding -->
<style type="text/css">
.code-folding-btn { margin-bottom: 4px; }
</style>
<style type="text/css">
#TOC {
margin: 25px 0px 20px 0px;
}
@media (max-width: 768px) {
#TOC {
position: relative;
width: 100%;
}
}
@media print {
.toc-content {
/* see https://github.com/w3c/csswg-drafts/issues/4434 */
float: right;
}
}
.toc-content {
padding-left: 30px;
padding-right: 40px;
}
div.main-container {
max-width: 1200px;
}
div.tocify {
width: 20%;
max-width: 260px;
max-height: 85%;
}
@media (min-width: 768px) and (max-width: 991px) {
div.tocify {
width: 25%;
}
}
@media (max-width: 767px) {
div.tocify {
width: 100%;
max-width: none;
}
}
.tocify ul, .tocify li {
line-height: 20px;
}
.tocify-subheader .tocify-item {
font-size: 0.90em;
}
.tocify .list-group-item {
border-radius: 0px;
}
.tocify-subheader {
display: inline;
}
.tocify-subheader .tocify-item {
font-size: 0.95em;
}
</style>
</head>
<body>
<div class="container-fluid main-container">
<!-- setup 3col/9col grid for toc_float and main content -->
<div class="row">
<div class="col-xs-12 col-sm-4 col-md-3">
<div id="TOC" class="tocify">
</div>
</div>
<div class="toc-content col-xs-12 col-sm-8 col-md-9">
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-bs-toggle="collapse" data-target="#navbar" data-bs-target="#navbar">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">Gannet</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="index.html">
<span class="fa fa-home"></span>
Home
</a>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false">
Documentation
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="getting-started.html">Getting started</a>
</li>
<li>
<a href="reference.html">Reference</a>
</li>
<li>
<a href="output-structure-attributes.html">Output structure attributes</a>
</li>
<li>
<a href="preprocessing.html">Preprocessing</a>
</li>
<li>
<a href="signal-modeling.html">Signal modeling</a>
</li>
<li>
<a href="data-quality-metrics.html">Data quality metrics</a>
</li>
<li>
<a href="quantification-tissue-correction.html">Quantification & tissue correction</a>
</li>
<li>
<a href="additional-tools.html">Additional tools</a>
</li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" data-bs-toggle="dropdown" aria-expanded="false">
Examples
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="gannetpreinitialise-settings.html">GannetPreInitialise settings</a>
</li>
<li>
<a href="batch-script.html">Batch processing script</a>
</li>
<li>
<a href="joining-data-files.html">Joining data files</a>
</li>
</ul>
</li>
<li>
<a href="faq.html">FAQ</a>
</li>
<li>
<a href="release-notes.html">Release notes</a>
</li>
<li>
<a href="https://forum.mrshub.org/c/mrs-software/gannet/9">Forum</a>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li>
<a href="https://github.com/markmikkelsen/Gannet/issues">Report an issue</a>
</li>
<li>
<a href="https://github.com/markmikkelsen/Gannet">
<span class="fa fa-github fa-lg"></span>
</a>
</li>
<li>
<a href="mailto:[email protected]?subject=[Gannet]%20<Please enter the subject of your query here>">
<span class="fa fa-envelope fa-lg"></span>
</a>
</li>
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container -->
</div><!--/.navbar -->
<div id="header">
<div class="btn-group pull-right float-right">
<button type="button" class="btn btn-default btn-xs btn-secondary btn-sm dropdown-toggle" data-toggle="dropdown" data-bs-toggle="dropdown" aria-haspopup="true" aria-expanded="false"><span>Code</span> <span class="caret"></span></button>
<ul class="dropdown-menu dropdown-menu-right" style="min-width: 50px;">
<li><a id="rmd-download-source" href="#">Download Rmd</a></li>
</ul>
</div>
<h1 class="title toc-ignore">Preprocessing</h1>
<h4 class="date">Last updated: October 14, 2024</h4>
</div>
<button onclick="topFunction()" id="myBtn" title="Go to top">
Back to top
</button>
<script>
//Get the button
var mybutton = document.getElementById("myBtn");
// When the user scrolls down 20px from the top of the document, show the button
window.onscroll = function() {scrollFunction()};
function scrollFunction() {
if (document.body.scrollTop > 20 || document.documentElement.scrollTop > 20) {
mybutton.style.display = "block";
} else {
mybutton.style.display = "none";
}
}
// When the user clicks on the button, scroll to the top of the document
function topFunction() {
document.body.scrollTop = 0;
document.documentElement.scrollTop = 0;
}
</script>
<style type="text/css">
table {
margin: auto;
}
table thead th {
border-bottom: 1px solid #ddd;
}
th, td {
padding: 5px;
}
</style>
<p><br></p>
<p>Listed below are the preprocessing routines that are run in
<code>GannetLoad.m</code>. Whether a particular routine or subroutine is
run will depend on the format of the inputted data and the options set
in <code>GannetPreInitialise.m</code>.</p>
<div id="rf-coil-combination" class="section level2 hasAnchor">
<h2 class="hasAnchor">RF coil combination<a href="#rf-coil-combination"
