fitting 3d model on 2D image

Hello, I am developing an algorithm to align to recover the 3D pose of a head of person. The 3D head is modeled by an ellipsoid and the 2D head is modeled by an ellipse To recover the pose, I project the ellipsoid which gives us an ellipse and I try to align the two ellipses by changing the 3D position of the ellipsoid. So the aim is to obtain the optimal transformation T=[thetax,thetay,thetaz,tx,ty,tz] (rotation+translation) of the ellipsoid that align the two ellipses. So the problem can be seen as an optimization problem that I will use the gradient descent or gauss newton or....to resolve it. I note that I have the parameter of projection As a cost function I need to use the intersection of the two image. My problem is how to write the cost function. Cost function= -(I +P(T(ellipsoid)) )/2 where I the ellipse binary image given by the camera and P is the projection. My problem is how to transform each pixel in the second image as a function of T. Thanks?