About

Albert Fannjiang is a Professor of Mathematics at the University of California, Davis. He earned his Ph.D. in Applied Mathematics from the Courant Institute at New York University in 1992. His research focuses on applied mathematics with particular emphasis on wave propagation, imaging, and transport phenomena in complex and turbulent media. Over the years, he has been the principal investigator or co-principal investigator on multiple National Science Foundation grants and Defense Advanced Research Projects Agency projects, addressing topics such as propagation, focusing, and imaging in complex media, transport and wave propagation in turbulence, and scalar and wave transport in random flows. Fannjiang has also contributed to the mathematical sciences computing research environments and diffusion in fluids. He has been recognized as a Centennial Fellow by the American Mathematical Society and served as a Chancellor's Fellow at UC Davis. Since 2007, he has been an associate editor for the journal Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal and, since 2012, for Mathematics & Mechanics of Complex Systems (MEMOCS). His expertise and contributions have led to numerous invited talks and lectures at international conferences and workshops worldwide, reflecting his active engagement in advancing the fields of applied mathematics and mathematical modeling in complex systems.

Research topics

• Artificial Intelligence
• Computer Science
• Mathematics
• Data Mining
• Mathematical analysis
• Algorithm
• Physics
• Applied mathematics
• Quantum mechanics
• Optics
• Discrete mathematics
• Statistics
• Combinatorics

Selected publications

• Structured Approximation of Toeplitz Matrices and Subspaces

  ArXiv.org · 2025-11-21

  preprintOpen access1st authorCorresponding

  This paper studies two structured approximation problems: (1) Recovering a corrupted low-rank Toeplitz matrix and (2) recovering the range of a Fourier matrix from a single observation. Both problems are computationally challenging because the structural constraints are difficult to enforce directly. We show that both tasks can be solved efficiently and optimally by applying the Gradient-MUSIC algorithm for spectral estimation. For a rank $r$ Toeplitz matrix ${\boldsymbol T}\in {\mathbb C}^{n\times n}$ that satisfies a regularity assumption and is corrupted by an arbitrary ${\boldsymbol E}\in {\mathbb C}^{n\times n}$ such that $\|{\boldsymbol E}\|_2\leq αn$, our algorithm outputs a Toeplitz matrix $\widehat{\boldsymbol T}$ of rank exactly $r$ such that $\|{\boldsymbol T}-\widehat{\boldsymbol T}\|_2 \leq C \sqrt r \, \|{\boldsymbol E}\|_2$, where $C,α>0$ are absolute constants. This performance guarantee is minimax optimal in $n$ and $\|{\boldsymbol E}\|_2$. We derive optimal results for the second problem as well. Our analysis provides quantitative connections between these two problems and spectral estimation. Our results are equally applicable to Hankel matrices with superficial modifications.

  PublisherOA PDFDOI

• Noise-robust one-bit diffraction tomography and optimal dose fractionation

  Inverse Problems · 2025-06-30

  articleOpen accessSenior author

  Abstract This study presents a noise-robust framework for 1-bit diffraction tomography, a novel imaging approach that relies on intensity-only binary measurements obtained through coded apertures. The proposed reconstruction scheme leverages random matrix theory and iterative algorithms to effectively recover 3D object structures under high-noise conditions. A key contribution is the numerical investigation of dose fractionation, revealing optimal performance at a signal-to-noise ratio near 1, independent of the total dose . This finding addresses the question: how to distribute a given level of total radiation energy among different tomographic views in order to optimize the quality of reconstruction?

  PublisherDOI

• Noise-Robust One-Bit Diffraction Tomography and Optimal Dose Fractionation

  arXiv (Cornell University) · 2023-10-09

  preprintOpen accessSenior author

  This study presents a noise-robust framework for 1-bit diffraction tomography, a novel imaging approach that relies on intensity-only binary measurements obtained through coded apertures. The proposed reconstruction scheme leverages random matrix theory and iterative algorithms to effectively recover 3D object structures under high-noise conditions. A key contribution is the numerical investigation of dose fractionation, revealing optimal performance at a signal-to-noise ratio near 1, {\em independent of the total dose}. This finding addresses the question: How to distribute a given level of total radiation energy among different tomographic views in order to optimize the quality of reconstruction?

  PublisherOA PDFDOI

• 3D tomographic phase retrieval and unwrapping

  Inverse Problems · 2023-12-02 · 1 citations

  articleOpen access1st authorCorresponding

  Abstract This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) Unique determination of (phase) projections from diffraction patterns —General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) Uniqueness for 3D phase unwrapping —General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) Uniqueness for projection tomography —Unique determination of an object of n 3 voxels from generic n projections or n + 1 coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.

