[Insight-users] Q on Jacobian in "Insight into Images: Principlesand Practice for Segmentation, Registration, and Image Analysis"

[Andreas Keil]

Hi [Pixel.to.life],

I don't know the book but from your quote it's clear that you found a
typo. The Jacobian is always the matrix of first derivatives of a
vector-valued function:

    J_ij = d f_i / d x_j  (where "d" indicates partial derivatives)

(In case of a scalar-valued function this would reduce to the gradient
vector.)

And as you already said, the Hessian is the matrix of second derivatives
in case of scalar-valued functions:

    H_ij = d^2 f / d x_i d x_j

(In the case of vector-valued functions this would get three-dimensional -
a so-called tensor, I think.)

Therefore, I totally agree with you and you should report the typo if
possible.

Regards,
Andreas.


-----Original Message-----
From: insight-users-bounces+andreas.keil=cs.tum.edu at itk.org
[mailto:insight-users-bounces+andreas.keil=cs.tum.edu at itk.org] On Behalf
Of Pixel Life
Sent: Wednesday, April 25, 2007 06:42
To: insight-users at itk.org
Subject: [Insight-users] Q on Jacobian in "Insight into Images:
Principlesand Practice for Segmentation, Registration, and Image Analysis"

Hi, 

I am looking for clarification on a topic published in the book "Insight
into Images: Principles and Practice for Segmentation, Registration, and
Image Analysis". I use it in conjunction with itk examples. (if the
question is not relevant here, please advise). 

>From page 31, page 2.4.8, I quote: : "...Where the first derivative
(gradient) is represented as a vector, the second derivative is a matrix,
known as the Jacobian.."

However, on page 249, Jacobian is presented as a matrix of first order
partial derivatives.

Even in all the literature I can find, the term 'Jacobian' is used to
represent a matrix of first order partial derivatives. I did see 'Hessian'
as a representation of 2nd order partial derivatives in a matrix form. 

The question is: am I missing some information here (e.g. the
terminology?), or the book on page 31 has a typo?

Thank you for the help.
 
[Pixel.to.life]