The Relationship of Mathematics to Image Processing

Starting from this week, we will start our discussion on mathematical models and their role in image processing. Before we proceed with technical details such as what is Markov Random Field, total-variation regularization or wavelet shrinkage, it is important to understand why we need mathematical models in the first place. If you have not read Wigner’s article on unreasonable effectiveness of mathematics in physical sciences, I suggest you to go through that one (it is available in my homepage) before reading this blog.

To understand the relationship of image processing to mathematics, we need to know what is mathematics. To scientists, mathematics is the language of logical reasoning. It is important to recognize the language role of mathematics because we surely start to see papers with tons of mathematical equations but little motivation or ideas. Why are some people obsessed with equations?  We cannot blame mathematicians for doing so because this is their profession; but to scientists or engineers, blind worship of the power of mathematics is unfortunate and unnecessary.

I can give several examples to support the above claim. We are going to talk about wavelets later in this class and if you do some research, Daubechies’ wavelet paper is among the most cited work in applied math. What is less known is that engineers such as Vetterli and Vaidyanathan have also worked out a theory called filter bank which is essentially equivalent to wavelet. One might argue Daubechies’ construction is still the best (for certain applications) but the point is that the idea of time-frequency analysis originated from engineering applications such as seismic signal analysis and multi-rate signal processing. Sometimes problem formulation is more important than the solution.

Another example is so-called sparse representations which are also related to wavelets. The idea of heavy tail distribution or scale-invariance has been known by TV engineers in 1970s but their scientific implications are not articulated until 1980s-1990s. Ted Adelson, who used to work at RCA Lab, was the one who turned many empirical knowledge in TV industry into scientific findings in psychology (in particular visual perception). Therefore, the morale of this story is: as long as you have a keen eyesight for science, it does not matter where you work. Albert Einstein created his ground-breaking theories when working as a patent clerk (ans he is famous for being not so good at mathematics).

The last example is directly out of my own experience. How is it possible for someone trained in an engineering field like myself to compete with so many talented mathematical minds around the world? After reading many autographies (there are several links in my homepage), I conclude that the only way to “beat” mathematicians is to think like physicists (did I mention that Einstein once said “I do not believe mathematics”?). The beauty of nature is manifested by its simple organizational principles underlying many complex phenomena and objects. If one truly believes that the world is created by a universal principle, he/she will start to see the connection between turbulence and intelligence. Image processing, which can be viewed as the art of modeling either intensity data or vision system, is not some game of constructing mathematical basis or norm functions but the same game of probing into the fundamental laws of nature as physicists play. From this perspective, mathematics is at most the language one chooses to articulate nature’s principle and it is likely that the same principle can be described in different languages (e.g., deterministic vs. statistical).

To further clarify the point of this blog, I think it is helpful to consider the relationship of math to physics. Here is the best answer I can find on the web:

1) “There is a difference between asking whether physical theories have mathematical proof and whether physical theories can be expressed mathematically. By design, truly physical theories are based on postulates that can’t be proven mathematically themselves; all that can be done is to derive consequences that can be compared to the results of physical experiments/observations. As long as a theory survives this way, it remains viable. Tomorrow’s results might change the situation. As for expressing physical theories mathematically, I know of no limits to the expressibility of physical relationships using mathematical vocabulary.”

2) “Much of mathematics itself is somewhat an abstract concept. Science, particularly physics, assumes that everything can be described logically and mathematics is very much the language of logic. So if the universe is logical then it would make sense that everything can be described mathematically. However from there on it is very much a matter of perspective and opinion to whether or not maths is fundamental to the universe, proof of the axioms would be nice but it’s very strange that there is a logical proof that you can’t prove them.”