Math is lame. When will I ever use it anyway?
I have heard this sentiment during every math class I have ever taken. And, not being the best math student myself, I didn’t speak up. But the reality is that you will need math in the real world if you want to succeed at anything important. My own life and business is enriched by math. IMO, as a culture we should work to get over the “hard subject” mentality, much less being proud of ignorance. Math is actually quite neat and useful. This is part of the story of how modern computing power is going to change almost everything.
The math I want to discuss this time is deconvolution. It may sound complex, but the idea is simple. It is in essence a process of reversing changes that happen to a signal or image as it passes through a medium or channel and/or is mixed with other images/signals. I can hear you thinking, but prononymous you deal with biology what do signals and images have to do with animals and plants? Well, there are some interesting examples.
This is not just useful to me either, there are a host of applications to this idea that could benefit many users. I’d like to mention a few here. Though I’m by no means an expert in these fields so if my understanding is bad please correct me.
Plants actually tell you a lot by just looking at them. Of course I mean you can see if their leaves are drooping, or turned yellow. But they also tell you something on a very subtle level that we can’t see with our naked eye. The chlorophyll in a plant’s tissues responds to light in some particularly interesting ways. The molecules in chlorophyll respond to various types of light by re-emitting their own fluorescence, a particular form of light.
You could imagine that it would be useful to know more about the state of chlorophyll in a plant. Though the fluorescence could be very useful it is hard to interpret multiple signals from the chlorophyll. There are many factors that contribute to the level of fluorescence you record in a leaf, or in my case a sample of single celled algae suspended in seawater. So what you can do is to hold as many factors steady as possible and only change a singe factor such as light or nutrients to record the changes in the fluorescence signal. Then the signal can easily be separated from the overall signal using math. Computers are really awesome at this step so for big jobs just get a coffee. But for chlorophyll fluorescence it is fairly easy to resolve the single recorded signal into useful information about how a “plant” is doing. The more signals you know as a baseline, the better your analysis can be. I could record the data about my algae into a log and deconvolve it into useful information available online. This could theoretically help spot nutrient deficiencies before they visibly or odorously change cultures. All because of cheap computing power I could theoretically view my algae reactor status across the world on my phone and actually modify various inputs in real-time.
How else can deconvolution be useful? Just about any time you know about an original signal but don’t know much about the transmission medium, or don’t know about the original signal but know about the medium, or don’t know either. I realize that encompass a rather wide range of uses, lol. Let’s say you want to know about the inside of the earth. Well, we know about the signals traveling through the earth. We can easily record earthquakes, nuclear bombs, demolition, etc before and after their waves have passed through the earth. We know what the signal looked like when it started, because we created it with dynamite or were close to the source earthquake for example. We can look at the signal that was recorded at the other end after it passed through the earth as being “messed up” (convolved) by the earth. The properties of the earth such as viscosity, composition, etc change how the wave propagates. If we know something about the propagation of waves through various materials we can separate the wave we know from the factors that are unknown such as deep geology. Thus, with the power of computers, we can figure out what something looks like that we can’t get to easily.
This last way could save me a bunch of money. I deal with a lot of microorganisms. To see what is going on in that tiny world I need a microscope, of course. The microscope I got, the quite nice Leitz Orthoplan, came with some very nice lenses. Though not all of them seemed immediately useful for my purposes. That changed though while thinking about how to use deconvolution for improving images with the “good” lenses. The process of passing through a lens convolves the image being perceived because of the refractive properties of the glass and coatings. If we can mathematically describe that convolution added by the lens then we can reverse the process and hence clarify the picture. In practice I will probably accomplish the measurement by taking reference pictures of suspended colloidal solids to obtain a value for the point spread function, and various aberrations, across the visible range. The computer does a huge amount of work in the next step, deconvolution. The deconvolution process removes noise from the image such as out of focus areas. But it can also make lenses I couldn’t use functional again because their aberrations can be removed during deconvolution. So extra computing power will be able to save me money in lenses. I could even buy lenses that others don’t want because of undesirable aberrations. And since the input to the process is a 3d stack of images, I could process the result into one of those neat volume renderings.
There are many other uses such as for csi-style image cleanup, audiophiles, camera enthusiasts, astronomy, etc. But I’ll stop before I bore the audience to death :). If you wonder about anything mentioned, want to know more, correct me, or have an idea for another scientific page, or just comments on the format/content let me know.