Highs and Lows

I was at an arts and music festival helping out Lori with her pottery booth. I had a chance to wander around and shoot a few pictures. I brought the little Canon G9 and a monopod to get something other than the straight-on shots we instinctively go for.



One thing you can do is attach the camera to one end of the monopod on the side and then just rest that end on the ground and have the camera shoot upwards to get a ground-level view of the world.



Another easy way to get a different viewpoint it to fully extend the monopod and raise it high overhead to get a top-down view.

In both cases I find it convenient to simply set the camera at the hyperfocal distance (about 6.5 feet or 2 meters for the G9 at 7.4 mm set at f/5.6). Set the mode to aperture-preferred, put on a 5-second timer, click the shutter and either set the camera down or raise it high and wait for your 5-count to let you know the shutter has released.

It's not going to get you precision framing but you can get decent quality images from a seldom-used perspective. Also, if you're using a pocket camera, it's quite easy to move the camera into position at the end of the monopod and you won't attract quite as much attention as you would trying to wrestle a big DSLR into position to do the same thing.

(The only disturbing comment I got was from a woman who came from behind and, as she passed, said "Oh, thank God; I thought you had a gun." Maybe I should put some nice friendly colored electrical tape on my gun-metal grey monopod to reduce such anxieties.)

Part 7: Final Thoughts

To wrap things up on this series of posts, here are some personal observations on images created with point-and-shoot consumer pocket cameras like the Canon G9. This photo is a ground-level view of the Lackawanna Train Station, now a historical museum, in Vestal, NY. It was shot with G9 and processed using the techniques we've been going through. Shot at 7.4 mm and f/4, the processing removed the mustache distortion (combination barrel and pincushion distortions often seen in shorter focal length zooms), we've reduced the transverse chromatic aberration, and we compensated for the light fall-off (vignetting) at the edges.



Looking at a detail of the original image, we can see the chromatic aberration along the high-contrast edges of an eave ornament. After processing we see the chromatic aberration cleaned up and image details are sharper.



So let's summarize what we're working with:

1. Small-sensor pocket cameras have great depth of field. In many cases this is good but it substantially removes one of the creative controls you have with a DSLR and fast lens: no chance of getting nice gauzy out-of-focus backgrounds with these cameras.
2. Pocket cameras all have some amount of transverse and longitudinal chromatic aberration. We can reduce longitudinal (axial) chromatic aberration by using a smaller aperture; we can reduce transverse chromatic aberration using red and blue channel corrections. There are other chromatic aberrations inherent in the small sensors and inexpensive lenses that are beyond our ability to correct but these tend to be secondary.
3. These small cameras (and many lenses on the big boys) have issues with geometric distortion. Barrel distortion, pincushion distortion, and even the more complex compound distortions can be corrected effectively. Not all pictures need this attention but ocean horizons, building edges, and other straight-line picture elements leap out when geometric distortion goes uncorrected.
4. Small sensors have noise issues (at least as of this writing). Sadly the manufacturers still feel obligated to push the sensor resolution to ever-higher levels for marketing advantage but the price is paid in increased noise. Many cameras, like the Canon G9, would have far better images if the sensor sites were a bit larger even if it means a lower megapixel count.
5. Looking up close at images created by any of these cameras will be clear to the experienced eye that they come from a smaller inexpensive point-and-shoot, and not one of the bigger pricier DSLRs or medium format digital cameras with good lenses.
6. Almost any current point-and-shoot pocket camera is far more than adequate for images destined for the Web or for small family album prints. Only with extra care, as we've been exploring, will the images survive closer scrutiny and larger printing. However, you will not see an image taken with one of these class of cameras on the cover of Architectural Digest.

So, although it's tempting to say we can, with the tools and techniques described here, enhance a point-and-shoot image up into DSLR level of quality, that's only true if the shooter using a DSLR doesn't apply equal skill and similar tools to nudge those images to the next level of quality. The point-and-shoot images start off at a substantial disadvantage, can be improved greatly, but won't ever quite reach the quality of a carefully crafted DSLR image.

