Don't ask me why there's no apostrophe, it looks possessive to me. Perhaps it's (the) Gibbs artifact rather than Gibbs (his) artifact. Most people simply refer to the effect as ringing anyway, so let's move on. This post concerns a phenomenon that, like aliasing last time, isn't unique to EPI but is a feature of all MRIs that are obtained via Fourier transformation.
In short, ringing is a consequence of using a period of analog-to-digital conversion in order to apply a (discrete) FT to the signals and produce a digital image. Or, to put it another way, we are using a digital approximation to an analog process and thus we can never properly attain the infinite resolution that's required to fully represent every single feature of a real (analog) object. Ringing is an artifact that results from this imperfect approximation.
We had already encountered one consequence of digitization in the Nyquist criterion in PFUFA Part Six. However, for our practical purposes, ringing isn't a direct consequence of digitization like the Nyquist criterion, but instead results from the duration of the digitization (or ADC) period relative to the persistence of the signals being measured. In principle, a signal decaying exponentially decays forever, which is rather a long time to wait for the next acquisition in a time series, so we instead enable the ADC for a window of time that coincides with the bulk - say 99% - of the signal, then we turn it off. This square window imposed over the exponentially decaying signal causes some degree of truncation, and it's this truncation that leads to ringing. (See Note 1.)
An example of ringing in EPI of a phantom
Let's start with an unambiguous example of ringing by looking at the artifact in a homogeneous, regular phantom. Below is a 64x64 matrix EPI acquired from a spherical gel-filled phantom. You're looking for the wave-like patterns set up inside and outside the edges of the main signal region:
In the left image, which is contrasted to highlight ringing artifacts within the signal region itself, the primary ringing artifact appears as a series of concentric circles, each with progressively smaller diameter and lower intensity as you move in from the edge of the phantom. One section of the bright bands is indicated with a red arrow, but you should be able to trace these circles all the way around the image. Also visible is a strong interference pattern (blue arrows) that arises between the aforementioned ringing artifact and the overlapping N/2 ghosts. This is because the ghosts maintain the contrast properties of the main image; they are, after all, simply weak (misplaced) clones of the main image.
The ringing pattern in the ghosts is very easily discerned once the background contrast is brought up, in the right image above. The orange arrow indicates a clear ringing artifact in the ghost, and a little bit of mental interpolation should be sufficient to comprehend how that artifact will persist in those sections of the ghosts that overlap with the primary phantom image, leading to the interference pattern just discussed.
With the background contrasted high we can now also see that the ringing artifact extends outside of the main phantom image; it's not just circles and patterns within the signal region. The top yellow arrow on the right image highlights one clear region of ringing extending into what should be noise. There are other similar regions easily discerned on all sides of the main signal region. What's more, the ringing artifact that extends away from the sample also generates its own N/2 ghosts, as indicated by the lower yellow arrow. (The N/2 ghosts will replicate all the features of the parent image.)
So, what's happening here? As a general rule you can expect to see ringing artifacts whenever spatial resolution is low and/or there is a strong contrast boundary, as is the case for the bright circular signal region in the image above. However, in the absence of any sort of filtering (see below), it is actually the case that all images, whether 2D or 3D, will exhibit ringing. The question isn't whether it's there or not, it's whether you can see it with the naked eye or not. The reason is that although we think of the individual pixels within an image as being square (or rectangular), this isn't actually the spatial response that comes out of the Fourier transform.
What causes ringing?
As we saw in PFUFA Part Six with an FID, if we're going to use a digitizer (an analog-to-digital converter, or ADC) to acquire k-space samples then at some point it's going to have to be turned on, and at some later point it's going to have to be turned off. It's a window in time. However, in principle, once a coherent magnetization (an observable MR signal) has been generated in your subject's brain, this signal will persist (theoretically) for infinity as it decays with time constant T2*. I'm sure you see the potential conflict. As we saw in the fifth cartoon in PFUFA Part Six, turning off the ADC "too soon," while the signal is still appreciably above the noise level, results in wiggles that spectroscopists call feet; hence the term apodization for getting rid of them.
