SoylentNews is people
SoylentNews
Submitted via IRC for Bytram
The Math Trick Behind MP3s, JPEGs, and Homer Simpson's Face
Over a decade ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn't a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, MP3 and JPEG compression, X-ray crystallography, voice recognition, and PET or MRI scans. This mathematical tool—named the Fourier transform, after 18th-century French physicist and mathematician Joseph Fourier—was even used by James Watson and Francis Crick to decode the double helix structure of DNA from the X-ray patterns produced by Rosalind Franklin. (Crick was an expert in Fourier transforms, and joked about writing a paper called, "Fourier Transforms for birdwatchers," to explain the math to Watson, an avid birder.)
You probably use a descendant of Fourier's idea every day, whether you're playing an MP3, viewing an image on the web, asking Siri a question, or tuning in to a radio station. (Fourier, by the way, was no slacker. In addition to his work in theoretical physics and math, he was also the first to discover the greenhouse effect.)
So what was Fourier's discovery, and why is it useful?
The story provides great visual examples of how even complex waves can be approximated by a series of sine waves summed together. Further, the parameters to the sine waves and a much more concise description of the approximated item. Examples are given of a roughly-square wave. Another example uses circles instead of sine waves. A great YouTube video shows these in action.
Wish I had this available to me before I was taught FT and FFT in college!
(Score: 5, Interesting) by bzipitidoo on Tuesday June 11 2019, @05:02PM
> While I'm here, why is it lossy?
The Fourier Transform Is not lossy. It wouldn't be a transform it if was. An example of a trivial transform is a simple change of units, like switching from inches to centimeters, or Fahrenheit to Celsius. A bit more involved is transforming from Cartesian coordinates (x and y distance) to polar coordinates (angle and radius).
What is inherently lossy is measurement. You talk as if PCM has no loss, and the sine waves can only be an approximation of the PCM. That is not so. The sine waves can contain exactly the same data as the PCM format. The inverse transform can take those sine waves and reproduce the data in PCM format, exactly as it was originally.
Where the loss in lossy compression enters is after the transform. The transform changes the data into a representation that is easier to work with. In the case of images, it's been noted that high frequency data is less likely to be missed. Lossy image and audio methods reduce the precision of those frequencies. For example, instead of storing any value between 0 and 255, which takes 8 bits, it could make do with values between 0 and 31 (multiplied by 8), which takes only 5 bits. Turns out, the precision can be reduced a great deal before quality is low enough that people begin to notice.