For the G9 in particular, here's a summary of its characteristics:

1. Images suffer from moderate geometric distortions, edge fall-off, and chromatic aberrations as expected. None are show-stoppers and a carefully shot image can be cleaned up substantially using the techniques we've explored here.
2. The aperture "sweet spot" for the G9 is between f/4 and f/5.6. Any lower and axial chromatic aberration becomes an issue. Any higher and diffraction starts turning detail to mush.
3. Many of us who own the G9 wish the zoom range went a little lower, say to 24 mm equivalent rather than 35 mm. There is an after market for adapters to extend the zoom range optically. I've mentioned the Raynox wide-angle adapter which, when added to the G9, will yield a focal length down at the 24 mm equivalent range. The image, however, suffers greatly with uncorrectable vignetting at the corners at short focal lengths, bad geometric distortion and substantial chromatic aberration. Some, but not all, of this is correctable. My advice to anyone wanting quality 24 mm equivalent images from their G9 is don't consider the Raynox adapter. Maybe others are better but I have my doubts.

Bottom line: don't bring a knife to a gunfight. A consumer pocket camera does not compete with any reasonably good DSLR with a good lens. The pocket camera does, however, work great when you're not carrying a DSLR. The pocket cameras have some other strengths compared to DSLRs which we'll visit in future posts, not the least of which it attracts less attention than the flamboyant DSLRs. (For some reason some uninformed power-tripping Barney Fifes of law enforcement don't find pocket cameras the grave threat to national security that a SLR is, especially one mounted on a tripod.)

Part 6: Image Correction Alternatives

After enduring the math and tedium of developing our own corrections for images with vignetting, geometric distortion, and chromatic aberration, it's worth looking at some alternatives not involving such a high pain level.

First, if you're a user of a recent version of Photoshop, you have the Lens Correction filter available. This filter has adjustments for the 3 areas we've been exploring. It works fine for estimating simple adjustments in any of these 3 areas but it doesn't have any kind of built in knowledge of adjustments required for particular lens/camera combinations. If you only need approximate corrections and like the easy-to-use interface, this is a good way to go. If, on the other hand, you have stricter requirements or you have short focal length lenses with complex mustache distortion or other more complex problems, the Lens Correction filter won't handle it.

Another option to consider is DxO Optics Pro which is a sophisticated product handling a number of image problems including those we've been discussing plus many others. This software is tied directly to your camera/lens combination by way of modules that have very specific correction parameters built in that are the result of DxO's expert lab testing. Results can be very good, the product comes with standalone and Photoshop plugin versions, and can be used to batch process large quantities of images fairly easily. On the minus side, it's not inexpensive (although certainly not overpriced for what it does), it's a bit daunting to learn the quirky interface, and it only supports a subset of camera and lens combinations. For example, at the time of this writing, it doesn't support the Canon G9 we've been using as our example in this series.

Finally, there's PTLens which is a much less ambitious effort in terms of functionality but which has a much broader camera/lens coverage than DxO. It comes with a standalone version (handling only JPEG and TIFF images) and a Photoshop plugin. In terms of cost (about 10% of DxO), simplicity, and effectiveness for its strong point, geometric distortion correction, this is a gem. It also handles vignetting and chromatic aberration but you must make those adjustments manually, unlike the corrections for geometric distortion which cover a very broad range of cameras and lenses. PTLens had its genesis from Panorama Tools roots and handles geometric distortion under the covers in the same way that we discussed using the third-order polynomial radial corrections—except you don't have to be aware of it. By the way, the manual corrections for chromatic aberration in PTLens are also analogous to what we developed using corrections for the red and blue channels. In fact, the PTLens sliders show the numeric radial deviation from 1.0 (no change) as you visually tweak the image to your satisfaction. You can interchangeably use the PTLens numbers and the d coefficients we used in hugin parameters. I routinely will use PTLens to get in the ballpark and then transfer the corrections to hugin to refine the correction to as close as I can get to eliminating transverse chromatic aberration. The PTLens Photoshop plugin can be used to handle large numbers of images simply by making a Photoshop Action to run during ACR processing for example. The one thing I don't like about PTLens is it doesn't handle very large images well, having a tendency to bomb out with memory errors. But, other than that, you'll have a hard time finding a bigger "bang for the buck".

Both DxO Optics Pro and PTLens have free trial versions which should be exploited if you're at all interested in looking at their capabilities.

Given you can buy good solutions for correcting image problems we've been discussing, you may wonder why all the trouble doing it ourselves with all those measurements and mathematical manipulations. There are a few reasons. First, none of the products mentioned above handles all camera/lens combinations. For example, DxO doesn't handle the Canon G9 and PTLens doesn't handle the Canon G9 with the Raynox wide angle adapter we mentioned back in the post about geometric distortion. Second, by doing your own measurements you can be more confident that your custom-built corrections are as accurate as possible for your particular images. And third, there's nothing like grinding your nose into the pixel-level image flaws to get a real understanding of what your equipment capabilities and limitations are.

