somewhere to talk about random ideas and projects like everyone else

# #small

## Cooley-Tukey FFT + DCT + IDCT in under 1K of Javascript30 May 2015

Technically this is a Hough transform and isn't at all related to the FFT*, but it looks a lot cooler than any of the actual FFT/DCT pictures I have, so I might as well stick it here

Almost a year ago, I wrote a simple script for computing perceptual hashes in Javascript (it’s going to be part of a forthcoming Naptha feature which allows it recognize source images for memes and do perfect inpainting). One step of the process is computing the 2D Discrete Cosine Transform of a shrunken version of the image.

The problem is that step was kind of ridiculously slow, and it’s not really hard to see why: there’s a series of four nested for loops— because that’s the obvious naive way for computing the DCT.

In practice, it’s actually not all that bad because you only have to do it once per image, and the dimensions of the source image are a fixed 32x32. But it still felt pretty bad, so when I was looking over the code again earlier this week, I tried doing some research to figure out a faster way to do the DCT.

Surely you’d imagine that there have to be extraordinarily efficient implements of the DCT out there, given that it’s the very foundation of JPEG out there. And you’d be right, but they’ve generally unrolled the butterfly into a series of bit-shifts which are quite unlikely to generalize beyond the 8x8 block size.

I’m reasonably confident that FFTW3 has a fast DCT implementation which also is capable of generalization, but it seems buried within quite a lot of code.

## MiniFFT

This isn’t a super performant implementation, but it’s not horrendously slow and it doesn’t use recursion. It’s probably pretty trivial to make it quite a bit faster by precomputing sine and cosine tables. This uses the standard Cooley-Tukey radix-2 decimation-in-time algorithm, which means it only works for one dimensional arrays with a length which is a power of two.

It doesn’t check that the length is a power of two, it’ll just give you the wrong answer. You should definitely check that it’s true beforehand.

``````function miniFFT(re, im) {
var N = re.length;
for (var i = 0; i < N; i++) {
for(var j = 0, h = i, k = N; k >>= 1; h >>= 1)
j = (j << 1) | (h & 1);
if (j > i) {
re[j] = [re[i], re[i] = re[j]][0]
im[j] = [im[i], im[i] = im[j]][0]
}
}
for(var hN = 1; hN * 2 <= N; hN *= 2)
for (var i = 0; i < N; i += hN * 2)
for (var j = i; j < i + hN; j++) {
var cos = Math.cos(Math.PI * (j - i) / hN),
sin = Math.sin(Math.PI * (j - i) / hN)
var tre =  re[j+hN] * cos + im[j+hN] * sin,
tim = -re[j+hN] * sin + im[j+hN] * cos;
re[j + hN] = re[j] - tre; im[j + hN] = im[j] - tim;
re[j] += tre; im[j] += tim;
}
}
``````

The transformation happens in place, so it’ll modify whatever arrays you pass in.

``````var re = [1, 2, 3, 4],
im = [0, 0, 0, 0];
miniFFT(re, im);
console.log(re, im); // [ 10, -2, -2, -2 ] [ 0, 2, 0, -2 ]
``````

Minified, it’s only 359 bytes. Unlike this other contender for the dubious prize of smallest javascript FFT implementation, it doesn’t require a special complex number library three times its size (though this doesn’t fit in a tweet— I’ll be really impressed if someone manages to do that).

I haven’t really golfed the code to the smallest size possible, but I think it’s at a reasonable balance of brevity while still being somewhat comprehensible.

It’s based on an in-place FFT from Princeton’s intro CS class Project Nayuki’s FFT.

## MiniIFFT

Turns out that you can invert the FFT modulo a scaling factor just by swapping the real and imaginary arguments.

``````function miniIFFT(re, im){
miniFFT(im, re);
for(var i = 0, N = re.length; i < N; i++){
im[i] /= N;
re[i] /= N;
}
}
``````

## MiniDCT

This is an implementation of a Type-II DCT using MiniFFT. For details about Makhoul’s algorithm for transforming a DCT into an FFT, see this post

``````function miniDCT(s){
var N = s.length,
K = -Math.PI / (2 * N),
re = new Float64Array(N),
im = new Float64Array(N);
for(var i = 0, j = N; j > i; i++){
re[i] = s[i * 2]
re[--j] = s[i * 2 + 1]
}
miniFFT(re, im)
for(var i = 0; i < N; i++)
s[i] = 2*re[i]*Math.cos(K*i)-2*im[i]*Math.sin(K*i);
}
``````

