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Correction for radial lens distortion &
chromatic aberration using a computer

By David Knight

Geometry before
Geometry after

Corner detail before
Corner detail after

Whenever a lens designed for use in air is used underwater, some degree of image distortion is inevitable due to the water-air boundary (which is, in effect, a crude lens). If the boundary is flat, there will be pincusion distortion, the severity of which increases with the angle of coverage. This distortion (being classed as a type of 'radial distortion') occurs because the image magnification increases with distance from the centre of the picture.
     In addition to the obvious large-scale effect, the boundary has the property that the resulting image magnification (and to some extent, the degree of radial distortion) is a function of wavelength. This means that the magnification will be different in the red green and blue colour channels. The result will be colour fringing in off-centre detail, otherwise known as 'chromatic aberration'.
     If a lens is perfectly rectilinear in air, i.e., it has been designed to preseve straight lines and right angles in the picture; its rectilinearity can be maintained underwater by mounting it behind a dome port and placing the entrance pupil at the centre of curvature of the dome. In practice however, the placement of the entrance pupil is unlikely to be exact; because dome port extensions are only manufactured in fixed lengths and the entrance pupil may be difficult to locate. Moreover, if a zoom lens is used, the entrance pupil position will change with focal length. If the entrance pupil is too close to the port, some pincushion distortion will be introduced; and if too far away, the converse effect, 'barrel distortion' will occur. Furthermore, although the dome port can give some measure of geometric compensation overall, it will still introduce a small amount of chromatic aberration.
     Notwithstanding the effect of ports, the camera lens itself may be far from perfect. This is particularly the case with zoom lenses of the type which offer a wide focal length variation range. Such lenses are usually only rectilinear at one particular focal length setting (if at all), and the overall geometry is sometimes actually improved when the lens is used with an underwater port.
     The unwitting introduction of chromatic aberration is highly undesirable because it degrades image quality. The uncontrollability of image geometry is also annoying in general, and deleterious to the scientific and technical applications of photography. With the advent of digital photography however, these privations are completely avoidable by the use of software radial-correction techniques.
     Various image editing programs now offer radial correction facilities.  Most of the image adjustments demonstrated below were carried out using the 3rd party Photoshop plugin 'Panaroma Tools'; which permits very accurate image correction but is complicated to use. Also discussed later is Adobe Photoshop Lightroom, which is considerably cheaper than the full Adobe Photoshop program, has a lens correction facility with simple slider controls, and is suitable for dealing with all but the most inadvisable of lens and port combinations.

Panorama Tools
'Panorama Tools' (or 'PanoTools' for short), is a set of software utilities written by Prof. Helmut Dersch and released under the GNU Public Licence (GPL). These programs were primarily designed to aid in the construction of panoramas from multiple photographs, but provide numerous general-purpose image processing functions. The functionality of relevance to this discussion lies in a module called 'correct', which provides a radial-shift facility for the removal of barrel and pincushion distortion. 'Correct' also works on red, green, and blue channels independently, and so can remove chromatic aberration. This means that even a far-from-ideal optical arrangement such as a lens with a field of view (FOV) of 97° (35mm equivalent focal length = 19mm) placed behind a flat port can be made to perform as a fully water-corrected lens. The correction determined for a particular lens + port combination and zoom setting moreover can usually be applied to all photographs so produced (focusing and aperture changes do not significantly alter the reguired corrections) and so after an initial experimentation phase the correction process is reasonably quick and easy. Thus the physical measures which might be taken to preserve image rectilinearity are no-longer strictly necessary (although dome-ports have other advantages as we will see).
     PanoTools is available for various operating systems and can be used on its own or as a set of plugins for popular image-editing packages. To find out more, read the articles at and For a 16-bit PanoTools installer for Windows, including automatic installation of Photoshop plugins, see the website of Jim Watters: The discussion to follow refers to the use of PanoTools with Adobe Photoshop on a Windows platform.