Next up, some final thoughts on compact consumer cameras generally and the Canon G9 in particular, given what I've learned going through these image correction exercises. After that, maybe we'll get back to some real photography discussion instead of all this techie pixel-bending stuff.

Part 5: Putting It All Together

In previous posts, we've looked at the Canon G9 point-and-shoot camera's vignetting, geometric distortion, and chromatic aberration. We've also looked at strategies for addressing these issues, primarily using the fulla command from the hugin panorama photo stitcher package.

The fulla command doesn't have a nice friendly GUI interface but it does allow us to put together arbitrary corrections to be applied to large numbers of images without a great deal of effort. Each of the fulla corrections we've explored in previous posts can be combined into a single command addressing vignetting, geometric distortion, and chromatic aberration.

Your workflow sequence is very important. The corrections we've been discussing should be applied to the images before any cropping, resizing, etc. have been done. I always shoot raw so I use ACR or some other tool for "developing" the image to set the correct initial tonality, color temperature, etc. and then export the image in 16-bit TIFF format (lossless unlike JPEG). I apply the image corrections and then bring the corrected version into Photoshop for subsequent processing.

Let's start with a hypothetical file shot on the Canon G9 at 7.4 mm and f/4, and processed from the raw CR2 file into a 16-bit TIFF file called example.tif. To apply my standard corrections for vignetting, geometric distortion, and chromatic aberration, I would use the following command:

fulla -c 1.0398:-0.1155:0.1954:-0.1605 \ 
      -g 0.028:-0.0871:0.0521:1.007 \
      -r 0:0:0:1.00024 -b 0:0:0:1.00041 example.tif

(The '\' characters indicate arbitrary line breaks for formatting here. This is actually all one command line.)

The "-c" option gives the polynomial coefficients for correcting vignetting, the "-g" option gives the polynomial coefficients for geometric distortion correction, and the "-r" and "-b" options provided the polynomial coefficients for transverse chromatic aberration correction. (But you already knew that.)

When fulla is done crunching the numbers, it outputs a file with a "_corr" filename suffix. So our example correction would create a new file called example_corr.tif. It will look substantially better than the original with the corrections applied.

Naturally you don't want to have to type in that command every time you process a file so you can create a MSDos batch command file (on Windows) or a Bash script (on Unix systems) to generalize it. Let's say we want to correct arbitrary numbers of files with one command. We can create a batch file like this:

for %%f in (%1) do \
       fulla -c 1.0398:-0.1155:0.1954:-0.1605 \ 
      -g 0.028:-0.0871:0.0521:1.007 \
      -r 0:0:0:1.00024 -b 0:0:0:1.00041 %%f

(Once again, this should be all on one line in the real batch file.) Assuming you named this something like G9_74_4.bat (because it's only useful for the G9's 7.4 mm f/4 images), you can process all the TIFF files in a directory with a single command:

g9_74_4 *.tif

Unix, Linux, Mac users can create analogous script files using for iterations to do precisely the same thing.

I must mention something about the fulla command here: it's flaky and not mature in some regards. One frustration is that, when it outputs a corrected image, that corrected image has its metadata stored in an unconventional way that is not visible to Photoshop and many other applications. If you're running any kind of responsible workflow, this is not good, but there is a way around the problem.

ExifTool by Phil Harvey is a superbly-done high-function image metadata management tool. It has an extraordinary number of capabilities and it handles them well. If you download the tool, you can use it to save your EXIF (and other metadata) from your image, apply your fulla corrections, and then restore the original metadata in standard format. For example, you can save the metadata for all TIFF files in a subdirectory with this command:

exiftool -o %f.mie -ext tif .

Process your files using the fulla corrections and then restore the metadata to the corrected files.

exiftool -overwrite_original -tagsfromfile %-.5f.mie \
                       -ext tif .

You can even skip the intial save to MIE file (Meta Information Encapsulation) and simply rewrite the original file's metadata over the fulla output file:

exiftool -overwrite_original -tagsfromfile %-.5f.%e \ 
                       example_corr.tif

Finally, if you're really ambitious and have Perl programming skills, you could use the Perl Image::ExifTool package to determine the focal length and f-stop for the image and then do a table lookup to determine which of the fulla option parameters to use. I've done a rudimentary job of this just for the subset of camera settings I almost always use for the Canon G9.

