Single Pixel shift in HDR alignment

In my recent statement at hdrcreme.com I stated that multiple-image HDR was difficult to do and, for that reason, had a limited application future. Specifically I said this:
“Multiple-image HDR is difficult to do right. It’s almost impossible to align multiple digital exposures correctly as a look at most multiple-image HDR photos will clearly show. Even one pixel off will degrade the picture.”
I instanced pictures of people, of sports, and even of landscape where there was anything moving such as foliage or clouds. My statements elicited a reply from ColdwaterJohn (CWJ) whom I know to be a great exponent of multiple image HDR and I know that he is a solid and capable worker in this area. CWJ heartily defends multiple image HDR (as he should) and supposes that a ‘rock-solid’ tripod is the answer in most instances. I don’t share CWJ’s sanguine view of this technique but let’s look at it a bit more rigorously. So here’s a question: What difference does a single pixel shift make for a subject at any given distance? In other words, if two images, A and B, are shifted by a single pixel with respect to each other then how much of a difference does that make? Let us turn to our famous conceptual camera and stipulate that it has a viewing angle of two radians (to make the math easy, this is about 114 degrees) and that it supports 4000 pixels horizontally. Now imagine that our camera is at the center of a circle and we make two exposures. The first is a reference exposure but the second exposure is misaligned in the horizontal by one pixel. How large an error is that? Let’s take 100 yards as the radius of our circle. Simple trigonometry says that the width of such a pixel at 100 yards is the tangent of the angle times 3600 (inches in 100 yards). This will give us our answer in inches. So tan(1/2000 radians) * 3600 = 1.8 inches which is accurate to five places. So a misalignment of 1 pixel for our conceptual camera at 100 yards is a misalignment of 1.8 inches. This means that an object of width 1.8 inches at 100 yards will appear to be 3.6 inches in width under ideal conditions. Thus the following table:

Distance                Error
---------------------------------------
1 inch                   0.0005 inch
12 inches              0.006 inch
100 inches            0.05 inch
1000 inches          0.5 inch
2000 inches          1.0 inch
3600 inches          1.8 inch
1 mile                 31.68 inches or 2 feet and 7.68 inches
10 miles            316.8 inches or 26 feet and (about) 5 inches

The first thing this suggests is that the smaller the linear distance covered by a single pixel the smaller the error so, for any given subject and at any given distance, the HDR photographer should attempt to box the intended subject as tightly as possible consistent with his or her purpose. That way the number of pixels (which is fixed) covers the smallest angle and any misalignment is minimized. If our conceptual camera had a zoom lens that would let me box in only one radian at 100 yards the error would be immediately halved.  Another thing is that you should shoot at the highest number of pixels that your camera supports.  The error for 4000 pixels is half the error for 2000 pixels.

The second thing is that you should absolutely keep the number of exposures to the bare minimum to fulfill your artistic goals. (see Consoli’s conjecture in a previous post). I think, (but I’m not sure), that the probability of a single misalignment grows only arithmetically with the increase in the number of separate exposures. For two exposures let’s say that the probability m of a misalignment is 1*m. For three exposures the probability of one misalignment is 2*m. For four exposures the probability of one misalignment among the four should be 3*m. If I’m right then that’s good news for multiple image practitioners. But I’m not sure and I invite my readers to consider this problem.
What is the probability of making such a misalignment given that the subject is absolutely still? I don’t know. Whatever it is it’s a function of the stability of the camera, whether the camera has a mirror or not, whether the mirror locks up or not, whether the shutter is released manually or by radio release etc., etc. (CWJ has already pointed out some of these factors) I confess that I have no way of approaching that probability using numbers.

If the camera is absolutely still but the subject is moving then (for our conceptual camera and for any given distance) the subject has to move more slowly than the distances given from the time of the first shutter release to the time of the final shutter closure. Let’s say that the time to make all these exposures is 10 seconds. An object a mile away must be moving more slowly than 3.168 inches per second in order not to stutter after 10 seconds. This is 3600 * 3.168 inches or 11, 404 inches per hour or 0.18 mph. The average cloud probably moves much faster than that. That’s why we see so many HDR landscapes with stuttered clouds.
So far I’ve only considered yaw. The numbers for pitch would be of the same magnitude (since it’s my conceptual camera I can specify that it supports 4000 pixels vertically). Each practitioner should work out the details for his or her own camera. Since I’ve forgotten all my spherical trigonometry I can’t work out what the error for roll would be but I do know that any error in roll would be more severe at the edges than at the center. I sincerely hope that my readers can help me out in this area. And although each of you should, as I said, work out the details for your own setup I think you get the idea about the kinds and sizes of errors that we’re talking about. Can multiple-image HDR be done at all? Of course it can. Practitioners like CWJ are teaching the rest of us how it’s done and sharing their knowledge. For that we are definitely grateful. But the multiple-image HDR perspective is skewed. The thing is, HDR is not an end in itself. It’s a means of extending the tonal range of camera capture beyond the pitifully narrow range that it currently supports. The magic was never multiple exposures (which are as old as photography itself. See Michael Freeman’s book on Digital Black and White Photography for a good and effective non-HDR approach using multiple images); the new magic is the tone-mapping algorithms themselves. They can be widely applied; they don’t require multiple exposures. There are other means to this goal of tonal extension. Support by the camera manufacturers of more capable sensors is the ultimate winner. There’s some evidence that they’re starting to realize this. Multiple image HDR will always be the province of the enthusiast but I sincerely believe that due to its own severe limitations it doesn’t have much of a future. (There are too many subjects that just can’t be approached using multiple exposures and I find CWJ’s remarks about Photoshop unconvincing – although perhaps he can tell us in more detail exactly how that’s done.) It could be, however, that multiple image HDR does have a future in the studio. I’ve seen demonstrations (you probably have too) where exposures are made, lights are shifted, more exposures are made, etc. and all the exposures combined using Photomatix or the like. Great results can be achieved like this.
For a future post I (or we) need to consider sharpening as a cure for small misalignments.

Best to all and I look forward to CWJ’s reply,

(A version of this was published at http://hdrcreme.com/forums/4/topics/53?page=1#posts-198.)

Bob Consoli