Image Files
Best results will be obtained by working with image files which have 48 bit-per-pixel (bpp) colour depth, i.e., converted RAW files saved as 16 bit-per-channel RGB TIFF or PSD files. If the camera will only output JPEG files, use the highest available output resolution and quality settings and convert to TIFF or PSD for intermediate adjustment.
     Be careful not to save adjusted versions over the original file, always use 'Save As...' and change the filename. There is a quality loss on applying JPEG compression, so only save as JPEG as the final step in working on an image. The GIF format is unsuitable for high quality photographs, due to restricted colour depth.
    The required correction for a particular image is largely dictated by the focal length of the lens used. It is therefore useful to have some means for reading EXIF data (i.e., the camera settings information embedded in the image file). Most RAW file conversion utilities provide this.

Video Monitor Performance
You will not be able to evaluate your corrections properly if your video monitor suffers from geometric distortion or poor colour convergence (i.e., inaccurate superimposition of the red, green, and blue pictures). Since such issues relate only to CRT type monitors, it is recommended that corrections are performed only with the aid of a flat panel monitor (such as an LCD).

Using Radial Shift
verboten You should only apply radial shift correction to uncropped images. The point is that the correction causes the image to expand or contract about its centre point, and if you crop the image, the centre will no longer correspond exactly to the lens axis. Therefore apply radial correction first, crop the image later.

Images must be in RGB mode (not CMYK). Assuming that you have loaded an image which requires correction, when you select the 'Filter / Panorama Tools / Correct' menu item you will see an applet box like this: 
Correct options

Select 'Radial shift' (tick the box as per the illustration), and click the Options button. The applet shown below will then appear.

Polynomial coeffs This box invites you to enter polynomial coefficients for the correction, and the initial values presented when you first use the tool are those which do nothing at all to the image. You don't need to understand the maths to use the tool, but a knowledge of what each of the coefficients does will ensure that you adjust them in a sensible way.

The coefficients, from left to right are known as a, b, c, and d. d is the first-order correction coefficient, c is the second-order coefficient, b is the third-order coefficient, and a is the fourth-order coefficient (the 'order' is the power to which the quantity rdest is raised).
The first-order coefficient d changes only the size of the image without affecting the geometry. By expanding or contracting the red, green, and blue images independently about the lens axis you can perform a first-order correction for chromatic aberration. This is usually all you need.

By changing the values of the higher-order coefficients, you can cause the image to expand or contract about its centre by an amount which depends on the distance from the centre to the pixel in question. Changing second and higher order coefficients therefore allows you to correct for barrel or pincushion distortion.

Making the sum of the coefficients a+b+c+d=1 conserves the original image height at the centre. Making the sum greater than 1 reduces the height of the image, and making the sum less than 1 increases the height of the image.

To correct for pincushion distortion, start by using positive values for the third-order (b) coefficients. To correct for barrel distortion, use negative values.

Interpolation Quality
If you click the 'Prefs' button of the Correct Options applet, and then click 'More', you will be given the various interpolator options shown right. Use polynomial interpolation for speed when determining correction coefficients, and use sinc interpolation for maximum quality when applying the correction finally.

