Rendering Fine Detail with the Distance Estimation Method

In this post we'll learn about the Distance Estimation Method (DEM) which can reveal fine detail not visible with the standard method. We'll also try to understand intuitively what the method is doing, as well as derive the maths behind the algorithm too.

As a taste, the following is a Julia set rendered using the DEM method.

Two Problems

As a reminder, the standard method for rendering Mandelbrot and Julia sets is to repeatedly iterate $z_{n+1} = z_n^2 + c$ and see if the orbit of $z_n$ diverges. 

We can use an escape condition to check for divergence. If $|z_n|>2$ then we can guarantee the orbit of $z_0$ diverges. For Julia sets, the escape condition has two tests, $|z_n| >2$ and $|z|>|c|$.

However, orbits close to, but just outside, the Mandelbrot and Julia set boundaries can take a long time to escape and we can only run the iterations a finite number of times. Similarly, points inside the sets don't diverge, but since we only run a finite number of iterations, we can't be sure they won't escape later.

This makes the standard algorithm for rendering the Mandelbrot and Julia sets imperfect - but generally good enough for most purposes. We trade off accuracy with the maximum number of iterations we test a point for, in effect, the time it takes to complete a rendering.

There is a second problem.

The points on the complex plane that we test correspond to the pixels we want to colour. Those pixels form a grid. The points we actually test are also a grid, usually corresponding with the centre or corner of a pixel. The problem with this is that very fine structure is entirely missed. 

Have a look at the default view of the Mandlebrot. The following shows "blobs" away from the main body. The appear to be disconnected, but we know they are connected to the Mandelbrot set. 

We know that in deep zooms around the edge of the Mandelbrot and Julia sets, there are intricate and quite abundant structures made of very fine structures - filaments - but they are invisible to the standard algorithm because these filaments fall between the grid of points being tested.

Now, we could colour the outside of the Mandelbrot set in an attempt to reveal this hidden structure. The very common "escape speed" method should be appropriate because it colours pixels according to how rapidly they reach the escape criteria. The following shows the same zoomed in region of the Mandelbrot set, on the left without the colouring, and on the right with this escape time colouring.

We can see on the left several "mini-mandelbrot" blobs but not the connecting structure. On the right the colouring has overwhelmed the image making it difficult to see the actual set.

The following shows the same but for a zoomed in region of a Julia set.

The image on the left is not too bad, but only because the collection of tiny blobs do hint at an intricate structure.

We need a new method.

Distance Estimation (by Gradient) Method

You will have noticed that away from the Mandelbrot and Julia sets, the escape-time coloured regions are larger. Their isolines are more widely spaced out. Closer to the boundary, the regions are thinner,  the isolines are pressed more closely together. The following image highlights these isolines.

If you are familiar with topographic maps, you'll know that altitude (height) isolines close together means a steeper rise or fall,  and widely spaced isolines means a shallow rise or descent. The same is true here.

The gradient of the magnitude of $z$ closer to the edge is higher than the gradient further away from the edge.

This is the key realisation that is behind the Distance Estimation Method. It uses the gradient to estimate the distance from the boundary (any boundary) of the Mandelbrot or Julia sets. 

The benefit of this method is that the points we test, the points that correspond to the grid of pixels, only need to be near the edge of the Mandelbrot or Julia sets to detect that edge. For very fine filaments, again, the points only need to be near them to detect them.

 

Results!

Further below, we'll talk about the algorithm and necessary calculations, but for now let's see how this method improves on the above Mandelbrot and Julia zooms.

The following shows the exact same zoom into the Mandelbrot set as above. The newly revealed structure is very impressive!

The following shows the same zoom into the Julia set as above. Again the resulting detail is amazing.

 

The Algorithm

Before we write down the algorithm, let's first derive how the gradient is calculated.

The primary calculation is the iteration:

$$z_{n+1} = z_n^2 + c$$

For the Mandelbrot set, $z_0=0$ and $c$ is the point being tested. For Julia sets, $c$ is fixed, and $z_0$ is the point being tested.

Now let's work out the gradient (with thanks to Claude for helpful guidance on math stackexchange).

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For the Julia set, we have fixed $c$ and are testing each $z_0$. Therefore the gradient of $z_n$ is with respect to $z_0$.  

