In the summer of 2012, my young son had a brain MRI to check for a chiari malformation. (He has a long medical history, due to inheriting a craniofacial syndrome from me.) After having been through several CT scans in his short life, I was accustomed to worries about sedation so that he would be perfectly still for the imaging. I was very surprised when we arrived at the hospital, for they did not need to sedate, and the scan was not going to take very long. He needed the short version. (The long version would have still required sedation.) We went through the scan and waited about an hour for the results. Thankfully, there was no chiari malformation. He did have to stay in the hospital, as we were also there for ICP (intracranial pressure) monitoring and a skull surgery. He “just” needed a bone graft and not a full cranial-vault remodeling, and we now knew this from the MRI results.

In the months and year that followed, I didn’t think too much about the speed of the MRI, until a colleague came into my office and shared about Terry Tao and compressed sensing. Terry Tao is a person who appears frequently in mathematical news, as he is the top mathematician of our age. Typically, a mathematician picks one small area in which to be an expert and focuses on that for a career. Terry Tao is rare in that he can pick up a topic of interest, quickly become an expert, and then make large contributions to that area of mathematics. This just gives you a hint of why my colleague was sharing about Terry Tao. Anyhow, I have digressed and must get back to compressed sensing.

What is compressed sensing, and how was it possibly related to my son’s MRI? Compressed sensing involves an algorithm that utilizes l1 minimization. It takes too much time and storage space to collect every pixel of data for an image. So, a camera or other device collects a fraction of these pixels. The l1 algorithm starts arbitrarily picking effective ways of filling in the missing pixels. Then, the algorithm starts putting in layers of colored shapes over the randomly chosen pixels while seeking sparsity. It wants to use the simplest kinds of shapes to closely match the existing pixels. Each new layer will have smaller and smaller shapes. Eventually, with enough layers, the resulting image will be extremely close to the original. The simplest or sparsest image is the closest to the original.

So, collecting the data during the MRI scan does not take as long, as fewer pixels are needed. Processing takes a little while, because the l1 algorithm needs time to work. I don’t know if this discovery was in place for my son’s MRI, but this story did give me something to relate to. His scan took only a few minutes, and we had to wait about an hour to see the results. All but one picture was clear, and we were relieved that he does not have a chiari malformation.

I am eager to see what all developments will come out of this discovery. Emmanuel Candes, Justin Romberg, and Terry Tao have laid groundwork and proven mathematically that the resulting image after running the l1 algorithm will be extremely close to the original. Now, people are looking at all kinds of applications. Besides constructing medical images quickly, people may restore old files, construct images of space, eliminate the need for compression software, and accomplish much more with speed and accuracy.

If you would like to read more, here are 3 articles that helped me to gain better understanding.

Fill in the Blanks: Using Math to Turn Lo-Res Datasets into High-Res Samples

Better Math Could Make Medical Diagnostics 6X Faster

Compressed Sensing Makes Every Pixel Count