Adjustment Strategy
Ideally, you should take a photograph of a rectangular test-card with white-on-black detail in at least one of the corners.
     Start by correcting only the image geometry, i.e., use the same coefficient values for the red, green, and blue channels. Simple barrel or pincushion distortion is an aberration which depends on the cube of the distance from the image centre, and so it is best to start by adjusting the 'b' (third order) coefficient. If the image has barrel distortion, try b= -0.1 and adjust 'd' so that the sum of coefficients is 1, i.e., d=1.1. If the image has pincushion distortion, try b=0.1 (and hence d=0.9). Look at the result and see if more or less correction is needed and adjust 'b' and 'd' accordingly. Hold down the [control] key and hit the Z key to revert to the original image before applying a new correction. To retain maximum image quality, the correction should always be carried out in a single operation, not incrementally.
     You can assess the straightness of straight lines by laying a clear plastic ruler against the monitor screen. Photoshop guides are only useful if the lines in the picture are exactly horizontal or vertical, which is unlikely. You can place a diagonal line on the picture with the Marquee tool, but a ruler is quicker. Note that it is important to use some instrumental means for determining straightness because an optical illusion occurs on comparing a distorted and an undistorted image, such that the corrected image may sometimes appear to be distorted in the opposite manner to the original. Adjust the coefficients until straight lines are on average straight. If a line appears to undulate after a third-order correction, then some second or fourth-order correction may be needed. Hence increase the magnitude of 'a' or 'c' while decreasing the magnitude of 'b' by a similar amount, and so-on until you have geometry as near perfect as you can be bothered to obtain. In general, even the most appalling lens-port combinations will succumb to a correction involving both 'b' and 'a' or 'c', and it is rarely necessary to use non-zero values for all three.
     With your geometrical correction parameters now determined, you can apply a first-order correction for chromatic aberration. For this, the green channel, being the middle colour in terms of wavelength, should be treated as the reference channel (i.e, the green channel is assumed to be correct and the others are brought into convergence with it). Hence adjust only the 'd' coefficients of the red and blue channels, leave green alone.
     Having reverted to the uncorrected image ( [ctrl] Z ), look for a white-on-black detail near one of the corners of the picture, magnify it, and and note any lack of convergence of the red, green, and blue images. If you are just correcting for the effect of an air-water boundary (i.e., assuming that the camera lens does not make a major contribution to the aberration) you will find that the blue image is slightly too large, and the red image is slightly too small. If this is the case, then increase the 'd' coefficient for blue slightly, and reduce the 'd' coefficient for red. Note also that, for water, the difference in refractive index for green and blue light is about twice the difference for green and red, and the deviation observed is roughly proportional to this difference. Hence, presuming that you are correcting mainly for the underwater port, you will probably need to apply about twice as much correction to the blue channel as to the red channel. Start by increasing the blue 'd' coefficient by 0.003 and reducing the red 'd' coefficient by 0.0015. Fiddle with the parameters until you have exact convergence in your chosen corner feature. Toggle between the corrected and uncorrected versions of the image by hitting [ctrl] Z repeatedly. Notice that when convergence is obtained, there may be a slight blue haze around the feature: this is because lenses in general focus short-wavelength (blue-violet) light slightly less sharply than they focus red or green light (but the effect also depends on the filtration system used to separate red, green, and blue light in the camera). Once you have a first-order correction, inspect the image all over to see if chromatic aberration has reappeared in some regions. If it has you will need to make a second or higher-order correction, i.e., you will need to make tiny adjustments to the blue and red 'c' coefficients and so on; but such corrections are time consuming and rarely lead to noticeable practical improvement. 
     Always apply any correction you make to the completely uncorrected image, i.e., hit [control] Z after every trial. PanoTools stores your last attempt in a preferences file, and gives it to you as a starting point for the next go. Hence you will quickly home in on a set of coefficients which performs both geometric and chromatic corrections in a single operation; and this formula will usually work for all subsequent photographs taken using the same lens, port, and zoom-setting combination. Use the 'Save' and 'Load' buttons at the bottom of the 'Correct Options' dialogue box to store and retrieve previously determined coefficients.

Tip (Windows OS): If you hold down the [Alt] key and hit the [PrtSc] key when the correction coefficients dialogue box is on the screen, it will be saved to the clipboard. You can then create a new file (File / New) and paste the clipboard into it (Edit / Paste), then save this file with the images concerned. This is useful if you want to create documents explaining what you have done (and is the method used to show dialogue boxes here). Also, after a few years, sereral new computers, and various software upgrades; you might lose your preset data, in which case the applet box snapshot will come in useful if you want to work-up more images from the set.


Olympus C5060 camera in PT020 housing 27mm lens and flat port
Olympus C-5060 camera (5.1Mp 1/1.8" RAW) in PT-020 housing with PPO-01 standard port. Zoom setting = maximum wide (35mm equiv: 27mm). Coverage: 77° in air, 56° underwater.

ppo01_27 Pincushion distortion before ppo01_27corr
Geometry after

Chromatic aberration before
Residual aberration after

Radial correction coefficients: ppo01_27coeffs

To show the extent to which the optical effects demonstrated above are due to the water-air boundary, a photograph was taken of the test-card in air, using the bare C-5060 camera (no port) and the same zoom setting (max wide).