If all these seems like a lot of trouble to correct the occasional image, you're right. It's most worthwhile if you routinely have large numbers of images you wish to optimize; then the use of fulla in batch or script files makes more sense.

If you only do occasional corrections, want something a bit easier to deal with, or want a commercial solution, stay tuned. Next up we'll glance at a couple of alternatives for correcting images that don't require all the up-front work we've been slogging through here.

Part 4: Chromatic Aberration

There are many reasons why an image may contain chromatic aberrations, the unexpected color casts or fringes in certain parts of the image. Lenses, particularly inexpensive ones such as found in point-and-shoot cameras, can contribute two main types of chromatic aberration: longitudinal (or axial) and transverse (or lateral).

Longitudinal chromatic aberration is caused by the lens system's failure to focus all wavelengths of light to the same focal plane. If we look, up close, at a piece of an image with sharp high-contrast edges, we can see evidence of longitudinal chromatic aberration in the unnatural blue and purple casts of leaves and branches in this example. This is part of a Canon G9 image shot at 7.4 mm at f/2.8.



The solution to longitudinal chromatic aberration is often quite simple: stop the aperture down, causing a greater overall depth of field and reducing the effect of the wavelength focus differences. When we close the aperture down to f/5.6, we notice a much better color rendition on this same subject.



From a series of tests on this subject, I notice the longitudinal chromatic aberration is mostly gone at f/4, and almost totally gone at f/5.6. There is no noticeable improvement if you stop down further. However, due to diffraction limitations, the image turns to mush, particularly at f/8. So, the rule of thumb is to stick between f/4 and f/5.6 to limit longitudinal chromatic aberration in images.

Transverse chromatic aberration is caused by the lens system enlarging the image on the focal plane to differing amounts depending on the wavelength of light. The effect is more noticeable the farther from the center of the image you look.

To evaluate transverse chromatic aberration, I taped a large (6 feet by 10 feet) sheet of white paper to the side of the garage and placed black electrical-tape X's all over it. This provided a target with a few handy properties for this exercise: relatively neutral tones (not favoring a particular color strongly), high contrast sharp edges between the black tape and white paper, and daylight illumination covering the whole visible spectrum (unlike, for example, fluorescent lights which have spikes at various wavelengths).



I set the Canon G9 camera to aperture-preferred f/4, framed the target to get useful X's to the corners, and shot at several focal lengths from the shortest, 7.4 mm, to the longest, 44.4 mm.

If we take a close look at the upper left corner of the 7.4 mm shot, we can see evidence of significant chromatic aberration around the X. (This sample of the full image is blown up to twice size and the saturation boosted way up to make the color aberrations quite clear.)

To correct this, we're going to use the fulla command mentioned frequently in previous posts. It's part of the freely-available hugin panorama photo stitcher package. The approach taken by the fulla command is to leave the green channel of the image alone and adjust the blue and red channels to line up correctly with the green channel. This makes a lot of sense since green appears brightest to our eyesight, there are twice as many green photo sites on a Bayer matrix sensor as there are either red or blue photo sites, and usually the green channel is the cleanest compared to the other two. Like we've seen in the discussions of vignetting and geometric distortion, fulla uses a polynomial to accurately represent adjustments to be made to each of the red and blue channels depending on radial distance from the center of the image. The polynomial adjustment, once again is of the form

 a * r3 + b * r2 + c * r + d 

There are a few ways we can approach finding the values for a, b, c, and d to use for correction. First, we can break the image into separate R, G, and B channel images and use one of the panorama stitching programs to compare the red and green channel images, select control points, and optimize the corrections for the red channel image required to make it line up with the green channel. Do the same thing comparing the blue channel image with the green channel, and we can get the set of polynomial coefficients that will shape the red and blue channels to line up with the green.

To break an image into separate channels, you can use Photoshop, go to the Channels tab, and select Split Channels. This will create 3 grayscale images, one for each color channel, with "_R", "_G", or "_B" appended to the filename to indicate which color channel it represents.

Another alternative to the tedium of fitting control points in a panorama stitching tool, is to use the tca_correct tool supplied in the hugin package. This command-line program also needs the separate channels broken out into separate image files with each beginning with "red_", "green_", or "blue_" denoting the channel it represents. An additional requirement is that each of the channel images must be an RGB image, not grayscale so that additional step would need to be done in Photoshop if you're going that route.