class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Certain raw MRS data formats store data without coil combination;
specifically, GE P-file (.7), NIfTI-MRS (if the source data were raw),
Philips .raw, and Siemens TWIX (.dat) data. Gannet uses generalized
least squares<span class="citation"><sup><a
href="#ref-An2013">1</a></sup></span> to optimally combine the signal
from the multiple RF channels. If water files are provided, these data
will be used as references for signal weighting and phasing of the coil
data.</p>
</div>
<div id="eddy-current-correction" class="section level2 hasAnchor">
<h2 class="hasAnchor">Eddy-current correction<a
href="#eddy-current-correction" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<div class="info">
<p><i class="fa fa-info-circle" style="color: white"></i> Eddy-current
correction can only be applied if water reference data are provided.</p>
</div>
<p>In <code>GannetPreInitialise.m</code>, users have the option to apply
eddy-current correction (ECC) to metabolite and water data. If applied,
Gannet uses the method described by Klose (1990)<span
class="citation"><sup><a href="#ref-Klose1990">2</a></sup></span>. The
code for the ECC routine can be found in
<code>EddyCurrentCorrection.m</code>.</p>
</div>
<div id="phase-correction" class="section level2 hasAnchor">
<h2 class="hasAnchor">Phase correction<a href="#phase-correction"
class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>It is common for unprocessed spectra to be out of phase. Gannet
applies a global zero-order phase correction to all transients by
fitting the real-valued 3 ppm Cr and 3.2 ppm Cho signals in the
frequency domain and correcting the phase to be 0° phase (i.e., to make
all the Cr and Cho peaks have positive phase). This is performed in
<code>PhaseCorrection.m</code>.</p>
</div>
<div id="line-broadending-apodization" class="section level2 hasAnchor">
<h2 class="hasAnchor">Line-broadending (apodization)<a
href="#line-broadending-apodization" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>FID data are multiplied by a time-varying exponential weighting
function where the weighting constant is set in
<code>GannetPreInitialise.m</code> (3 Hz is the default).</p>
</div>
<div id="zero-filling" class="section level2 hasAnchor">
<h2 class="hasAnchor">Zero-filling<a href="#zero-filling"
class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Gannet zero-pads all raw FIDs (i.e., adds zeros to the end of each
FID) to obtain a nominal spectral resolution (the resolution between
each frequency-domain data point) of 0.061 Hz/point. Differences in
spectral width and number of complex data points of the raw FIDs are
accounted for to obtain this nominal spectral resolution.</p>
</div>
<div id="frequency-and-phase-alignment"
class="section level2 hasAnchor">
<h2 class="hasAnchor">Frequency and phase alignment<a
href="#frequency-and-phase-alignment" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>During acquisition, spectral data are affected by frequency and phase
offsets as a result of biophysical, electronic, and participant factors.
Gannet has several algorithms to correct for these errors during
preprocessing. Users can choose which method to use in
<code>GannetPreInitialise.m</code>:</p>
<table>
<colgroup>
<col width="20%" />
<col width="20%" />
<col width="40%" />
<col width="20%" />
</colgroup>
<thead>
<tr class="header">
<th align="left"><u>Method</u></th>
<th align="left"><u>Option</u></th>
<th align="left"><u>Description</u></th>
<th align="left"><u>Function</u></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">Spectral registration<span class="citation"><sup><a
href="#ref-Near2015">3</a></sup></span></td>
<td align="left"><code>SpecReg</code></td>
<td align="left">A time-domain-based alignment method that uses
nonlinear least-squares optimization to align each individual transient
to a reference transient.</td>
<td align="left"><code>SpectralRegistration.m</code></td>
</tr>
<tr class="even">
<td align="left">Multi-step frequency and phase correction<span
class="citation"><sup><a
href="#ref-Mikkelsen2018">4</a></sup></span></td>
<td align="left"><code>SpecRegHERMES</code></td>
<td align="left">A method originally developed to align multiplexed
edited HERMES data. This approach is based on spectral
registration.</td>
<td align="left"><code>SpectralRegistrationHERMES.m</code></td>
</tr>
<tr class="odd">
<td align="left">Robust spectral registration<span
class="citation"><sup><a href="#ref-Mikkelsen2020">5</a></sup></span>
(the default and recommended)</td>
<td align="left"><code>RobustSpecReg</code></td>
<td align="left">A method based on spectral registration that is robust
against spectral distortions caused by unstable residual water peaks and
lipid contamination.</td>
<td align="left"><code>RobustSpectralRegistration.m</code></td>
</tr>
<tr class="even">
<td align="left">Peak alignment</td>
<td align="left"><code>Cr</code>, <code>Cho</code>, <code>NAA</code>,