Exactly the same phenomenon occurs in imaging. Now, to comprehend the situation for imaging we first need to recall that each signal in the EPI echo train is a gradient echo, i.e. a signal peak. Either side of the signal peak the signal changes (exponentially) as some function of the dephasing imparted by the imaging gradient plus the inherent T2* relaxation. (See PFUFA Part Eight, first figure, for a review of a single gradient echo.) Still, the signal hasn't necessarily been driven down into the noise level at either side of the echo. Any signal that persists at either side of the ADC period will be "clipped."
Here's an illustration of ringing for a one-dimensional image (a readout profile) through a homogeneous cubic phantom, resulting in what should be a square 1D profile. For the purposes of the illustration I've also included cartoon noise so as to differentiate the ringing effect from random wiggles. In the left column are the k-space (or time) domain representations while in the right column are the image (or frequency) domain responses produced from 1D FT of the left partners. The top row shows the ideal case, the bottom row shows the response from the ADC period by itself, and the middle row shows the experimental result: a time domain signal that has been "clipped" at either end by the ADC period, resulting in a 1D profile that has sinc-like wiggles at edges:
|Please note that the slight discontinuity in the center of the top-right profile was a slip of my pen! It's supposed to be a flattish profile with noise. Consider the discontinuity as a large noise spike!|
You will remember from PFUFA Part Six that multiplication in one domain - here the k-space domain - is equivalent to convolution in the conjugate domain (image space). The sinc response from the ADC period convolves with the ideal profile to generate pronounced waves at the edges of the signal, both extending into the signal itself and into the adjacent "noise" (or no signal) regions around the phantom.
Now, there's something subtle going on here that can help you understand why you see ringing in certain regions of an image and not others. In PFUFA Part Eleven you saw how different spatial frequencies within an object, such as a brain (or a Hawker Hurricane aeroplane), "live" in different parts of reciprocal, or k-space. High spatial frequencies - edges, fine details - live in the periphery of k-space. Now think about the k-space trajectory for the 1D gradient echo that you saw in PFUFA Part Nine. The center of k-space - where all the low spatial frequencies live - is by definition in the center of the k-space trajectory, and in the center of the ADC period (if we're being smart and digitizing the entire k-space line!). What lives at the either end of the k-space trajectory, the parts that are going to get chopped off (clipped) first, should there be appreciable signal as the ADC switches on or off? That's right, the high spatial frequencies! Thus, the act of clipping the signal affects the high spatial frequencies - edges - more than it does the low spatial frequencies. And so the strongest effects of ringing tend to be detectable where there are fine details or boundaries between high and low intensity signals.
An analogous situation arises for the phase encoding dimension. We define an effective digitization window and, as in the frequency encoding axis, increasingly large values of k produce signal attenuation as well as spatial encoding. If infinite k values in the phase encoding dimension were obtainable then, as for the readout direction, there would be no "clipping" and hence no ringing. But that isn't feasible in practice and so we have ringing in the phase encoding dimension in just the same way as in the readout dimension. That's why the real example of ringing at the top of this post was so symmetric.
Can ringing be reduced or fixed?
In PFUFA Part Eleven we saw how extending k-space increased image resolution, and that restricting the extent of k-space resulted in low resolution images. It's important to remember that the k-space matrix is, as its name implies, a discrete space but we are using it to map something that is continuous in space. Real brains aren't made up of pixels!
Increasing image resolution makes the absolute distance affected by ringing lower, because the pixels are getting smaller. But this doesn't actually fix the problem; it tends to restrict the spatial extent (which may be all you need to do). Let's take a look at the effects of resolution on the spatial extent of ringing by comparing three different resolutions of EPI in a spherical phantom:
|Left: 64x64 matrix. Middle: 96x96 matrix. Right: 128x128 matrix. (Click to enlarge.)|
Parameters other than the resolution were held constant as far as possible, but to get to the higher resolution it was necessary to increase the echo spacing (thereby increasing the distortion) and also increase the TE (thereby decreasing the signal-to-noise ratio) slightly.