Another way to get the separate channel images broken out is through the use of ImageMagick a wonderfully powerful set of command-line tools available on all major platforms. Using ImageMagick's convert command we can extract each of the RGB channels into separate RGB image files:

convert %1 -channel R -separate -depth 8 -type TrueColor red_%1
convert %1 -channel G -separate -depth 8 -type TrueColor green_%1
convert %1 -channel B -separate -depth 8 -type TrueColor blue_%1

(This example uses a MSDos batch command file for Windows; the analogous thing can be done with the same parameters on the other platforms.)

The tca_correct command to evaluate your image and determine the polynomial coefficients is

tca_correct -o abc filename

where filename is the root file name without the color prefixes. After crunching things for a bit, it will show the fulla options to be used to correct the transverse chromatic aberration.

This is ingenious, simple, and a great time saver. However, there are a couple of problems with this approach. First, the tca_correct command is not being distributed reliably with the various hugin updated packages. Second, it doesn't work. At least for me, it didn't work well—it seemed to substantially under-correct the test images I used.

Another way to go, the method I chose, is to use the old reliable eyeball-it approach. For the G9 test images I was looking at, the transverse chromatic aberration appeared to be approximately linear with the radius. Logically this makes sense given that, by it's nature, transverse chromatic aberration is caused by the failure to cast different wavelengths on the image plane with the same magnification. We need to shrink or expand the red and blue channels to line up with the green. I used the fulla chromatic aberration correction command options on my test images adjusting only the d polynomial coefficient which is the linear term of the expression and thus causes a linear expansion or contraction of the channel.

Fulla wants 2 options provided for the red and blue channel adjustments ("-r", and "-b" respectively) providing the polynomial coefficients in the usual syntax:

fulla -r a:b:c:d -b a:b:c:d filename

I set the a, b, and c coefficients to zero and start with a small adjustment. By examination of the test image above, it's clear both the red and blue channels need to be pushed out so we start with something like this:

fulla -r 0:0:0:1.0003  -b 0:0:0:1.0003 filename

We examine the output, and adjust blue and red channels up or down as needed. For the 7.4 mm f/4 test image above, I ended up with the following adjustments:

fulla -r 0:0:0:1.00024 -b 0:0:0:1.00041 filename

As you can see in the processed image (magnified and saturation-enhanced the same as the original above), we've eliminated the most egregious transverse chromatic aberrations.

With luck, the rest of the image will be similarly corrected. If not, there may be secondary adjustments required where you have to start tweaking the c coefficients, for example. It's not something to look forward to.

Now that we've addressed longitudinal and transverse chromatic aberrations, our images should be free of chromatic aberrations, right? Sadly, no. There are other chromatic aberrations originating in other parts of the camera system including the photo sites themselves and the microlenses on top of them. Once again, the point-and-shoot cameras tend to be worse, particularly due to the tiny photo sites and their very high density. These aberrations, unfortunately, can not be fixed as easily as what we've been talking about. However, usually these are not nearly as noticeable as the aberrations we do have some control over.

Next, we'll put vignette correction, geometric distortion correction, and transverse chromatic aberration correction together in a more organized way now that the hard work has been done. We'll also look at a couple of commercial alternatives to our labor-intensive solution, one modestly priced and the other more expensive and robust.

Part 3: Geometric Lens Distortion Correction

All lenses, particularly cheaper mass-produced lenses have some amount of geometric distortion. Some primary lenses, particularly expensive ones, can be highly corrected. Others, particularly with large zoom ranges, are characterized by mixes of barrel or pincushion distortion (or both). The result is images where straight lines appear curved, particularly noticeable in the outer portions of the images.

In 1998 Helmut Dersch created a set of software tools that could be used to, among other things, correct lens distortions so that they could be more effectively stitched together in panoramas. His library of tools caused a cottage industry to grow with many GUI front-ends developed to use the tools. Not everyone is interested in creating panoramas, but many are interested in correcting noticeable distortions for things like ocean horizons, architectural features, and many other distortion-sensitive subjects.



The Canon G9, which we've been using as the guinea pig for our series discussion, has a zoom range of 7.4 mm to 44.4 mm (equivalent to roughly 35 - 210 mm in 35 mm terms). At the short end of the range, it has moderate inner barrel and outer pincushion distortion (the combination commonly called mustache distortion). As the lens zooms out, images acquire slight pincushion distortion. The compound distortions in the lens don't correct well using simplistic generic tools so we'll develop specific corrections to flatten the images at the various zoom settings.

