2009 Blair Bohannan - Revision history

imported>Psych 204: /* Appendix 1: Code */

2009-12-11T07:42:55Z

Appendix 1: Code

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	= Appendix 1: Code =		= Appendix 1: Code =
	[[File: ~~imageProc~~.zip \| imageProcessing.zip]]		[[File:imageProcessing.zip \| imageProcessing.zip]]

	= Appendix 2: Acknowledgements =		= Appendix 2: Acknowledgements =

imported>Psych 204: /* Appendix 1: Code */

2009-12-11T07:42:10Z

Appendix 1: Code

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	= Appendix 1: Code =		= Appendix 1: Code =
	[[File: ~~imageProcessing~~.zip \| imageProcessing.zip]]		[[File: imageProc.zip \| imageProcessing.zip]]

	= Appendix 2: Acknowledgements =		= Appendix 2: Acknowledgements =

imported>Psych 204 at 06:58, 11 December 2009

2009-12-11T06:58:16Z

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	[[File:2dFFT_shifted.jpg \| Figure 4]]		[[File:2dFFT_shifted.jpg \| Figure 4]]
	<br>		<br>
	Note: fftshift should be used ~~''after'' performing~~ the ~~FFT~~. Example:		Note: fftshift should be used on the frequency-domain signal. Example:
	:X=fftshift(fft(x));		:X=fftshift(fft(x));

	=== Using ifftshift ===		=== Using ifftshift ===
	The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ifftshift.html ifftshift] function reverses the effect of fftshift (use this instead of performing fftshift twice). Like fftshift, ifftshift is always applied to the frequency-domain signal in this tutorial:		The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ifftshift.html ifftshift] function reverses the effect of fftshift (use this instead of performing fftshift twice). Like fftshift, ifftshift is always applied to the frequency-domain signal in this tutorial, as follows:
	:x=ifft(ifftshift(X));		:x=ifft(ifftshift(X));

imported>Psych 204 at 06:56, 11 December 2009

2009-12-11T06:56:44Z

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	[[File:sinusoidFftshift.jpg \| Figure 2]]		[[File:sinusoidFftshift.jpg \| Figure 2]]
	<br>		<br>
	In the 2-dimensional case, fftshift swaps quadrants 1 and 3, and also 2 and 4 of the Fourier transform. Thus, the low-frequency information is moved to the center of the image, with the [0,0] frequency point moving from matrix index [1,1] to [nPix/2+1,nPix/2+1]. The fftshifted Fourier transform of an image is what we are used to looking at when we talk about [http://en.wikipedia.org/wiki/K-space_%28MRI%29 k-space]. For the purpose of this tutorial, [http://en.wikipedia.org/wiki/Time_domain ''time domain''] is the 1-dimensional case of ''image domain'', and ~~[http://en.wikipedia.org/wiki/K-space_%28MRI%29~~ ''k-space''] is specifically the 2-dimensional case of [http://en.wikipedia.org/wiki/Frequency_domain ''frequency domain''].		In the 2-dimensional case, fftshift swaps quadrants 1 and 3, and also 2 and 4 of the Fourier transform. Thus, the low-frequency information is moved to the center of the image, with the [0,0] frequency point moving from matrix index [1,1] to [nPix/2+1,nPix/2+1]. The fftshifted Fourier transform of an image is what we are used to looking at when we talk about [http://en.wikipedia.org/wiki/K-space_%28MRI%29 k-space]. For the purpose of this tutorial, [http://en.wikipedia.org/wiki/Time_domain ''time domain''] is the 1-dimensional case of ''image domain'', and ''k-space'' is specifically the 2-dimensional case of [http://en.wikipedia.org/wiki/Frequency_domain ''frequency domain''].
	<br>		<br>
	Here is a 2-d Fourier transform without fftshift. Note the bright spots (high amplitudes) in the outer corners of the image; these are the low frequencies.		Here is a 2-d Fourier transform without fftshift. Note the bright spots (high amplitudes) in the outer corners of the image; these are the low frequencies.

