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Help with a certain derivation in a paper

  1. Dec 7, 2009 #1
    Hello everyone,

    I am currently reading a paper on medical imaging and it is talking about linear and higher order transformations in 3D.
    So, we can represent a transformation for each 3D position (say in a discrete image) as follows:

    T(x) = x + u(x)
    Here x is (x1, x2, x3) coordinates in 3D space and u(x) is the displacement field over the image. So basically, it just represents a displacement.

    And if T is an affine transformation, we can represent this as:
    T(X) = F(x) + t(x)

    where F is a linear transformation matrix and t is the translation bit.
    Now, the paper goes on to say that if we differentiate both these equations in terms of x, we can see that

    F = I + J(u) where J(u) is the Jacobian of the displacement field. I is the identity matrix.

    I am really struggling to see this derivation. I would be really grateful if someone could help me understand this...

    Many thanks,

  2. jcsd
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