Mathematical Imaging

Maybe you’re an avid traveler. Maybe you’ve tried to order a Big Mac combo in English in Quebec. Maybe you’re just a student that makes his way around this multicultural campus enough. One way or the other, you know that in every place where there are people, language is the key to communication.

So what if there was one language that enabled you to communicate with people of every tongue? In an interview with his research team, Dr. Edward Vrscay, a professor in the department of Applied Mathematics, gave Imprint a glimpse of just that. Beyond mere multilingualism, Vrscay and his graduate students have uncovered the secret to universal communicationimages.

“The images speak for themselves,” Vrscay said very simply.

And it’s the mathematics behind the .jpg that allows it to be universally interpreted.

“When you download an image from the internet,” Vrscay explained, “that .jpg is based on what’s called the Fourier series.”

The Fourier series is a branch of mathematics involving the decomposition of periodic functions into a sum of simple oscillating functions, such as the sine and cosine. The discrete cosine transform allows images to be compressedwhich is how we see them on the Internetreducing file size without sacrificing too much of the quality.

Still, Vrscay and his team are not just talking the languagethey’re developing it, too. Combining theoretical models with applications, his graduate students are finding concrete ways to make use of their math.

Nataliya Portman, a graduate student under Vrscay’s supervision, has been modeling biological growth based on image data, a sequence of images depicting a growing organism. The ultimate goal, she told Imprint, is to classify the observed growth pattern as either healthy or unhealthy.

“For this, I need quantitative characteristics of growth,” Portman said. “And that requires calculations of changes in the area or volume of a growing organism, [both of which] can be extracted from images.”

Later, Portman elaborated on the mathematical basis of such an extraction and its ability to actually give an idea as to the actual cell location of abnormal growth.

“We are trying to figure out the location of this active cell that has caused the change that we observe,” she said.

She said it is the inverse problems based on shape changes that tell something about the source of the change.

“What has caused this change? It’s a hidden source, basically. We do a [grey-scale value] seed in an image, and based on this grey-scale information, as well as the shape of the organism, [we can extract the shape information] from the image using image-processing tools.

“The locations of these cells are being modeled as random seeds, and if I see one is active, that basically means that this gene is turned off at this location and causes a certain structural change.”

In terms of biological accuracy, the beauty of Portman’s model lies in its ability to demonstrate the one-to-one correspondence between gene expression pattern and the observed shape change. The probabilistic model also takes into account genetic variability.

“If you’re looking at the development of a mouse brain, you don’t look at just one brain in particular. You look at a family of brains at a certain developmental stage,” she explained. “The shape of the brain is subject to variability. The brains are not identical.

“This variability is taken into account in our model [and is] the stronger point of our model.”

Greg Mayer is also a graduate student under Dr. Vrscay’s supervision, and is on what Vrscay calls his “super-resolution team”. Like Portman, Mayer’s research in mathematical models is also versed in medical applications. His work focuses on image processing algorithms, working on various spatial resolution enhancement algorithms to improve the image quality of Magnetic Resonance Imaging (MRI) data.

Explained Dr. Vrscay: “Greg wants to work with the raw data that comes out of the machine, which is frequency data, as opposed to the already-altered image that comes out of the MRI machine.”

“In MRI, you have limitations on what the machine hardware can do in terms of resolution,” explained Mayer. “And there’s only so much time and so much the machine can produce in terms of spatial resolution.

“There are only so many pixels that can be put into these digital images, and what we’re trying to do is get more pixels.

“Without the understanding of how the physics determines [the creation of] the image, you lose a lot of information,” stressed Mayer. “In just treating it as a general image, without that knowledge of where it’s coming from, you throw out a lot of information and constraints that you can use to determine whether or not an algorithm is even going to work.”

Ultimately, from modeling biological growth to improving MRI data quality, Portman spoke to the need to develop mathematical models that “serve a certain purpose”.

“For biomedical research it is important to diagnose a certain developmental disease, a deformed pattern, a pathology,” she emphasized. “How is it possible to recognize pathological growth based on imaging? We are addressing this question.”

“[As applied mathematicians], we have to understand pure math but we also have to find a concrete application, proving theorems, finding algorithms,” said Dominique Brunet, Applied Mathematics graduate student and new addition to Vrscay’s team.

“We’re straddling the line between theory and application,” said Vrscay.

And Vrscay’s team is well aware that this “straddling” is not immune to difficulties.

“One of the challenges in applied math, is that you’re either speaking to people who are experts in the mathematics, not necessarily in your application, or vice versa, the application but not the mathematics,” said Mayer. “So you have to prepare your presentation in a way that people can understand.”

For Portman, the challenges lie in “choosing an appropriate language of communication that both sides understand really well.”

“I make big efforts to communicate with biologists and I’m still learning how to do this in an effective way,” she admitted. “But mostly, I can feel the distance between another professional biologist. There’s still a long way to go.”

Still, Vrscay’s team remains optimistic about the future.

“The thing that’s good with imaging is that you can at least show the image you get,” Brunet pointed out. “If you do pure math, abstract math, then it’s really impossible to show what you are doing.”

“Mathematics is really the language of Science,” said Vrscay.

“A picture speaks a thousand words. An image speaks a thousand theorems.”