Image rotation is a lossy operation, but rotating an image once then rotating it back likely loses very little detail, especially compared to typical JPEG compression.

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Image rotation works like this mathematically:

A grey level image consists of luminance values L_(x,y) at integer pixel positions x, y. First a real-argument function f(x,y) is constructed that reproduces the values L_(x,y) at the same x,y positions, but will also give values at non-integer x,y. Hence it's interpolating between integer x,y positions (this is where interpolation methods come in---there are several possible choices for f(x,y)).

Then construct a rotated version g(x,y) = f( x cos(a) - y sin(a), x sin(a) + y cos(a) ) by angle a. This operation is mathematically lossless, not counting the numerical roundoff errors when performing the calculation on a finite precision computer.

Finally g(x,y) values are calculated at integer x,y positions again to create an image.

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At this point you can ask: why is this lossy? Can't we just reverse all these calculations to reconstruct the original unrotated L_(x,y), if we know what interpolation method was used to construct f?

Theoretically that is possible, but this is not what happens when you do a rotation of the same angle in the opposite direction. Instead of reversing the original rotation operations precisely, an opposite sign rotation is performed using the same sequence of operations that the initial rotation used. The back-rotated image will not be precisely the same as the original.

Furthermore, if the values of g(x,y) were rounded to a low precision (8-bit, 0..255), the information loss is even greater.

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Lots of accumulated rotations will effectively blur the image. Here's an example of rotating a 500 by 500 pixel Lena image 30 times by 12 degrees, amounting to a full 360 degree rotation:

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There's another reason why blur-type operations will be lossy. One might naively think that mathematically reversing a blur should give us back the original unblurred image. This is theoretically true for as long as we're working with infinite precision. The procedure is called deconvolution and it is being used in practice to sharpen images that are blurry due to motion blur or optical reasons.

But there is one catch: blurring is insensitive to small changes in the source image. If you blur two similar images, you get even more similar results. De-blurring is very sensitive to small changes: if you de-blur two only slightly different images, you get two very different results. We're usually not working high precision (8 bit is quite low precision actually), and the roundoff errors will get magnified when attempting to reverse a blur.

This is the reason why blurring is "irreversible" and why it is losing detail.