A Longitudinal CA Metric for Photographers

While perusing Jim Kasson’s excellent Longitudinal Chromatic Aberration tests[1] I was impressed by the quantity and quality of the information the resulting data provides.  Longitudinal, or Axial, CA is a form of defocus and as such it cannot be effectively corrected during raw conversion, so having a lens well compensated for it will provide a real and tangible improvement in the sharpness of final images.  How much of an improvement?

In this article I suggest one such metric for the Longitudinal Chromatic Aberrations (LoCA) of a photographic imaging system: the linear spatial resolution lost to LoCA at MTF50 by the camera+lens under review as setup[2], in the location and direction under examination, expressed as a negative percentage.  For instance based on the last post on CA we could say that the MTF50 of  Jim’s a7RII+FE55mm at f/1.8 near the center was about 7.7% lower than it could have been – as a result of Longitudinal Chromatic Aberrations.   Clearly a smaller number is better.

Longitudinal CA = Color Plane Defocus

Taking that system as an example, this is how one of Jim’s LoCA capture sets presents itself.  Recall that he takes many raw captures of a slanted edge chart in manual focus mode by moving the camera 4 mm towards the target each time without touching focus, thereby effectively walking it through back focus, perfect focus and front focus.  The distance to the target is of the order of a few meters, not infinity.  He uses MTF Mapper to extract the MTF50 performance of the individual raw color planes for each capture.   I am grateful to Jim for generously sharing his data which in this case results in the following plot :

LoCA As Measured
Figure 1. Longitudinal Chromatic Aberrations become evident as a camera’s focus is shifted, Jim Kasson captures.

As the camera is moved closer to the target slanted edge each of the three color planes comes into focus in turn.  If there are Longitudinal Chromatic Aberrations in the imaging system the three color planes do not come into focus at the same time, as shown by the colored curves achieving peaks in different frames.  The black curve is instead the combined grayscale response calculated as discussed in a related article, representing overall system performance:

MTF50_{LoCA} = 0.25 MTF50_r + 0.5 MTF50_g + 0.25 MTF50_b

Spatial Resolution Performance Without LoCA

If this imaging system had been perfectly corrected for Longitudinal CA the three r,g,b planes would have come into best focus at the exact same time, something that can be simulated by shifting the color curves above so that their peaks align.

Shifted LoCA
Figure 2. The same system as above if it did not have LoCA. It was obtained by arbitrarily shifting curves of the three color planes so that their peaks aligned.

A combined grayscale MTF50 curve of the color plane data as shifted would then be indicative of the performance of the system in the absence of Longitudinal Chromatic Aberrations:

MTF50_{NoLoCA} = 0.25 MTF50_{rs} + 0.5 MTF50_{gs} + 0.25 MTF50_{bs}

Loss of System MTF50 Due to Longitudinal CA

Figure 3 plots both composite grayscale curves (with and without LoCA) to show how much linear spatial resolution is lost in this system to Longitudinal CA .  I aligned the curves approximately by hand but their relative position doesn’t really matter:  what we are after is the magnitude difference of the two peaks, which represent perfect focus.  In this case the grayscale curve with LoCA zeroed out peaks at an MTF50 of 1118 lp/ph, while the original grayscale curve with LoCA achieves only 1032 lp/ph, a loss attributable  to Longitudinal Chromatic Aberrations of 7.7% of total system performance as setup, in the location and direction tested.

We could say that the LoCA score for this system at f/1.8 near the center of the field of view  is -7.7%.

LoCA Composites
Figure 3. Composite grayscale MTF50 near perfect focus, with and without LoCA.

A difference of 7.7% in MTF50 is typically just noticeable by a pixel peeper.

Calculating the Metric

Therefore to calculate this LoCA metric all one needs to do is

  1.  calculate the original system grayscale MTF50 curve from the individual color plane values and the first formula above as shown by the black curve in Figure 1 ;
  2. find the MTF50 at which this curve peaks;
  3. for each of the three color planes obtain its peak MTF50 value;
  4. combine the r,g,b peak MTF50 values by applying the coefficients of the grayscale formula to obtain the peak System MTF50 in the absence of LoCA;
  5. the difference between the two peak MTF50 values in 2. and 4. above expressed as a percentage of 4. is the linear spatial resolution lost to Longitudinal CA.