Geometry before correction.
Geometry after correction.

c5060_air19dt Chromatic aberration before correction. c5060_air27cordt
Chromatic aberration after correction.

Radial correction coefficients: c5060_air27coeff

In this case, note that the lens on its own produces some barrel (fisheye) distortion at the widest setting, and so actually compensates for the distortion introduced by the water-air boundary when the camera is used underwater. The chromatic aberration is also minor, and quite different from that caused by an underwater port. Here we find that a white object has a magenta fringe on the outside, and a green fringe on the inside. The outer magenta fringe means that both the red and the blue images are too large. The inner green fringe means that the green image is too small, which is the same as saying that the red and blue images are too large, i.e., the inner green fringe is simply the colour complement of the outer magenta fringe. The upshot is that a small compensation can be achieved, in this case, by slightly reducing the sizes of both the blue and the red images.

aberrat You will, of course, see different colour fringes with other lenses, and the fringe colours will change while you are working towards an optimum correction. Consequently, to work out the required direction of adjustment, you may find it helpful to memorise the complementary colours. These are as follows:

Secondary Colour Complementary Colour
Cyan = Green + Blue Red
Magenta = Red + Blue Green
Yellow = Red + Green Blue

The wedge-shape of the test-card in the final image above, incidentally, is simply due to the fact that the camera was not pointing directly at the card, i.e., it is an effect of perspective not lens distortion (you can pull-out this effect using the Photoshop Free-Transform tool). A slightly oblique camera angle does not affect the usefulness of a test shot because determining a geometric correction is merely a matter of making straight lines come out straight

Epoque DCL20 Epoque DCL-20 wide converter and 27mm lens
Epoque DCL-20 0.56x wide-angle converter attached to Olympus C-5060 camera in PT-020 housing with standard port . Zoom setting = maximum wide.

Barrel distortion before
Geometry after

Chromatic aberration before
Residual aberration after

Radial correction coefficients: dcl20_27coef

Note that the unprocessed image has a circular vignette, and that the correction process reduces its effect so that only a small amount of final cropping will be required. The vignette is due to the fact that the maximum angle of coverage of the camera lens exceeds that for which the DCL-20 conversion lens was designed.

Olympus PT020 housing with PPO02 wide port 19mm lens and flat port
Olympus C-5060 camera with WCON-07C 0.7x wide-angle converter in PT-020 housing with PPO-02 flat wide-port. Zoom setting = maximum wide, Coverage: 97° in air (35mm equiv: 19mm), 68° underwater.

Pincushion distortion before
Geometry after

Chromatic aberration before
Residual aberration after.

Radial correction coefficients: ppo02_19coeff

Under normal circumstances, the use of a 19mm equivalent lens behind a flat port is something no serious underwater photographer would consider. The PPO-02 packaging even had warnings about lens-distortion printed on it, and true to the laws of optics it gives chromatic aberration so bad that it can even be seen in the de-magnified image given above (top left picture). After correction however, the results are perfectly acceptable.

Once again, we can separate the effect of the underwater port from the performance of the optical system overall by photographing the test card in air using just the C-5060 camera and the WCON-07C wide converter.

Geometry before correction.
Geometry after correction.

Aberration before correction.
Aberration after correction.

Radial correction coefficients: c5060_air19coeff

As with the camera on its own, the camera with the Olympus wide-converter also produces fisheye distortion and magenta-out, green-in, chromatic aberration. Once again, a small amount of chromatic compensation is possible, but the initial aberration is by no means problematic.

From the above we can see that, when used in conjunction with a wide-angle lens, a flat port introduces pincushion distortion and pronounced blue-out, red-in, chromatic aberration, both of which can be corrected fairly easily using software. A camera lens which exhibits a certain amount of barrel distortion when used in air will to some extent offset the pincushion due to a flat port, but it will usually have little effect on the aberration.

