[Insight-users] PASHA algorithm in ITK

[Anja Ende]

Hello,

I have been trying to understand the PASHA algorithm from INRIA and there
are some local cross correlation filters in ITK that implement it. I was
going through the paper and trying to understand the gradient expression
for LNCC. In particular, we have a function of the following form:

F = W * (I.J(t)) - (W * I).(W*J(t))

where I and J are two images. J depends on some transformation parameters t
and W is a gaussian kernel with some fixed standard deviation and zero
mean. * represents the convolution operator. Now, I want to compute the
derivative of F wrt to the transformation parameters 't'.

So, I try the following:

dF/dt  = d/dt [(I . W*J(t)) - (W*I)(W*J(t))]

I can talk 'I' out as it can be treated as a constant. This gives (I think):

DF/dt = (I. W*J'(t)) - (W*J'(t)) . (W*I)

Can I treat the convolution operators this way or is this wrong? I ask
because this expression is a bit different from the one in the paper and I
am sure their version is correct. But I am not able to convince myself as
to what I have done wrong here. The convolution kernels are fixed width
Gaussians and do not depend on the parameters 't'. Perhaps I am treating
the convolution operators incorrectly in the gradient expression?

Thanks for any help you can give me.

Anja
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