Former DTU Compute student, Henriette Dyhr Rahbek, has been awarded the Danish Mathematical Society's Thesis Prize sponsored by Edlund A/S.
If you've heard of the Hubble Space Telescope, launched by NASA in 1990, you might recall that the initial images it transmitted back to Earth were blurry. The reason was a flaw in the telescope's primary mirror design.
The cause of this optical error was identified and rectified using a mathematical model known as phase retrieval. This technique enabled researchers to deduce the original appearance of the images and determine how the mirror needed to be adjusted for sharper results. Three years later, NASA replaced two instruments, correcting the telescope's flaw.
Phase retrieval, a process of mathematically reconstructing or refining data from images, microscope captures, CT scans, or sound waves, constitutes a vital field within mathematics. This is the area in which Henriette Dyhr Rahbek immersed herself while composing her thesis at DTU Compute. Her work is of such high quality that she has been honored with the Danish Mathematical Society’s Edlund Thesis Prize for 2022, awarded to the best mathematics thesis submitted by a student at a Danish university last year.
In the justification for granting the award to Henriette, the Danish Mathematical Society states: “The thesis is mathematically accurate, clear, detailed, and comprehensive, yet it remains pedagogical, employing examples and illustrations to engage the reader. This thorough study facilitates the understanding of modern theoretical aspects of the phase retrieval problem by others.”
Henriette Dyhr Rahbek was presented with the Edlund Thesis Prize of 15,000 DKK at a reception held at DTU Compute. She reciprocated with a brief lecture on her thesis titled 'Analytical and Numerical Treatment of the Phase Retrieval Problem,' which she submitted in January 2022.
"I am extremely proud and delighted that the selection committee highlighted my pedagogical approach and the way I presented the problem in a manner accessible to others with a similar background. I dedicated myself to crafting something understandable, including all the intermediate steps, so that readers could follow the process," says Henriette Dyhr Rahbek.
At DTU, Henriette Dyhr Rahbek pursued a bachelor’s degree in 'Mathematics and Technology' before completing her master’s degree in 'Mathematical Modelling and Computing.' Her advisor, Mirza Karamehmedovic, believes the award underscores the profound mathematical knowledge students can gain at DTU.
"At DTU, we explore and solve challenging technical-scientific problems, which demand advanced mathematical modeling, solution development, and characterization. Our outstanding thesis students at DTU Compute are often motivated by a technical application and are simultaneously captivated by the mathematical elegance underlying it. Hence, they can certainly earn the most esteemed mathematics award for students, even if they come from an engineering background," he remarks.
Much of the knowledge Henriette Dyhr Rahbek gained during her mathematics studies at DTU is applicable in her current role at Cerius-Radius in Virum. The company owns the power grid and ensures electricity supply to network customers across most of Zealand. While light and sound phases have been replaced with electrical phases, she employs data analysis and modeling for three-phase alternating current.
Brief on Phase Retrieval
The phase retrieval problem arises in various fields, including sound signal compression, microscopy of nanostructures, and troubleshooting complex optical systems. Intense global research is dedicated to this topic.
The problem involves estimating the phase of a complex signal given the amplitude of the signal's Fourier-transformed (or other linear transformation of the signal) and any supplementary information or prior knowledge about the signal.
Phase retrieval can be used to recover missing information from measurements of light, for instance. Light, characterized by frequency, intensity, phase, polarization, and propagation direction, can be described as waves. Optical sensors usually cannot measure the phase of light, leading to substantial loss of information about the objects or structures that emitted the measured light. Consequently, the so-called inverse problem—calculating these objects or structures from the measured light—becomes significantly more challenging.
Phase retrieval is a complex problem because it generally lacks a unique solution and even slight changes in the measured amplitude can profoundly impact the outcome. Through mathematical modeling and advanced solution methods, researchers can minimize the disparity between the original data and the computed phase.
In her thesis, Henriette Dyhr Rahbek delved into more theoretical aspects of mathematics and specifically addressed the issue of multiple solutions. She investigated the signal processing required to ensure that only a specific input signal could have produced a certain output signal.