The discrete wavelet transform (DWT) compresses an image by repeatedly downscaling it, and storing the information which was lost during downscaling. Here's an image which has been downscaled twice, with its difference images (residuals): https://commons.wikimedia.org/wiki/File:Jpeg2000_2-level_wav.... To decompress that image, you essentially just 2x-upscale it, and then use the residuals to restore its fine details.

Wavelet compression is better than the block-based DCT for preserving sharp edges and gradients, but worse for preserving fine texture (noise). The DCT can emulate noise by storing just a couple of high-frequency coefficients for a 64-pixel block, but the DWT would need to store dozens of coefficients to achieve noise synthesis of similar quality.

The end result is that JPEG and JPEG 2000 achieve roughly the same lossy compression ratio before image artefacts show up. JPEG blurs edges, JPEG 2000 blurs texture. At very low bitrates, JPEG becomes blocky, and JPEG 2000 looks like a low-resolution image which has been upscaled (because it's hardly storing any residuals at all!)

FFmpeg has a `jpeg2000` codec; if you're interested in image compression, running a manual comparison between JPEG and JPEG 2000 is a worthwhile way to spend an hour or two.

I found a jpeg2000 reference PDF somewhere. It may as well have been written in Mandarin. I got as far as extracting the width and height. Its much more advanced than jpeg. Forget about writing a decoder.