Introduction

 Many functions in mathematics are reversible: for example, if you add seven to a number, you can then subtract seven to recover the original number. Other functions are not reversible: for example, if you multiply a number by zero, there is nothing you can do to recover the original value. In general, functions are only reversible if there is a one-to-one mapping between their inputs and outputs.

If I take a number and square it, I can then try and reverse the process by taking a square root. Unfortunately, however, an ambiguity exists over the original sign of the number. Since both positive and negative numbers have the same square, the squaring process is not reversible.

In this thesis, I am interested in taking square roots not of single numbers, but of multi-dimensional wavefields and operators. The wavefields are very large, containing tens of millions of pixels, and the operators represent wave-propagation through very complicated structures. However, a fundamental problem remains the same: when taking the square root of nine, do we choose three, or minus three?