Still, increasing the resolution to 128x128 hasn't eliminated the ringing, just reduced the spatial extent and made it harder to detect by eye. If you wanted to reduce the ringing further then even higher resolution would help, or you could apply a smoothing function to the data. I don't want to get side-tracked today with smoothing because this post is on recognizing the artifact, not fixing it. I may do an exhaustive post on smoothing another day but for now see Note 2 for some thoughts on smoothing.
What does ringing look like in real data?
Compared to the slice through a spherical phantom above, detecting ringing in even low-resolution (64x64 matrix) EPI of brains can be quite a challenge, perhaps leading you to wonder why I'm bothering to do an entire post on the subject. Well, it's because as I've already said: ringing is ubiquitous and I want you to be able to differentiate the relatively benign ringing artifact from other, more important artifacts for fMRI, those where timely intervention might make the difference between good and not so good data quality.
To ease the transition from EPI of phantom to EPI of brain, I'm going to take a short detour via regular "spin warp" imaging of brain. In these anatomical scans the SNR is generally a lot higher than for EPI with the same matrix (because TE is much lower and each phase encoding line is acquired after a new RF excitation, generating a signal averaging effect for the final image) and this tends to make the ringing artifact more easily detected. (There is also less inherent smoothing arising from T2* than for EPI, but I'll deal with this fact below in more detail.)
Here is an anatomical (gradient echo spin warp) 128x128 matrix 2D image through a brain with contrast to highlight anatomy (left) and background noise (right):
|(Click to enlarge.)|
The ringing both inside and outside of the brain is pretty easily recognized without the yellow arrows. The most prominent ringing artifacts arise from the black-white contrast border at the sides of the brain and from the scalp fat around the head.
Now you're tuned into seeing the wave-like patterns, see if you can make out similar patterns within the brain in these 64x64 matrix axial slices of EPI:
|(Click to enlarge.)|
It's considerably harder to see in the EPIs than in the (higher resolution) anatomicals above! Why should this be, given that earlier on in this post I showed that the low resolution should make the effects of ringing more visible, not less? There are two major reasons. Firstly, the SNR in the EPIs is five to ten times lower than in the anatomical scan. Thus, all other factors being equal, our ability to detect such a small feature is greatly reduced. We often don't have the SNR in EPI to tease out fine details, not even those that are artifacts! Secondly, in the phase encoding dimension of the EPI scans there is an effective smoothing function being "applied" by virtue of the T2* decay that is occurring simultaneously with the readout echo train. Recall from PFUFA Part Twelve that the practical limit for the number of echoes in our train is set by T2*; we're in a race to recycle and reuse the magnetization as many times as we can until T2* renders the signal level too low. The phase encoding dimension might well experience reduced ringing as a result. But not always.
Inferior regions of the brain suffer extensively from magnetic susceptibility gradients, leading to signal dropout as well as higher ghosting and distortion of any remaining signal. For this remaining signal, then, the presence of the susceptibility gradients will tend to interact with the desired, or ideal, k-space matrix and may have concomitant effects on ringing. We know that the signal maximum should occur at the center of k-space as defined by the readout and phase encoding imaging gradients. But, if a region experiences high magnetic susceptibility gradients these additional dephasing processes can cause the signal for that local region to attain its maximum early (if the susceptibility gradient adds to the phase encoding gradient) or late (if the susceptibility gradient tends to offset some of the phase encoding gradient). Furthermore, the displacement causes one side of the 2D signals to shift towards the edge of the assumed (ideal) k-space matrix until they start to "fall off" the plane, like this:
|Illustration of the effects of background magnetic susceptibility gradients on k-space. In this example the susceptibility gradients have caused the early refocusing of signals along the phase encoding dimension, ky.|
It should be fairly obvious that signals located towards the bottom of this k-space matrix have been "pushed off" the sampled (ideal) k-space plane by magnetic susceptibility gradients and would therefore be truncated (clipped) more severely than they should have been, had the signals been properly centered at the k-space origin. This will cause parts of the resulting image - those having appreciable k-space representation in the clipped signal - to exhibit increased ringing.
