Still checking Gauss-Newton
Though Levenberg-Marquardt works I’m still trying to save Gauss-Newton, especially as I’ve read paper saying that Gauss-Newton with dogleg trust-region works well for bundle adjustment. I’ll probably try direct substitution with Cholesky rank-1 update and constrained optimization.
Solution – free gauge
Looks like the problem was not the large Gauss-Newton residue. The problem was gauge fixing.
Most of bundle adjustment algorithms are not gauge invariant inherently (for details check Triggs “Bundle adjustment – a modern synthesis”, chapter 9 “Gauge Freedom”). Practically that means that method have one or more free parameters which could be chosen arbitrary (for example scale), but which influence solution in non-invariant way (or don’t influence solution if algorithm is gauge invariant). Gauge fixing is the choice of the values for that free parameters. There exist at least one gauge invariant bundle adjustment method (generalization of Levenberg-Marquardt with complete matrix correction instead of diagonal only correction) , but it is order of magnitude more computational expensive.
I’ve used fixing coordinate of one of the 3d points for gauge fixing. Because method is not gauge invariant solution depend on the choice of that fixed point. The problem occurs when the chosen point is “bad” – error in feature point detector for this point is so big that it contradict to the rest of the picture. Mismatching in the point correspondence can cause the same problem.
In my case, fixing coordinate of chosen point caused “accumulation” of residual error in that point. This is easy to explain – other points can decrease reprojection error both by moving/rotating camera and by shifting their coordinates, but fixed point can do it only by moving/rotating camera. It looks like if the point was “bad” from the start it can become even worse next iteration as the error accumulate – positive feedback look causing method become unstable. That’s of cause only my observations, I didn’t do any formal analysis.
The obvious solution is to redistribute residual error among all the points – that mean drop gauge fixing and use free gauge. Free gauge is causing arbitrary scaling of the result, but the result can be rescaled later. However there is the cost. Free gauge means matrix is singular – not invertible and Gauss-Newton method can not work. So I have to switch to less efficient and more computationally expensive Levenberg-Marquardt. For now it seems working.
PS Free gauge matrix is not singular, just not well-defined and has degenerate minimum. So constrained optimization still may works.
PPS Gauge Invariance is also important concept in physics and geometry.
PPPS While messing with Quasi-Newton – it seems there is an error in chapter 10.2 of “Numerical Optimization” by Nocedal&Wright. In the secant equation instead of 
Bundle Adjustemnt on the Mars with Rover
Just found out – Mars Rovers used bundle adjustment for its localization and rocks modeling:
“Purpose of algorithm:
To perform autonomous long-range rover localization based on bundle adjustment (BA) technology.
Processing steps of the algorithm include interest point extraction and matching, intra- and inter- stereo tie point selection, automatic cross-site tie point selection by rock extraction, modeling and matching, and bundle adjustment”
Marker vs markerless (bundle adjustment)
#augmentedreality
Here is a sample of image registration with fiduciary marker (actually the marker I used in my games) vs registration with bundle adjustment. Blue lines are points heights (relatively to marker plane) calculated using marker registration and triangulation. White lines are the same using bundle adjustment(modified). Points extracted with multiscale FAST and fitted with log-polar Fourier descriptors for correspondence (actually SURF descriptor produce the same correspondence).
As you can see markerless is in no way worse then markers, at least on this example ))).
Why 3d markerless tracking is difficult for mobile augmented reality
I often hear sentiments from users that they don’t like markers, and they are wondering, why there are so relatively few markerless AR around. First I want to say that there is no excuse for using markers in the static scene with immobile camera, or if desktop computer is used. Brute force methods for tracking like bundle adjustment and fundamental matrix are well developed and used for years and years in the computer vision and photogrammetry. However those methods in their original form could hardly produce acceptable frame rate on the mobile devices. From the other hand marker trackers on mobile devices could be made fast, stable and robust.
So why markers are easy and markerless are not ?
The problem is the structure , or “shape” of the points cloud generated by feature detector of the markerless tracker. The problem with structure is that depth coordinate of the points is not easily calculated. That is even more difficult because camera frame taken from mobile device have narrow baseline – frames taken form position close one to another, so “stereo” depth perception is quite rough. It is called structure from motion problem.
In the case of the marker tracker all feature points of the markers are on the same plane, and that allow to calculate position of the camera (up to constant scale factor) from the single frame. Essentially, if all the points produced by detector are on the same plane, like for example from the pictures lying on the table, the problem of structure from motion goes away. Planar cloud of point is essentially the same as the set of markers – for example any four points could be considered as marker and the same algorithm could apply. Structure from motion problem is why there is no easy step from “planar only” tracker to real 3d markerless tracker.
However not everything is so bad for mobile markerless tracker. If tracking environment is indoor, or cityscape there is a lot of rectangles, parallel lines and other planar structures around. Those could be used as initial approximation for one the of structure from motion algorithm, or/and as substitutes for markers.
Another approach of cause is to find some variation of structure from motion method which is fast and works for mobile. Some variation of bundle adjustment algorithm looks most promising to me.
PS PTAM tracker, which is ported to iPhone, use yet another approach – instead of using bundle adjustment for each frame, bundle adjustment is running in the separate thread asynchronously, and more simple method used for frame to frame tracking.
Tracking planes in the city
In relation to tracking cityscape I did some planar segmentation test. Segmented FAST generated corners with simple 5-points projective invariant.
In some cases 5-point give some rough approximation:
In some cases outliers are quite bad – some point have very close projective invariant but still are in diffferent planes.
So simple method not quite work…
Tracking cityscape
One of the big problem in image registration/structure from motion/3d tracking is using global information of the image. Feature/blob extraction, like SIFT, SURF or FAST etc using only local information around the point. Region detector like MSER using area information, but MSER is not good at tracking textures, and not quite stable at complex scenes. Edge detection provide some non-local information, but require processing edges. That could be computationally heavy, but looks promising anyway. There are a lot of methods which use global information – all kind of texture segmentation, epitome, snakes/appearance models, but those are computationally heavy and not suitable for mobiles. The question is how to incorporate global information from the image into tracker, and make it with minimal amount of operations. One way is to optimise tracker for specific environment – for example use the property of cityscape, a lot of planar structures and straight lines. Such multiplanar tracker wouldn’t work in the forest or park, but could be a working compromise.
Region Tracking
Experimenting with MSER region tracking
The problem is that regions seems not stable enough and way too big.
MSER of downsampled images, original images, MSER of mean shift filtered images, MSER of smoothed images:
Mean shift filtering seems capture more feauters, but it’s too computationally expensive.
