Today’s post is on the topic of motion contamination of fMRI data and how subject motion, motion correction and receive field contrast can conspire to create erroneous image-space signal and decrease temporal SNR (signal-to-noise ratio) of the EPI (echo planar image) time-series data used in fMRI (functional magentic resonance imaging). We have recently written a paper which reports on simulations of this RFC-MoCo (receive field contrast and motion correction) effect and we welcome feedback pertaining to the methods and results of the paper.

Today I will take a look at the basics of parallel magnetic resonance imaging (MRI) from a frames perspective. I know of only one publication [1] discussing frame-based encoding for MRI nevertheless I think that the mathematics of frames provides a solid foundation for understanding parallel imaging. In my opinion all practical or currently used methods of doing parallel imaging should be understood in terms of how well they can approximate a frame-based image reconstruction.

Back in the day there was but one coil element in a receiver designed for magnetic resonance imaging of the head. Modern commercial MRI scanners are now equipped with head-imaging receivers comprised of an array of 8 to 32 coil elements. These extra elements are primarily used for two purposes: (1) To improve signal-to-noise ratio (SNR), or (2) To decrease the time required to collect the imaging data (accelerated imaging). The term *parallel imaging* is usually associated with the second purpose but I will use the term to define imaging methods which simply make use of a receive array for either purpose (with or without acceleration). So what is a frame and what does this mathematical concept have to do with MRI?

This is my first official post on this new blog. As the name of the blog partially implies, the topic will be the mathematics and physics of neuroimaging. In particular the focus will be upon functional imaging of the brain. I will loosely define *functional imaging* as the spatial and temporal representation of any quantity associated with the activity of the brain’s neurons. I will therefore speak of images . The spatial and temporal scale (resolution) associated with the image will hopefully be adequate to form meaningful models of brain function or meaningful dynamic markers of disease.

The style of this blog will range from tutorial-like expositions of present functional neuroimaging technology to whimsical explorations of how we might create better functional neuroimaging technology. It is my hope that this blog will generate lots of detailed discussion in the comments section. My choice of the WordPress publishing platform was largely based upon this hope and the fact that mathematical symbols can be included with relative ease in both the posts that I make as well as the comments that readers may make. WordPress allows one to insert mathematical symbols in the comments section in the form of LaTex code. The simple directions for including LaTex code in comments can be found by following this link.

At the core of the mathematics behind any imaging technology lies an inverse problem. The goal of an *inverse problem *is to determine the source, in this case the image , from measurements of a signal that depends upon the source. The measurements and the source are related by some mathematical operator such that . Often is a linear operator. Some topics related to neuroimaging that I imagine will surface repeatedly will be:

- The well-posedness (in the Hadamard sense) of the inverse imaging problem,
- The robustness of the inverse problem to perturbations (like subject motion),

- Imaging artifacts and how they might be corrected,

- imaging resolution and its limits,

- imaging signal-to-noise ratio.

There will understandably be a heavy focus upon magnetic resonance imaging (MRI) and in particular upon echo-planar imaging (EPI) since the EPI sequence is the workhorse of functional MRI (fMRI). But I will also be interested in magnetoencephalograhy (MEG) and electroencephalography (EEG) even though these technologies can not, according to the definition given above, be considered to be functional imaging technologies. In addition, as mentioned above, I hope to look beyond the present neuroimaging technologies and entertain fledgling ideas that may someday allow us to measure neuronal activity on spatial and temporal scales that may be more conducive to forming models of brain function.

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