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In case you're still curious about it, the effect in the image you posted is not made using polar coordinates. It is a stereographic projection of a panoramic image.

This is visually similar to the polar coordinate method, but using polar coordinates creates a circular image with a hole in the middle. Stereographic projection creates a square image that covers the entire 360 degree x 180 degrees of the panorama.

Here is a link with more information about different kinds of projections.

Not sure about that lens, but I use PTGui for stitching, and a Nodal Ninja tripod head (my spherical panos: www.panoramiceye.com). Might want to try /r/panorama too.

http://www.ptgui.com/

http://shop.nodalninja.com/

PTgui.

If you're using a tripod, you're gonna have to mess with it a bit no matter what software you use. However if you have a nodal tripod head, you can calibrate it to your setup specifically, shoot your brackets, and just pipe your raw images into PTgui and get a pretty reasonable result. Plus if you are happy with what you get, you can save out a template, and just shoot your images in the same order and automate things in the future.

It's a panorama made up of many photos.

There is no way for the whole photo to look "normal" so there are a number of options for how to warp the images. This one looks like equirectangular.

Check out http://www.ptgui.com/man/projections.html
There are also many other resources look up "panorama projections".

Cycles can only use mirror-ball and equirectangular images for environment textures. If you want to use that particular skybox, you'll have to transform the cubemap from the format it's in into an equirectangular map. You can convert it from a cubemap using the Flexify Photoshop filter, or, if you also want the flexibility to stitch photographs that you've taken yourself into an equirectangular panorama as well, get PTGui.

You can also look for some FOSS panorama stitchers, although they probably aren't as fast as the ones I listed.

1. Shoot RAW with a constant focal length, white balance, aperture, and ISO. If shooting HDR, only shutter speed should change.
2. Some prefer to process images as a batch. I prefer to create the panorama first to ensure no issues. I use PTGui.
3. Edit the final panorama in Lightroom, Photoshop, or your preferred application.

Edit: As /u/Matterchief pointed out, no settings should change for a regular panorama. Shutter speed should only change when shooting HDR.

If you're using a full frame fisheye lens, shoot six overlapping images in portrait orientation for a 360° view and then one for the zenith and another for the nadir. A circular fisheye lens can get this done in four frames. Of course you'll want to try and rotate from the lens' no-parallax point.

In the panoramic stitching software choose a stereographic projection and use the sliders on the preview window to zoom out. Or you could create an equirectangular panorama and make a "little planet" using the polar coordinates filter in Photoshop.

Hope that helps!

I tried different kind of panorama stitching software at work and this delivered by far the best results: http://www.ptgui.com/

You can see some 360 interior shots of some planes we did here: http://www.netjetseurope.com/Your-fleet/Cassna-Citation-Bravo/360-Tour/

Sorry no theta experience at all. But about 15 years' experience of 360ºx180º panos with a fisheye lens on my dSLR and mirrorless cameras with a panohead and tripod.

Just me, but a monopod with feet is going to be tippy by nature, and my guess is if it's stable enough to use, then the feet are still going to show up in your pano. But I'm sure there are cheaper options out there you can find. Just don't expect to use it on a windy day outside or on uneven ground. I think youtube might be your friend here.

Full tripod/photographer erasure, for me with mirrorless/dSLR and a fisheye lens means masks & layers tricks, but how you'd do that with a 360º camera, I've got no clue, other than to take two shots, with the camera at the same height, but at 90º (i.e., pointed at the ground, horizontally) orientation, with the photographer moving from where they were, and then reorienting the panos to have the same viewpoint, and using mask/layers to do erasure. Assuming you can get equirectangulars from a theta. Or, better yet, the original fisheye images to work with.

But there's no way around it being a complete and total PITA, post-processing wise, because getting exactly matching viewpoints is unlikely, and the only software I know of that does viewpoint correction with masking is PTGui. But with parallax, etc. you still only have to get the tripod feet areas to match to do erasure.

Personally, I use an 8mm fisheye lens (because it's very wide). I only need 6 vertical pictures around the horizon, then one each for the nadir and zenith. I combine all of them except the nadir picture in PTGui. Then I manually blend the nadir shot because it never lines up otherwise, since it's the one image I have to take without using the tripod. (How exactly I do that is beyond the scope of this comment, but it's a combination of PTGui and Photoshop.) The result will be an equirectangular image like this. Then I take that final image back into PTGui and transform it to stereographic.

What do you mean? Lightroom will fix fisheye effect with one click, PTGui automatically corrects lens distortion.

Any difficulties that may arise trying to stitch such images are because of viewpoint changes, not lens distortions.

There are free versions for stitching images together (http://www.ptgui.com). Unfortunately, stitching video is a bit more complicated. The two main software packages that can do this are Kolor AVP, and VideoStitch. Both of these are expensive as you mention. I believe they both have free demo versions available that will let you try them out and see what works best for you. I use AVP and find it very easy and intuitive. I think VideoStitch costs a bit more and can do a few fancier features like live streaming and stuff.

I was looking at a few similar pano projects with amazing results a a few years ago and researched some software, I downloaded one awhile back but never gotten around to creating one, but I hear it's pretty good:

PTGUI

To me it doesn't look like he has a full 360 x 180 degrees worth of shots to output an equirectangular. Hugin is great, I used to use it but haven't quite a while now. PT Gui is a very similar program (of not the same, I couldn't tell the difference when I used it).

Happy cake day, have an upvote :).

I made the image using a combination of hardware and software. The final image is a stereographic projection of a 360-degree panoramic image.