  PublisherDOI

• 3D Tomographic Phase Retrieval and Unwrapping

  arXiv (Cornell University) · 2022-08-09

  preprintOpen access1st authorCorresponding

  This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) {\em Unique determination of (phase) projections from diffraction patterns} -- General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) {\em Uniqueness for 3D phase unwrapping} -- General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) {\em Uniqueness for projection tomography} -- Unique determination of an object of $n^3$ voxels from generic $n$ projections or $n+1$ coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.

  PublisherOA PDFDOI

• Uniqueness theorems for tomographic phase retrieval with few coded diffraction patterns

  Inverse Problems · 2022-06-10 · 4 citations

  articleOpen access1st authorCorresponding

  Abstract 3D tomographic phase retrieval under the Born approximation for discrete objects supported on a n × n × n grid is analyzed. It is proved that n projections are sufficient and necessary for unique determination by computed tomography with full projected field measurements and that n + 1 coded projected diffraction patterns are sufficient for unique determination, up to a global phase factor, in tomographic phase retrieval. Hence n + 1 is nearly, if not exactly, the minimum number of diffractions patterns needed for 3D tomographic phase retrieval under the Born approximation.

  PublisherOA PDFDOI

• Super-Resolution Limit of the ESPRIT Algorithm

  IEEE Transactions on Information Theory · 2020 · 107 citations

  Senior authorCorresponding
  • Computer Science
  • Artificial Intelligence
  • Algorithm

  The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like SRF <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2λ-2</sup> ×Noise, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and λ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of M in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments.

  PublisherDOI

• Fixed Point Analysis of Douglas--Rachford Splitting for Ptychography and Phase Retrieval

  SIAM Journal on Imaging Sciences · 2020 · 21 citations

  1st authorCorresponding
  • Mathematics
  • Algorithm
  • Applied mathematics

  Douglas--Rachford splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas--Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Fixed point analysis also leads to a simple, explicit expression for the optimal step size in terms of the spectral gap of an underlying matrix. When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, alternating minimization with the DRS inner loops, even with a far from optimal step size, converges geometrically under the nearly minimum conditions established in the uniqueness theory.

  PublisherDOI

• Fixed Point Analysis of Douglas-Rachford Splitting for Ptychography and Phase Retrieval

  arXiv (Cornell University) · 2019-09-18 · 1 citations

  preprintOpen access1st authorCorresponding

  Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Fixed point analysis also leads to a simple, explicit expression for the optimal step size in terms of the spectral gap of an underlying matrix. When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, Alternating Minimization with the DRS inner loops, even with a far from optimal step size, converges geometrically under the nearly minimum conditions established in the uniqueness theory.

  PublisherOA PDFDOI

• Raster Grid Pathology and the Cure

  Multiscale Modeling and Simulation · 2019-01-01

  preprintOpen access1st authorCorresponding

  Blind ptychography is a phase retrieval method using multiple coded diffraction patterns from different, overlapping parts of the unknown extended object illuminated with an unknown window function. The window function is also known as the probe in the optics literature. As such blind ptychography is an inverse problem of simultaneous recovery of the object and the window function given the intensities of the windowed Fourier transform and has a multiscale set-up in which the probe has an intermediate scale between the pixel scale and the macroscale of the extended object. The uniqueness problem for blind ptychography is analyzed rigorously for the raster scan (of a constant step size $\tau$) and its perturbations. The block phases are shown to form an arithmetic progression and the complete characterization of the raster scan ambiguities is given, including, first, the periodic raster grid pathology of degrees of freedom proportional to $\tau^{2}$, and, second, a nonperiodic, arithmetically progressing phase shift from block to block. Finally, irregularly perturbed raster scans are shown to remove all ambiguities other than the inherent ambiguities of the scaling factor and the affine phase factor under general requirements, including roughly the minimum overlap ratio 50%.

  PublisherOA PDFDOI

Recent grants

• Theory and Computation of Mask-Aided Phase Retrieval

  NSF · $213k · 2014–2018

• Propagation, Focusing and Imaging in Complex Media

  NSF · $195k · 2009–2012

• Transport and Wave Propagation in Turbulence

  NSF · $109k · 2003–2007

Frequent coauthors

• Tomasz Komorowski

  25 shared

• Wenjing Liao

  14 shared

• Pengwen Chen

  10 shared

• Lech Wołowski

  8 shared

• Gláucio H. Paulino

  8 shared

• Knut Sølna

  7 shared

• Youn-Sha Chan

  University of Houston - Downtown

  7 shared

• Stéphane Nonnenmacher

  Université Paris-Saclay

  6 shared