We can differentiate the primary iteration expression using the chain rule, noting that $c$ is constant so its differential is zero.

$$ \frac{d}{dz_0} z_{n+1} = 2 \times z_n \times \frac{d}{dz_0} z_{n} + 0$$

We can write this more simply as

$$z_{n+1}' = 2 z_n  z_n'$$

Whilst this may look unhelpful, this form actually works well with the iterative algorithm. That is, at each iteration:

• as usual, we calculate $z_{n+1}$ from the previous $z_n$ using $z_{n+1} = z_n^2 + c$
• we also calculate $z_{n+1}'$ from the previous $z_n'$ using $z_{n+1}' = 2 z_n  z_n'$

But what should the initial value of the gradient $z_0'$ be set to? Well,

$$z_0' = \frac{dz_0}{dz_0} = 1$$

This tell us to initialise $z_0'$ to 1, in the case of Julia sets.

---

For the Mandelbrot set, we have fixed $z_0 = 0$ and are testing different $c$. Therefore the gradient of $z_n$ is with respect to $c$.

The differentiation is slightly different now.

$$ \frac{d}{dc} z_{n+1} = 2 \times z_n \times \frac{d}{dc} z_{n} + 1$$

We can write this more simply as

$$z_{n+1}' = 2 z_n  z_n' + 1$$ 

Again, what should the initial value of the gradient $z_0'$ be set to? This time,

$$z_0' = \frac{dz_0}{dc} = 0$$

So we initialise $z_0'$ to 0, in the case of the Mandelbrot set.

---

DEM Mandelbrot Set Pseudocode

• or each test point, calculate $c$
• initialise $z_{0}=(0+0i)$
• also initialise the gradient $dz_{0}=(0+0i)$
• for the maximum allowed iterations
• calculate next $z$ using $z\mapsto z^{2}+c$
• calculate next gradient $dz$ using $dz\mapsto2\cdot z\cdot dz+1$
• break out of iteration loop if $|z|>4$ escape condition met
• calculate the distance estimate as $d=\left(|z|\cdot\log|z|\right)/|dz|$
• colour the pixel based on distance estimate, for example $255\times\tanh(d\times\text{resolution}/\text{size})$

Because we're primarily using this technique to explore fine detail, we'll boost the maximum allowed iterations to 4096. We'll also increase the escape radius from 2 to 4, to give the gradient a chance to develop. Both of these will cause the rendering to take longer.

The suggestion for colouring the pixel uses the hyperbolic tangent $\tanh()$ function which squishes all values, no matter how large, into the range 0 to 1, ready for scaling up to 0-255.

Notice that the pixel colour is not based on escape speed, the iterations it took to meet the escape condition, but the distance estimate, which itself is based on the gradient of the orbit of $z$. 

The distance estime $d=\left(|z|\cdot\log|z|\right)/|dz|$ is the only thing we haven't explained here. Your can find out more here. The key is that the larger the gradient the smaller the distance, the rest is scaling.

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DEM Julia  Set Pseudocode

The algorithm changes slightly for Julia sets.

• $c$ is chosen and fixed
• for each test point, calculate $z_{0}$
• also initialise the gradient $dz_{0}=(1+0i)$
• for the maximum allowed iterations
• calculate next $z$ using $z\mapsto z^{2}+c$
• calculate next gradient $dz$ using $dz\mapsto2\cdot z\cdot dz$
• break out of iteration loop if $|z|>4$ escape condition met
• calculate the distance estimate as $d=\left(|z|\cdot\log|z|\right)/|dz|$
• colour the pixel based on distance estimate, for example $255\times\tanh(d\times\text{resolution}/\text{size})$

Notice the gradient is initialised to 1, not 0, and is updated as $dz\mapsto2\cdot z\cdot dz$, not $dz\mapsto2\cdot z\cdot dz+1$.

Art

The following is a zoom into the Mandelbtot set rendered using this Distance Estimation method.

I love the different shapes and balance of positive and negative space, and also the contrast between the spiral and patterned regions.

I love it so much, I made it the cover image of the Second Edition of Make Your Own Mandelbrot !

Second Edition is out!

 The Second Edition is out - as ebook, paperback and hardback.

This book is written specifically for anyone who is curious about what the famous Mandelbrot Set fractal is, and how to recreate it, but has struggled with the traditional textbooks and online guides.

