2017 marked the 100th anniversary of the Radon transform

1917-2017-100-years-of-the-Radon-transform

God certainly inhabits the honest work of believers and unbelievers alike and uses it to ends far beyond the original vision of those workers! And how often this has happened in mathematics!

— Dale McIntyre, PhD

The year 2017 celebrated the 100th anniversary of an important and unique yet somewhat obscure mathematical operator that came to be known as the Radon transform — aptly named for its discoverer, the Austrian mathematician Johann Radon. In 1917, Radon introduced this transform early in his career in a paper titled “On the Determination of Functions from Their Integral Values along Certain Manifolds.”

On a technical note, transform denotes special meaning in math. Intuitively, however, we may describe a mathematical transformation as an operation that changes the mathematical nature of some shape, entity, or thing. With this concept in mind, the Radon transform is defined as an integral transform, which means it makes use of integration to achieve this purpose.

Now, when we think of integration (and an integral), our thoughts lean us towards calculus (with its two main divisions, differentiation and integration, divisions which constitute differential calculus and integral calculus). And as the case is in most areas of math, in calculus each function or equation we work with can itself be represented as a graph. And when we consider the most basic approach to integration, we teach students to calculate the area under a given curve. This basic premise gives us a number, which in this case represents area under the curve. Integration is not, however, capable of producing the actual equation whose graph is the given curve. On the other hand, the Radon transform offers more detail. Notably, when integration is carried out within the principal constraints and context of the Radon transform, the process finds the equation.

The Radon transform uniquely offers us the ability to integrate, and as a result, get back the equation and not simply a number.

To be clear, transforms in the mathematical sense were not new in 1917, but what Radon offered mathematics was. What Johann Radon discovered was a pure mathematical construct, and he applied it that way — purely within mathematics. In fact, most of the early applications were in a field of advanced math called differential equations.

But are there physical applications? On this note, a distinguishing feature of the Radon transform is its ability to take one thing or body in space and represent it elsewhere (for example, transforming some entity that exists spatially to a graphical representation, such as a picture or image). Outside of mathematics, the Radon transform famously underpins medical x-ray computed tomography (also called CAT scans, or the preferred designation today, CT scans). However, it would be a half century — following Radon’s 1917 published paper — until the transform’s application and usefulness to CT theory became recognized. And with this realization, a new era in medical imaging emerged, subsequently allowing CT scanners to leap out of the minds of engineers and become a possibility (rather than existing merely as thought experiments). According to Wininger (2012), the CT scanner consequently became one of the most significant inventions witnessed in the field of radiology and medicine since the discovery of x-rays in 1895 (p. 2).

Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator [Lawgiver].

— C. S. Lewis

Now when giving this mathematically-related discussion honest thought, it is actually quite interesting the way in which abstract mathematical concepts end up having significance in the real world, especially since mathematical abstractions so often lead us to muse over how and why mathematics plays any role in our lives. Or we may keenly ask what it is about science that promotes mathematics as its language. It is for these reasons, then, that we humbly consider the thoughts of Christian apologist C.S. Lewis (1947) from his book Miracles: A Preliminary Study, who said, “Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator [Lawgiver]” (p. 140). This quote points to the inherent order in the universe and in nature — as opposed to naturalistic or materialistic chaos.

And continuing this line of rationale we claim that a Christian worldview imparts order and meaning on mathematical abstractions and their bearing in the physical world (i.e., cause-and-effect and the laws of physics). For example, the Bible records two pertinent and related questions that God directed to Job: “Do you know the laws of heaven? Can you impose its authority on earth?” (Job 38:33). Here God is saying that He is the Creator, so He understands. Therefore, the more we learn about God, the more we get to know and learn about our universe — including the role of mathematics — and our place in it.

According to noted mathematician (and Christian apologist) John Lennox (2009), Professor Emeritus of Mathematics at the University of Oxford, the real world and mathematics are traceable to the mind of God, the Creator of the universe and our human minds (p. 61). Thus, it ought not to surprise us that mathematical theories can be spun and woven within the tapestry of our thoughts. After all, we are each made in the image of God (including our minds which reflect His image). Yet, God’s thoughts supersede our own immeasurably.

Order is the fingerprint of God upon all His works.

— Jim Berg

But perhaps Jim Berg (2008) sums it up best in Essential Virtues: Marks of the Christ-Centered Life, when he wrote, “It is the predictability of order within the creation that allows scientists to theorize, research, and experiment. It is this predictability that is the foundation of the creation’s stability” (p. 63).

In closing, the framework embedded in the transform that Johann Radon introduced, and intended purely as a mathematical operator, ended up to be instrumental for medical x-ray CT scanning. In fact, it is equally astounding and worth celebrating that the Radon transform found similar usefulness in other areas of medical imaging, as well as importance in the fields of astronomy and geophysics. Ultimately, in the Christian worldview, the well-ordered concurrence and explanation found in the universe is grounded in the ultimate rationality of God — in His stability and trustworthiness. As a result, we also discover that questions that exercise the heart the most, such as why we are here and what our purpose is, make the most sense when we have a personal relationship with the Creator, the God of the Bible.

References

Berg, J. (2008). Essential virtues: Marks of the Christ-centered life. Greenville, SC: JourneyForth Books – BJU Press.

Lewis, C.S. (1947). Miracles: A preliminary study. New York, NY: Macmillan.

Lennox, J. (2009). God’s undertaker: Has science buried God? Oxford, England: Lion Hudson.

Wininger, K.L. (2012). On the foundations of x-ray computed tomography in medicine: A fundamental review of the ‘Radon transform’ and a tribute to Johann Radon. History of Mathematics Special Interest Group of the Mathematical Association of America: Student Writing Contest Winners Archive. Retrieved from http://historyofmathematics.org/archive/.