Best representation/model of a 3D object from multiple observations

I have a sensor that detects the corners of a 3D object. There are 9 corners for that rigid object.

The goal is to create the most accurate representation of that 3D object, which can be defined as a polyhedron with 9 corners.

Lets call these corners as c1, c2, c3, c4, c5, c6, c7,c8 and c9.

Each observation from the sensor can detect either 3 or 6 corners with a noise of few millimeters. And each corner detection from sensor has a 3D position (x,y,z) in the sensor coordinate frame. The object is rotated so that the sensor can detect many observations with 3 or 6 corners to cover the total of 9 corners. We have captured a few hundred observations.

Goal: To estimate the polyhedron model with 9 corners from sensor data

Problem formulation till now:
As the object can move between observations, I cannot take the raw sensor detections of corners. The distance between a pair of corners is meaningful. Hence, from a sensor observation of 3 corners (c1, c2, c3), we can create distance pairs (D12, D23, D13), where D12 is distance between c1 and c2.

Required Output:
Now, for the final result of estimating polyhedron with 9 corners, we need to estimate eight distance pairs, namely D12, D13, D14, D15, D16, D17, D18 and D19. From resulting eight distance pairs, fixing the corner c1 at origin, will give the 3D model of the object.

Available Input:
The inputs to the system are basically every combination pair of 9 corners, which will be 36 pairs, such as D12, D12, D14, D15 — D23, D24, D25, D26 — D34, D35, D36 —- and so on till D78, D79, D89. And we have many such observations.

Question:

Can anyone guide me through the process of accurately estimating those eight distances, from the 36 distances ?

The simplest way is to choose those eight distance pairs and take the mean of all observations

Can we do better than this ??

Thank you once again for your time and help 🙂