Accurately measuring relative distance between a set of fiducials (Augmented reality application) - math

Question

Let's say I have a set of 5 markers. I am trying to find the relative distances between each marker using an augmented reality framework such as ARToolkit. In my camera feed thee first 20 frames show me the first 2 markers only so I can work out the transformation between the 2 markers. The second 20 frames show me the 2nd and 3rd markers only and so on. The last 20 frames show me the 5th and 1st markers. I want to build up a 3D map of the marker positions of all 5 markers.
My question is, knowing that there will be inaccuracies with the distances due to low quality of the video feed, how do I minimise the inaccuracies given all the information I have gathered?
My naive approach would be to use the first marker as a base point, from the first 20 frames take the mean of the transformations and place the 2nd marker and so forth for the 3rd and 4th. For the 5th marker place it inbetween the 4th and 1st by placing it in the middle of the mean of the transformations between the 5th and 1st and the 4th and 5th. This approach I feel has a bias towards the first marker placement though and doesn't take into account the camera seeing more than 2 markers per frame.
Ultimately I want my system to be able to work out the map of x number of markers. In any given frame up to x markers can appear and there are non-systemic errors due to the image quality.
Any help regarding the correct approach to this problem would be greatly appreciated.
Edit:
More information regarding the problem:
Lets say the realworld map is as follows:
Lets say I get 100 readings for each of the transformations between the points as represented by the arrows in the image. The real values are written above the arrows.
The values I obtain have some error (assumed to follow a gaussian distribution about the actual value). For instance one of the readings obtained for marker 1 to 2 could be x:9.8 y:0.09. Given I have all these readings how do I estimate the map. The result should ideally be as close to the real values as possible.
My naive approach has the following problem. If the average of the transforms from 1 to 2 is slightly off the placement of 3 can be off even though the reading of 2 to 3 is very accurate. This problem is shown below:
The greens are the actual values, the blacks are the calculated values. The average transform of 1 to 2 is x:10 y:2.

Answer 1

You can use a least-squares method, to find the transformation that gives the best fit to all your data. If all you want is the distance between the markers, this is just the average of the distances measured.
Assuming that your marker positions are fixed (e.g., to a fixed rigid body), and you want their relative position, then you can simply record their positions and average them. If there is a potential for confusing one marker with another, you can track them from frame to frame, and use the continuity of each marker location between its two periods to confirm its identity.
If you expect your rigid body to be moving (or if the body is not rigid, and so forth), then your problem is significantly harder. Two markers at a time is not sufficient to fix the position of a rigid body (which requires three). However, note that, at each transition, you have the location of the old marker, the new marker, and the continuous marker, at almost the same time. If you already have an expected location on the body for each of your markers, this should provide a good estimate of a rigid pose every 20 frames.
In general, if your body is moving, best performance will require some kind of model for its dynamics, which should be used to track its pose over time. Given a dynamic model, you can use a Kalman filter to do the tracking; Kalman filters are well-adapted to integrating the kind of data you describe.
By including the locations of your markers as part of the Kalman state vector, you may be able to be able to deduce their relative locations from purely sensor data (which appears to be your goal), rather than requiring this information a priori. If you want to be able to handle an arbitrary number of markers efficiently, you may need to come up with some clever mutation of the usual methods; your problem seems designed to avoid solution by conventional decomposition methods such as sequential Kalman filtering.
Edit, as per the comments below:
If your markers yield a full 3D pose (instead of just a 3D position), the additional data will make it easier to maintain accurate information about the object you are tracking. However, the recommendations above still apply:
If the labeled body is fixed, use a least-squares fit of all relevant frame data.
If the labeled body is moving, model its dynamics and use a Kalman filter.
New points that come to mind:
Trying to manage a chain of relative transformations may not be the best way to approach the problem; as you note, it is prone to accumulated error. However, it is not necessarily a bad way, either, as long as you can implement the necessary math in that framework.
In particular, a least-squares fit should work perfectly well with a chain or ring of relative poses.
In any case, for either a least-squares fit or for Kalman filter tracking, a good estimate of the uncertainty of your measurements will improve performance.