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I currently have a PIL Image object which I store as an array of bits (1's and 0's). However I now would like to be able to rotate that image 45 degrees. One way to do it is take the original PIL image, apply a transform matrix on it, then convert that to an array of bits. Problem with this approach is that it's computationally expensive, especially if I want to start doing more than just one rotation. It would be faster to just modify the array of bits directly.

I tried using numpy.roll:

numpy.roll(bits, 45) # rotate 45 degrees

Unfortunately, this just does a circular shift, not an actual angular rotation.

What algorithm can I use on the array of bits to give me the desired output without having to go through the original image? Even though my application is in Python, your answer can be in whatever language you feel comfortable with, I'm more interested in the algorithm itself not the syntax :)

  • It's called rotation-by-shearing. I'll let others write a more explanatory answer.
    – rwong
    May 27, 2013 at 19:37
  • Note that if you want to perform several transforms, you can compose them into a single operation and only do the decode/encode step once.
    – Useless
    Oct 20, 2014 at 15:45

4 Answers 4


If you always want to rotate an image by a specific angle each time, you only need to do the heavy math once (and really, it's not that heavy). The best way to do this is by working the process backwards. For every pixel in the destination image, you calculate the corresponding location in the starting image. This "source" pixel coordinate is saved in what is called a Lookup Table or LUT. Once you build this LUT for the angle and origin of location that you want, then it is very quick to use each entry of the LUT to "move" the pixels of a source image to the correct pixel of the destination image. Where this gets tricky and involves more math is whn deciding how to weight the source date to come up with the destination data. When you work backwards from the integer destination image coordinates, you will end up with floating point coordinates in the source image. These floating point coordinates need to be interpolated. The simplest method is to use the "Nearest Neighbor" integer pixel coordinate to the calculated floating point values, but there are other algorithms such as "Bi-Linear" or "Bi-Cubic" that will provide a better quality of the finished image (but are of course slower than Nearest NEighbor"


To see that it can't be just a question of shifting points around, just draw a frame around your image: In the rotated version, the image will have diagonal edges and so won't even fit in the dimensions of the original image.

So, to rotate it you'll have to compute where each pixel goes. Note that the top left point (0,1) will not end up at (1,1) but at a point on the diagonal whose distance from the origin is 1: namely, at (sqrt(2)/2, sqrt(2)/2). Since sin(45) = cos(45) = sqrt(2)/2, the algorithm you're looking for is multiplication of each point's coordinates by this transformation matrix:

|  sqrt(2)/2  sqrt(2)/2 |
| -sqrt(2)/2  sqrt(2)/2 |

See the wikipedia page on Transformation Matrices for the details. So, you have to do a lot of multiplications but you don't have to calculate any sines or cosines (I just did that for you from memory). You can take some other shortcuts, as the other answers suggested, but one way or another you have to compute where each pixel must go.

Of course you'll also have to crop or pad to create a normal rectangular image.

PS. Better yet, instead of implementing this solution, outsource it to a graphics library that does rotations. Or check out this SO question and the solution suggested in the comments under the question:

Consider scipy.ndimage.interpolation.shift() and rotate() or skimage.transform.fast_homography() for interpolated translations and rotations of 2D numpy arrays.


there's no way to do rotation by arbitrary angles without sines and cosines, but it is possible to optimize the calculation depending on what other constraints you're prepared to impose. This can get very complicated, but can result in huge improvements for the "special case" you've defined.

If your images are all the same size and reasonably small, like your butterfly, and you were planning to do a lot of them, you could preform the slow calculation once, and simply remember the x,y of each pixel of the source image that became the x',y' of the destination image

  • But a particular rotation only requires one sine and cosine call, for the required angle; for 45 degrees it's just sqrt(2)/2. The time-consuming work is elsewhere.
    – alexis
    Jun 2, 2013 at 22:33
  • Correct, but you still have several multiply, adds, bounds checks and so on per pixel in a straightforward implementation, and strategies to optimize that can get very messy. Even more so if you want your rotated image to be even slightly anti-aliased.
    – ddyer
    Jun 3, 2013 at 1:20

If you only want to limit rotation to 45 degrees, you don't need to use sin and cos, for that particular case I would advice you to use a fast isometric diamond drawing algorithm and adapt it to your matrix. Usually a couple nested loops and 4 variables (x, y, width, height) will do it.

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