# Theoretical limits of IBIS away from the center of an image

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Theoretical limits of IBIS away from the center of an image
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IBIS attempts to correct for camera motion by moving the sensor. This cannot do a perfect job of correcting for camera motion. Indeed, the whole reason that some photographers bother with shift-tilt lenses is that shifting the sensor doesn't have the same effect as changing the direction that the lens is pointing.

In this post, I will use a simple model of what IBIS does. To keep things simple, I will assume that whatever is being photographed is extremely distant, for example stars in the night sky. That means that IBIS has to compensate for changes in the orientation of the camera but not changes in its position. It also means we don't have to worry about changed to the focal plane, since everything is so far away that it will be in focus if the camera is focussed at infinity.

Here is our camera with a lens and image sensor. The colored lines are two rays from the subject being photographed.

During the course of the exposure, the direction that the camera is pointed changes:

The rays now hit different points on the sensor than they did in the first image.

The camera is rotating, but it will be easier to understand what is happening if we pretend that the camera is standing still and the universe is rotating around it. This allows us to combine the two diagrams above into a single diagram:

If you look closely, you may note that tilting the camera has a larger effect near the edge of the sensor that it does at the sensor, both because the distance between the lens and where the ray strikes the sensor is larger, and because the ray is hitting the sensor at an angle.

To put some numbers on the above effect, we need to choose a few parameters. We make the point where the blue ray hits the censor 18mm away from the center of the sensor. (On full frame cameras, the corners are about 21.6mm away from the center, so this distance will cover most of a full frame sensor). We set the level of acceptable motion blur to 0.012mm, which is two pixels on a 24 MP full frame camera.

Now I will present the equations I used to calculate the positions where the rays hit the sensor. If you don't like math, you can skip down to the table below where I provide numerical results.

In the following equations,
f is the focal length of the lens,
x is the amount that the camera is rotated,
r0 and b0 are where the red and blue rays strike the sensor before the rotation,
r1 and b1 are where the red and blue rays strike the sensor after the rotation.

r0 = 0
r1 = f * tan(x)
b0 = 18
b1 = f * tan(atan(b0/f) + x)

The initial angle of the blue ray is atan(b0/f), and we add x to that angle to compute the position after the rotation is performed.

If IBIS is enabled, then we assume that the IBIS does a perfect job of shifting the sensor so that the center of the image does not move. r2 and b2 are where the rays strike the sensor after rotation when IBIS is enabled.

r2 = r0 = 0
b2 = b1 - r1

What we are actually interested in is not the positions, but how big a rotation we can perform without exceeding our limit of 0.012mm on motion blur. To determine that value at the center of the sensor, we set the difference in position before and after the rotation (i.e. r1 - r0, b1 - b0, or b2 - b0) to 0.012 and solve for x. This gives us the following values, which I've placed in an image because the comment system doesn't seem to do tables:

The second and third columns are calculations with IBIS disabled. The second column shows the maximum angle the camera can be tilted during the shot if we want to limit motion blur to 0.012mm. The third is the maximum amount of tilt if we want to limit blur to 0.012mm everywhere within 18mm of the center. The column is labeled “at 18mm,” but 18mm is the worst case, so anything within 18mm from the center will have no more than 0.012mm of motion blur.

In our model, IBIS cancels out all motion blur at the center of the image, so with IBIS enabled we only have a column for 18mm away from the center. This improves with longer focal length.

The next column (column 5) is the one that inspired this exercise. It shows the ratio of the two preceding columns, indicating how much improvement IBIS can provide. I had expected the ratios for 20mm and 50mm to be larger.

The table shows that IBIS can be quite effective in principle with a 200mm lens, but the IBIS system would have to work fairly hard to achieve that. The final column shows how far the IBIS system has to move the sensor to compensate for the change in angle shown in column 4.

At this point, I should mention two ways an IBIS system could do better than the above table shows. First, I assume that the IBIS system attempts to eliminate all motion blur at the center of the sensor. As far as I have been able to determine, this is what all current IBIS implementations do. In principle, an IBIS system could compromise between the center and the edges, allowing some motion blur at the center in exchange for less motion blur near the edges.

Second, I have assumed that the lenses have no geometric distortion. A certain amount of barrel distortion should improve matters, while pincushion distortion will make things worse. I haven't investigated the magnitude of this effect.

In the table, I list improvement in terms of angle, but IBIS performance is normally specified in number of stops improvement in shutter speed. It's unclear how these two measures relate, but I'd question whether IBIS can achieve shutter speed improvements that are vastly better that the angle improvements that I calculate in this post.

As an anecdotal data point, let's take the Nikon 14-30mm f/4 on a Z7. The model shows an improvement of 1.604x for the 14mm focal length and 3.769x for the 30mm focal length. Ken Rockwell's review of this lens reports 1/3 of a stop improvement at 14mm and 2 stops improvement at 30mm. In this case, there is less than 1/3 of a stop difference between the angle improvements shown by the model and the actual observed improvement.

So what about the CIPA numbers showing 5 stops improvement on the Fuji X-H1 with a 14mm f/2.8 (full frame equivalent 21mm), or 7 stops improvement on the Canon R5 with the 50mm f/1.2? The first thing we note about the CIPA test is that their threshold for unacceptable blur is 0.063mm on a full frame camera, or 10.5 pixels on a 24 Mpixel camera. This less stringent standard actually makes it harder for IBIS to improve things, although the difference is minor except at 200mm:

The CIPA measurement process uses software developed by CIPA. Unless I missed it, CIPA doesn't say which portions of the image it examines, but the test chart provides some clues. The chart includes copies of a “natural image” (a picture of some fruits and vegetables in a basket), but I assume that the software ignores these, based on a comment that these are to make focusing easier. Ignore these, and the chart is a black and white checkerboard pattern, but only four boxes are guaranteed to be in the field of view (depending on the aspect ratio of the camera). So it would appear that the software is analyzing the blur of a horizontal and a vertical line, both of which pass through the center of the image. This means that the software is going get a result that is a lot closer to the blur at the center of the image than to the worst case blur anywhere on the image.

In other words, the CIPA stabilization scores are more a measure of how well the center of the image is stabilizes than a measure of how well the entire image is stabilized.

Conclusion:

1) Unless you only care about motion blur near the center of your images, the ability of IBIS to compensate for motion blur is less than you might expect on both wide angle and normal angle (50mm full frame equivalent lenses).

2) Again, unless you only care about motion blur near the center of your images, you shouldn't use the CIPA numbers when deciding which stabilized wide angle lens to buy. Lens based stabilization (unlike IBIS) may do a good job of eliminating motion blur across the frame rather than just in the center, but the CIPA numbers won't tell you this.

Kenneth Almquist's gear list:Kenneth Almquist's gear list
Nikon D7200 Fujifilm X-H1 Nikon AF-S DX Nikkor 35mm F1.8G Nikon AF-S DX Nikkor 18-300mm F3.5-6.3G ED VR Fujifilm 16-55mm F2.8R LM WR +2 more
Canon EOS R5 Fujifilm X-H1 Nikon Z7
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