• Explore what a fractal is, artificial and natural.
• Learn about functions, iteration, and chaos.
• Discover the (not so) complex numbers.
• Find out what the Mandelbrot and Julia Sets really are.
• Follow a hands-on tutorial to make your own Mandelbrot and Julia Sets in python.
• Discover different ways to render the fractals - escape time, image filters, and 3d landscapes.
• Bonus chapter on distance estimation to reveal very fine detail.

 

This book is aimed at complete beginners, and only requires a browser and internet connection to follow the tutorials. No software installation is required.

This Second Edition has been updated to python 3, and all the tutorials now run completely in web-based notebooks with no software to install. To keep the price down, this second edition has been redesigned to be attractive in monochrome print.

I'm particularly pleased that the bonus chapter on Distance Estimation Methods (DEM) worked so well. The following is an example of a zoom into the Mandelbrot Set rendered by DEM, and used as the book cover.

All the code for the Second Edition is freely available online:

• https://github.com/makeyourownmandelbrot/Second_Edition
  

Code for the Second Edition is on GitHub

All the code for the Second Edition is now on Github at:

• https://github.com/makeyourownmandelbrot/Second_Edition
  

 

The python notebooks are as follows.

The basics:

• Make_Your_Own_Mandelbrot_black_and_white.ipynb - the very basic Mandelbrot set, no colouring
• Make_Your_Own_Mandelbrot_grey_scale.ipynb - the Mandelbrot set with the outside region coloured according to escape speed
• Make_Your_Own_Julia_grey_scale.ipynb - Julia sets

 

Experimenting with Sobel edge detection filters:

• sobel_mandelbrot.ipynb
• sobel_mandelbrot.ipynb

And finally, creating 3D landscapes from the fractal images:

• 3d_Mandelbrot_landscape.ipynb
• 3d_Julia_landscape.ipynb
  

 

 

Second Edition

I've started work on a second edition of Make Your Own Mandelbrot.

The key differences will be:

• monochrome to minimise the price of the books
• updated for python 3
• using cloud-based notebooks (Google Colab) to avoid the need to install any software
• updated library for generating 3d fractal landscapes
• better quality diagrams and ollustrations
• more concise text - but not too concise because a priority is to maximise understanding
  

Escape Condition for Julia and Mandelbrot Fractals

This post is reproduced from the blog for my next book, Make Your Own Algorithmic Art.

---

This post isn't a tutorial on creating the Julia or Mandelbrot fractals - you can learn about that here. Here we'll focussed on a specific bit of mathematics about the "escape condition" that many guides state but fail to explain.

Basic Idea

The basic idea behind both Julia and Mandelbrot fractals is to take a starting complex number $z_0$, apply a simple function $z^2 + c$, and feed the output $z_1$ back into this function as input. Doing this repeatedly we get a sequence of output complex numbers, $z_1$, $z_2$, $z_3$ ...

$$ z_{n+1} = z_{n}^2 + c$$

These output values can behave in one of three ways:

• They can get larger and larger, towards infinity. This is called escaping.
• They can get smaller and smaller, towards zero.
• They can orbit around, but without getting ever larger. In some cases they can approach a finite constant.

The fractal patterns are coloured to show which points escape, and which don't. Those that escape are outside the fractal, and those that don't are inside.

The difference between Julia and Mandelbrot fractals is only how we choose $z_0$ and $c$.

Computational Problem

The behaviour of complex numbers under $z^2 + c$ is chaotic. That is:

• The output sequence is very often irregular, and predicting future values is very difficult without actually working out the sequence. They seem random, even if they're not.
• The sequence is very sensitive to starting conditions. That is, even a tiny change in the starting conditions can drastically change how the sequence behaves.

We can't derive a mathematical formula which tells us which points are inside the fractal, and which are outside (they escape). And this chaotic behaviour suggests we can't truly know even if we run the feedback iterations many many times, because the sequence can suddenly escape after a long period of orbiting around.

Practical Compromise

So we have to compromise - we agree to generate a fixed number of output values, so we can render an approximate pattern. Perhaps a sequence of 50 output values is sufficient? Maybe 100? Maybe even 1000?

By experimenting, it becomes clear that 50 or 100 iterations is fine, except when we are zooming into a very small area of the fractals, where we need more iterations to be able to separate out the inside and outside regions. If we don't do this, the details of the fractal don't emerge.