Clearly the smaller the number, the better corrected the system.    Peeking at some of the other excellent prime lenses that Jim has tested on the a7RII[1] we can see that LoCA scores typically, but not always, get better as the f-number gets larger.   As a reference, the range for those lenses at f/2.8 near the center is between about  -1 and -8%.

Caveats

Since chromatic aberrations vary throughout the field of view those results are only valid for the given test conditions in the position and direction measured.  Having played with a few combinations, if noise is controlled (good technique, properly illuminated edge, long enough) I think that these results should be repeatable within about +/-0.25%, all other things being equal.

Intuitively I would think that results should not change much if an AAless sensor with differently sized pixels is used aotbe, because MTF50s are multiplicative in a cascaded linear system and all else should be about the same.  But I do not have the data to verify this thought, yet.

Notes and References

1. A7RII LOCA & AF VS COLOR PLANE. The Last Word, Jim Kasson.
2. The context of this post are raw captures of neutral (hueless,achromatic) slanted edges under a uniform illuminant by Bayer CFA digital cameras for the purpose of measuring by the slanted edge method the linear spatial resolution (‘sharpness’) of photographic equipment.

8 thoughts on “A Longitudinal CA Metric for Photographers”

  1. Axial Color (which you are referring to LoCa) does not reduce the performance when a given color is correctly focused. It is simply a change in focus position, by adjusting the focus it is removed. Same as for petzval (aka field curvature).

    If these results are along the optical axis and the lens is well centered, the drop in the MTF may be attributed purely to spherochromatism, a variation in the quantity of spherical aberration with respect to color. Because spherical aberration is the only aberration that is present on axis for a rotational symmetric system, its child aberration (spherochromatism) exists in the color axis in isolation as well.

  2. Thanks for your input Brandon. Yes, when a given wavelength is correctly focused that in itself does not reduce the performance of the system. If it’s not correctly focused, however, it does – and MTF50 values peaking at different distances from the relatively on-axis target show that wavelengths in the three raw color planes do not come to focus simultaneously in the tested configuration. That’s what photographers call Longitudinal (or Axial) Chromatic Aberrations.

    Spherical Aberrations (including spherochromatism) affect the shapes and heights of the curves above at this wide aperture but I haven’t attempted to measure their effect separately. I am interested in a reasonable MTF model for SA if you can suggest one.

    Jack

    1. Take your figure 2. Take whatever color has the best correction, in this case green, then look at your drop. From ~1200 to ~900, -300lp/ph is the drop associated with spherochromatism, if the data is gathered from an on-axis point.

      There is not perfect association, but the fit is perhaps 90% good.

      1. Sure Brandon, qualitatively one can say that Spherical Aberrations dominate at f/1.8, as mentioned in the previous article. I would be interested in a practical quantitative model if you can suggest one.

        Jack

        1. In a rotationally symmetric system absent of axial color, spherical aberration is the only aberration present on axis.

          Child aberrations, such as Spherochromatism, are also allowed. If there is a chromatic dependence to the MTF on-axis for a well-centered lens and the axial color is accounted for, all that is left is spherochromatism. Any change to the resolution has to do with spherochromatism.

          The 10% error I allow for is margins in the test, odd sensor behaviors, and other miscellaneous issues.

          The level of spherical aberration cannot be modeled from polychromatic resolution measurements precisely, because spherical aberration exists in many forms. Key is that there is a 3rd order spherical aberration, as well as a 5th order spherical aberration. Their relative amounts tell you a great deal about many aspects of the lens’ behavior as it is stopped down, as well as the energy distribution of the spot at full aperture (i.e. if it tends towards “contrasty” or “high resolution”). Without a Shack-Hartmann, interferometer, or other wavefront sensor you can’t tell the exact aberrations – only approximate.

          “Aberrations from MTF” are an active area of my academic research. There is a large degree of complication.

          1. Right, thanks. I was hoping for a simplified formula or two, like those for diffraction or defocus. I’ve sent you an email for future reference in case you can suggest one.

            Jack

            1. An excellent discussion. Not included in this discussion is the behavior of the autofocus system on system MTF. Since the MTF metric is weight towards green, the focus should be determined by the green channel. Jim showed that the Sony autofocus seems to bring the red in focus rather than the green.

              1. Yes, good point Bill. Jim’s Batis seems to lock on red but fortunately some of his other lenses lock correctly around the peak of the system curve, typically close to the green channel’s.

                Jack

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