Like MiniFFT, it operates in-place (well, not really, but it writes its output to the source array).

``````var arr = [3, 4, 1, 7];
miniDCT(arr);
console.log(arr); // [ 30, -5.09493566, 7.0710678118, -8.604744653 ]
``````

It produces the same output as scipy’s fftpack, but Matlab’s DCT uses orthogonal normalization and produces a different result. However, it’s pretty simple to do it the Matlab way— just scale everything by 1/sqrt(2 * N) and divide the first element by sqrt(2).

``````function matlabDCT(arr){
miniDCT(arr)
arr[0] /= Math.sqrt(2);
for(var i = 0, N = arr.length; i < N; i++) arr[i] /= Math.sqrt(2 * N);
}
``````

## MiniIDCT

I really considered renaming this series of functions into TinyFFT and friends because of the predicament of MiniIDCT— yeah, exactly— the horror— repeating I’s.

This is an implementation of a Type-III DCT, also known as the IDCT because it happens to be the inverse of a Type II DCT (frequently refered to as “The DCT”).

``````function miniIDCT(s){
var N = s.length,
K = Math.PI / (2 * N),
im = new Float64Array(N),
re = new Float64Array(N);
re[0] = s[0] / N / 2;
for(var i = 1; i < N; i++){
var im2 = Math.sin(i*K), re2 = Math.cos(i*K);
re[i] = (s[N - i] * im2 + s[i] * re2) / N / 2;
im[i] = (im2 * s[i] - s[N - i] * re2) / N / 2;
}
miniFFT(im, re)
for(var i = 0; i < N / 2; i++){
s[2 * i] = re[i]
s[2 * i + 1] = re[N - i - 1]
}
}
``````

This is based on Tom Grydeland’s IDL implementation of the IDCT.

Let’s start with a 8 element array of real numbers (DCT values).

``````[A, B, C, D, E, F, G, H]
``````

From that it generates the corresponding complex values. Note that the real part is intact, and the complex part is the original array with its head chopped off and the rest of it flipped around and attached to a water balloon.

``````[A, B, C, D, E, F, G, H] - j[0, H, G, F, E, D, C, B]
``````

Then the complex values are rotated by `exp(i * pi / (2 * N))` to get spectral components. After an IFFT that gets transformed into:

``````[a, h, c, f, e, d, g, b]
``````

It took me a while to see what was going on there, but the even indices are map to the first half of the results, and the odd ones map to the second half in reverse

``````Scrambled      a h c f e d g b
Even Indexed   a   c   e   g
Odd Indexed      h   f   d   b
Odd Reversed     b   d   f   h
Unscrabmled    a b c d e f g h
``````

## HeapQueue.JS28 January 2014

I have a habit of writing blog posts in Sublime Text in lieu of legitimate READMEs. This project was simple and short enough that I think I did manage to write something with a passable semblance to a functional README before initiating this quest to craft a blog post.

This is hardly a project that I’d expect to go down in the annals of my blog as particularly interesting or notable. Instead, it’s a rather boring low-level computer sciencey thingy who would feel perfectly comfortable amidst homework assignments.

In fact, it’s virtually assured that documenting this endeavor– the combination of writing the words which comprise this post and the documentation already written on how to use this rather simple library– will inevitably consume more of my time than actual coding (people who are more familiar with highly intricate professional software tapestries may claim that such is merely the ideal and typical experience of a programmer).

Notice that I’ve managed to spend three paragraphs (well, my paragraphs are on the order of two sentences, so it doesn’t even qualify for the fourth-grade definition for what a paragraph is) talking about the uninteresting character of the project, its relative diminutive size, and the process of writing about the subject, without actually describing what exactly the project is. This, more than perhaps any other project of mine deserves such a diversion, for it is unlikely to yield “much amaze” or even a modest “such wow”.

It is a binary heap priority queue, a teensy bit shy of 50 lines, a mere kilobyte minified. It’s very loosely based off of Adam Hooper’s js-priority-queue but considerably smaller.

Now that I’ve gotten the actual informative and useful part of the blog post off my shoulders, I can take you along for the unproductive plunge into the rather lame backstory for the project. I was working on a port of the Telea inpainting algorithm and needed a priority queue, because the little incremental insertion sort that I hacked together in two lines was taking a few orders of magnitude longer than was acceptable.

