A great tool

I’ve started using latex2wp to convert my lecture notes to wordpress format, and it is great.  I spend far less time futzing with wordpress, and I get nicer looking posts. I also get to write in standard LaTeX,  and can produce pdfs from the same document I use to produce blog posts. In that mythical future when I have free time, I may start seeing if I can tweak it or contribute a bit.

In the mean time, I hope to rewrite all of the earlier posts using this tool to get a more consistent look and an easier to manage set of source documents.  In other news: next week brings a new quarter, and a set of lectures on electrical impedance tomography.

Inverse Problems Course Notes — Numerical Methods for X-Ray Transform Reconstruction

These notes are based on Gunther Uhlmann’s lectures for MATH 581 taught at the University of Washington in Autumn 2009.

An index to all of the notes is available here.

The X-Ray transform is particularly interesting because we can use it for tomography. We can send X-Rays through a patient, over many different lines, and measure the accumulated attentuation. Once this is done, we know the X-Ray transform of our object, so we can simply pass this in to the inversion formula and reconstruct a three-dimensional model. There are two substantial problems with this:

  • We cannot collect data for every line, we can only sample a finite number of lines.
  • Our measurements will have error.

There are in fact even more complications — beam hardening comes to mind — but these two must be dealt with first. Loosely speaking, the solution to the first problem relies on the stability estimates for the X-Ray Transform, and the solution to the second problem depends on the range characterization. In this post we will discuss the first problem, finite sample size, in detail. The second problem is manageable too: the range characterization of the X-Ray Transform lets us find the image function most likely to have produced the error-filled data. It turns out that the filtered backprojection algorithm we introduce below is an important step in this process. The rest of this post is organized as follows: we will start by showing that for an infinite, discrete set of samples we have unique reconstruction. Then we will consider finite sets of samples and see that reconstruction is hopeless without a priori assumptions. From there we will introduce the Filtered Backprojection algorithm and state and Natterer’s theorem about filtered backprojection error estimates. Continue reading

Inverse Problems Course Notes — The Range of the X-Ray Transform

These notes are based on Gunther Uhlmann’s lectures for MATH 581 taught at the University of Washington in Autumn 2009.

An index to all of the notes is available here.

Now we are prepared to study the range of the X-Ray Transform. We will will look at the image of {\mathcal{S}({\mathbb R}^n)} in detail — once we understand this we can easily use our understanding of the Radon transform to characterize the image of {X} for smooth, compactly supported functions.

So let {f \in \mathcal{S}({\mathbb R}^n)}. Recall that

\displaystyle  Xf(x,\theta) = \int_{\mathbb R} f(x + t\theta) dt

It will be convenient to extend {Xf} to all of {{\mathbb R}^n\times({\mathbb R}^n - \{0\})} as follows

\displaystyle  \begin{array}{rcl}  Xf(x,\xi) &=& \int_{\mathbb R} f(x + t\xi) dt, \quad \xi \in {\mathbb R}^n - \{0\}) \\ \\ &=& \frac{1}{|\xi|}\int_{{\mathbb R}}f(x + t\frac{\xi}{|\xi}) dt \\ \\ &=& \frac{1}{|\xi|} \int_{\mathbb R} f(x - \langle x, \frac{\xi}{|\xi|}\rangle\frac{\xi}{|\xi|} + t\frac{\xi}{|\xi|}) dt \\ \\ &=& \frac{1}{|\xi|}Xf(x-\langle x, \frac{\xi}{|\xi|}\rangle\frac{\xi}{|\xi|}, \frac{\xi}{|\xi|}) \end{array}

So {Xf} is said to be positive homogeneous of degree -1 in {\xi}.

Claim 1 For {x \in {\mathbb R}^n}, {\xi \in {\mathbb R}^n - \{0\}} The partial differential equations

\displaystyle  \frac{\partial^2}{\partial x_i\partial \xi_j}Xf - \frac{\partial^2}{\partial x_j\partial \xi_i}Xf = 0

hold for all {1 \leq i, j \leq n}.

To show this, check the following

\displaystyle  \begin{array}{rcl}  \frac{\partial^2}{\partial x_i\partial \xi_j}Xf &=& \int_{\mathbb R} \frac{\partial^2}{\partial x_i\partial \xi_j} f(x + t\xi) dt \\ &=& t\int_{\mathbb R} \partial^2_{ij}f(x + t\xi) dt \end{array}

which is symmetric in {i} and {j}.

These equations are called John’s Equations, named for Fritz John who was the first to point this fact out. The main result is that these are the only conditions we need to characterize the range of {X} Continue reading