Computational Shortcut

For many starting points in a fractal pattern, the output values will get larger and larger very quickly. We want to stop calculating the sequence for two reasons:

• If the numbers get too large, this can cause our code to crash with an error, or worse, continue with incorrect calculations. The root cause of this is that the largest size of number we can store and calculate with is fixed.
• We don't want to waste time and computational effort continuing to calculate the sequence which is only every going to get larger and larger. This shortcut can be important because the calculations for detailed fractals can take a long time.

So many guides will state that we can stop calculating the sequence when the magnitude $|z_n|$ gets larger than 2.

That works well, and makes intuitive sense. If $z_n$ has got so large that it is more than 2 away from the origin, then it will forever get larger and larger. But the statement is rarely explained.

Next is a simple proof - and also a demonstration that the popular escape condition $|z_n| > 2$ is incomplete.

Simple Proof

First let's remind ourselves of the triangle inequality, which basically says that a direct path is always shorter than an indirect path between two points.

$$ | a + b | \leq |a| + |b| $$

Let's contrive to artificially expand out $|z^2|$,

$$ |z^2| = |z^2 +c -c| $$

If we use that triangle inequality, with $a = z^2 +c$ and $b = -c$, we have

$$ |z^2 +c -c| \leq |z^2 +c| + |-c|  $$

but because $|-c|$ is just $|c|$ we have

$$ |z^2 +c -c| \leq |z^2 +c| + |c|  $$

Now, remember the next value in a sequence is $z^2 + c$ because that's the function we keep applying. Let's bring that to the left of that last inequality.

$$ |z^2 +c| \geq |z^2| - |c| $$

Also, $|z^2|$ is the same as $|z|^2$ so we have

$$ |z^2 +c| \geq |z|^2 - |c| $$

Now is the interesting part. If we say that $|z|$ is bigger than $|c|$, that would mean

$$ |z|^2 - |c| \gt |z|^2 - |z| $$

That means we can also say,

$$ |z^2 +c| \gt |z|^2 - |z| $$

Which can be factorised as

$$ |z^2 +c| \gt |z|(|z| -1) $$

Let's rewrite that previous expression as a ratio of the sizes of the current $z_n$ and the next $z_{n+1} = z_n^2 + c$,

$$ \frac {|z_{n+1}|} {|z_n|} \gt (|z| -1)$$

Now another interesting part. If we say $|z|$ is greater than 2, that means $|z| -1 \gt 1$. So we finally have

$$ \frac {|z_{n+1}|} {|z_n|} \gt 1$$

Which is saying that $|z_{n+1}|$ is always greater than $|z_n|$ as long as:

• $|z| \gt |c|$, and 
• $|z| \gt 2$

So we've shown that two conditions need to be true to prove that the sequences escapes, not just the traditional one. However in practice, the traditional $|z| \gt 2$ seems to work well enough.


Reference
One text that does try to cover this is Chaos and Fractals: New Frontiers of Science.

Python 3 Code on GitHub

The code for the book Make Your Own Mandelbrot is now on github:

• https://github.com/makeyourownmandelbrot/makeyourownmandelbrot

And whilst I was doing this, I updated the code to Python 3.

This required only minor changes: the xrange() becomes the simpler range() function.

(sadly the 3d code requires the mayavi libraries which are not yet ported to Python 3)

Republished for Better Formatting

I've republished the kindle and print book.

The main reason is that a few people with older kindle devices didn't have great experiences with the formatting. For some images didn't show properly, for others, the margins were wonky, etc

This is sad because it shouldn't be that hard to get right in 2016. The core problem is that ebook file formats are not open, stable and implemented in an interoperable way. It's like the web 20 years ago - with big companies not implementing web standards properly, and deliberately trying to pervert them to their own ends. Thankfully after 20 years - that's all settled down and usable.

I took the decision to publish the ebooks using Amazon's new Kindle Textbook format. That promises to have much greater certainty over layout, even for complex content, ... like a PDF. This will be great for people who want to see a page more or less as it was intended., and certainly not mashed up.

The following are screenshots from my Android phone's Kindle app - and it looks fantastic!

There is a down side - those with older Kindles that can't support this new Textbook format, won't be able to buy the book. So happier readers, but fewer of them. I didn't take that decision lightly but not having unhappy readers was a priority for me.