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I am looking for ideas on how to mathematically go about the following situation:

Given are $n$ $SE(3)$ Lie groups, i.e. $n$ 6-DegreesOfFreedom (DoF) poses. For example, compare the following picture where I have created $n=6$ poses each depicted by coordinate systems. Case1: Spatially even distributed poses

As you can see, in case 1 all poses are different in location, but share the same orientation. However, this may not always be the case. The next case might show a different $SE(3)$ pose configuration: Case2: Spatially grouped poses

Case 2 depicts that the configuration may form two distinct clumps of poses. Among many other cases, the poses may also be almost identical in position but vary in orientation. This is illustrated in case 3: Case3: Spatially almost identical poses

The example cases 1-3 above are supposed to demonstrate the situation I am faced with. Now I am looking for ideas how to utilize statistical methods to characterize the different cases. Do you know any mathematical way to, for example, distinguish cases 1, 2 and 3 from each other? Ideally, I am looking for a scalar value, where $0.0$ means "case 3", $0.5$ means "case 2", $1.0$ means "case 1", to begin with. But I am very open to any suggestions as my question is not so much about a perfect solution but more about ideas / methods to try and investigate here.

Of course, I have also thought about several approaches myself:

  1. Probability theory: Sample mean and covariance: Good as a starting point, but does not capture the individual clusters of poses
  2. Clustering methods such as k-means: The challenge is to correctly choose the number of clusters but this could be a nice indicator of the different cases
  3. Information theory: Measures such as mutual information and (relative) entropy could quantify the similarity of clusters

In essence, I am lacking ideas how to approach this problem. Do you know any research papers that tackle similar problems or do you have suggestions? I would highly appreciate if you could provide numerical examples on how your approach would be applied to this problem in case it is not straight-forward.

Thank you for your time and kind regards!

Blender Ei
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  • Quick approach. Position can be expressed as a point in $\mathbb{R}^3$. Orientation can be expressed as a unit quaternion, which is a number of the form $a + bi + cj + dk$, with $a^2+b^2+c^2+d^2 = 1$. That constraint means that although quaternions are four-dimensional, the unit quaternions can be thought of as occupying the three-dimensional surface of a hypersphere. The difference between two position vectors is normally just Euclidean distance; the difference between two unit quaternions $q_1, q_2$ is usually $2 \arccos |q_1 \cdot q_2|$. – Brian Tung Oct 17 '23 at 23:00
  • So with this in mind, you could calculate dispersion in space, and dispersion in orientation, and hopefully characterize the distribution of poses in your set. ¶ If that sounds like something that might work for you, I can elaborate in an answer at a later time. – Brian Tung Oct 17 '23 at 23:01
  • Thank you @BrianTung for your very fast answer. This sounds nice and helpful. How would you extend your idea from pairs of poses to the actual set of $n$ poses? Would you calculate combinations of pairs and take the mean? Probably you have something better in mind, right? Would love to see an example with eight example poses (x,y,z,a,b,c,d):
    • (0,2,0,1,0,0,0)
    • (1,1,z,1,0,0,0)
    • (2,0,0,1,0,0,0)
    • (1,-1,0,1,0,0,0)
    • (0,-2,0,1,0,0,0)
    • (-1,-1,0,1,0,0,0)
    • (-2,0,0,1,0,0,0)
    • (1,1,0,1,0,0,0)

    I would then compare your approach on various other pose configurations to see if it works.

     – Blender Ei Oct 17 '23 at 23:20
  • The positions could be dealt with using standard statistical approaches on $\mathbb{R}^3$. The orientations are more of a special case. There is a reasonable way to calculate the average of a bunch of unit quaternions (average in an average direction sense, not the component-wise average), which is generally regarded as not an easy problem; unfortunately, I don't remember offhand what it is. I do remember seeing a paper on this recently, out of the mathematical mainstream, and if I can find it, I'll post a link. – Brian Tung Oct 17 '23 at 23:31
  • Once you compute the average, use the inverse cosine formula above to calculate the difference between the average and each input quaternion, and then you can decide on the basis of those differences whether the collection is dispersed in orientation or not. – Brian Tung Oct 17 '23 at 23:32
  • I think Luke Hutchinson's answer to this question contains a link to the paper in question. It looks familiar. – Brian Tung Oct 17 '23 at 23:34

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