Is rotation an intrinsic lossy operation (for angles not multiple of 90 degrees)?

Question

mattdm's statement "that blur is largely reversible if you do the exact same thing in reverse" raised the question in my mind. The rotation is a geometric transformation of the image that consist of a spatial transformation of the coordinates, and an intensity interpolation. If the rotation is not a multiple of 90 degrees then the interpolation algorithm plays a crucial role.

In such cases, if we use an averaging interpolation algorithm (e.g., bicubic) the operation is lossy^*. But can we use a different approach (e.g., nearest neightbor interpolation) instead and have our rotated image "un-rotated"?

_{(*) It is only my feeling (I still cannot support it with a mathematical proof): as we cannot know for sure which pixel contributed to wich value, we cannot reverse the averaging. But I'm not confident that we cannot use probabilistic methods to accurately estimate the original values.}

---

While I lack the required math skills, I have performed some tests myself (with gimp), but after anti-rotating, the images differ:

Test 1

Figure 1 - Source image (256x256)

Figure 2 - From left to right: a) image rotated 9,5 degrees clockwise; b) image rotated again 9,5 degrees anti-clockwise; and c) difference between images. For these operations I have used nearest neightbor interpolation. The images are downscaled after the operations to better fit this website layout.

Figure 3 - From left to right: a) image rotated 9,5 degrees clockwise; b) image rotated again 9,5 degrees anti-clockwise; and c) difference between images. For these operations I have used bicubic interpolation. The images are downscaled after the operations to better fit this website layout.

Test 2

Following @unapiedra's sugestion, I did a simpler test: rotating a 2x2 matrix. This case is uninteresting because depending upon the angle either all cells are rotated by the same angle, or no cell is rotated. That is, the rotation is always lossless.

So I tried again with a 3x3 matrix and a 30 degrees rotation:

Figure 4 - From left to right: a) source image; b) image rotated 30 degrees clockwise; c) image rotated 30 degrees anti-clockwise; d) difference. The images are upscaled to fit this website.

In this case, the differences are evident. The rotation is clearly lossy... But what happens if I upscale before the rotation?

Test 3

Figure 5 - From left to right: a) source image; b) image upscaled by a 6x factor; c) upscaled image rotated 30 degrees clockwise; d) image rotated 30 degrees anti-clockwise; e) downscaled transformed image; and f) diference (no difference). The images are upscaled to fit this website.

In this case, I'm upscaling by a 6x factor. Reasoning for choosing this factor (unfortunately incorrect as I have seen with a counter-example):

A 30-degrees rotated pixel has coordinates bottom-left to top-right: [0,0]-[0.3660, 1.3660]. That is, the shortest projected side has a 0.36 pixels length. The sampling theorem requires that we sample at a double rate.

Thus, to accurately sample a 30 degrees rotated image, I must sample each 0.17 pixels, yielding a x5.88 resizing factor; 3 x 5.88 = 17,64, thus I resample the source image to a 18x18 image.

Answer 1

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.

---

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.

---

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.

---

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:

---

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.

Answer 2

Rotating isn't lossy in a general context but in combination with images it is.

The reason is binning of pixel values together. If an image is rotated most pixels' location will not alighn perfectly with the image's grid structure. Interpolation is now needed to decide where to place that pixel's value.

In nearest neighbour interpolation the pixel value is moved to the closest grid cell (pixel location).

In bicubic interpolation the pixel value has effect on all close grid cells in the vicinity, in essence the value is distributed to the close pixels.

Answer 3

If you increase the resolution sufficiently, I believe you should be able to make it lossless, but I think it would have to be quite a bit larger. You would effectively need to "fill out" enough pixels that the error is reduced to under the rounding error of reversing the operation and reducing the resolution back to the original. At this point though, you might as well simply store the original image in a hidden layer of the image since you are adding substantially more data points than a copy of the image would take.

So practically speaking, it is required to be lossy since it is averaging values to create new points and would generally require more additional storage to upscale sufficiently than it would take to simply store another copy.

Answer 4

AJ's theory about increasing resolution is correct. The issue is the relative size of the cells (pixels) to the details we are able to make out. The larger the cells, the more we have to shove results into bins. Here I rotate with bicubic interpolation with +10 and -10 and compare with abs(I1-I2). Secondly, I do a lanczos resize x10 , rotate with bicubic interpolation with +10 and -10 and resize back to original size, and compare with abs(I1-I2).

A significant difference in the distribution of errors:

Answer 5

Rotation to "non-cardinal" angles will always be lossy due to the finite precision in the floating-point numbers (and methods) used in performing the rotation.

Note that "lossy" does not always equate to "noticeably lossy".