Limitations of the Correction Process
One issue which must be understood from this discussion is that the procedure outlined will turn a good air-corrected lens into a good underwater-corrected lens, but it cannot turn a bad lens into a good lens. The point here is that if the lens behind the port can produce sharp pictures when used in air, then the correction process will restore its ability to produce sharp pictures underwater; but if it gives fuzzy pictures in air, it will also produce fuzzy underwater pictures.
     At risk of repetition, we should also discuss the fact that the coefficients required for a particular lens and port vary according to the zoom setting. This however, will trouble old-school underwater photographers very little; since for pictures other than macro, they will all automatically wind the zoom to its widest angle and leave it there. The simplest operational policy is therefore to make test-card shots and determine coefficients for a set of easily repeatable focal-length settings, and stick to these settings when taking real pictures. Alternatively, make a set of test-card shots at reasonably closely spaced focal-length intervals and plot a graph of the way in which the coefficients vary with focal length. You can then use the zoom at will, and provided that you have a way to record the zoom setting with the image data, you can interpolate the graph for corrections at your randomly chosen focal lengths. If this sounds like hard work, observe that it is always advisable to try out a camera system in a swimming pool before venturing into the ocean, and a once-and-for-all, acquisition of calibration shots will take about 10 minutes.

Unsharp Masking
Some digital cameras (particularly, but not exclusively, inexpensive ones) apply unsharp masking to the image by default. You should turn this feature off if at all possible, since it will interfere with any corrections you make for chromatic aberration and lead to an unsatisfactory result. The effect of unsharp masking is to increase the contrast at brightness transitions in the image (edges); sometimes with overshoot which creates black or white fringes around objects. If you apply a correction for chromatic aberration, these fringes will blur, and white fringes will split into three colours. The result is a picture which is notionally corrected, but has more colour fringing and softer edges than before; i.e., correction becomes pointless and reduces the subjective image quality. If you must use unsharp masking, use it only on the final image, use it only after the image has been re-sized to its final resolution, and never do it over a radius of more than about 0.7 pixels. The general rules for unsharp masking are very simple and easy to remember:
Rule #1: Don't do it.
Rule #2: If you must do it; make it the final editing operation.

Why Aperture and Focus Settings Don't (usually) Matter
The correction procedure described above works regardless of the lens aperture setting because, for a reasonably well designed lens, the geometry of the image is not affected by the aperture. Likewise the geometries of the separated R G and B images are not affected by aperture, which is why you can't reduce the chromatic aberration of a lens and port combination by stopping down. What the aperture does is change the size of the circle of confusion (the extent to which a point is reproduced as a fuzzy circle), and so while the sharpness of a feature in the red green and blue channels may vary with aperture, its centre-point should always land on the film or sensor in the same place. Hence, once you have obtained exact convergence of the red green and blue images, changing aperture may alter the amount of coloured haze around a feature, but changing the correction will not result in better convergence. It is of course, possible to make a lens in which this convergence will wander, by failing to place the iris at the nodal point, but in the days of computer-aided design, such abominations are unlikely to be encountered.
     The focus setting, incidentally, does have an effect; but for a wide-angle lens, the difference between closest-focus and infinity is likely to be too trivial to warrant any adjustment of the correction coefficients. For close-focusing macro lenes however (e.g., 1:1), particularly when the image format is relatively large; there can be a significant change in both aberration and geometry as the object distance is varied. For more information on the behaviour of macro port systems, see the flat port theory article

Why Dome Ports are Still a Good Idea
Although the techniques outlined here make the expense of a dome-port system less necessary, a dome port has some very compelling advantages, which software correction cannot hope to match. The first point is that a flat-port increases the effective focal length of a lens, due to refraction at the air-water boundary, and hence reduces the angle of coverage. Since the idea in underwater photography is to use a wide-angle lens in order to put the minimum amount of water between the camera and the subject, a flat-port somewhat defeats this intention. The second point is that the dome-port was introduced in the 1960s as a way of avoiding port-vignetting with very wide-angle lenses (its optical advantages were actually discovered by accident) and it will obviously still fulfil this purpose. The third point is that a flat-port introduces pincushion distortion, and radial correction applies a compensatory barrel distortion. This means that there is barrel-shaped vignette in the corrected picture, which will have to be cropped-off before the picture is ready for presentation. The upshot is that you will lose up to about 10% of the format area in correcting a flat port, reducing the effective number of camera pixels and so causing a small reduction in the maximum available resolution. Thus the lens-port combination which gives the least distortion is the best starting point for radial correction, because it maximises the usable format area.