We need to apply a corrective distortion to the image to compensate for the lens-induced distortions. The image distortions are radially symmetric so we'll use a function that determines image distortion based on radial distance from the center of the image. In short, we'll be using a 3rd-order polynomial to correct the radial distance each part of the picture needs to be moved, based on its original radial distance from the center:

 a * r3 + b * r2 + c * r + d

Our goal is to find the values for a, b, and c to correct for our lens's distortion at a specific focal length. The d value is only used to adjust the total image size to compensate for the other corrections. It has a value of

 d = 1 - (a + b + c)

There are a few approaches to finding the coefficients for the polynomial. One way is to shoot several images and stitch them together into a panorama using one of the commonly-available tools and then note the corrections the software made to fit the images together. In essence, the process of stitching the images accurately requires the software to correct for distortions to complete its job.

We'll use a more targeted approach using one image. In this technique, you shoot an image that contains several horizontal and vertical lines that you know are straight. A large office building is ideal. The objective is to have several lines available in the image at varying distances from the center of the image and stretching all the way to the edges of the image. You then use a panorama stitching tool such as the freely-available hugin panorama photo stitcher package. Using hugin to do this, you load the target image twice, cancel the automatic control point detection, and manually add control points along the lengths of each of the lines you know should be straight. All control points on a line must belong to the same line labeled tn where n is a line number starting at 3 for the first line. (Lower numbers are reserved for other types of lines.)

Once you go through the tedium of identifying a few hundred control points on 6 to 12 lines covering various parts of the image, you run the optimizer to determine what the coefficients should be to cause the distorted lines to become straight.

For the G9, I created test images through the zoom range and here are the corrections I came up with which give me images with the lens distortions largely removed.

Canon G9 Lens Distortion Correction Coefficients

Focal Length	a	b	c	d
7.4 mm	0.028	-0.0871	0.0521	1.007
8.2 mm	0.0313	-0.089	0.0474	1.0103
9.0 mm	0.0082	-0.0186	-0.0046	1.015
12.7 mm	0.0187	-0.0558	0.0528	0.9843
16.8 mm	-0.0172	0.0541	-0.0477	1.0108
22.0 mm	-0.0053	0.0196	-0.0201	1.0058
25.0 mm	-0.0038	0.0131	-0.0133	1.004
29.2 mm	-0.0065	0.0184	-0.0111	0.9992
36.8 mm	0.0102	-0.0355	0.0391	0.9862
44.4 mm	-0.0159	0.0626	-0.0674	1.0207
Special bonus: 7.4 mm with Raynox 0.7X wide angle conversion lens attached:
5.2 mm	0.0513	-0.1509	0.0835	1.0161

To apply corrections to an image, you can use the hugin fulla command we've mentioned previously. This time we use the -g option to pass our correction coefficients:

 fulla -g 0.028:-0.0871:0.0521:1.007 image_file_name

The output will be our distortion-corrected image. (In this case the image of the building was shot with the G9 set at its shortest focal length, 7.4 mm, so we use the coefficients corresponding to that focal length.)



This technique will go a long way toward correcting obvious geometric distortion but it's not perfect. It doesn't correct to perfection although in most cases it's nearly impossible to tell without careful measurements of the resulting image. It also doesn't handle radially asymmetric distortions such as might occur in misaligned lenses.

Next up, we'll take a look at chromatic aberration and see how we can correct one type of it.

Part 2: Vignette Correction

All camera/lens systems vignette the produced images, some much more than others. Vignetting (or more-correctly light falloff as we're discussing here), the darkening of the created image toward the edges, is caused by a number of factors. We are so used to seeing vignetting in images, we usually don't notice it; in fact many think that vignetting can add to the appeal of an image and will add more in during post-processing. Vignetting in some applications is not desirable: scientific and technical images, cartographic images, repro work, and panorama stitching are among the applications where vignetting is to be avoided.

In this image, the upper left portion is a flat evenly-lit surface showing the effects of vignetting causing light fall-off radiating from the center. It becomes noticeable when compared to the true flat field uniform gray half of the image in the lower right.

