Determining the shape of a human vocal tract from pressure measurements at the lips

TL;DR

This paper investigates the inverse problem of reconstructing the human vocal tract shape from pressure measurements at the lips, revealing conditions for uniqueness and nonuniqueness based on lip curvature.

Contribution

It provides a mathematical framework for determining vocal-tract shape from pressure data, including criteria for uniqueness and multiple candidate solutions.

Findings

Nonuniqueness arises when the upper lip curves downward, with up to M solutions.

Unique solutions are guaranteed if the upper lip is horizontal or curves upward.

The theory is supported by illustrative examples.

Abstract

The inverse problem of determining the cross-sectional area of a human vocal tract during the utterance of a vowel is considered in terms of the data consisting of the absolute value of sound pressure at the lips. If the upper lip is curved downward during the utterance, it is shown that there may be up to an $M$-fold nonuniqueness in the determination, where $M$ is the maximal number of eligible resonances associated with a related Schr\"odinger operator. Each of the $M$ such distinct candidates for the vocal-tract area corresponding to the same absolute pressure is uniquely determined. The mathematical theory is presented for the recovery of each candidate for the vocal-tract area, and the admissibility criterion for each of the $M$ candidates to be a vocal-tract radius is specified. On the other hand, if the upper lip is horizontal or curved upward during the utterance, then the inverse problem has a unique solution. The theory developed is illustrated with some examples.