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1---
2type: article
3identifier: opengl-fractal-explorer
4title: GPU-Accelerated Fractal Explorer
5description: Using OpenGL's compute shaders to dispatch fractal computation to the GPU and render in realtime.
6datestring: 2023-12-07
7banner_image: /static/images/mandelbrot.png
8links:
9 Source Code: https://github.com/JoshuaS3/zydeco/tree/fractal
10 The Mandelbrot Set: https://en.wikipedia.org/wiki/Mandelbrot_set
11 IEEE 754: https://en.wikipedia.org/wiki/IEEE_754
12 OpenGL Compute Shaders: https://www.khronos.org/opengl/wiki/Compute_Shader
13 Adam7 Algorithm: https://en.wikipedia.org/wiki/Adam7_algorithm
14 OpenGL Memory Model: https://www.khronos.org/opengl/wiki/Memory_Model
15---
16
17<style>
18svg {
19 display: block;
20 margin: 0 auto;
21 color: var(--text-color);
22 transform: scale(0.9);
23}
24</style>
25
26I've been toying around for a while with an idea for a procedural world
27generation + simulation project as an experiment in C++ and graphics
28programming to teach myself more about computer science and rendering
29techniques. Part of this is, of course, setting up the infrastructure for input
30handling, world logic, debug menus, and rendering. When writing the initial
31code, I used the Mandelbrot set for testing. This led me down a rabbit hole of
32improving my rendering techniques for this application, as well as trying out
33different fractals, ultimately culminating in this GPU-accelerated fractal
34explorer (transform, zoom, pan) with progressive refine:
35
36<iframe style="max-width:720px;max-height:405px;display:block;margin:0 auto 1em auto" src="https://www.youtube-nocookie.com/embed/Zqfeut60Qbc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>
37<figcaption style="text-align:center">Video compression doesn't allow for demonstration of the progressive refine element very well; this is explained in detail later here.</figcaption>
38
39**Outline**
401. [Fractal Sets](#fractal-sets)
41 1. [The Mandelbrot Set](#the-mandelbrot-set)
42 2. [The Tricorn Set](#the-tricorn-set)
43 3. [The Burning Ship Fractal](#the-burning-ship-fractal)
442. [Notes on Fractal Computation](#notes-on-fractal-computation)
45 1. [Divergence](#divergence)
46 2. [Iteration Count](#iteration-count)
47 3. [Floating-Point Precision](#floating-point-precision)
483. [Rendering on the GPU](#rendering-on-the-gpu)
49 1. [Using a Fragment Shader](#using-a-fragment-shader)
50 2. [Using a Compute Shader](#using-a-compute-shader)
51 3. [Progressive Refine](#progressive-refine)
52
53# Fractal Sets
54
55## The Mandelbrot Set
56
57The [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set) is defined
58to be the set of all numbers *c* in the complex plane for which the following
59sequence (what I call a "z-transform" here — not related to the signal
60processing Z-transform) *does not* diverge to infinity:
61
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63
64Note that the z-*squared* term is squaring a complex number, given by the
65following:
66
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68
69where *a* is the real term and *b* is the imaginary term.
70
71When rendering, we take the real axis to be *x* and the imaginary axis to be
72*y*. Points (numbers) in the set are colored black, and points not in the set
73are colored with a brightness corresponding to the number of iterations
74required until divergence.
75
76The above unassuming sequence and rules of complex algebra result in perhaps
77the most popular fractal shape, which exhibits infinite complexity at the
78boundary of the set and yields new patterns—including copies and variations of
79the set itself!—wherever you zoom in, forever.
