Photo-acoustic tomography in a rotating setting

Photo-acoustic tomography is a coupled-physics (hybrid) medical imaging modality that aims to reconstruct optical parameters in biological tissues from ultrasound measurements. As propagating light gets partially absorbed, the resulting thermal expansion generates minute ultrasonic signals (the photo-acoustic effect) that are measured at the boundary of a domain of interest. Standard inversion procedures first reconstruct the source of radiation by an inverse ultrasound (boundary) problem and second describe the optical parameters from internal information obtained in the first step. This paper considers the rotating experimental setting. Light emission and ultrasound measurements are fixed on a rotating gantry, resulting in a rotation-dependent source of ultrasound. The two-step procedure we just mentioned does not apply. Instead, we propose an inversion that directly aims to reconstruct the optical parameters quantitatively. The mapping from the unknown (absorption and diffusion) coefficients to the ultrasound measurement via the unknown ultrasound source is modeled as a composition of a pseudo-differential operator and a Fourier integral operator. We show that for appropriate choices of optical illuminations, the above composition is an elliptic Fourier integral operator. Under the assumption that the coefficients are unknown on a sufficiently small domain, we derive from this a (global) injectivity result (measurements uniquely characterize our coefficients) combined with an optimal stability estimate. The latter is the same as that obtained in the standard (non-rotating experimental) setting

Unique continuation for water waves and dispersive multiplier equations

We show that if a solution to the water wave equation, for an arbitrary short time interval, is flat on an open set and the horizontal fluid velocity at the surface is zero on the same open set, then the wave must vanish everywhere for all times. In addition, we use a result from non-harmonic Fourier analysis to show that (1 + 1d) linear dispersive PDE with Fourier multipliers also have this unique continuation property, subject to a natural asymptotic growth condition on the multiplier symbol.Comment: 12 pages, 1 figur

Feynman's inverse problem

We analyse an inverse problem for water waves posed by Richard Feynman in the BBC documentary Fun to Imagine. The problem can be modelled as an inverse Cauchy problem for gravity-capillary waves on a bounded domain. We do a detailed analysis of the Cauchy problem and give a uniqueness proof for the inverse problem. This results, somewhat surprisingly, in a positive answer to Feynman's question. In addition, we derive stability estimates for the inverse problem both for continuous and discrete measurements, propose a simple inversion method and conduct numerical experiments to verify our results

A metaphor called "Mozart"

In the following essay I shall venture on the ocean of metaphor, reducing the rashness of this project by the use of the well-worn boat of philosophy. I shall ask four questions: 1. Is it possible to realise metaphor through thought, action or emotion? 2. What is the opposite of metaphor? 3. Does an alternative to metaphorical thinking exist? 4. Does an alternative metaphorical thinking exist? However, should my project fail, perhaps the raft of metaphor itself might carry me safe to the island ruled by Ariel and Prospero, and where other shipwrecked once were met with soft music