# fourier analysis Published articles for fourier analysis. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## The Complexity of Communication DevFeed: [The Complexity of Communication]() Original publisher: [Read original article]() Published: 2014-11-10T09:00:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computer science](), [Algorithms, Complexity](), [Streaming](), [circuit]() Tags: [communication](), [communication-complexity](), [computational-complexity](), [fourier-analysis](), [information-theory](), [log-rank-conjecture](), [lower-bounds](), [matrices](), [streaming-algorithms](), [theory]() ### AI overview This tutorial introduces communication complexity: how much information two parties must exchange to jointly compute a function of separate inputs. It presents the basic two-player model and explains the subject's use in proving lower bounds, including applications to circuit design and streaming algorithms. ### Source excerpt satellite One of the most interesting questions posed in the last thirty years of computer science is to ask how much "information" must be communicated between two parties in order for them to jointly compute something. One can imagine these two parties living on distant planets, so that the cost of communicating any amount of information is very expensive, but each person has an integral component of the answer that the other does not. ## Making Hybrid Images DevFeed: [Making Hybrid Images]() Original publisher: [Read original article]() Published: 2014-09-29T10:00:06Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Computing](), [Programming](), [Code]() Tags: [albert-einstein](), [art](), [dali](), [design](), [fourier-analysis](), [hybrid-images](), [image-manipulation](), [images](), [marilyn-monroe](), [mathematics](), [mona-lisa](), [programming](), [python](), [salvador-dali](), [signal-processing]() ### AI overview This article explains hybrid images, a visual technique in which different images appear at different viewing distances. It connects examples such as the Mona Lisa, Salvador Dali's work, and the Marilyn Einstein image to Fourier analysis and the use of mathematics, science, and programming. ### Source excerpt The Mona Lisa Leonardo da Vinci's Mona Lisa is one of the most famous paintings of all time. And there has always been a discussion around her enigmatic smile. He used a trademark Renaissance technique called sfumato, which involves many thin layers of glaze mixed with subtle pigments. The striking result is that when you look directly at Mona Lisa's smile, it seems to disappear. But when you look at the background your peripherals see a smiling face. ## The Two-Dimensional Fourier Transform and Digital Watermarking DevFeed: [The Two-Dimensional Fourier Transform and Digital Watermarking]() Original publisher: [Read original article]() Published: 2013-12-30T19:24:05Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [fourier transform](), [watermarking](), [Algorithm](), [Code]() Tags: [algorithm](), [animations](), [big-o-notation](), [calculus](), [dimension](), [fft](), [fourier-analysis](), [fourier-transform](), [github](), [graphics](), [image-manipulation](), [images](), [james-hance](), [mathematics](), [matrices](), [programming](), [python](), [star-wars](), [up](), [watermarking]() ### 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. ## The Fast Fourier Transform DevFeed: [The Fast Fourier Transform]() Original publisher: [Read original article]() Published: 2012-07-18T08:00:54Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithm](), [processing]() Tags: [algorithm](), [analysis](), [audio](), [divide-and-conquer](), [fourier-analysis](), [fourier-transform](), [graphics](), [mathematics](), [processing](), [programming](), [python](), [signal-processing](), [sound]() ### AI overview A tutorial on the Fast Fourier Transform explains its historical development, the improvement from O(n^2) to O(n log n) computation for the discrete Fourier transform, and a derivation and implementation approach. It also explores audio denoising by filtering a noisy signal's frequency spectrum. ### Source excerpt It's often said that the Age of Information began on August 17, 1964 with the publication of Cooley and Tukey's paper, "An Algorithm for the Machine Calculation of Complex Fourier Series." They published a landmark algorithm which has since been called the Fast Fourier Transform algorithm, and has spawned countless variations. Specifically, it improved the best known computational bound on the discrete Fourier transform from $ O(n^2)$ to $ O(n \log n)$, which is the difference between uselessness and panacea. ## The Discrete Fourier Transform -- A Primer DevFeed: [The Discrete Fourier Transform -- A Primer]() Original publisher: [Read original article]() Published: 2012-06-23T14:13:53Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Programming]() Tags: [analysis](), [course](), [fourier-analysis](), [fourier-transform](), [function](), [mathematics](), [primer](), [vector-spaces]() ### AI overview This primer explains the intuitive connections between continuous and discrete Fourier transforms. It covers discrete approximations of functions and transforms, the transition between discrete representations, and the role of sampling as motivation. ### Source excerpt So here we are. We have finally made it to a place where we can transition with confidence from the classical continuous Fourier transform to the discrete version, which is the foundation for applications of Fourier analysis to programming. Indeed, we are quite close to unfurling the might of the Fast Fourier Transform algorithm, which efficiently computes the discrete Fourier transform. But because of its focus on algorithmic techniques, we will save it for a main content post and instead focus here on the intuitive connections between the discrete and continuous realms. ## Generalized Functions -- A Primer DevFeed: [Generalized Functions -- A Primer]() Original publisher: [Read original article]() Published: 2012-06-06T20:57:05Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [abstraction](), [Programming]() Tags: [abstraction](), [fourier-analysis](), [fourier-transform](), [functional-analysis](), [generalized-functions](), [linear-algebra](), [mathematics](), [primer](), [programming](), [theory]() ### AI overview This primer develops a more rigorous mathematical framework for Fourier transforms. It motivates the search for a class of functions that remains well behaved under the Fourier transform and its inverse, while avoiding divergent integrals. ### Source excerpt Last time we investigated the naive (which I'll henceforth call "classical") notion of the Fourier transform and its inverse. While the development wasn't quite rigorous, we nevertheless discovered elegant formulas and interesting properties that proved useful in at least solving differential equations. Of course, we wouldn't be following this trail of mathematics if it didn't result in some worthwhile applications to programming. While we'll get there eventually, this primer will take us deeper down the rabbit hole of abstraction. ## The Fourier Transform -- A Primer DevFeed: [The Fourier Transform -- A Primer]() Original publisher: [Read original article]() Published: 2012-05-27T19:00:15Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [fourier transform](), [function]() Tags: [analysis](), [convergence](), [convolution](), [duality](), [fourier-analysis](), [fourier-transform](), [function](), [limit](), [mathematics](), [primer]() ### AI overview This primer introduces the Fourier transform as a limiting case of the Fourier series for functions without periodic behavior. It develops the transform from intuitive definitions and then addresses convergence more rigorously using distributions. ### Source excerpt In our last primer we saw the Fourier series, which flushed out the notion that a periodic function can be represented as an infinite series of sines and cosines. While this is fine and dandy, and quite a powerful tool, it does not suffice for the real world. In the real world, very little is truly periodic, especially since human measurements can only record a finite period of time. Even things we wish to explore on this blog are hardly periodic (for instance, image analysis). ## The Fourier Series--A Primer DevFeed: [The Fourier Series--A Primer]() Original publisher: [Read original article]() Published: 2012-04-25T21:43:06Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [function]() Tags: [analysis](), [classification](), [complex-analysis](), [foundation](), [fourier-analysis](), [heat-equation](), [mathematics](), [orthogonality](), [primer]() ### AI overview This primer introduces the mathematics of Fourier series, beginning with periodic functions and explaining how sine and cosine functions serve as building blocks for representing functions. It establishes foundational concepts for later work involving Fourier transforms, sound and image analysis, classification, and machine vision. ### Source excerpt Overview In this primer we'll get a first taste of the mathematics that goes into the analysis of sound and images. In the next few primers, we'll be building the foundation for a number of projects in this domain: extracting features of music for classification, constructing so-called hybrid images, and other image manipulations for machine vision problems (for instance, for use in neural networks or support vector machines; we're planning on covering these topics in due time as well).