[Insight-users] Question about workings of ExponentialDeformationFieldImageFilter

[Tom Vercauteren]

On Fri, May 20, 2011 at 09:04, Anja Ende <anja.ende at googlemail.com> wrote:
>> This addition is need to compute the composition. Let u, v and w be
>> displacement fields, we want w such that
>>  Id+w = (Id + u) o (Id + v)
>>          = Id + v + u_warped_by_v
>>
>> meaning
>>  w = v + u_warped_by_v
>
> Thanks for this Tom. How is this coming about though?
>
> By distributive law, we have:
>
> (Id+u)o(Id+v) = (Id+u)o(Id) + (Id+u)o(v)
>
> I am guessing the second bit generates the u_warped_by_v bit. How does
> (Id+u)o(Id) results in Id.

The composition and addition are not distributive. Try to apply the
composed transformation to a point p:
(Id + u) o (Id + v)(p) = (Id + u)(p+v(p))
                            = p+v(p)+u(v(p))
                            = (Id + v + u_warped_by_v)(p)

Tom