[Insight-users] Quaternions and Versors

[Rupert Brooks]

Hi Tom,  Luis,

This is an interesting conversation.  It led me to spend the morning
scribbling quaternions in my notebook :-)

I worked through a lot of the QuaternionsII.pdf, and theres one thing
left that troubles me.  Versor multiplication (ie composition of 3D
rotations) is non-commutative.  In the notes there is
q_(n+1)=(p/q_(n))^a * q_(n)

where q_(n+1) and q_(n) are the new, and the current versors, and p/q
is the update step.  This agrees with my intuition.  We have a current
position in the parameter space, q, and now we are taking an
additional step.  However, in the code there is

//
  // Composing the currentRotation with the gradientRotation
  // produces the new Rotation versor
  //
  VersorType newRotation = currentRotation * gradientRotation;

To me, this composition seems to be in the wrong order.  (I did a
quick check, and the versors compose in the same order their assocated
rotation matrices do.)  Did i miss something?

> In order to minimize with respect to q, we form the differential
> according to all the rules in Hamilton's book:
>
>   dfq =  df(q,dq) = dq . v . q^(-1)  -  q . v . q^(-1) . dq . q^(-1)
>
> So how do I get from this expression to the Jacobian implemented in the
> itk::VersorTransform?

I believe you can get there by expressing the fourth component of the
quaternion as a function of the first three, writing out the matrix
representation, and taking the derivative in the usual way.  I did
this once, and i convinced myself that the Jacobian was correct.  This
considers the function M(q) which generates the rotation matrix from
the quaternion q.  Then you can write

Xnew=M(q) x Xold+T

M(q) can be written in closed form, and the derivative taken in the
usual way.  However, you could write

Xnew=M(q) x M(dq) x Xold +T

and take the derivative with respect to dq.  This would be truly a
compositional step - and because the derivative of dq would be taken
around zero, it would be valid everywhere.  (At zero rotation, the w
component is at a maximum, so its derivative is zero - only the x,y,z
components will have derivative values)

>> A.3) "Is there any literature on the subject from this century,
>>       preferably focusing on the given problem of image registration?"
>>
>>
>>      Modern books on Quaternions are actually of very inferior
>>      quality to Hamilton's one. They are more focused on the
>>      notation details, and are usually handicaped by the obsesive
>>      compulsion for describing quaterions as "even harder than
>>      complex numbers".

I have not read Hamiltons original work, so i cannot comment, but i
found Andrew Hansens "Visualizing Quaternions" to be a helpful book.

Another way to think about it is that the space of versors / 3D
rotations is a Lie group.  The derivative at a point is an element of
its tangent space - that is, it is a logarithm of a rotation.  To move
along a geodesic of the group, we move along the exponential map
induced by this tangent.  This is another way to get to the
exponentiation in the update step.

q_(n+1)=dq^a . q_(n)

There is definitely work out there on optimization on Lie groups.  I'm
no expert, so i'll stop writing here before i embarrass myself.

Anyway, coming back to the practical side, if your angles that you are
optimizing over do not approach 90 degrees, you can quite happily use
Euler angles with additive optimizers.  Its an easy way to get a
general additive optimizer to work on the problem, without worrying
about compositional update steps.  However, if your problem does
approach the area where Euler angles break down - watch out.

Alternatively, like you already said, if you are using similarity
transforms, then you can use a quaternion transform with an additive
optimizer.

Cheers,
rupert
-- 
--------------------------------------------------------------
Rupert Brooks
McGill Centre for Intelligent Machines (www.cim.mcgill.ca)
Ph.D Student, Electrical and Computer Engineering
http://www.cyberus.ca/~rbrooks