imported>Psych 204 at 06:53, 11 December 2009

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	The tutorial uses cells, and each section below corresponds to a cell in the tutorial. To evaluate a cell in the tutorial, first navigate into it. Ctrl+Enter evaluates the current cell, and Ctrl+Shift+Enter evaluates and moves to the next cell. Variables are re-used throughout the tutorial, so start at the first cell and work through the tutorial in order.		The tutorial uses cells, and each section below corresponds to a cell in the tutorial. To evaluate a cell in the tutorial, first navigate into it. Ctrl+Enter evaluates the current cell, and Ctrl+Shift+Enter evaluates and moves to the next cell. Variables are re-used throughout the tutorial, so start at the first cell and work through the tutorial in order.
	<br>		<br>
	The tutorial package contains a selection of image files. Those titled "stimulus##.tif" (including the one used on this page) are from an image set prepared by Roozbeh Kiani. The associated publication can be found [http://jn.physiology.org/cgi/content/full/97/6/4296 here]~~. Used with permission~~.		The tutorial package contains a selection of image files. The "brain.jpg" image is from a k-space tutorial that was included with the materials for this class. Those titled "stimulus##.tif" (including the one used on this page) are taken, with permission, from an image set prepared by Roozbeh Kiani. The associated publication can be found [http://jn.physiology.org/cgi/content/full/97/6/4296 here].

	= About this page =		= About this page =
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	= Appendix 1: Code =		= Appendix 1: Code =
			[[File: imageProcessing.zip \| imageProcessing.zip]]

	= Appendix 2: Acknowledgements =		= Appendix 2: Acknowledgements =

imported>Psych 204 at 06:34, 11 December 2009

2009-12-11T06:34:02Z

imported>Psych 204 at 06:11, 11 December 2009

2009-12-11T06:11:35Z

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	= Appendix 2: Acknowledgements =		= Appendix 2: Acknowledgements =
	I received help from Jonathan Abel (Consulting Professor, CCRMA) with some of the derivations~~; he~~ also ~~proposed~~ the ~~use~~ of ~~inerp2 for simulating~~ the jitter artifact.		I received help from Jonathan Abel (Consulting Professor, CCRMA) with some of the derivations, and also with the details of implementing the jitter artifact.
	<br>		<br>
	The Psych 204 instructors and TA offered helpful and practical suggestions toward improving and expanding the tutorial. Thank you!		The Psych 204 instructors and TA offered helpful and practical suggestions toward improving and expanding the tutorial. Thank you!
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	= References/Further reading =		= References/Further reading =
	Huettel, Scott A., Andrew W. Song, and Gregory McCarthy. ''Functional Magnetic Resonance Imaging'', 2nd Edition. Sunderland, Massachusetts: Sinauer Associates, 2009.		Huettel, Scott A., Andrew W. Song, and Gregory McCarthy. ''Functional Magnetic Resonance Imaging'', 2nd Edition. Sunderland, Massachusetts: Sinauer Associates, 2009.
			<br>
	Hornak, Joseph P. [http://www.cis.rit.edu/htbooks/mri/inside.htm ''The Basics of MRI''].		Hornak, Joseph P. [http://www.cis.rit.edu/htbooks/mri/inside.htm ''The Basics of MRI''].
			<br>
	Oppenheim, Alan V., and Ronald W. Schafer. ''Discrete-Time Signal Processing'', 2nd Edition. Prentice Hall Signal Processing Series. Upper Saddle River, New Jersey: Prentice Hall, 1999.		Oppenheim, Alan V., and Ronald W. Schafer. ''Discrete-Time Signal Processing'', 2nd Edition. Prentice Hall Signal Processing Series. Upper Saddle River, New Jersey: Prentice Hall, 1999.
			<br>
	Smith, Julius O. [https://ccrma.stanford.edu/~jos/mdft/ ''Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications''], 2nd Edition. W3K Publishing, 2007.		Smith, Julius O. [https://ccrma.stanford.edu/~jos/mdft/ ''Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications''], 2nd Edition. W3K Publishing, 2007.

