As the human mind does to all concepts and creatures, the transformation is not an exception to be classified with respect to some world view too. Here is my classification in which I use my insights gained from the research I have conducted about the topic. The transformations can be abstract, which means we are talking about changes in psychological, philosophical and artistic concepts; whereas the other class is about changes in basic sciences such as math, physics and their applications.

Coming to the mathematical transformations, I came across the fact that they are the “passive” transformations in which there is no change relative to the referenced coordinate frame. This means we new apply a mathematical transformation, the result does not change. The way we follow to the result changes for gaining convenience in computing or thinking. For example the Mellin transformation is beneficial in determining the asymptotic expansions of functions, often harmonic sums, in computer sciences. The same result can be obtained with using Drichlet series, number theory, etc. where the theorems and tools appeal with respect to the problem in question. The idea is to look from a different angle for faster and reliable result. In the case of Mellin transforms, the regions where the image function which is the input of the transformation process, can be known by considering its Mellin transform. An interesting application of the Mellin transform is that the scaling of an object moving away from/to the camera. The image is processed with the help of Mellin transform digitally.

So the mathematical transformations are classified, too. There are integral transformations such as Mellin transforms, whereas discrete transformations like Binomial transformations exist too. The Binomial transform deals with series and produces another series. It is called as a generating function, which implies that it produces a closed form for the elements of sequences. This particular transformation is invaluable for detecting how series behave and eventually, how systems behave. I remember from my combinatorial classes that we can easily “generate” some new functions for capturing the elements of sequences that seem too cumbersome to compute.

As I went through my findings, I connected the idea of transforms to one of the most notable notions that I have learned in my education. That is the concept of “duality”. Duality enables us to gain insights for a problem by looking from the opposite perspective. When the dual of a problem is solved, the original problem is solved too. The same principle works for the transformations: When we take the transform of a function and perform some operations in the new domain, we have a result that says the same thing when the computations are done in the original domain. This transformation-mapping- is one-to-one. As the duality does, transformations also relate concepts and facts in two distinct areas. For example, the dual of the maximizing profits is minimizing costs, that is the problem of maximizing profits is transformed to minimizing costs.

References:

http://www.websters-online-dictionary.org/definition/Transformation
http://en.wikipedia.org/wiki/Mellin_transform
http://www.josleys.com/articles/printgallery.htm
http://en.wikipedia.org/wiki/Binomial_transform