 Dome Port Examples

ikelite DP60 on compact housing 19mm lens with 6" dome port
Olympus C-5060 + WCON-07C 0.7x wide converter, in Ikelite 6130.61 housing with DP60 dome port (3" internal radius). Zoom setting = max. wide. Angle of coverage in air and water: 97° (nominal).

Geometry before correction
Geometry after correction

full-size detail before correction (aperture=f/8)
full-size detail after correction

Radial correction coefficients: dp60wc7coeff

Test pictures for the bare camera lens in air were given earlier, and show that the dome port introduces no additional geometric distortion. The barrel distortion in the uncorrected picture above is due entirely to the zoom lens (the centre of curvature of the dome is at the entrance pupil). The dome port does however introduce a small amount of chromatic aberration, this being a consequence of the extremely wide angle of coverage (97° nominal), but the result after correction is virtually perfect in this respect. The distance from the front of the dome to the test card was 256mm (calculated) to record a subject field width of 610mm after correction. The camera was set in macro focusing mode to cope with the proximity of the virtual image produced by the curved air-water boundary. The aperture setting was f/8.

27mm lens with 6" dome port
Olympus C-5060 camera in Ikelite 6130.61 housing with DP60 3" radius dome port. Camera zoom setting = max wide (35mm equiv. f=27mm). Angle of coverage: 77° in air (nominal), 73° underwater (actual, measured).

Geometry before correction
Geometry after correction

full-size detail before correction (aperture=f/8)
full-size detail after correction
Radial correction coefficients: dp60coeff

Although the DP60 dome port was designed for the WCON-07C, it can still be used with the bare camera lens. In this case however, the lens entrance pupil ends up slightly behind the centre of curvature of the dome, and so the in air coverage of the lens (77°) is not exactly conserved. Actual coverage underwater (measured using a method described in the angle of coverage article), turned out to be 73.4±1.1° at a lens pupil to subject distance of 0.75m (but the fact that this figure is less then 77° is actually due to expansion of the diagonal in the correction for barrel distortion, rather than misconvergence of the entrance pupil and dome centre). Using a dome port on its own with the camera therefore gives wider coverage than when the camera is used with the WCON-07 and a flat port (only 68°); and the amount of chromatic aberration produced by the dome is effectively negligible at 73° coverage. A dome port for wide angle photography is evidently a very good idea.

Using Radial Correction with Film Cameras
Although this article has so-far been about radial correction of digitally produced images, there is no reason why it cannot be applied to images scanned from film. Operational points to note are firstly: that the slide or negative must be scanned full-frame and cropped to the frame border (so that the lens-axis corresponds reasonably accurately to the centre of the picture); and secondly: any dirt specks on the film should be removed before correction (using the Photoshop cloning stamp or its equivalent) because they will acquire colour fringes after correction. Also note that the correction for chromatic aberration may not be quite so effective (it depends on the film), the reason being due to the spectral-bandwidth (wavelength spread) of the filters used to separate the three colour images. The filters used in digital cameras are usually fairly sharp, giving three almost discrete sampling wavelengths, which makes chromatic correction extremely effective. The dye-filters used in film, on the other hand, are often rather broad and prone to spurious responses, and the scanned RGB image must be synthesised from a CMY image; all of which means that there will be some dispersion within each colour channel, and hence more coloured haze around image features after correction.