imported>Psych 204 at 06:10, 11 December 2009

2009-12-11T06:10:37Z

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	The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/interp1.html interp1] function can accomplish this. For our purposes, it will take in the original input vector (time-index integers), original output vector (the sinusoid), and the modified input vector (jittered time indices), outputting the sinusoid values at those jittered time indices.		The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/interp1.html interp1] function can accomplish this. For our purposes, it will take in the original input vector (time-index integers), original output vector (the sinusoid), and the modified input vector (jittered time indices), outputting the sinusoid values at those jittered time indices.
	<br>		<br>
	In the figure below, we can see the original sinusoid (in blue) and the sinusoid at the jittered time values (in red). The jittered samples are being displayed at the time points ''at which they were sampled'', which is why the ~~overall~~ shape of the sinusoid ~~remains unchanged~~.		In the figure below, we can see the original sinusoid (in blue) and the sinusoid at the jittered time values (in red). The jittered samples are being displayed at the time points ''at which they were sampled'', which is why the shape of the sinusoid still looks normal.
	<br>		<br>
	[[File:interpJitter.jpg \| Figure 20]]		[[File:interpJitter.jpg \| Figure 20]]
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	[[File: scaledLPFnoise.jpg \| Figure 26]]		[[File: scaledLPFnoise.jpg \| Figure 26]]
	<br>		<br>
	Here is the reconstructed image with this random-walk noise. The noise is harder to detect; ~~if you look closely you will see that it~~ is concentrated around the figure of the zebra.		Here is the reconstructed image with this random-walk noise. The noise is harder to detect; is is also more concentrated around the figure of the zebra.
	<br>		<br>
	[[File: reconstrRandW.jpg \| Figure 27]]		[[File: reconstrRandW.jpg \| Figure 27]]
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	= References/Further reading =		= References/Further reading =
	~~Heuttel~~		Huettel, Scott A., Andrew W. Song, and Gregory McCarthy. ''Functional Magnetic Resonance Imaging'', 2nd Edition. Sunderland, Massachusetts: Sinauer Associates, 2009.
	Hornak		Hornak, Joseph P. [http://www.cis.rit.edu/htbooks/mri/inside.htm ''The Basics of MRI''].
	Oppenheim		Oppenheim, Alan V., and Ronald W. Schafer. ''Discrete-Time Signal Processing'', 2nd Edition. Prentice Hall Signal Processing Series. Upper Saddle River, New Jersey: Prentice Hall, 1999.
	Smith		Smith, Julius O. [https://ccrma.stanford.edu/~jos/mdft/ ''Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications''], 2nd Edition. W3K Publishing, 2007.

imported>Psych 204 at 01:32, 11 December 2009

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	The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/interp1.html interp1] function can accomplish this. For our purposes, it will take in the original input vector (time-index integers), original output vector (the sinusoid), and the modified input vector (jittered time indices), outputting the sinusoid values at those jittered time indices.		The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/interp1.html interp1] function can accomplish this. For our purposes, it will take in the original input vector (time-index integers), original output vector (the sinusoid), and the modified input vector (jittered time indices), outputting the sinusoid values at those jittered time indices.
	<br>		<br>
	In the figure below, we can see the original sinusoid (in blue) and the sinusoid at the jittered time values (in red). The jittered samples are being displayed at the time points ''at which they were sampled'', which is why the shape of the sinusoid ~~still looks normal~~.		In the figure below, we can see the original sinusoid (in blue) and the sinusoid at the jittered time values (in red). The jittered samples are being displayed at the time points ''at which they were sampled'', which is why the overall shape of the sinusoid remains unchanged.
	<br>		<br>
	[[File:interpJitter.jpg \| Figure 20]]		[[File:interpJitter.jpg \| Figure 20]]
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	[[File: scaledLPFnoise.jpg \| Figure 26]]		[[File: scaledLPFnoise.jpg \| Figure 26]]
	<br>		<br>
	Here is the reconstructed image with this random-walk noise. The noise is harder to detect; is ~~is also more~~ concentrated around the figure of the zebra.		Here is the reconstructed image with this random-walk noise. The noise is harder to detect; if you look closely you will see that it is concentrated around the figure of the zebra.
	<br>		<br>
	[[File: reconstrRandW.jpg \| Figure 27]]		[[File: reconstrRandW.jpg \| Figure 27]]