80
81<figure class="full">
82 <img width="700px" src="/static/images/mandelbrot.png">
83</figure>
84
85Needless to say, I've been pretty fascinated by it. This isn't the only
86fractal set though. You can generate more interesting shapes and
87patterns by simply modifying the original sequence, or just coming up with
88something new. You can also add an additional parameter to play around with,
89transforming fractals. I don't get very scientific with it. You can see this
90used in the video to transform between fractals. Most random variants however
91are relatively boring in that they 1. don't produce more than one or two
92patterns, 2. produce patterns that are just the Mandelbrot set (this by itself
93is an interesting pattern of emergence), or 3. devolve into noise when zooming
94in most places. There are a couple exceptions of note:
95
96## The Tricorn Set
97
98The Tricorn set is a variant of the Mandelbrot set that uses the *conjugate* of
99z, which inverts the sign of the imaginary term.
100
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102
103<figure class="full">
104 <img width="700px" src="/static/images/tricorn.png">
105</figure>
106
107## The Burning Ship Fractal
108
109A more well-known variant of the Mandelbrot set is the Burning Ship fractal,
110which takes the *absolute value* of z before squaring it.
111
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113
114<figure class="full">
115 <img width="700px" src="/static/images/burning_ship1.png">
116</figure>
117
118The most interesting part about this one is actually the figure to the left,
119which is what the fractal is named after.
120
121<figure class="full">
122 <img width="700px" src="/static/images/burning_ship2.png">
123</figure>
124
125
126# Notes on Fractal Computation
127
128I want to talk about some details of computing and rendering fractals. As you
129would expect, for a full quality render you will need to compute iterations for
130every pixel in the image. **<i>This is very computationally expensive.</i>**
131Even more troublesome is having to calculate this for *every frame* when
132panning around and zooming in if you're writing a realtime explorer.
133
134## Divergence
135
136Let's first define what is meant by "diverge" when iterating over a
137z-transform. Mathematically, this means the point is unbounded, or transforms
138off to infinity. We can discard a point from the set long before infinity
139though—in fact, for the three fractals mentioned above, any complex number with
140a distance from the origin *greater than 2* will diverge during a z-transform.
141Storing the square of this—to prevent having to compute square roots when
142applying the Pythagorean theorem—in a `discard_threshold_squared` constant or
143parameter, we can speed things up by stopping before unneeded iterations in our
144compute code:
145
146```c
147const int discard_threshold_squared = 4;
148```
149
150```c
151// [inside z-transform loop]
152if ((a*a + b*b) > discard_threshold_squared)
153{
154 // [store current iteration count for purpose of
155 // coloring, indicating point is not in set]
156 break;
157}
158```
159
160Points in the set will not exit the sequence early. An implication of this is
161that *the more points in the set the frame contains, the longer the frame will
162take to render.*
163
164## Iteration Count
165
166We also need to define a maximum iteration count, the number of iterations it
167takes to confidently say "this point does not diverge." This makes for another
168design consideration, though. Note in the screenshots above how points closer
169to the set are brighter; this means it takes more iterations for those points
170to diverge. From this, it should follow that **<i>increasing the maximum number
171of iterations will lead to greater detail at the bounds of the set</i>**. If we
172set the iteration count too low, we get undetailed renders like the following
173(compare to previous screenshot).
174
175<figure class="full">
176 <img width="700px" src="/static/images/burning_ship3.png">
177</figure>
178
179Not only that, but zooming in only makes the boundary seem coarser. To
180compensate for this, I define my maximum iteration count to be a function of
181zoom level, where `n0` is a "base" iteration count parameter and `s` is the
182scale (decreases when zooming in).
183
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185
186This largely fixes the coarseness of the shape when zooming in, but poses a new
187issue. At a certain point when zooming in, the iteration count will become so
188large that framerate begins to drop. In the [Rendering on the
189GPU](#rendering-on-the-gpu) section I detail a rendering method called
190*interlacing* (or *progressive refine*) that lets us split up the work of a
191render across multiple frames.
192
193## Floating-Point Precision
194
195The primary limitation with a realtime fractal renderer like this is computer
196hardware architecture. For most applications, computers store decimal numbers
197according to standard [IEEE 754](https://en.wikipedia.org/wiki/IEEE_754) (or
198variants thereof), which, in essence, represent decimal numbers in scientific
199notation form, comprising a significand ("mantissa") and an exponent. On
200modern CPUs and GPUs, there are floating-point arithmetic units (FPUs) built
201into hardware that make computation with floating-point types significantly
202faster than it would be with a software-only implementation. FPUs nowadays come
203in sizes of 16 bits (half-width/FP16), 32 bits (single-width/FP32), 64 bits
204(double-width/FP64), and 128 bits (quad-width/FP128). As you might be able to
205guess, more bits means larger values means greater precision.