Now, in my cartoon example above I shifted the signals for the entire slice earlier (down) in ky. That may happen in an experimental situation, but it's not common. What is more likely to occur is the shift of signals from just a localized region, with the remainder of the signals staying closer to the intended k-space origin. This results in some parts of the slice exhibiting pronounced ringing but not others, as in this example:
|An inferior axial oblique EPI slice showing localized ringing.|
Understanding why this ringing should be localized should be relatively straightforward. After all, dropout is usually localized, too. The magnetization residing in these poorly shimmed regions has managed to survive the aggressive dephasing of the susceptibility gradients to the point where it will yield appreciable (non-zero) signal in the final image, but the environment is so contaminated by poor magnetic field homogeneity that the signals may be very disfigured indeed, often being characterized by excessive distortion, high N/2 ghosting and localized ringing.
Is ringing a problem for fMRI?
As with many artifacts in fMRI, the answer to this question depends on where you are in the brain, but it also depends on what you subsequently do with your EPI time series in the fMRI analysis.
We have already noted that the ringing effects are quite subtle, both because of the inherent smoothing and because of the relatively low SNR (compared to anatomical scans). But we have been considering (raw) EPI images, not the spatial response that you use in your fMRI analysis necessarily. For example, there are several reasons to smooth EPI data that have nothing to do with ringing. It is common to want to coalesce "active" pixels in order to get reasonably sized activation "blobs" that pass a statistical threshold. (We usually assume that truly active pixels don't appear in isolation in image space, a reasonable assumption given the broad spatial response inherent to BOLD imaging.) Another common reason to smooth is to reduce the total number of pixels in the data set, thereby allowing a valid reduction in the aggressiveness of a Bonferroni correction for multiple comparisons. Whatever the motivation, even moderate smoothing will likely eliminate the mild ringing that might be barely visible in the raw EPIs.
For the most part, then, we tend not to worry specifically about Gibbs artifact in fMRI. If you're smoothing your data then you almost certainly have no need to be concerned with ringing further. But if you find that your EPIs are exhibiting substantially worse ringing than the examples presented here then it's time to go talk to your friendly facility physicist. There could be a problem with your acquisition parameters or even a problem with the imaging gradients (the hardware) themselves. (See Note 3.)
Ringing and motion
In this post I've dealt with the static effects of Gibbs ringing. In later posts we will be looking at the way motion tends to interact with artifacts such as ringing. Until those in-depth posts I will leave you with a brief observation. Contrast of all sorts, whether it's "real" anatomical contrast arising from T1/T2*/spin density differences in the brain tissues, or artifact banding from Gibbs ringing, will tend to exacerbate the effects of motion to some extent. So, if you wanted to be pedantic (and why not?), you could argue that the banding exhibited along the inside edges of the brain in the axial EPIs above would lead to enhanced degradation of temporal SNR compared to a situation where those artifacts weren't present. True enough. Except, of course, there are lots of other valid edges - most obviously the margins of the brain - that must be present and that will exhibit similar degradation of TSNR in the presence of motion. All you can do is apply motion correction, perhaps, and hope that the TSNR is returned.
Thus, it seems to me that to worry about ringing as a source of motion sensitivity is to overdo it. Motion is a concern for other, more critical reasons than the way it interacts with ringing. At least, that is my opinion today. If I find any information to the contrary I shall be sure to post it immediately!
1. There is another way to comprehend the ringing phenomenon: as a series of sinc functions building up ever sharper features of the real object. This is a perfectly valid, and technically correct, way to approach the phenomenon. However, I decided against using it as the description here because I've always thought that consideration of the square ADC window and the relative duration of the MR signals is a more useful approach for experimental technique. Once you have a sense of the relative durations of the sampling and signal periods then you can use that intuition to predict qualitatively the effects of increasing the magnitude of a spatial encoding gradient, say (it causes more rapid MR signal decay), or the effects of magnetic field heterogeneity (which also causes accelerated signal decay). There is a good explanation of the so-called "point spread function" and the true spatial resolution of an image in Rick Buxton's book, Introduction to Functional Magnetic Resonance Imaging, 2nd edition pages 220-231.
