# Help with a certain derivation in a paper

• pamparana
In summary: I hope this helps you in your understanding of linear and higher order transformations in 3D medical imaging.In summary, a transformation in 3D space can be represented as T(x) = x + u(x), where x is the original position and u(x) is the displacement field. For affine transformations, this can be represented as T(X) = F(x) + t(x), where F is the linear transformation matrix and t is the translation component. By taking the derivative of both equations with respect to x, we can see that F = I + J(u), where I is the identity matrix and J(u) is the Jacobian of the displacement field. This means that for affine
pamparana
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,

Luc

y

Hello Lucy,

Thank you for sharing your question with us. As a scientist with expertise in medical imaging, I am happy to assist you with understanding the derivation of the equations you mentioned.

First, let's define some terms. A transformation in 3D space is a mathematical operation that maps one set of coordinates to another set of coordinates. In the context of medical imaging, this can be used to describe the movement of structures within the body, such as organs or tissues, or the movement of the imaging equipment itself.

Now, let's look at the equation T(x) = x + u(x). This equation represents a transformation where the new position (T(x)) is equal to the original position (x) plus the displacement field (u(x)). The displacement field represents the distance and direction of movement for each point in the image.

Next, we have the equation T(X) = F(x) + t(x). This equation represents an affine transformation, which is a type of transformation that preserves straight lines and parallelism. In this equation, F represents the linear transformation matrix, which describes how the image is rotated, scaled, and sheared. t represents the translation component of the transformation, which describes the movement of the image as a whole.

To understand the derivation of F = I + J(u), we need to take the derivative of both equations with respect to x. For T(x) = x + u(x), the derivative is simply 1, since the displacement field u(x) is constant. For T(X) = F(x) + t(x), the derivative is F. This is because the translation component t(x) is constant, and the derivative of F(x) is F itself.

Now, let's combine the two equations by setting them equal to each other:

1 = F

We can see that F is equal to the identity matrix (I), which is a matrix with 1's on the diagonal and 0's elsewhere. This makes sense because the identity matrix represents no rotation, scaling, or shearing.

Finally, we can substitute this value for F into the equation T(X) = F(x) + t(x). This gives us T(X) = I(x) + t(x). But we know that I(x) is just x, so we can rewrite this as T(X) = x + t(x).

This is the same equation we started with, T(x) = x + u(x), but now we have a

## What is a derivation in a paper?

A derivation in a paper refers to the process of logically deriving or explaining a certain conclusion or result. It typically involves using established principles, theories, and data to support a new argument or claim.

## Why is a derivation important in a scientific paper?

A derivation is important in a scientific paper because it adds credibility to the findings and conclusions presented. It shows that the author has followed a logical and systematic approach in their research and can support their claims with evidence.

## How do I know if my derivation is correct?

There are a few ways to check the correctness of a derivation. First, make sure all the steps are clearly explained and logically connected. You can also review any assumptions made and check if they are valid. Lastly, double check your calculations and make sure they are accurate.

## What should I do if I am stuck on a derivation in a paper?

If you are stuck on a derivation in a paper, try breaking it down into smaller steps and reviewing any relevant background information or equations. You can also consult with colleagues or the author for clarification. It may also be helpful to take a break and come back to it with a fresh perspective.

## How much detail should I include in a derivation in a paper?

The level of detail in a derivation may vary depending on the specific paper and audience. However, it is important to provide enough information for the reader to follow your reasoning and understand the derivation. This may include relevant equations, assumptions, and calculations.

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