I think there are some confusions here about the mind's eye, and the way the visual cortex works.

First of all, I suggest you do the selective attention test. Here will do. Selective attention test

This video illustrates the difference between looking at a scene and actually seeing it. Do pay attention closely or you might miss something important!

The bottom line is that when you look at the outside world, thinking that you see it, your brain is converting the external world of light images into an internal coding of that image. It cheats, royally, when it tells you that you're seeing the world as it is. You're perceiving a coded version of it, and that code is actually optimised for usefulness and the ability to respond quickly. It's not optimised for completeness - it's more about enabling you to pay attention to one main thing, and ignore everything else in your visual field that doesn't currently matter.

And that's where your comparisons later fall down. The computers rendering Avatar have to create images of the fictional world. Your own internal mind's eye doesn't have to do that - it only has to generate codes that stand for visual scenes. Where Avatar's computers had to render the dragon pixel by pixel, your internal eye only has to create a suitable symbol standing for "Dragon in visual field, dead centre." It doesn't bother to create nearly all of the rest of your imagined world in the same way as some people ignore the important thing in the attention test. Because you only EVER see the coded versions of the world, the two look the same to you. But it is a much cheaper operation as it's working on small groups of codes, not millions of pixels.

The human brain is a very nice machine - but I also suspect it's not as fast as many people think it is. Time will tell.

Yes, your brain has much more processing power than anybody's laptop. But I think the people you quote are referring to the available general-purpose processing power.

Your visual cortex, for instance, packs a huge amount of processing power, but it's specialized for one task - processing visual information. You can't just tell it to crunch numbers and get a useful output, because it isn't built for that. Yes, there are clever tricks for employing your visual cortex in calculating big numbers, but even those don't take full advantage of it - the author of the linked paper says he was able to multiply two random 10-digit numbers together over a period of 7 hours. If you could use all the power in your visual cortex, you'd get the result instantly.

The brain packs a huge amount of power, but most of it is very specialized and hard to take advantage of, unless your task happens to be one similar to the domain that your brain has specialized circuitry for. Yes, with enough practice and the right tricks, people can learn to perform impressive feats of arithmetic and memory, so the circuitry is to some extent reconfigurable. But no human yet has managed to outdo modern computers in pure number-crunching. And getting even that far takes a lot of practice, so the circuitry isn't usefully available for most tasks.

We do have some processing power available for truly general-purpose computing. Given any (non-quantum) algorithm, a pen and some paper, you can simulate that algorithm with enough time. We had people do that before coming up with digital computers. But it's going to take a while, because your usefully available general-purpose computing power is very limited. In contrast, practically all of the laptop's processing power is general-purpose and usefully available - give it the same algorithm, and it's going to run through it a lot faster than you will.

But it seems to me, maybe naively so, that most of my human abilities involve massive amounts of number crunching that no desktop computer could do.

I think it's an unfair comparison because you are allowed to cheat. You don't have to produce 1500×1000 pixels, 25 times per second, consistently, with correct lights and reflections, etc. Different minds may work differently, but I suspect there is a lot of cheating. A bad result may seem OK, because you are allowed to fix any detail at the moment you start paying attention to it; you don't even have to notice that you are doing this.

On the other hand, it wouldn't surprise me if human brain would be really faster in some tasks -- those we are optimized for by evolution, such as image processing. Visual part of brain does some massive parallel computation, and if a task allows parallel computation, many slow computers can outperfom one fast computer (or a smaller group of fast computers).

Both effects may work together -- we have parallel hardware optimization for image processing and we are allowed to give imprecise answers and even cheat.

When you create a movie in your mind's eye, then firstly, it's not clear that it actually has the amount of detail that a computer-rendered movie does. But secondly, there is (probably) no structure in your visual cortex that would straightforwardly map to a number. (This is very unlike the case in a computer, where every voltage in some sense represents a number.) Instead your neurons are taking shortcuts that a human engineer would simulate by using explicit arithmetic. Recall the famous pitch-detecting circuit; it is clearly doing the equivalent of some arithmetic, but there's certainly no explicit representation within it of any numbers.

Just because something can be achieved by explicitly representing numbers, doesn't mean that your brain does it that way - not even under the hood. Consequently the comparison to things that were achieved by massive number-crunching capacity is not strong evidence of such capacity in your brain.

I once thought that mathematical geometry worked by a kind of detail crunching.

If a line is just a systematic set of infinite number of points checking whether two lines intersect would just be a "simple" operation to check whether they contain a point in common. Take points from one line and check whether it is a part of the other line. Doing this with literally infinite number of points would amount to a supertask. So you could only do so to an arbitrary precision but not exactly.

However a very simple math problem like "find the intersection of lines y=2x and y=3x+5" can be done exactly in a finite small number of symbol operations. And actually the description of the infinite number of points on the first line can be done by a very finite expression of "y=2x". There are also an infinite number of such lines but finding each of their intersection doesn't include attending them pair by pair. The procedure of solving the descriptions as a equation pair can be expressed in a expression more meta and "more finite".

So instead of just a big fleet of lowest level comparisons what really happens is a tiny amount of work on different levels. If one would count each symbol manipulation as a single number crunching operation the supertask of point comparisons would seem to be the most demanding. However using multiple levels of symbols means supporting a wider array of symbol manipulation operations.

So while I appears that I compare infinite numbers of points when I am doing simple geometry, it's just that I am bypassing one kind of calculations limits by using another kind of calculation.

Yet any computer can outperform me at simple calculations.

Humans aren't built to do math. This is true on both a lower level, in that an imperfectly designed neural network wouldn't be very efficient at precise calculations, and a higher level, in that humans do not have any sort of specialized part of their brain for math.

It is actually pretty interesting: the software that tries to do something highly parallelizable that brain does, always ends up requiring within the ballpark of how many operations per second neurons do in parallel. If we are to talk Avatar, more interesting is the fact that this much computing power is necessary to fool you, within the ballpark of the power of visual cortex, 'naively' computed. Likewise for other things. Even for things that are pretty narrow, where brain performs badly, like Chess, the computing power required is quite formidable. Much moreso for something like Go.

The belief that your abilities are doable without immense number crunching is the notion of overly optimistic AI researchers of the 1960s. It's been obsoleted everywhere but here. Here you have this notion that brain hardware is somehow very 'badly designed' by evolution, never mind that it packs immense operations per second into small volume and small power consumption, with the only major failure being the use of organic materials rather than silicon. And you have the notion that the software is equally very bad. And total lack of awareness why world doesn't think its true any more.