imported>Psych 204 at 01:30, 11 December 2009

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	== Cell XII. Artifact 5: Sampling jitter ==		== Cell XII. Artifact 5: Sampling jitter ==
	The final artifact presented in the tutorial involves the effect of [http://en.wikipedia.org/wiki/Jitter jitter]. For imaging purposes, this refers to the scenario of the image being acquired (sampled) at frequencies that differ from the expected integer frequencies, by small amounts. Rather than being sampled at frequencies <math> 1, 2, \dots</math>, it may have in fact been sampled at frequencies <math>1+\delta_1, 2+\delta_2, \dots</math>, where the <math>\delta</math> values refer to small offsets (like -0.1, 0.05, etc). This will cause problems during image reconstruction as the ifft function does not know that the image was sampled at these non-integer frequencies, and ~~assumes~~ integer ~~frequency indices~~.		The final artifact presented in the tutorial involves the effect of [http://en.wikipedia.org/wiki/Jitter jitter]. For imaging purposes, this refers to the scenario of the image being acquired (sampled) at frequencies that differ from the expected integer frequencies, by small amounts. Rather than being sampled at frequencies <math> 1, 2, \dots</math>, it may have in fact been sampled at frequencies <math>1+\delta_1, 2+\delta_2, \dots</math>, where the <math>\delta</math> values refer to small offsets (like -0.1, 0.05, etc). This will cause problems during image reconstruction as the ifft function does not know that the image was sampled at these non-integer frequencies, and evaluates each sample as though it were on an integer index.
	<br>		<br>
	=== One-dimensional time-domain example ===		=== One-dimensional time-domain example ===
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	<br>		<br>
	In the figure below, we can see the original sinusoid (in blue) and the sinusoid at the jittered time values (in red). The jittered samples are being displayed at the time points ''at which they were sampled'', which is why the shape of the sinusoid still looks normal.		In the figure below, we can see the original sinusoid (in blue) and the sinusoid at the jittered time values (in red). The jittered samples are being displayed at the time points ''at which they were sampled'', which is why the shape of the sinusoid still looks normal.
			<br>
	[[File:interpJitter.jpg \| Figure 20]]		[[File:interpJitter.jpg \| Figure 20]]
	<br>		<br>

← Older revision		Revision as of 06:34, 11 December 2009
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	= Project Title - MATLAB image processing tutorial =		= Project Title - MATLAB image processing tutorial =
	This page accompanies a MATLAB tutorial which simulates some artifacts that might be found in MR images. Their causes, and in some cases corrections, are demonstrated in the code. One- and two-dimensional examples are provided.		This page accompanies a [http://www.mathworks.com/ MATLAB] tutorial which simulates some artifacts that might be found in MR images. Their causes, and in some cases corrections, are demonstrated in the code. One- and two-dimensional examples are provided.
	<br>		<br>
	Image artifacts in this tutorial include DC Offset, Quadrature Ghosting, RF Noise, Gradient, and Frequency-Sampling Jitter. Information on the first four topics can be found in [http://www.cis.rit.edu/htbooks/mri/inside.htm Hornak, Joseph P. ''The Basics of MRI''], Chapter 11. More information on the mathematical derivations can be found in [https://ccrma.stanford.edu/~jos/mdft/ Smith, Julius O. ''Mathematics of the Discrete Fourier Transform (DFT)'']. ~~A full reference list can be found~~ at the ~~end~~ of this page.		Image artifacts in this tutorial include DC Offset, Quadrature Ghosting, RF Noise, Gradient, and Frequency-Sampling Jitter. Information on the first four topics can be found in [http://www.cis.rit.edu/htbooks/mri/inside.htm Hornak, Joseph P. ''The Basics of MRI''], Chapter 11. More information on the mathematical derivations can be found in [https://ccrma.stanford.edu/~jos/mdft/ Smith, Julius O. ''Mathematics of the Discrete Fourier Transform (DFT)'']. These and other references are listed at the bottom of this page.

