<div dir="ltr">Hello,<div><br></div><div>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:</div>
<div><br></div><div>F = W * (I.J(t)) - (W * I).(W*J(t))</div><div><br></div><div>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'.</div>
<div><br></div><div>So, I try the following:</div><div><br></div><div>dF/dt = d/dt [(I . W*J(t)) - (W*I)(W*J(t))]</div><div><br></div><div>I can talk 'I' out as it can be treated as a constant. This gives (I think):</div>
<div><br></div><div>DF/dt = (I. W*J'(t)) - (W*J'(t)) . (W*I)</div><div><br></div><div>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?</div>
<div><br></div><div>Thanks for any help you can give me.</div><div><br></div><div>Anja </div></div>