# james hance

Published articles for james hance.

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## The Two-Dimensional Fourier Transform and Digital Watermarking

DevFeed: [The Two-Dimensional Fourier Transform and Digital Watermarking](<https://devfeed.tech/articles/the-two-dimensional-fourier-transform-and-digital-watermarking-40336.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/12/30/the-two-dimensional-fourier-transform-and-digital-watermarking/>)

Published: 2013-12-30T19:24:05Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [fourier transform](<https://devfeed.tech/topics/fourier-transform.md>), [watermarking](<https://devfeed.tech/topics/watermarking.md>), [Algorithms](<https://devfeed.tech/topics/algorithms.md>), [Code](<https://devfeed.tech/topics/code.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [animations](<https://devfeed.tech/tags/animations.md>), [big-o-notation](<https://devfeed.tech/tags/big-o-notation.md>), [calculus](<https://devfeed.tech/tags/calculus.md>), [dimension](<https://devfeed.tech/tags/dimension.md>), [fft](<https://devfeed.tech/tags/fft.md>), [fourier-analysis](<https://devfeed.tech/tags/fourier-analysis.md>), [fourier-transform](<https://devfeed.tech/tags/fourier-transform.md>), [github](<https://devfeed.tech/tags/github.md>), [graphics](<https://devfeed.tech/tags/graphics.md>), [image-manipulation](<https://devfeed.tech/tags/image-manipulation.md>), [images](<https://devfeed.tech/tags/images.md>), [james-hance](<https://devfeed.tech/tags/james-hance.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [matrices](<https://devfeed.tech/tags/matrices.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>), [star-wars](<https://devfeed.tech/tags/star-wars.md>), [up](<https://devfeed.tech/tags/up.md>), [watermarking](<https://devfeed.tech/tags/watermarking.md>)

### AI overview

This tutorial introduces the multidimensional Fourier transform, explains its relationship to the one-dimensional transform, describes an FFT-style algorithm for computing it, and applies it to digitally watermarking images.

### Source excerpt

We've studied the Fourier transform quite a bit on this blog: with four primers and the Fast Fourier Transform algorithm under our belt, it's about time we opened up our eyes to higher dimensions. Indeed, in the decades since Cooley & Tukey's landmark paper, the most interesting applications of the discrete Fourier transform have occurred in dimensions greater than 1. But for all our work we haven't yet discussed what it means to take an "n-dimensional" Fourier transform.