	= About the tutorial =		= About the tutorial =
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	== Cell II. Using fftshift ==		== Cell II. Using fftshift ==
	The [http://www.mathworks.com/access/helpdesk_r13/help/techdoc/ref/fftshift.html fftshift] function is helpful for ~~visualing~~ Fourier transforms in a familiar format. In the 1-dimensional N-point Fourier transform (like the transform of an audio signal), it takes points N/2+1 to N (containing the Nyquist and negative frequency components) and moves them to the beginning of the array. See Oppenheim and Schafer, Chapter 4, if you want to know ~~why this can happen~~. Thus, the fftshifted function causes the ~~poisitive~~ and negative frequencies to be symmetric about the zero-frequency point. This being the case, it's helpful to keep in mind that the zero-frequency sample is at sample N/2+1.		The [http://www.mathworks.com/access/helpdesk_r13/help/techdoc/ref/fftshift.html fftshift] function is helpful for visualizing [http://en.wikipedia.org/wiki/Fourier_transform Fourier transforms] in a familiar format. (Note that for this tutorial we will be using the [http://en.wikipedia.org/wiki/Discrete_Fourier_transform discrete Fourier transform].) In the 1-dimensional N-point Fourier transform (like the transform of an audio signal), it takes points N/2+1 to N (containing the Nyquist and negative frequency components) and moves them to the beginning of the array. See Oppenheim and Schafer, Chapter 4, if you want to know more about repeated spectrums due to sampling. Thus, the fftshifted function causes the positive and negative frequencies to be symmetric about the zero-frequency point. This being the case, it's helpful to keep in mind that the zero-frequency sample is at sample N/2+1.
	<br>		<br>
	Here is a 1-dimensional example. The red dot is the zero-frequency component of the transform of the sinusoid in the top subplot.		Here is a 1-dimensional example. The red dot is the zero-frequency component of the transform of the sinusoid in the top subplot.
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	[[File:sinusoidFftshift.jpg \| Figure 2]]		[[File:sinusoidFftshift.jpg \| Figure 2]]
	<br>		<br>
	In the 2-dimensional case, fftshift swaps quadrants 1 and 3, and also 2 and 4 of the Fourier transform. Thus, the low-frequency information is moved to the center of the image, with the [0,0] frequency point moving from matrix index [1,1] to [nPix/2+1,nPix/2+1]. The fftshifted Fourier transform of an image is what we are used to looking at when we talk about [http://en.wikipedia.org/wiki/K-space_%28MRI%29 k-space]. For the purpose of this tutorial, ''time domain'' is the 1-dimensional case of ''image domain'', and ''k-space'' is specifically the 2-dimensional case of ''frequency domain''.		In the 2-dimensional case, fftshift swaps quadrants 1 and 3, and also 2 and 4 of the Fourier transform. Thus, the low-frequency information is moved to the center of the image, with the [0,0] frequency point moving from matrix index [1,1] to [nPix/2+1,nPix/2+1]. The fftshifted Fourier transform of an image is what we are used to looking at when we talk about [http://en.wikipedia.org/wiki/K-space_%28MRI%29 k-space]. For the purpose of this tutorial, [http://en.wikipedia.org/wiki/Time_domain ''time domain''] is the 1-dimensional case of ''image domain'', and [http://en.wikipedia.org/wiki/K-space_%28MRI%29 ''k-space''] is specifically the 2-dimensional case of [http://en.wikipedia.org/wiki/Frequency_domain ''frequency domain''].
	<br>		<br>
	Here is a 2-d Fourier transform without fftshift. Note the bright spots (high amplitudes) in the outer corners of the image; these are the low frequencies.		Here is a 2-d Fourier transform without fftshift. Note the bright spots (high amplitudes) in the outer corners of the image; these are the low frequencies.
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	=== Using ifftshift ===		=== Using ifftshift ===
	The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ifftshift.html ifftshift] function reverses the effect of fftshift (use this instead of performing fftshift twice). Like fftshift, ifftshift is always be applied the frequency-domain signal in this tutorial:		The [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ifftshift.html ifftshift] function reverses the effect of fftshift (use this instead of performing fftshift twice). Like fftshift, ifftshift is always applied to the frequency-domain signal in this tutorial:
	:x=ifft(ifftshift(X));		:x=ifft(ifftshift(X));

