The laboratory specializes in developing advanced mathematical and computational methods for challenging inverse problems in signal processing, with particular focus on cryo-electron microscopy, phase retrieval, and computational imaging.

Key Capabilities:
> Advanced Cryo-Electron Microscopy Reconstruction: Developing mathematical methods to reconstruct 3D molecular structures from extremely noisy microscopy images. The laboratory’s breakthrough algorithms require significantly fewer images than traditional methods while achieving better accuracy in determining protein and molecular structures. These techniques enable scientists to visualize biological molecules that were previously too challenging to study.
> Phase Retrieval Theory: Creating mathematical frameworks to recover complete signal information when only partial measurements are available, with applications in X-ray crystallography and medical imaging. The laboratory establishes theoretical foundations that guarantee unique solutions to these challenging inverse problems. These advances enable better reconstruction methods across multiple imaging technologies.
> Computational Signal Processing Methods: Building advanced algorithms that solve complex signal recovery problems across different scientific domains. The laboratory develops optimization techniques that handle unknown rotations and transformations in data while providing mathematical guarantees for finding correct solutions. These general-purpose tools improve reconstruction quality in various imaging and signal processing applications.

Applications:
• Structural Biology: Revolutionary computational tools for determining three-dimensional molecular structures from cryo-EM data
• Medical Imaging: Advanced reconstruction algorithms for various imaging modalities where phase information is lost
• Computational Photography: Mathematical frameworks for super-resolution and image recovery problems

References and links:
1. Bendory et al. (2023) PNAS
2. Bendory and Edidin (2024) SIAM Journal on Mathematics of Data Science
3. Bendory et al. (2020) IEEE Signal Processing Magazine

Key algorithms:
1. Approx-EM-cryo-EM
Code for “A Stochastic Approximate Expectation-Maximization for Structure Determination Directly from Cryo-EM Micrographs” by Shay Kreymer, Amit Singer, and Tamir Bendory.
2. self_fourier_shell_correlation
An algorithm for computing the Fourier shell correlation from a single measurement.
3. CryoEMSignalEnhancement
Code to enhance the SNR of Cryo-EM images, using 2-D classification and expectation-maximization.
4. unrolling_synchronization
Unrolled algorithms for group synchronization – Python & TensorFlow implementation for experiments in synchronization over the group of 3-D rotations, including applications to cryo-EM.