Current Research Topics

Advanced Imaging Modalities
We focus on two separate modalities.

  1. Nanoscale Imaging using optical microscopy beyond the diffraction limit.  We have developed deep learning based algorithms to perform 3D nanoscale imaging using a fairly standard imaging setup with a spatial light modulator.  These algorithms allow for the 3D imaging of the inner workings of a cell - at the level of individual proteins - in high resolution.

  2. Full Sound-Speed Inversion Ultrasound.  While ordinary (b-mode) ultrasound imaging essentially captures the boundaries of objects, our new technique allows for the dense computation of material properties at each point in space.  Such an imaging modality effectively allows ultrasound to mimic aspects of much more expensive CT imaging (albeit while imaging different biophysical properties), and has broad medical applications.​

Automated Medicine

Our goal is to automate various aspects of detection and diagnosis of disease.  Artificial Intelligence offers the possibility of providing both detections and diagnoses which are more accurate and consistent than those provided by doctors.  This research is the core focus of the group of scientists I lead at Google Research; we have focused primarily on applications in endoscopy and minimally invasive surgery.  Separately, I have published academic research on the use of weak supervision, which will allow for better scaling (cheaper annotation) of AI-based medical efforts; as well as the integration of ordinary differential equation modelling into deep learning pipelines to improve EKG classification.

Physics and Deep Learning

This is a new and exciting area of research.  The idea is to use neural networks less in the traditional learning sense, and more as a function class with desirable properties from the Approximation Theory point of view.  We are working on several problems:

  1. Generation of desired quantum correlations between photon-pairs via design of  nonlinear  photonic crystals and pump beams.

  2. Solution of the Non-Linear Schrodinger Equation in optics.

  3. Ab-initio electronic structure computation for quantum chemistry.

  4. New techniques for solutions to PDEs, with applications to fusion.

We have recently started our journey in this area, with some results available and others forthcoming shortly.

Computer Vision and Artificial Intelligence

I continue to work in the more traditional fields of computer vision and artificial intelligence.  Recent work has focused on transfer learning, noisy labels, unsupervised learning, and image dehazing.  While I’m currently most interested in using deep learning techniques for the above mentioned applications (advanced imaging modalities, automated medicine, and quantum mechanics), I still retain an interest in the mathematical foundations and basic techniques of AI as well.

Older Research Topics

Computational Algebraic Topology

Algebraic topology is often considered to be one of the purer areas of mathematics, but in the last two decades researchers have considered both computational aspects of the field, as well as outright applications.  In previous work, we were interested in both.  On the computational side, we examined the problem of finding the optimal basis of a homology group, under various definitions of optimality; under some definitions, the problem admits a solution with low-degree polynomial complexity, while under other definitions it is NP-complete.  On the applications side, we incorporated persistent homology into various types of segmentation problems, based both on differential equations and Markov Random Fields.

Mathematical Techniques for Computer Vision

We were primarily interested in two areas here: geometric partial differential equations and combinatorial optimization.  While still fascinating from the mathematical point of view, these techniques seem to have been crushed underfoot by the deep learning juggernaut.