Convolution A Memoir With A Future

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       Convolution; from the mathematical definition is a relationship or expression involving a  quantity that may change within the context of a problem or experiment, which is derived from other two given relationships by integration that expresses, how the shape of one is modified by the other. In other words let’s say you have two sets of ingredients, one is bananas and the other is milk, then you blend them and you have a banana shake. That is convolution; the math is in the blending result, but you don’t need to be aware of it.

     Why am I telling you something about such a weird word?

      If you look in the dictionary, the  grammar definition of convolution means coil or twist, especially one of many.

      We can say; “Crosses adorned with elaborate convolutions.”

         In  engineering we use a lot of convolution as a tool, for example;  large electric power networks today are highly interconnected through high voltage transmission lines; convolution helps to reduce costs and improve reliability, and also reduce operating costs.

      In analytical  Chemistry it is  used for the analysis of spectroscopic data. In Acoustics, reverberation is the convolution of the original sound with echoes from objects surrounding the sound source. In radiotherapy treatment planning systems and most part of all modern codes of calculation applies a convolution algorithm. (algorithm is instructions in equations that will be used in a process).

     A common output from automatic inspection systems is pass/fail decisions and it is based in convolution analysis. Other common outputs from this marvelous tool include object position and orientation information for robot guidance systems.

         Additionally, convoluted output types include numerical measurement data, data read from codes and characters, counts and classification of biological cell samples, displays of the process or results, stored images, alarms from automated space monitoring MV systems, and process control signals. We are surrounded by convolution!

              Following the money, because we know it’s important, let’s say that even talking about just a single use of convolution, societies will have a large monetary impact, i.e., The global Machine Vision market is expected to reach USD 15.46 billion by the end of 2022.

      The most fascinating thing is that nowadays everybody can play with convolution. Let me explain; I like to take photos, I have many thousands of them, also in the internet are much more. Lately I discovered a site where you can upload your pictures and they put the convolution algorithm to work, and in a few minutes you have a new image. This new one can be totally fantastic, or totally absurd, or totally unexpected, it can be very beautiful. One of my favorites convolutions is when I put together one image of a Bengal Tiger and the other image had colorful yarns. I uploaded them and together, the convolution algorithm constructed an image of a tiger made of yarn in the most amazing way.

     I have been playing with many images with all kind of results, I can see that in the future we will have the ability to educate next generations with a set of tools, mathematical tools that go beyond the elemental ones that are taught today. The new teachings will be very intuitive, and with a ludic or playful approach. We will use this approach in  Calculus, Algebra and Probability to say the least, they will be in the minds of children in a simply beautiful way. Have you seen the new toys about automation? They are made out of wood, yet they simulate mechanisms that are automated, and the kids love them. Or the new toys about electronics or computing? They are not only for children but for adults too.

       Can you put more Math than Arithmetic in a recipe?  Today we can; like the banana smoothie I mentioned at the beginning. Cooks and Mathematicians have a lot in common, they just need to cross the language bridge to discover each other.

        Are we twisting, curling, coiling images? Yes, we are! In our cell phones when we put filters to enhance our pictures, we are using convolution to make areas smooth. Photoshop has convolution, it is sometimes like “ironing the wrinkles in a face,” or  have an instant “electronic facelift;” or enhanced a model feature(s) for advertisements.

      Convolution  is very old, it was invented more than a hundred years ago; as an engineer I have used it several times. As a human being I have been around convolution all my life as all people. This memoir is  about how we have lived in a mathematical world sometimes without being aware of that; but pretty sure I am, that we will play much more with math and convolution in years to come. It’s a memoir in the making, it’s a memoir with a future.


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