Research

My research focuses on inverse problems and its imaging applications. In general, I am interested in developing robust computational methods and efficient algorithms for solving applied science problems, espcially with big data. Here are a few topics I have spent significant time on:

Figure: A blurred QR code obtained from the band limited reconstruction with incomplete discrete Fourier transform coefficients (left), with its exact binary reconstruction (right).

Reconstruction Algorithms for Binary Images

Can one accurately reconstruct an image with underdetermined information if the image is known to be binary (has only two different values). With incomplete access to the discrete Fourier transform coefficients, we have shown that there is indeed a super-resolution effect for binary images. However, even if it is known that there is a unique solution, finding the correct binary entry locations is an NP-hard problem. We have developed efficient combinatorial and nonconvex optimization algorithms to exactly reconstruct the original binary image. Improving these methods to allow for larger computational images is key ongoing work. These algorithms have applications to binary signal processing and recovering incomplete or blurred QR codes.

Figure: A classic compressed sensing problem: recover the sparse vector x from measurements y in this underdetermined system. What can we do when Φ is a nonlinear mapping? Or what if we have additional information about the structure of the sparsity of x?

Sparsity and Compressive Imaging

Using sparsity information often allows one to reduce the number of necessary data measurements to accurately reconstruct some unknown. I am interested in building upon these ideas by developing computational methods and designing algorithms to solve nonlinear compressive imaging problems. Can we reliably recover a sparse vector from (noisy) measurements that are sensed nonlinearly? How many measurements do we need? This is a general problem which has specific application to sparse inverse scattering problems. I am also interested in using additional information about the structure of the sparsity to improve algorithm convergence speed.

Figure: Model with localized fluorophores indicated as grey dots (left), with its inverse scattering problem reconstruction (right).

Fluorescence Microscopy Applications

Obtaining optical resolution past the limit of diffraction has been of great interest for centuries. In optical microscopy, diffraction limited resolution is capped at roughly 250 nanometers, whereas the cellular structures one wants to image can be as small as 10 nanometers. However, high-resolution images past the diffraction limit (so called “super-resolution”) can be obtained using photoactivated localization microscopy (PALM). In PALM, time-sequenced imaging of sparsely illuminated fluorophores allows for individual source localization with high precision, on the order of 10 nanometers. We have proposed the complementary use of an inverse scattering problem following an intial PALM experiment. This can produce high-resolution images of the entire surrounding medium, not just the locations of the fluorophores, all while potentially reducing the necessary amount of experimental work.

Figure: Diagram of an example inverse scattering problem. The goal is to reconstruct the function η inside Ω from measurements of the scattered field u_s outside of Ω.

Nonlinear Inverse Scattering Problems

While many imaging applications are based on linearizations of the scattered field, there are many interesting applications where such approximations are extremely limiting. An example of this is diffuse optical tomography, where strong multiple scattering occurs. Reconstructing images with multiple scattering is a difficult task, both mathematically and computationally. We developed an algorithm (the Data Compatible T-Matrix Completion algorithm) aimed at solving nonlinear inverse scattering problems in general, with a wide range of applications. A key feature of the algorithm is that its computational requirements are limited by the number of unknowns to be reconstructed, as opposed to the size of the data set. This is especially important in nonlinear (and linearized) scattering experiments, where many data measurements are needed to extract information.

Figure: General flowchart of the iterative algorithm to reconstruct source positions and the surrounding medium.

Source Localization and Super-resolution

Source localization typically relies on a linearization of the scattered field. It is possible to more accurately locate sources by adding nonlinear corrections, in the form of optical properties of the surrounding medium. If one knew how light scattered through the medium, one would have a better understanding of how light would propagate from a specific source, and could retrace measurements back to the source. The main issue is that one rarely knows these internal properties. To address this issue, we propose an iterative optimization algorithm to recover both the source locations and the surrounding medium, all at subwavelength resolution. From an initial source localization (say, from PALM), one can iteratively solve an inverse scattering problem to both reconstruct the medium and update the source locations.