206
207This pertains to computing fractals because the points needed on the complex
208plane are decimals, not integers. **<i>Zooming into one point—effectively
209increasing the number of decimal places encoded by each pixel's location—only
210increases the precision required in computing.</i>**
211
212Limiting ourselves to hardware floating-point implementations caps our
213precision at FP128. In reality, if we're running this on the GPU, we're capped
214to FP64, since most GPU architectures don't support FP128. (OpenGL's shader
215language doesn't even provide a quad-width type, e.g. `long double`. Even for
216CPU architectures, hardware support for FP128 is
217[iffy](https://en.wikipedia.org/wiki/Quadruple-precision_floating-point_format#Hardware_support).)
218Also, since graphics applications generally don't need more than 32 bits of
219floating-point precision, GPUs tend to have only 32-bit wide FPUs, with a slow
220processing path for FP64 (about 1/64 the speed of FP32 according to some
221benchmarks). Despite this, GPUs have significantly more floating-point
222execution units than CPUs, so we're *still* running faster on the GPU.
223
224With 64-bit precision on the GPU, we can zoom in by a factor of about 14 times
225before we hit our precision limit.
226
227<figure class="full">
228 <img width="700px" src="/static/images/fractal_precision.png">
229</figure>
230
231This isn't ideal but it's the best I could come up with (or cared to,
232considering this was not a planned project) for a realtime renderer. Fractal
233dive renderers use high-level CPU software implementations for
234*arbitrary-precision* floating-point computation, like
235[BigFloat](https://github.com/nicowilliams/bigfloat), but this would be
236disastrously slow for a realtime application (and be incompatible with GPU
237acceleration).
238
239# Rendering on the GPU
240
241## Using a Fragment Shader
242
243The 10,000-foot view of the basic OpenGL rendering pipeline for an object is as
244follows:
245
2461. You give the graphics card mesh data and some arbitrary program-defined
247 render settings
2482. A vertex shader interprets this mesh data as primitive shapes, e.g.
249 triangles, and applies perspective transformations to scale, rotate, and
250 position them relative to the screen or "camera"
2513. A fragment shader uses the geometry of the primitive plus the given
252 arbitrary render settings (including textures) to fill in the colors of
253 fragments (pixels, basically) within the primitive
2544. The graphics card does some linear algebra magic to combine the computed
255 data for all objects into a rendered scene, the framebuffer
256
257<figcaption>This is the explain-like-I'm-five version. If you actually know
258OpenGL you know this is so insanely simplified it could just be called "wrong,"
259but it's a good enough overview for the purposes of this writeup.</figcaption>
260
261Knowing my CPU would be too slow for realtime rendering, rather than doing
262fractal computation on the CPU and loading the result into a texture to be
263displayed by the GPU, my initial attempt used an OpenGL fragment shader to do
264computation directly on the GPU. It seemed like a good idea; given two
265triangles filling the entire screen, the fragment shader is executed for every
266pixel of the frame and outputs a color for that pixel. However, there were some
267drawbacks:
268
2691. Everything was redrawn every frame, even if the user hadn't moved,
270 unnecessarily consuming GPU resources
2712. This was extraordinarily slow at high iteration counts (high zoom)
2723. Iterative refine—either splitting iterations across frames or breaking the
273 frame up into chunks—wasn't feasible (or wouldn't be efficient/elegant
274 enough)
275
276Clearly, a different strategy was needed.
