A Taste of Fourier Transforms and JPEG Compression

Hello, take a seat. I'd like to introduce you, just very briefly, to the beautiful world of Fourier transforms.

When Mr. Fourier first proposed this idea, he was laughed at by some. His work was criticized for decades. He fought for this idea, for taking a look at our world in a very different way than we're used to.

He said anything can be represented as a combination of circular waves, also known as sine waves.

Do you store images in the JPEG file format on your computer? If you do, you're using work based on this research. It's living proof that maybe, indeed, we really can represent anything as a combination of circular waves-- even images of the world around us!

I wish I could go back into the past, take Mr. Fourier out for coffee, and tell him it'd all turn out alright.

Even this cat is shocked he can be represented by the summation of circular curves and still look this good

Even this cat is shocked he can be represented by the summation of circular curves and still look this good

JPEG's actually use a variant of Fourier transform known as the Discrete Cosine Transform. Take a look at this picture below us. This is a 2D visualization of circular waves.

And then take a look at this picture of a JPEG set to very poor quality:

Look closely at these images and think about it. Do you see how that image is split into blocks, and each block tries to approximate its content by combining those different circular wave blocks together?

Fourier transforms have all kinds of other applications as well. Get creative when you think of what can be represented as a function. Obviously all physics equations (this was originally invented to help solve the heat equation). We just talked about how we can use this concept to represent images. How could this help us understand radio or sound waves? Data on your computer?

In my opinion, mathematics is just a way of trying to view the world in a different, creative way. Maybe the way we view the world isn't really through little blocks of summed up circular curves. But that representation turns out to be very, very useful for computer science applications.

Don't think mathematics is a field that stays stagnant, a field of truths you just have to memorize. We're often very wrong, and you might be surprised at how useful your creative take on the world really is.

 

 

Some great resources to learn from:

An Interactive Guide to the Fourier Transform
Strengths and Weaknesses of the DCT
JPEG Visualization

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