geometry - Average and Measure of Spread of 3D Rotations -

i've seen several similar questions, , have ideas of might try, don't remember seeing spread.

so: working on measurement system, computer vision based.

i take n captures, , process them using library outputs pose estimations in form of 4x4 affine transformation matrices of translation , rotation.

there's noise in these pose estimations. standard deviation in euler angles each axis of rotation less 2.5 degrees, orientations pretty close each other (for case euler angles close 0 or 180). standard errors of less 0.25 degrees important me. have run problems endemic euler angles.

i want average these pretty-close-together pose estimates single final pose estimate. , want find measure of spread can estimate accuracy.

i'm aware "average" isn't defined rotations.

(for record, code in numpy-heavy python.)

i may want weight average, since captures (and axes) known more accurate others.

my impression can take mean , standard deviation of translation vector, , rotation can convert quaternions, take mean, , re-normalize ok accuracy since these quaternions pretty close together.

i've heard mentions of least-squares across quaternions, of research how implemented has been dismal failure.

is workable? there reasonably well-defined measure of spread in context?

without more info geometry setup hard answer. anyway rotations would:

1. create 3 unit vectors

   • x=(1,0,0),y=(0,1,0),z=(0,0,1)
   • apply rotation on them , call output x(i),y(i),z(i)
   • it applying matrix(i) position @ (0,0,0)

2. do measurements have

3. now average vectors

   • x=avg(x(1),x(2),...x(n))
   • y=avg(y(1),y(2),...y(n))
   • z=avg(z(1),z(2),...z(n))

4. correct vector values

   • so make each of x,y,z unit vectors again
   • take axis more closest rotation axis main axis
   • it stay is
   • and recompute remaining 2 axises cross product of main axis , other vector
   • to ensure orthogonality
   • beware of multiplication order (wrong order of operands negate output)

5. construct averaged transform matrix

   • see transform matrix anatomy
   • as origin can use averaged origin of measurement matrices