	== Cell III. Artifact 1: DC offset ==		== Cell III. Artifact 1: DC offset ==
	=== Derivation ===		=== Derivation ===
	The DC offset is a constant, positive shift in amplitude. It results in a large peak at frequency zero. This can be derived from the equation for the DFT. Recall that the DFT is defined as		The [http://en.wikipedia.org/wiki/Dc_offset DC offset] is a constant, positive shift in amplitude. It results in a large peak at frequency zero. This can be derived from the equation for the DFT. Recall that the DFT is defined as
	:<math>X_k=\sum_{n=0}^{N-1} x_n e^{-\frac{j 2 \pi}{N} k n}\quad \quad k = 0, \dots, N-1 </math>		:<math>X_k=\sum_{n=0}^{N-1} x_n e^{-\frac{j 2 \pi}{N} k n}\quad \quad k = 0, \dots, N-1 </math>
	where <math> x </math> is the time signal at times <math> n </math> and <math> X </math> is the Fourier transform at frequency indices <math> k </math>. The DFT of a signal <math>x</math> plus a constant DC offset <math> \Delta </math> would then be		where <math> x </math> is the time signal at times <math> n </math> and <math> X </math> is the Fourier transform at frequency indices <math> k </math>. The DFT of a signal <math>x</math> plus a constant DC offset <math> \Delta </math> would then be
	:<math>X_k=\sum_{n=0}^{N-1} (x_n+\Delta) e^{-\frac{j 2 \pi}{N} k n} \quad \quad k=0, \dots, N-1 </math>		:<math>X_k=\sum_{n=0}^{N-1} (x_n+\Delta) e^{-\frac{j 2 \pi}{N} k n} \quad \quad k=0, \dots, N-1 </math>
	Since this is a linear system, we can separate the DC component from the original signal.		Since this is a [http://en.wikipedia.org/wiki/Linear_system linear system], we can separate the DC component from the original signal.
	:<math>X_k=\sum_{n=0}^{N-1} x_n e^{-\frac{j 2 \pi}{N} k n } + \sum_{n=0}^{N-1} \Delta_n e^{-\frac{j 2 \pi}{N} k n} \quad \quad k=0, \dots, N-1 </math>		:<math>X_k=\sum_{n=0}^{N-1} x_n e^{-\frac{j 2 \pi}{N} k n } + \sum_{n=0}^{N-1} \Delta_n e^{-\frac{j 2 \pi}{N} k n} \quad \quad k=0, \dots, N-1 </math>
	We are interested in what is going on at frequency <math>k=0</math>. Assuming that the first term does not have some other DC artifact, we know that its zero-frequency component should be zero (this has to do with [https://ccrma.stanford.edu/~jos/mdft/Orthogonality_DFT_Sinusoids.html orthogonality of the DFT sinusoids]). Focusing on the second term, since we are considering frequency <math>k=0</math>, the exponential is equal to <math>1</math> for all <math>n</math>. Thus, the amplitude of the transform at this frequency is the sum of every term in the DC signal.		We are interested in what is going on at frequency <math>k=0</math>. Assuming that the first term does not have some other DC artifact, we know that its zero-frequency component should be zero (this has to do with [https://ccrma.stanford.edu/~jos/mdft/Orthogonality_DFT_Sinusoids.html orthogonality of the DFT sinusoids]). Focusing on the second term, since we are considering frequency <math>k=0</math>, the exponential is equal to <math>1</math> for all <math>n</math>. Thus, the amplitude of the transform at this frequency is the sum of every term in the DC signal.
