Wavelets: Analyzing Scale
Physics is littered with transforms: Fourier, Radon, Wavelet, Laplace, Hilbert and many more. The idea behind each transform is - almost exclusively - to represent data in a way which makes it easy to “see” hidden structures. For instance, the Fourier transform allows you to see if there is any periodicity within your signal quite […]
Neural Networks Tutorial, Part #1
This is the first part of a set of postings on neural networks (NNs). NN are constructs that attempt to mimic our brain’s activity to a degree. NNs have become so common that it’s useful to know what people are talking about, so I’ve decided to write a brief tutorial - in several parts - […]
Shearing your Data with a Fourier Transform
Here’s a neat little trick that has helped me out a few times.
It often happens, when acquiring 2D data, that the resulting data set seems slanted, or sheared in some way. This can be the result of hardware error or some other artifact in your system (it happens to me all the time when acquiring […]
The Mathematics of Picture Taking
or: The Logic Behind Convolutions
When I was taught about convolutions in my undergraduate degree I found them somewhat puzzling. Somehow they always got tied up with the Fourier Transform and the convolution theorem. But any convolution really has nothing to do with Fourier Transforms. They actually have a lot to do with the science of […]
On Factors of 2-pi in the Fourier Transform
Every physicist is exposed during his or her undergraduate degree to the Fourier Transform, defined as:
However, a source of great confusion is that of factors. Different books define the Fourier Transform using varying factors in front of the integral:
Are those definitions valid, and if so, how can we settle the conflict between them?
The […]