We're going to take a look at a couple of options for eliminating or reducing vignetting in images during post-processing, particularly using the fulla command available in the free hugin panorama photo stitcher package. The fulla command is a useful utility we'll explore for other purposes in future posts, as well as using it to address vignetting here.

Before we get started, a couple of things need to be understood. First, vignette characteristics are unique to a particular camera,lens, focal length, and aperture. Change any one of those and the vignette characteristics change. The rate of brightness falloff from center to edges of an image is not uniform; systems that may cause the same center-to-edge brightness falloff may do it in very different ways. It can happen in a gradual linear way or it may have relatively uniform brightness until getting closer to the edges when it drops of quickly. Vignetting is not necessarily radially symmetrical, although in most cases it is close enough for our purposes in this discussion. Finally, this overview is not for the faint of heart; it outlines the non-obvious aspects of what we're doing but if you wish to pursue this yourself, you must be willing to dig in and work through the details yourself.

Using a flat field image as a corrective for other images is a technique that has been around a long time and is still commonly used by astrophotographers. Essentially, you shoot an image of a perfectly uniform empty subject devoid of any detail and uniformly lit from edge to edge. A clear cloudless haze-free sky directly overhead would be an example. You then apply this image to target images using software that cancels out the brightness differences represented in the flat field image. The advantage to this approach is that you are directly using the systems recorded image to cancel out the brightness anomalies in the produced images. The fulla command can use a flat-field image to correct for vignetting by passing the flat-field image's name as a command option.

The trouble comes when you try to produce an accurate flat-field image for your camera, particularly if you aren't doing this for a telephoto lens. It's actually fiendishly difficult to get the featureless uniformly-lit subject you need to do this. The gray image above is an attempt to produce such a flat-field image (upper left portion) using the Canon G9 camera's 7.4 mm focal length and f/2.8 aperture. This is a relatively wide angle (about 50° on the long side), presenting quite a challenge for coming up with a suitable flat-field subject. I was not successful—I came close but never could get an image in which all corners were consistent, for example.

Another approach is to correct the vignetting mathematically. You provide a formula to software that can use to determine what brightness correction should be used at each radial distance from the center of the image. The fulla command provides 2 options for doing this, one in which you provide the formula for the value that should be added to the brightness depending on radial distance, the other method uses a correction in which the brightness is divided by a value provided by a formula. I found the latter worked better for me. The fulla command has very sparse and nearly unusable documentation so I'll fill in the details as I describe the process I used to profile my camera system for vignette correction.

The net result of what we're going to do is to provide a set of 4 numbers to a fulla option that describe an equation fulla uses to determine what adjustment needs to be made at each radial distance in the image. The four numbers are coefficients for an even-powered 6th order polynomial of the following form:

a + b * r2 + c * r4 + d * r6

where r is the radius from the center of the image (0) to the farthest corner (1.0). This formula is applied at each radial distance by dividing the image's RGB values by the calculated factor to determine what the new adjusted values should be. Assuming you haven't left already, our job is to find the values for a, b, c, and d that work best for the particular camera, lens, focal length, and aperture we're interested in correcting.

The first step is to measure the brightness fall-off from the center to the corners of an image produced by your system. Because there's so much trouble trying to get a single flat-field image, we'll shoot multiple images of a uniformly-lit small target, panning the camera from the center to a corner between shots. Our goal is to get about 15 or 20 shots of the target ranging in position from the center to the lower right corner of the image.

I used a gray card illuminated by an incandescent lamp that was turned on about 20 minutes before starting to make sure the light has stabilized. The camera was mounted on a tripod about 15 feet away and set on the focal length and aperture I wanted to profile (7.4 mm at f/2.8 for this example). I adjusted the tripod head so the camera was angled to the left in such a way that I could pan over the subject from the center to the lower left corner without adjusting anything else on the camera. Then I made the series of about 20 shots panning a little more to the left each time until I had images of the target all the way into the farthest reaches of the corner. I made sure I was using manual exposure and the target was approximately gray (midrange) in value. By the way, I was shooting raw, not JPEG, as I didn't want the internal JPEG processing mucking up the precision of my test images.

Next step is process the images precisely how you would normally do it. Do not apply any contrast adjustments to the images unless you use exactly the same adjustments on all your images all the time. I ended up generating about 20 TIFF files that I could load in sequence into Photoshop to sample the brightness of the target at each position.