277
278## Using a Compute Shader
279
280The [compute shader](https://www.khronos.org/opengl/wiki/Compute_Shader) is
281OpenGL's interface for general purpose GPU (GPGPU) programming—analogous to
282NVIDIA's CUDA, which lets you use a GPU for arbitrary floating-point-intensive
283computation. Unlike CUDA however, compute shaders provide cross-platform
284support (including integrated GPUs) and integrate with the rest of the graphics
285library, for things like accessing textures. Perfect!!
286
287As they're meant to be used for GPGPU programming, compute shaders aren't built
288into the core OpenGL rendering pipeline. They must be explicitly invoked
289(dispatched) by the application, separate of the rendering sequence. This is
290actually great for our renderer because being able to control when computes
291happen means we can have them run only when they're actually needed (when the
292user pans, zooms, or transforms).
293
294```cpp
295// inside application C++ "world logic"
296// once we determine we need to recompute after some user input, we can do it like this:
297
298// bind texture/image to save computed data in
299m_pTexture->BindAsImage();
300
301// upload arbitrary data/settings (x/y position, zoom level, screen size, iteration count)
302m_pComputeShaderUniformUploader->UploadUniforms(program);
303
304// do compute across the width and height of the screen split up into 32x32 pixel chunks
305glDispatchCompute((m_windowWidth+31)/32, (m_windowHeight+31)/32, 1);
306```
307
308The compute shader looks like this:
309
310```glsl
311#version 460 core
312
313// define the size of the local working group to be 32x32x1
314// main() runs every time for each pixel in the 32x32 region
315layout (local_size_x = 32, local_size_y = 32, local_size_z = 1) in;
316
317// arbitrary data/settings ("uniforms") uploaded by the application
318layout(r32f, binding = 0) uniform image2D texture0;
319uniform ivec2 screen_size;
320uniform dvec2 offset; // indicates x/y position within the fractal
321uniform double scale; // decreases when zooming in
322uniform float discard_threshold_squared;
323uniform int max_iteration_count;
324
325void main()
326{
327 // 1. convert screen pixel location to world/graph space
328 // 2. run z-transform
329 // 3. store iteration count into texture
330}
331```
332
333Splitting up the screen into 32x32 chunks is pretty arbitrary. GPUs are very
334heavily designed for parallelism, so, generally, splitting work up into chunks
335means things will get done more quickly, but there is an upper limit to this.
336In my experimentation, 32x32 chunks seemed to work best for whatever reason.
337
338The only data we're storing in the texture during computes is a single number
339per pixel: the number of iterations it takes for the pixel's corresponding
340point to diverge. The same texture is then fed into the fragment shader during the
341rendering stage, which reads these values and spits out colors to the screen
342accordingly. `-1` can be used to indicate a point that doesn't diverge (in the
343set, colored black).
344
345<figcaption>Note the texture is declared in the computer shader as having
346format <code>r32f</code>; this indicates a single channel <code>r</code> with
347type FP32, though <code>r32i</code> would work just as well here. See
348possible formats in OpenGL documentation for <a
349href="https://registry.khronos.org/OpenGL-Refpages/gl4/html/glTexImage2D.xhtml">glTexImage2D</a>.
350Texture creation is managed by the application.</figcaption>
351
352This is great and all, but it doesn't really solve the issue with high
353iteration counts slowing things down. When zooming in a lot, the application
354becomes so slow that doing anything has a more-than-noticeable input latency
355(see this in the video around 0:15). This is where we implement a method for
356*progressive refine*.
