In this article, you will learn how to flatten a 360 degree fisheye image back to its landscape panoramic form. But first, what’s a 360 degree fisheye image?

[WARNING: graphic-intensive article ahead]

[image by gcardinal]

It’s created by using fisheye lens in your camera, take a bunch of pictures using that, and use stereographic projection to get the final image. Or so I understand.

Basically, it’s like taking a normal (but panoramic is better) picture and “curling” it along its bottom or top (but much more complicated than that).

Here’s how to visualise the fisheye image. Hold your right hand, palm downwards, in front of you, thumb towards you, little finger away from you. The tips of your fingers form the “left end” of the image. Your wrist forms the “right end” of the image. Now form a fist with your right hand. There’s your fisheye image.

### The conversion algorithm explanation

So given the fisheye image, we want to get that landscape image back. And the way we do that, is to “uncurl” the fisheye image. Here’s the diagram to explain the logic:

You may have noticed that the corners of the source image will not be in the resulting image. You may also notice that the centre of the source image is very “squeezed”, and the pixels around there will be repeated in the resulting image.

The first problem can be solved by using a bigger destination image. But it doesn’t matter to me, and you will get a jagged top with unfilled pixels. I didn’t like that, so decided to give those up.

The second problem… I don’t know if there’s a solution. Because you’re trying to “guess” the pixels mapped in the destination image from the source image. But the source image has less pixel information. The simplest solution seems to be to get a higher resolution source image. But that only mitigate the problem, not solve it.

You may also notice that only the pixels within the inscribed circle of the source image is used. Well, what do you get when you curl up a line? A circle. *wink*

What happens when circles are involved? Radius and angles, that’s what.

So in the destination image, in raster coordinates, going from top to bottom is equivalent to going from outer inscribed circle of source image to centre of source image. Or the variable l slowly reduces to zero.

Going from left to right is equivalent to going from 2PI radians and decreasing to 0 radians on the inscribed circle. It’s also equivalent to going from 0 radians to -2PI radians. Sines and cosines are periodic functions.

Here’s another diagram to show what happens when we iterate over the destination image:

Before we get to the code, here are 2 assumptions to simplify the process:

- The source image is square
- The width of the source image is even

They’re not necessary, but they make the programming easier. And I’m mapping the quadrants to the standard Cartesian quadrants because they make the math easier. The centre of the image should be the centre of the “circle” (or that small planet, as it’s affectionately known).

[The original source image wasn’t square, and its centre wasn’t the centre of the planet. So I cut the image, and guessed as best as I could on the centre. More info on the centre later in the article.]

### Fisheye to landscape algorithm/code

I’m plagiarising my own code from the image rotation with bilinear interpolation article for the bilinear interpolating parts. There are 2 resulting images, one with and one without bilinear interpolation. And here’s the code:

// assume the source image is square, and its width has even number of pixels Bitmap bm = new Bitmap("lillestromfisheye.jpg"); int l = bm.Width / 2; Bitmap bmDestination = new Bitmap(4 * l, l); Bitmap bmBilinear = new Bitmap(4 * l, l); int i, j; int x, y; double radius, theta; // for use in neighbouring indices in Cartesian coordinates int iFloorX, iCeilingX, iFloorY, iCeilingY; // calculated indices in Cartesian coordinates with trailing decimals double fTrueX, fTrueY; // for interpolation double fDeltaX, fDeltaY; // pixel colours Color clrTopLeft, clrTopRight, clrBottomLeft, clrBottomRight; // interpolated "top" pixels double fTopRed, fTopGreen, fTopBlue; // interpolated "bottom" pixels double fBottomRed, fBottomGreen, fBottomBlue; // final interpolated colour components int iRed, iGreen, iBlue; for (i = 0; i < bmDestination.Height; ++i) { for (j = 0; j < bmDestination.Width; ++j) { radius = (double)(l - i); // theta = 2.0 * Math.PI * (double)(4.0 * l - j) / (double)(4.0 * l); theta = 2.0 * Math.PI * (double)(-j) / (double)(4.0 * l); fTrueX = radius * Math.Cos(theta); fTrueY = radius * Math.Sin(theta); // "normal" mode x = (int)(Math.Round(fTrueX)) + l; y = l - (int)(Math.Round(fTrueY)); // check bounds if (x >= 0 && x < (2 * l) && y >= 0 && y < (2 * l)) { bmDestination.SetPixel(j, i, bm.GetPixel(x, y)); } // bilinear mode fTrueX = fTrueX + (double)l; fTrueY = (double)l - fTrueY; iFloorX = (int)(Math.Floor(fTrueX)); iFloorY = (int)(Math.Floor(fTrueY)); iCeilingX = (int)(Math.Ceiling(fTrueX)); iCeilingY = (int)(Math.Ceiling(fTrueY)); // check bounds if (iFloorX < 0 || iCeilingX < 0 || iFloorX >= (2 * l) || iCeilingX >= (2 * l) || iFloorY < 0 || iCeilingY < 0 || iFloorY >= (2 * l) || iCeilingY >= (2 * l)) continue; fDeltaX = fTrueX - (double)iFloorX; fDeltaY = fTrueY - (double)iFloorY; clrTopLeft = bm.GetPixel(iFloorX, iFloorY); clrTopRight = bm.GetPixel(iCeilingX, iFloorY); clrBottomLeft = bm.GetPixel(iFloorX, iCeilingY); clrBottomRight = bm.GetPixel(iCeilingX, iCeilingY); // linearly interpolate horizontally between top neighbours fTopRed = (1 - fDeltaX) * clrTopLeft.R + fDeltaX * clrTopRight.R; fTopGreen = (1 - fDeltaX) * clrTopLeft.G + fDeltaX * clrTopRight.G; fTopBlue = (1 - fDeltaX) * clrTopLeft.B + fDeltaX * clrTopRight.B; // linearly interpolate horizontally between bottom neighbours fBottomRed = (1 - fDeltaX) * clrBottomLeft.R + fDeltaX * clrBottomRight.R; fBottomGreen = (1 - fDeltaX) * clrBottomLeft.G + fDeltaX * clrBottomRight.G; fBottomBlue = (1 - fDeltaX) * clrBottomLeft.B + fDeltaX * clrBottomRight.B; // linearly interpolate vertically between top and bottom interpolated results iRed = (int)(Math.Round((1 - fDeltaY) * fTopRed + fDeltaY * fBottomRed)); iGreen = (int)(Math.Round((1 - fDeltaY) * fTopGreen + fDeltaY * fBottomGreen)); iBlue = (int)(Math.Round((1 - fDeltaY) * fTopBlue + fDeltaY * fBottomBlue)); // make sure colour values are valid if (iRed < 0) iRed = 0; if (iRed > 255) iRed = 255; if (iGreen < 0) iGreen = 0; if (iGreen > 255) iGreen = 255; if (iBlue < 0) iBlue = 0; if (iBlue > 255) iBlue = 255; bmBilinear.SetPixel(j, i, Color.FromArgb(iRed, iGreen, iBlue)); } } bmDestination.Save("fisheyelandscape.jpg", System.Drawing.Imaging.ImageFormat.Jpeg); bmBilinear.Save("fisheyebilinearlandscape.jpg", System.Drawing.Imaging.ImageFormat.Jpeg);

### So what did we get from our fisheye image?

Let’s look at our results, shall we? First, let’s bring our source image back.

Here’s the straight-up result from the algorithm:

Here’s the result with bilinear interpolation:

And I’m fortunate that the original photographer had the original landscape version for comparison:

[image by gcardinal]

Hmm… better than I expected. The code was also easier than I expected. I think it’s the math that was harder…

The fisheye image I used is one of those where the bottom of the original landscape image is curled into the centre. The other type is where the *top* of the original image is curled into the centre.

In that case, using the algorithm provided results in an upside-down image. I’ll leave it as an exercise to you for flipping the resulting image right-side-up.

One note about the centre of the source image. I found that if the centre is off, the horizon of the resulting image won’t be flat. I’m just showing you how to get the landscape image from a “perfect” source fisheye image. I’ll leave the details of fixing missing corners and undulating horizons to you.

Oh, while figuring out the algorithm and writing the code, I needed a test image. So I made this monstrosity:

with this result:

It ain’t pretty, but I needed to test if the quadrants map correctly…

P.S. A reader submitted this in a recent survey. I hope I’m interpreting the question and answering it correctly.

The math might be even easier than you think, if you allow yourself to think in complex variables. There’s a type of function called a linear fractional transformation (http://mathworld.wolfram.com/LinearFractionalTransformation.html) that has the property that it maps lines and circles to lines or circles (depending on the specific constants used).

If you simply convert a two-dimensional coordinate system into a complex one-dimensional coordinate system by setting the y-component to the imaginary component, then do the linear fractional transformation, and then set the imaginary component back to the y-component, it might make the math even easier.

All you really have to do is to figure out which four constants you need for the particular transformation you’re doing. (Although I say “all you really have to do…”, this is really the heart of the matter, and may or may not do what you need, as I have to admit I don’t entirely understand the computer code you posted. I’m a math guy, not a computer guy.)

There would be some technical issues with the edges of the image, but from what I can tell, your method encounters those as well.

Hey Eric, thanks for the reference link.

I have to admit, after reading through the reference, and trying to find the 4 constants, I failed. Completely. Perhaps I misunderstood the process…

The fisheye-to-landscape conversion is basically mapping concentric circles of the source fisheye image to horizontal lines on the destination landscape image. I also realised that I didn’t explain the math of the conversion…

Hello ,

Thanks for your article .

I have question which my is be strange for you .

i would like to know what is this language you use ? is it Java , ActionScript or etc…

Thanks

Ali

Ali, the code is written in C#.