Next use your Photoshop (or other) eyedropper tool to measure the same relative location of your target in each image. I used the 5X5 averaging for more uniform readings. Record the location in the image that you sample (x and y values) and the readout of the brightness (either the green channel, for example, or gray value if you converted the images to grayscale during processing).

Plug these into a spreadsheet and start crunching the numbers into something more meaningful. First, you want to convert the x and y values into a radius value ranging from 0 at the center to 1.0 at the far corner. The Canon G9 I used makes images 4000 X 3000 so the spreadsheet formula for finding the radius based on x and y readings goes something like this:

=sqrt(($Xn-2000)^2+($Yn-1500)^2)/2500

where X and Y are, obviously whatever column numbers you used to plug in your x and y coordinate numbers.

Next, create a column of your radius numbers squared—this will be used in determining the magic coefficient numbers we're after.

Finally, create a column of normalized brightness levels. At this point you need to decide how you want to correct your image's brightness levels for vignetting. You could brighten the darker outer region, darken the lighter central region, or do something in between. I took the in-between route to minimize the damage I'm doing to pixels that are having their brightness adjusted. So, my normalized brightness values are calculated by dividing the recorded brightness value by the average of the brightest and darkest values. This would give you a range of numbers going from a little over 1.0 to a little less than 1.0 depending on how bad the vignetting is. For my f/2.8 test images, the brightest central reading was 153 and the darkest corner reading was 129 giving me a normalized range of 1.085 to 0.915, approximately.

Next we calculate a polynomial regression line through the values we've calculated. If you are using an Excel spreadsheet, this is easy. Create a XY (Scatter) graph of normalized brightness values corresponding to the radius squared. Then on the chart do Add Trendline and, in the Type tab, select Polynomial and Order 3; in the Options tab select "Display equation on chart" and "Display R-squared value on chart". Once you click OK, a regression line will be drawn through your scatter plot and an equation will show up on your graph. This equation has the a, b, c, and d coefficient numbers we've been working toward. In my f/2.8 case, the regression equation produced was

y = -0.1355x3+0.0443x2-0.0943x+1.0873

The terms are in the reverse order of the fulla format so our a, b, c, and d coefficients for fulla are 1.0873, -0.0943, 0.0443, and -0.1355 respectively. The R² (squared coefficient of variation) value was 0.9963 in my case, which is a very good fit. (A R² of 1.0 represents a perfect fit while 0 represents no correlation whatsoever; the closer to 1.0 we get, the more confidence the regression line is a good representation of our measured data.)

So, why did we plot against radius squared rather than radius? It's because we need the even-numbered sixth-order polynomial which isn't available as an option in Excel so we finessed it by using a 3rd-order polynomial regression line against r².

If you don't have Excel, you may have access to another spreadsheet program with a similar regression analysis capability. If not, there are other programs such as statistical analysis packages that can do it, or you can even use online tools to accomplish the same thing. See for example http://udel.edu/~mcdonald/statcurvreg.html and scroll down to the heading, "How to do the test" which has links to several online tools as well as a spreadsheet you can download for use when you don't have the integrated trend line feature.

Next we look at a quick test. This image is a composite of 2 images shot with a pan between to see how they match up. Notice the seam line where the two images line up. The vignetting is clearly visible. If we process the two images separately using the fulla tool, we can check how effective our calculations were. We use our calculated coefficients in fulla using the command like this:

fulla -c 1.0873:-0.0943:0.0443:-0.1355 image_file_name

When we composite the corrected images together in the same way as the originals, we see the seam is markedly less visible indicating our correction has improved the vignetting situation.

Is it perfect? Not at all. This method can cause color shifts because it's not actually correcting strictly luminance but rather just doing a crude adjustment of the RGB values in the image. It also uses assumptions of perfect radial behavior, an imperfect regression line, imperfect input test data, and so on. However, in most cases it is an improvement as we can see in the corrected image. Is it worth the trouble? Perhaps not for most images but once the calculations are done it can be applied nearly effortlessly to any image shot with the same camera, lens, focal length, and aperture combination such as what we're targeting here for our panoramas.

There are other more advanced techniques evolving that can guess at vignette correction, particularly in multi-image panorama environments and make adjustments accordingly. Of course there's the now-standard panorama tools Smartblend, Enblend, etc. that disguise vignette and other tonal anomalies.

Next up, we'll look at lens distortion and how we can correct that using fulla as well. We're working toward a single fulla command line to correct as many flaws as we can and when we're finished we'll put together some simple scripts to simplify correcting batches of images.