Film Examples

50mm Macro lens with Flat Port
Sigma 50mm f/2.8 macro lens (~46° coverage in air) with 3/8" thick acrylic flat port. Kodachrome 200 film. Port to subject distance (from memory) about 0.5m.
Raw scanned image, uncropped.
Corner detail before correction.
Corner detail after correction.
Radial correction coefficients
(first order only):

In this case no geometric correction was applied for the simple reason that no test-card shots were available (and the camera system passed out of service long ago). If you can't see distortion moreover (and you don't need to make measurements from the photograph) there is arguably no point in correcting for it. Chromatic aberration in the original is not severe, as should be expected for the optical system used (the detail images are considerably more magnified than for the previous examples); but if you want to blow an image up to poster size, the radial correction is evidently worthwhile.

Nikonos V with W-Nikkor 35mm f/2.5 35mm W-Nikkor Lens
Nikonos camera with W-Nikkor 35mm f/2.5 lens (air corrected underwater lens with flat glass front element), Coverage: 62° in air, 46.5° underwater. Kodachrome 200 film.
Uncropped image after correction.

Edge detail before correction.
Edge detail after correction.

Radial correction coefficients
(first order only):

No geometric correction was applied for the same reason as before.

The 35mm W-Nikkor used on its own is difficult to focus and consequently, for this author at least, produced a fair proportion of useless pictures. The picture above is not particularly sharp, but it is one of the better examples from the few occasions on which the author decided to give the lens a try without its close-up attachment.
     In the off-centre detail of the picture shown above, blue light is considerably more out-of focus than green, and green is more out of focus than red. Hence there is a blue haze around details after correction. The coefficients for radial correction were determined using the detail shown, but the final coefficients are a compromise (best average) obtained by looking all over the picture. The author was intrigued to discover that the picture 'sprang to life' after correction: coloured patterns on the bodies of the fish suddenly fell into register, indicating that the general fuzziness of pictures taken with the W-Nikkor 35mm is as much due to chromatic aberration as it is due to inaccurate focusing.

Collected Coefficients for a Flat Port
Shown in the table below are the shifts in the first order coefficient (d) which were used in order to correct for chromatic aberration in the flat port underwater picture examples given above. Where correction coefficient shifts for the lenses in air were known (27mm and 19mm) these have been subtracted from the values for the lens and underwater port combined.

Shift in d coefficient Angle of Coverage of Camera Lens (and 35mm equiv. focal length)
Red -0.003 -0.0025 -0.0013 -0.0015
Blue +0.0023 +0.0019 +0.004 +0.003

Obviously there is considerable scatter in these data, especially in the shifts for blue light, and this can be accounted for in two ways: Firstly there will be some deviation away from the value required for the port alone in the cases where the coefficient shifts for the lens in air are unavailable. Secondly, the final choice of coefficients is always something of a compromise, and the intention was to determine them without forcing them to agree to some predetermined scheme (and there is considerable latitude in choosing the blue coefficient when blue is less sharply focused than red or green). There is a trend nonetheless, and it is consistent with a theoretical analysis given elswhere (see flat port theory) which shows that chromatic aberration caused by the water-air boundary diminishes as the angle-of-coverage is reduced; and settles down, for long focal-length lenses, so that the radial displacements of the red and blue images relative to green are proportional to the differences is the refractive index of water for red and blue light relative to green. 

To clarify a point made briefly earlier; it is useful to note that, if the chromatic aberration seen is entirely due to an air-water boundary, then there is every reason to expect that the coefficient shift for blue will be about twice the shift for red, and that the shifts will be in opposite directions. This can be understood by presuming that the manufacturer of the sensor or film will have tried to choose primary colours at wavelengths corresponding reasonably closely to the peak spectral responses of the cone cells in the human eye. These peak response wavelengths are at 560nm (red), 530nm (green), and 424nm (blue), but most colour photography systems are based on old research which places than at about 600 (R), 540 (G) and 450nm (B) . Data from Kaye and Laby (ISBN 0-582-46354-8) give the refractive index for pure water at wavelengths close to the traditional primaries as follows:

Wavelength / nanometres Refractive index at 20°C Difference from green Difference /% 
435.8 1.340210 +0.005744 +0.430 %
546.1 1.334466 0 0
632.8 1.331745 -0.002721 -0.204 %

Since the deviation between the red, green, and blue images at a particular point in the image is small, the relationship between deviation and refractive index will be almost linear (i.e., directly proportional). Hence, since the magnitude of the refractive index difference between blue and green is about twice that between red and green, and one is positive while the other is negative; we expect the radial deviation to follow roughly the same pattern.

