Better DE estimate using orbit traps

[hobold]

One more thing just occurred to me. The mandelbulb is notorious for the need to scale down the distance estimates derived from the standard 2D Mandelbrot construction. Maybe the instability near the pole is the underlying reason for this. This would mean that the aforementioned polar orbit trap, when done the "right" way (i.e. with more rigorous and more stringent mathematical analysis) could end up accelerating the mandelbulb renderers, because it would replace the global fudge factors that reduce DE everywhere across the board (rather than just in the places where it is really needed). So you would no longer do ten times the number of required steps everywhere, only to still overstep near the polar features.

Dang, I think I need to abandon my voxel based renderer, and play with ray marching instead. :-)

[knighty]

Hi,

I absolutely agree.
I hope that some day, someone with good math background will study this stuff and give us the right answers. Meanwhile, the tricks I'm giving here really do the job. 

I was curious about what would give the full DE would give. Because an full analytic formulation is not yet available I've tried the numerical approache to compute the gradient (as described in the famous mandelbulb thread). Of cours, I do the tests on the mandelbulb variant that have the most adverse case: positive Z-component method.

The result without the correction trick:


With the correction trick:


The analytic DE I'm curently using (because it's simple and fast. See post:http://www.fractalforums.com/mandelbulb-implementation/realtime-renderingoptimisations/msg9072/#msg9072) without the correction trick:


With the correction trick:


Humm... maybe the full analytic DE will do the job! if the artifacts are due to the numerical approximations.

[hobold]

Did you use a cylindrical or a conical orbit trap for these test renders?

[knighty]

I used cylindrical traps. Have not yet modified it to conical.

[hobold]

I didn't mean to order you to get to work. :-) I just wanted these experiments to be documented well.

I still don't feel confident about my grasp of whats happening during the computation of the mandelbulb, in particular this whole idea of turning a smooth iteration function into a distance estimate. Nevertheless, I want to wager a guess as to what the "right" correction for pole artifacts would be.

Picture a globe with the standard spherical coordinate grid of latitude and longitude. At the equator, the grid cells are nearly perfect squares. Towards the poles, the grid cells turn rectangular, getting taller and taller. Until eventually, at the pole, the final row of grid cells is triangular, with the pole as the common tip of all triangles.

In the case of the 2d Mandelbrot, where the equations for distance estimation were originally derived, the situation is like an equator everywhere. No poles, no distortions anywhere. This is the assumption that the distance estimator is based on.

I believe the "correct correction" for the Mandelbulb distance estimates would be to shrink them proportionally to the shorter edge of a spherical grid cell at the respective latitude, to compensate for the rectangular distortion. So you would use a conical orbit trap, keeping track of the closest position to a pole. Then compute the latitude of that closest position, and shrink the distance estimate by cos(latitude). With a bit of luck, this compensation of all orbits (not just those that move close to a pole) should remove the need for global downscaling of distance estimates.

[knighty]

I didn't mean to order you to get to work. :-) I just wanted these experiments to be documented well.

LOL 
I said so just because I was about to try it.
My Provisional conclusion is that it doesn't make a big difference because the mandelbulb (I'm talking about power 8 mandelbulb) surface is near the radius 1 sphere.

I still don't feel confident about my grasp of whats happening during the computation of the mandelbulb, in particular this whole idea of turning a smooth iteration function into a distance estimate.

The DE formula usually used gives an estimate. It looks to me like the newton root finding method.

You can take a look at this:http://www.evl.uic.edu/hypercomplex/. It's about quaternion julia set BTW.
There is a PDF book with the distance estimate formulas and their proofs. If I've understood well, there exist a "perfect" DE for Mandebrot and connected julia sets (with complex and quaternion) but wich is practically difficult to use. The DE formula usually used (G/G') is an approximation. The DEs for the mandelbulb (and its relatives  ) have, to my knowledge, never been proved.

Nevertheless, I want to wager a guess as to what the "right" correction for pole artifacts would be.

Picture a globe with the standard spherical coordinate grid of latitude and longitude. At the equator, the grid cells are nearly perfect squares. Towards the poles, the grid cells turn rectangular, getting taller and taller. Until eventually, at the pole, the final row of grid cells is triangular, with the pole as the common tip of all triangles.

In the case of the 2d Mandelbrot, where the equations for distance estimation were originally derived, the situation is like an equator everywhere. No poles, no distortions anywhere. This is the assumption that the distance estimator is based on.