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	== Cell XII. Artifact 5: Sampling jitter ==		== Cell XII. Artifact 5: Sampling jitter ==
	The final artifact presented in the tutorial involves the effect of [http://en.wikipedia.org/wiki/Jitter jitter]. For imaging purposes, this refers to ~~the scenario of~~ the image being acquired (sampled) at frequencies that differ from the expected integer frequencies~~, by small amounts~~. Rather than being sampled at frequencies <math> 1, 2, \dots</math>, it may have in fact been sampled at frequencies <math>1+\delta_1, 2+\delta_2, \dots</math>, where the <math>\delta</math> values refer to small offsets (like -0.1, 0.05, etc). This will cause problems during image reconstruction as the ifft function does not know that the image was sampled at these non-integer frequencies, and evaluates each sample as though it were on an integer index.		The final artifact presented in the tutorial involves the effect of [http://en.wikipedia.org/wiki/Jitter jitter]. For MR imaging purposes, this refers to the image being acquired (sampled) at frequencies that differ by small amounts from the expected integer frequencies. Rather than being sampled at frequencies <math> 1, 2, \dots</math>, it may have in fact been sampled at frequencies <math>1+\delta_1, 2+\delta_2, \dots</math>, where the <math>\delta</math> values refer to small offsets (like -0.1, 0.05, etc). This will cause problems during image reconstruction as the ifft function does not know that the image was sampled at these non-integer frequencies, and evaluates each sample as though it were on an integer index.
	~~<br>~~
	=== One-dimensional time-domain example ===		=== One-dimensional time-domain example ===
	Here is the sinusoid from earlier, and its Fourier transform. The time index starts from 1.		Here is the sinusoid from earlier, and its Fourier transform. The time index starts from 1.
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	= Appendix 2: Acknowledgements =		= Appendix 2: Acknowledgements =
	I received help from Jonathan Abel (Consulting Professor, CCRMA) with some of the derivations, and also with the ~~details~~ of implementing the jitter artifact.		I received help from Jonathan Abel (Consulting Professor, CCRMA) with some of the derivations, and also with the conceptualization of implementing the jitter artifact.
	<br>		<br>
	The Psych 204 instructors and TA offered helpful and practical suggestions toward improving and expanding the tutorial. Thank you!		The Psych 204 instructors and TA offered helpful and practical suggestions toward improving and expanding the tutorial. Thank you!

	= References/Further reading =		= References/Further reading =
	Huettel, Scott A., Andrew W. Song, and Gregory McCarthy. ''Functional Magnetic Resonance Imaging'', 2nd Edition. Sunderland, Massachusetts: Sinauer Associates, 2009.		Huettel, Scott A., Andrew W. Song, and Gregory McCarthy. [http://www.amazon.com/Functional-Magnetic-Resonance-Imaging-Second/dp/0878932860/ref=sr_1_1?ie=UTF8&s=books&qid=1260512248&sr=1-1 ''Functional Magnetic Resonance Imaging'', 2nd Edition]. Sunderland, Massachusetts: Sinauer Associates, 2009.
	<br>		<br>
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