We often take it for granted that we compute with numbers, such that the area of a rectangle is its base times height: \(A = b \cdot h\) .
In practical settings, we don't know exactly what b and h really is - but we may know that they are bounded within some interval, e.g.: \(b^I = [b^L, b^U]\), where L and U, denote the upper and lower bounds of the b. If so, how do we determine the lower and upper bound for A? Questions like this is what interval analysis can help us with, but it can be extended to answer much more interesting questions in beamforming aswell.
Interval analysis has been used for bounding the beampattern of ultrasound arrays with element position errors, and for the localization of underwater acoustic sources. The following examples are to provide an understanding of the nature of this field and not to constrain the topic of the thesis:
"The framework of interval arithmetic (IA) and its extension to complex numbers has in the last decade been applied as a tool for finding robust tolerance bounds of antenna arrays. The IA framework complements statistical methods, as inclusive upper and lower bounds of the beampattern are obtained directly, only assuming error bounds on specifically chosen array parameters. [...] For this purpose, we have created an open-source MATLAB toolbox to calculate beampattern bounds for an array with bounded error tolerances. The toolbox features an object oriented library of interval classes and an interactive graphical user interface with easily configurable settings, where results for different interval representations are shown along with their corresponding bounds." (Arnestad et al.: Sonar array beampattern bounds: tolerance analysis using interval arithmetic, ICUA 2022, Southampton)
"The proposed approach to solve this problem is to use set theory. The method is to use simulated acoustic amplitude to solve the source localization problem. Using interval analysis, all possible source positions compatible with the recorded hydrophone signal can be enclosed. In addition, possible sets for source position compatible with each receiver can be intersected to increase the certainty of the source location. Besides requiring a good knowledge of the scene, this method requires simulating the acoustic propagation as well as possible to correctly solve this localization problem." (Brateau et al.: Acoustic source localization in underwater environment using interval analysis, ICUA 2022, Southampton)
In the project, you will do the following:
- Study the literature on interval analysis and understand its applications.
- Choose an application within ultrasound imaging such as
- Broadband array response tolerance analysis
- Beamformed image estimation using IA
- Effects of array errors on adaptive beamforming
- Guided elastic waves (mode tracing)
- Design and perform simulations to validate your interval computations
- Contribute to the Matlab based Beampattern Interval Analysis Toolbox
Qualifications:
- Excellent MATLAB programming skills (Object oriented MATLAB)
- Git Version Control
- Signal Processing (e.g. IN3015/4015, IN3190/4190, IN5450)
- Interest in applied mathematics (e.g. geometric algebra, algebra of sets)
Literature:
- [1] H. K. Arnestad, G. Geréb, T. I. B. Lønmo, J. E. Kirkebø, A. Austeng, and S. P. Näsholm, Sonar array beampattern bounds and an interval arithmetic toolbox, Proc. Mtgs. Acoust., vol. 47, no. 1, p. 055002, Jun. 2022, doi: 10.1121/2.0001613.