Adobe Photoshop Lightroom
Panorama Tools allows images to be corrected to a very high degree of precision, but the interface can hardly be described as intuitive. Also, we can observe, that a first-order (linear magnification) correction for chromatic aberration is usually sufficient; and most of the required geometric correction can be accomplished by adjusting only the third-order (cubic-law) coefficient. This means that a correction facility very nearly as good as has been required for the demonstrations given so far, can be provided by a software interface with simple slider controls. One such interface is provided by Adobe Photoshop Lightroom. 
     Lightroom is likely to annoy most professional computer users with its attempts to take charge of file management; but if you can come to terms with its workflow concepts (importing, libraries, development and exporting - rather than simply opening, editing  and saving files) and its unhelpfulness if the user should dare to manage files it thinks it owns using the operating system (it becomes necessary to delete moved or re-named files from the library and re-import them), it offers a large number of camera RAW format converters, an EXIF reader, and some good basic image adjustment tools. It is also aimed at the amateur user, and so is a great deal cheaper than Photoshop.
     Some examples of optical corrections performed using Lightroom are given below.

24mm lens with flat port
Canon PowerShot S100 camera in Recsea housing. Camera zoom setting = max. wide (35mm equiv. f = 24mm).  Aperture = f/5.6.  Lens FOV = 84° in air, 60° underwater.

Geometry and aberration before correction:
s100fp24 s100fp24dt

Geometry and aberration after correction.
s100fp24lrcor s100fp24lrcdt

The lens correction adjustments made using Lightroom are shown on the right.

Note that in earlier work with PanoTools, correction for the geometric distortion caused by a flat port required adjustment only of the 3rd-order coefficient. This basic cubic-law distortion has been removed perfectly by Lightroom.

The Lightroom (version 3) programmers were probably not expecting users to be dealing with systems quite as awful as underwater flat ports, and so the blue-yellow correction is not quite complete with the slider at +100.  Still, the result is an enormous improvement over the original image.
Lightroom lens correction interface

Here's what the picture looks like when corrected using Pano Tools.
s100fp24cor s100fp24cdt

Pano Tools
radial correction

Inon UWL-H100 wide-angle converter and 24mm lens
Inon UWL-H100 28LD 0.6× wide converter mounted on Recsea S100 housing with LD bayonet port adapter. Camera zoom setting: max. wide (equiv. f = 24mm).  Aperture = f/5.6.

Geometry and aberration before correction
uwlh100ld_24 uwlh100ld_24dt

Geometry and aberration after correction
uwlh100ld_24lrcor uwlh100ld_24lrcdt

The adjustments made using Lightroom are shown on the right.

Note that Lightroom (version 3) normalises the width and height transformations separately in such a way as to anchor the picture at the corners. This does not strictly preserve the aspect ratio of the original scene, but the error will not be noticeable in the majority of applications.

Also notice that Lightroom has not been able to make the ruler in the picture completely straight. This is because this particular image requires both 3rd and 4th order corrections (see below). 

Finally, once again, the blue-yellow fringing correction is insufficient and some aberration remains.
Lightroom lens correction interface

Here's the picture corrected using Pano Tools with finite 3rd and 4th order coefficients.
uwlh100ld_24cor uwlh100ld_24cdt

Radial correction

Overall, Adobe Lightroom is not as capable as Panorama Tools; but for photographers who want to apply relatively minor image corrections for non-technical purposes (i.e., the majority), the ease of use (of this part of the program interface at least) is reason to favour it. Technical users might also be inclined to try it first before resorting to Pano Tools.

Further information
Correcting Wide Angle Distortion, by Peter Schulz. Splashdown Divers
Eliminating color fringing, by Norman Koren (using Picture Window Pro):

© Cameras Underwater Ltd. 2004, 2006, 2011, 2012
David Knight asserts the right to be recognised as the author of this work.
Photographs by David Knight and Steve Knight.