357
358## Progressive Refine
359
360Progressive refine is the act of taking an intensive piece of work and breaking
361it down into multiple chunks *over time*. This is commonly done in dedicated
362renderers when you want a preview of a render that will take a while, or in
363networking when loading images over a slow connection; it quickly gives you an
364image at, for example, 1/64 full quality, then not-so-quickly an image at 1/32,
365then 1/16, and so on, with each step taking longer on average than the
366previous. Inspired by a friend's [use of the Bayer matrix for this
367purpose](https://jbaker.graphics/writings/bayer.html), I used a similar
368**<i>interlacing pattern</i>** defined by the **<i>[Adam7
369algorithm](https://en.wikipedia.org/wiki/Adam7_algorithm)</i>**, which splits
370work in an 8x8 grid across seven steps:
371
372```glsl
373const int ADAM7_MATRIX[8][8] = {
374 {1, 6, 4, 6, 2, 6, 4, 6},
375 {7, 7, 7, 7, 7, 7, 7, 7},
376 {5, 6, 5, 6, 5, 6, 5, 6},
377 {7, 7, 7, 7, 7, 7, 7, 7},
378 {3, 6, 4, 6, 3, 6, 4, 6},
379 {7, 7, 7, 7, 7, 7, 7, 7},
380 {5, 6, 5, 6, 5, 6, 5, 6},
381 {7, 7, 7, 7, 7, 7, 7, 7},
382};
383```
384
385<figcaption>For this pattern of interlacing, the 1st and 2nd steps actually
386always take the same amount of time because they do the same amount of work.
387Every step after that, though, takes twice as long as the
388previous.</figcaption>
389
390Incorporating this into the fractal renderer, what we can do is pass some
391number to the compute shader which indicates which step we're on, [1-7]. The
392compute shader then indexes the above matrix at the pixel's position, compares
393that value to the instructed interlace step, and only computes if the values
394are equal.
395
396```glsl
397uniform int interlace_step;
398```
399
400```glsl
401// [in main()]
402
403// 32x32 work group size means we'll have 16 internal 8x8 grids for interlacing.
404// Check where the current pixel is within whichever 8x8 grid it falls in:
405int relative_pixel_grid_pos_x = gl_LocalInvocationID.x % 8;
406int relative_pixel_grid_pos_y = gl_LocalInvocationID.y % 8;
407
408bool do_compute = (ADAM7_MATRIX[relative_pixel_grid_pos_y][relative_pixel_grid_pos_x] == interlace_step);
409if (do_compute)
410{
411 // z-transform
412}
413```
414
415For each pixel, the routine then stores either the new computed value or, if
416nothing was computed, does nothing. Correspondingly, the fragment shader must
417be updated for this behavior as well. Following step 1, most elements in the
418texture will still be empty; to prevent the screen from displaying mostly
419uncomputed pixels (undefined behavior?), we should check whether a pixel has
420been computed before trying to use it, and use the nearest computed pixel if
421the first hasn't been.
422
423The resulting behavior is this:
424
425<figure class="full">
426 <img width="700px" src="/static/images/fractal_refine.gif">
427</figure>
428
429Now, when zooming in at high iteration count, the first compute step is only
430doing 1/64 (~1.56%) of the computations it normally would, keeping things
431within the span of a single frame, i.e. preventing framerate drops. This is
432great for user actions, where zooming and panning happen for as long as the
433mouse input lasts—potentially hundreds of frames.
434
435Suppose, though, that a single step takes longer than a frame to compute. Well,
436that's no worry either since *compute shaders act independent of the rendering
437pipeline*, so rendering is not being held up by a compute shader still running.
438Furthermore, to prevent compute dispatches from overlapping, you can make use
439of asynchronous OpenGL interfaces like [fences and memory
440barriers](https://www.khronos.org/opengl/wiki/Memory_Model) to prevent
441dispatching a new compute for the next step until the previous has finished,
442framerate unaffected.
443
444My implementation can actually take the idea of progressive refine a step
445further, allowing the point's z-transform itself to be distributed across
446computes, resulting in *multiple* interlacing passes. It does this by storing
447another two elements per pixel in the texture, the real and imaginary
448components of the z-transform output, for it to pick up with on the start of
449the next compute. This poses the same precision issues as before, however,
450considering OpenGL textures only allow you to store elements up to FP32
451precision, so it doesn't work very well at high zoom levels. We can work around
452this by using a separate texture with four elements per pixel: two 32-bit
453elements for the real component, and two more 32-bit elements for the imaginary
454component. See
455[floatBitsToInt](https://registry.khronos.org/OpenGL-Refpages/gl4/html/floatBitsToInt.xhtml)
456and related GLSL functions for a way you might accomplish this.
457
458