Can Diffeomorphisms be Represented by Matrices and Used in Image Analysis?

[garrus]

I'm a complete rookie here, and i'd like some help.
For starters , can a diffeomorphic mapping be represented via a matrix , like say a transformation?
If so, how would it be parameterised?

 

[dx]

No you cannot represent it by a matrix, since it is not a linear transformation.

 

[dx]

A diffeomorphism is a linear transformation of all the tangent spaces, so you can give the infinite collection of matrices

∂Xⁱ/∂Y^j

at each point of the manifold

 

[WannabeNewton]

Hi garrus. dx already answered the crux of your question but let's specialize to the case of $\mathbb{R}^{n}$. Let $U\subseteq \mathbb{R}^{n}$ be open, $p\in U$, and $F:U\rightarrow \mathbb{R}^{m}$ a map differentiable at $p$. Recall that $F$ is differentiable at $p$ if there exists a linear map $DF(p)$ such that $\lim _{v\rightarrow 0}\frac{|F(p + v) - F(p) - DF(p)v|}{|v|} = 0$. As you may remember, we call $DF(p)$ the total derivative of $F$ at $p$. Now, as dx noted we may not be able to represent $F$ itself as a matrix if it isn't itself linear on $\mathbb{R}^{n}$ (in which case it agrees with its total derivative) but $DF(p)$ is linear and one can show that the standard matrix representation of $DF(p)$ is given by $[DF(p)]_{S} = (\frac{\partial F^{j}}{\partial x^{i}}(p))$. This is none other than the Jacobian matrix. You can think of the linear map $DF(p)$ as being the best linear approximation of $F$ in a neighborhood of $p$. As dx noted above, you can then develop such a formalism on arbitrary smooth manifolds.

 

[garrus]

Thanks for your responses, but i think I'm way out of my league :/
I want to apply a diffeomorphism in image analysis and I'm looking for a way to build a function to map pixel positions.

edit: disregard that.