I believe the "correct correction" for the Mandelbulb distance estimates would be to shrink them proportionally to the shorter edge of a spherical grid cell at the respective latitude, to compensate for the rectangular distortion. So you would use a conical orbit trap, keeping track of the closest position to a pole. Then compute the latitude of that closest position, and shrink the distance estimate by cos(latitude). With a bit of luck, this compensation of all orbits (not just those that move close to a pole) should remove the need for global downscaling of distance estimates.

This makes sens but using cos(latitude) gives me too small estimates. I think that the contribution of each trap should be weighted with a function of the magnitude. Will see what it gives.

[hobold]

Just to be clear: the cos(latitude) factor should be applied only once per orbit, not once per iteration. Only for the one iteration where the orbit came closest to a pole.

[knighty]

Taking the minimum z_i angle or a combination of the z_i angles doesn't make a big difference. The important thing here is that we want to do the correction near the rendered surface and to not scaling down the ED far from the actual surface. That's why I multiply the "star" correction term by a function of the estimated distance (yeah! it's like a snake bitting it's tail).
The trouble with using cos(latitude) is that for some z_i with the smallest cos(latitude), |z_i| may be very big. the region where the correction will occure will be unnecessarily large. I notice (now) that using cylindrical trap instead corrects somehow this situation. We have:
cos(latitude(z_i))=sqrt((x_i²+y_i²)/|z_i|²) for the angular trap.
The cylindrical trap is (x_i²+y_i²)=cos(latitude(z_i))²*|z_i|². It also seems that using squared cos(latitude) gives better results.
I'll try to post some pictures to illustrate what happens.

I want to add a comment about orbit traps in this particular case. taking the minimum is not the only choice we have. Moreover, it doesn't give the smoothest results. Some other possible formulas are:
1-product(1-exp(- positive_constant * x_i²)), and
1-product(1-1/(x_i²/positive_constant+1))

x_i is some property of z_i.
positive_constant acts on the zone of influence of each x_i.

These functions give smooth results. They are some sort of fuzzy minimum.

[hobold]

Well, one thing I learned is that I really don't understand this orbit trap business. But it's true; distance estimates far from the surface don't really need the correction. Hm ... so maybe the traps should be flattened spheres (ellipsoids) on both poles ... ?

[knighty]

Don't worry! I understood the orbit traps last november .

Here are two pictures to illustrate what I said in my previous post. The yellow and green areas are where the value (cos²(latitude) or x²+y²) is low:

Cos²(latitude) -squared because it looks better and works better for me-

A closeup:

Cylinder trap:

Closeup:


Edit: These are vertical slices along the x axis

[knighty]

Hm ... so maybe the traps should be flattened spheres (ellipsoids) on both poles ... ?

Good idea. I'll try it ASAP!

EDIT: wow! works well. Thanks


EDIT2: And you say you dont really understand that busness?   :surprise:

[hobold]

I guess I have relatively high standard for "to understand" ... let me elaborate. As I mentioned in another thread, I was allowed to enjoy the privilege of a thorough mathematical education. However, my brain does not seem to be a good match for the formal, disciplined, stringent, goal oriented methods in the mathematical toolkit. Don't get me wrong, I love math, there is much beauty in it, and it is a very powerful tool in the hands of a true master.

But I am more of a seeker, a wanderer. I never quite know where I will end up. In fact, I like it not to know where I will end up. That's one of the better ways to encounter surprises. :-)


There is a phrase that is often used as an euphemism for "I don't know what I am talking about". That phrase is: "I have an intuitive approach to <insert topic>". I think that phrase describes me fairly accurately ... including the occasional case where it assumes its euphemistic meaning. :-)

I sometimes have a knack for making good guesses, but then I tend to have a very hard time justifying those guesses with solid theoretical reasoning. But such reasoning is one of the best ways to explain an idea to somebody else. And the ability to explain is, in my book, the one relevant mark to understanding. Only one who really understood something will be able to truly explain it to somebody else. That's my standard.


Glad I guessed right. :-)

[knighty]

Only one who really understood something will be able to truly explain it to somebody else. That's my standard.

"Ce qui se conçoit bien s'énonce clairement - Et les mots pour le dire arrivent aisément."
"What is conceived well is clearly stated - and the words to say it come easily."

Nicolas Boileau-Despréaux - L'Art poétique (1674)
 

Glad I guessed right. :-)

And thank you for your valuable feedback.

[KRAFTWERK]

And both of you amaze me, great thread, love it!

Good "guessing" hobold! 

[hobold]

And thank you for your valuable feedback.

I expect to be rewarded with beautiful images and animations of the intricate Mandelbulb details which were overlooked so far.