<code>H2O</code></td>
<td align="left">Frequency-domain-based alignment using one of the
following peaks in each transient as the target signal: Cr, Cho, NAA, or
residual H2O.</td>
<td align="left"><code>AlignUsingPeak.m</code>,
<code>AlignUsingH2O.m</code></td>
</tr>
<tr class="odd">
<td align="left">No alignment</td>
<td align="left"><code>none</code></td>
<td align="left">Do not perform any frequency or phase alignment between
transients.</td>
<td align="left">n/a</td>
</tr>
</tbody>
</table>
</div>
<div id="signal-averaging" class="section level2 hasAnchor">
<h2 class="hasAnchor">Signal averaging<a href="#signal-averaging"
class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Gannet provides two methods for averaging individual transients
(selected in <code>GannetPreInitialise.m</code>): arithmetic averaging
(with outlier rejection) and weighted averaging (the default). The code
for the signal averaging routines can be found in
<code>SignalAveraging.m</code>.</p>
<div id="arithmetic-averaging" class="section level3 hasAnchor">
<h3 class="hasAnchor">Arithmetic averaging<a
href="#arithmetic-averaging" class="anchor-section"
aria-label="Anchor link to header"></a></h3>
<p>Arithmetic averaging is straightforward. All sequentially acquired
<span class="math inline">\(n\)</span> pairs of subspectra <span
class="math inline">\(x_i\)</span> (e.g., all edit-ON and edit-OFF
subspectra) are averaged using the arithmetic mean: <span
class="math inline">\(\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i\)</span>.</p>
<p>Note that before the arithmetic averaging of subspectra, individual
transients are excluded based on the outlier rejection algorithm used
during frequency and phase alignment.</p>
</div>
<div id="weighted-averaging" class="section level3 hasAnchor">
<h3 class="hasAnchor">Weighted averaging<a href="#weighted-averaging"
class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Weighted averaging down-weights individual difference subspectra that
are corrupted by signal artifacts — this is an important distinction
from traditional signal averaging. First, the difference between
sequentially acquired pairs (e.g., all edit-ON and edit-OFF subspectra)
is calculated. A similarity matrix <span
class="math inline">\(\mathbf{D}\in\mathbb{R}^{P{\times}P}\)</span> is
obtained by calculating the mean squared error between each real-valued
difference subspectrum <span class="math inline">\(p\)</span> and every
other real-valued difference subspectrum (in the range 1.8 to 3.4 ppm).
A similarity metric <span class="math inline">\(d_{p}\)</span> is
calculated as the column-wise median of <span
class="math inline">\(\mathbf{D}\)</span>. Normalized weights <span
class="math inline">\(w_{p}\)</span> are then derived, <span
class="math inline">\(w_{p} = d^{-2}_p/\sum{d^{-2}_p}\)</span>, and
applied to the difference pairs before summation.</p>
<p>Other algorithms to calculate weights can be found in
<code>SignalAveraging.m</code>. (Note: These have not been tested
extensively and should be considered experimental.)</p>
</div>
</div>
<div id="residual-water-removal" class="section level2 hasAnchor">
<h2 class="hasAnchor">Residual water removal<a
href="#residual-water-removal" class="anchor-section"
aria-label="Anchor link to header"></a></h2>
<p>If <code>water_removal</code> is set to 1 in
<code>GannetPreInitialise.m</code> (the default), the residual water
peak is removed from all difference spectra using an HSVD filter<span
class="citation"><sup><a
href="#ref-Barkhuijsen1987">6</a></sup></span>.</p>
<p><br></p>
<div id="references" class="section level3 unnumbered hasAnchor">
<h3 class="unnumbered hasAnchor">References<a href="#references"
class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body">
<div id="ref-An2013" class="csl-entry">
<div class="csl-left-margin">1. </div><div class="csl-right-inline">An
L, Willem van der Veen J, Li S, Thomasson DM, Shen J. <span
class="nocase">Combination of multichannel single-voxel MRS signals
using generalized least squares</span>. <em>Journal of Magnetic
Resonance Imaging</em>. 2013;37(6):1445-1450. doi:<a
href="https://doi.org/10.1002/jmri.23941">10.1002/jmri.23941</a></div>
</div>
<div id="ref-Klose1990" class="csl-entry">
<div class="csl-left-margin">2. </div><div
class="csl-right-inline">Klose U. <span class="nocase">In vivo proton
spectroscopy in presence of eddy currents</span>. <em>Magnetic Resonance
in Medicine</em>. 1990;14(1):26-30. doi:<a
href="https://doi.org/10.1002/mrm.1910140104">10.1002/mrm.1910140104</a></div>
</div>
<div id="ref-Near2015" class="csl-entry">
<div class="csl-left-margin">3. </div><div class="csl-right-inline">Near
J, Edden R, Evans CJ, Paquin R, Harris A, Jezzard P. <span
class="nocase">Frequency and phase drift correction of magnetic
resonance spectroscopy data by spectral registration in the time
domain</span>. <em>Magnetic Resonance in Medicine</em>.