With my considerable Google-Fu, I searched “javascript heap queue” and found js-priority-queue as well as PriorityQueue.js. They both happened to be implemented in CoffeeScript, which I love, except not really. I like the syntax, but creating a project using it usually involves setting up some compilation pipeline in order to get it to do the things that I want it to do. Overall, it’s a bit too much of a hassle for little projects.

I couldn’t actually find the source code for PriorityQueue.js so I settled for js-priority-queue. It was a bit annoying in that it required require.js to function and included several storage strategies that were totally useless. So I copied and pasted a few files together and stripped it of its AMD dependencies and created a somewhat dieted version.

But as 9th grade World History teaches us, appeasement never works, and this purely pragmatic minification leads inexorably to an irrational code golfing. At that point, I was overcome by this undeniable urge to cleanse it further, perhaps blinded by not-invented-here syndrome and software anorexia. Minutes later the file had been laid waste and in its place there existed its bare skeletal remains.

Now all I need to do is push it to github and write a blog post…

## Javascript SHA1 and SHA256 in03 January 2010

Not sure why I made this, but here’s some super tiny implementation of some cryptographic functions. There are some optimizations made for size rather than speed (especially with SHA256), for instance, there are some 71 constants which are used in SHA256 which in most implementations are stored precomputed (they take up around 1KB in hexadecimal), but with mine, they are computed at runtime, which adds a significant runtime penalty, taking twice as long as comparable implementations. It should also be noted that it doesn’t do any UTF8 encoding that some other implementations do, but it can be added by UTF8 encoding the text before running it through the functions.

SHA1

SHA256

## Migrating small stuff to GitHub03 September 2009

I’ll take the liberties of plagarizing this as who wouldn’t recognize its source?

So I’m moving the small totally unknown and 1-2 file things that I’ve spammed Google Code with previously. Small things like js-xdb, mental-interpreter, js-tpl-engine, js-xdomain, subleq2, vxjs-ajax (a whole project for a single function? crazy stuff).

So i’m shrinking my google code profile to reduce my spamminess, becasue I used to feel like it would be awesome to have a project for everything I spent more than 2 minutes on doing in hopes that someone would eventually find it interesting.

Hopefully someone would find it interesting on github.

I’m also adding some lost projects, like my backups of stuff that got lost when appjet shut down, a substition code cracker a few jetpacks and still adding more.

http://github.com/antimatter15/antimatter15/tree/master

Those are all the tiny projects not big enough to deserve a actual repo or google code project page.

## ShinyTouch is 1 month old28 July 2009

For as long as my notes show, ShinyTouch is now 1 month old.

So today, I added VideoCapture support, so it will now work better on Windows.

Auto calibration has been rewritten, and a few other small changes.

## Can Anyone Beat This?14 October 2008

The original vX function was 337b. Now, it’s been reduced down to 293 bytes, while adding a new feature (callback is now optional).

Apparently, the above stuff doesn’t work (Wordpress?)

It’s really quite amazing. The big things that reduced size were using lots of condititional things, really obfuscated unreadable stuff, and using !0 instead of true, and !1 instead of false.

If you want to use it, try building your own copy from.

The usage has signifigantly changed though, there, everything’s namespaced under an underscore, so it’s _.X(“url”)

## vX JS Library12 October 2008

Built on top of the vX Ajax Function, is the vX JS Library. It’s probably the world’s smallest JS Library, in total, about 1.45kb, with things like Animations, Ajax, JSON Serialization, URL Encoding, Cloning, Event Handling, Fade Effects, and more. It’s signifigantly less elegant than jQuery and others, but it is extremely lightweight and quite cross-platform. The code has been optimized down to each individual byte.

It’s not too useful. It may be useful for some tiny things, but it’s not really that useful.

It’s not good enough to make things really high-quality, or complex such as the Ajax Animator. It’s good only if your making like something small, where you might want some ajax, but still want it to load fast.

Also, another thing, not exactly part of the library is vXg, a Get-Only version of vX that’s only 221 bytes. http://vxjs.googlecode.com/svn-history/r26/trunk/ajaxget.js

vXg(URL, CALLBACK)

## Small Stuff Again28 July 2008

I’m just ironing out the little things now.

## Today's (Tiny) Updates30 April 2008

Really, nothing much. There are some corrections to the About page, and some more icons. There is also a non-working Bug-Reporting tool.

I will try adding OnlyPaths soon, but it still needs much work. (and it’s Build 40 now!!!)