2. To fully understand the effects of smoothing requires consideration of the "point spread function," the mathematical relationship that defines the actual shape of pixels in an MRI. We'd like pixels to have ideal, rectangular shapes but they aren't. In the absence of smoothing the pixels are actually sinc-shaped, i.e. the majority of signal is located at the pixel's nominal position, but some signal from that (real) position extends away from the pixel into neighboring pixels. There is a little bit of spatial overlap.
As mentioned in Note 1, there is a good explanation of the PSF and the true nature of spatial resolution of an image in Rick Buxton's book, Introduction to Functional Magnetic Resonance Imaging, 2nd edition pages 220-231. Other textbooks deal with the PSF in some fashion, too. I encourage you to find one and read the relevant sections because I'm afraid I can't find a nice overview (for MRI) online. Perhaps I'll try to write one and append it to the PFUFA series as Part Thirteen.
Until a more comprehensive treatment, then, let's quickly go through some of the ways that ringing can be manipulated beyond simply changing the acquired matrix (which isn't usually an option in EPI for fMRI). Another alternative is to apply the imaging equivalent of "apodization" that was mentioned in PFUFA Part Six. In that post the ringing was shown for a one-dimensional example. But the same principles can be applied to 2D images. When it's applied to images we tend to refer to it as smoothing because in addition to reducing the ringing the effect is to blur, or smooth, the image, i.e. to reduce the absolute resolution in the image plane. That's the price of the fix.
Now, it may well be that you want to apply smoothing to your fMRI time series data anyway, e.g. to increase functional SNR, or to reduce the multiple comparisons problem. Thus, if you know ahead of time what level of smoothing you're going to use then you could, if you insisted, include a smoothing filter at the data acquisition stage. My personal opinion is that this isn't a good idea; just apply the smoothing offline. (Multiplication of a smoothing function in the k-space domain is equivalent to convolution of the FT of the smoothing function in the image domain. You can do either.) Then you have the luxury of testing different smoothing kernels to find an optimum, or deciding to run some data-driven analyses that could benefit from the highest possible spatial resolution, etc.
"Zero filling" is another commonly used trick to reduce the apparent effects of ringing. It's a form of interpolation in the image domain. Here, instead of using a 64x64 matrix of k-space values, the k-space matrix is extended, or padded, up to some higher matrix, say 128x128, by appending matrix values having zero signal level (and zero noise) to the acquired data. This extension of zeros adds no new information to the matrix, and it doesn't change the nominal resolution of the resulting image (i.e. the maximum acquired k-space values). But it does produce (in this example) twice as many pixels with which to define the image plane, thereby lessening the ringing effects detectable to the naked eye. Here's an illustration of the effects of zero filling on an image:
Both images have the same nominal resolution, but the right image has the illusion of better resolution because it doesn't appear pixellated.
A final note about zero filling: it's applied in the k-space domain, prior to Fourier transformation. Thus, it's unlikely that you will have much facility to use zero filling on the scanner, unless your center has the capacity to pipe the raw (k-space) data off the scanner and you can do all the signal processing steps, including 2D FT, offline. That's not the case for the vast majority of routine fMRI labs, and I wouldn't think that most sites would want to install such a pipeline for the purposes of zero filling. Something a lot more useful would need to provide the motivation! But given that any sort of interpolation in the spatial domain isn't common (as far as I know) in fMRI, it's more likely that you will use a simple smoothing function, such as a Gaussian smoothing kernel, offline and be done with the issue.
Incidentally, the difference between smoothing and interpolation? With interpolation you're adding data points, with smoothing you're throwing them away.
3. This discussion concerns fully sampled EPI, not partial Fourier or parallel accelerated (e.g. GRAPPA) variants of EPI. I will deal with these options separately, noting in passing that increased Gibbs ringing is one possible negative consequence of these data reduction methods. Thus, if your EPI does exhibit severe Gibbs ringing the very first thing to check is whether you're using a partial k-space technique and it's actually working as it should.