2015;73(1):44-50. doi:<a
href="https://doi.org/10.1002/mrm.25094">10.1002/mrm.25094</a></div>
</div>
<div id="ref-Mikkelsen2018" class="csl-entry">
<div class="csl-left-margin">4. </div><div
class="csl-right-inline">Mikkelsen M, Saleh MG, Near J, et al. <span
class="nocase">Frequency and phase correction for multiplexed edited MRS
of GABA and glutathione</span>. <em>Magnetic Resonance in Medicine</em>.
2018;80(1):21-28. doi:<a
href="https://doi.org/10.1002/mrm.27027">10.1002/mrm.27027</a></div>
</div>
<div id="ref-Mikkelsen2020" class="csl-entry">
<div class="csl-left-margin">5. </div><div
class="csl-right-inline">Mikkelsen M, Tapper S, Near J, Mostofsky SH,
Puts NAJ, Edden RAE. <span class="nocase">Correcting frequency and phase
offsets in MRS data using robust spectral registration</span>. <em>NMR
in Biomedicine</em>. 2020;33(10):e4368. doi:<a
href="https://doi.org/10.1002/nbm.4368">10.1002/nbm.4368</a></div>
</div>
<div id="ref-Barkhuijsen1987" class="csl-entry">
<div class="csl-left-margin">6. </div><div
class="csl-right-inline">Barkhuijsen H, Beer R de, Ormondt D van. <span
class="nocase">Improved algorithm for noniterative time-domain model
fitting to exponentially damped magnetic resonance signals</span>.
<em>Journal of Magnetic Resonance</em>. 1987;73(3):553-557. doi:<a
href="https://doi.org/10.1016/0022-2364(87)90023-0">10.1016/0022-2364(87)90023-0</a></div>
</div>
</div>
</div>
</div>
<div id="rmd-source-code">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</div>
<!DOCTYPE html>
<br>
<hr>
<p style="text-align: right;">Built with <a href="https://rmarkdown.rstudio.com/" target="_blank">R Markdown</a> in <a
href="https://posit.co/products/open-source/rstudio/" target="_blank">RStudio</a>
<br>
</p>
<p style="text-align: right;">Copyright © 2020–2024, Mark Mikkelsen <a
href="https://creativecommons.org/licenses/by/4.0" target="_blank"><img id="cc"
style="height:22px!important;margin-left: 3px;vertical-align:text-bottom;" src="images/license/cc-xlarge.png"><img
id="cc" style="height:22px!important;margin-left: 3px;vertical-align:text-bottom;"
src="images/license/by-xlarge.png"></a>
</p>
</div>
</div>
</div>
<script>
// add bootstrap table styles to pandoc tables
function bootstrapStylePandocTables() {
$('tr.odd').parent('tbody').parent('table').addClass('table table-condensed');
}
$(document).ready(function () {
bootstrapStylePandocTables();
});
</script>
<!-- tabsets -->
<script>
$(document).ready(function () {
window.buildTabsets("TOC");
});
$(document).ready(function () {
$('.tabset-dropdown > .nav-tabs > li').click(function () {
$(this).parent().toggleClass('nav-tabs-open');
});
});
</script>
<!-- code folding -->
<script>
$(document).ready(function () {
window.initializeSourceEmbed("preprocessing.Rmd");
});
</script>
<script>
$(document).ready(function () {
// temporarily add toc-ignore selector to headers for the consistency with Pandoc
$('.unlisted.unnumbered').addClass('toc-ignore')
// move toc-ignore selectors from section div to header
$('div.section.toc-ignore')
.removeClass('toc-ignore')
.children('h1,h2,h3,h4,h5').addClass('toc-ignore');
// establish options
var options = {
selectors: "h1,h2",
theme: "bootstrap3",
context: '.toc-content',
hashGenerator: function (text) {
return text.replace(/[.\\/?&!#<>]/g, '').replace(/\s/g, '_');
},
ignoreSelector: ".toc-ignore",
scrollTo: 0
};
options.showAndHide = false;
options.smoothScroll = true;
// tocify
var toc = $("#TOC").tocify(options).data("toc-tocify");
});
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement("script");
script.type = "text/javascript";
script.src = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML";
document.getElementsByTagName("head")[0].appendChild(script);
})();
</script>
</body>
</html>