Learning a projection operator onto the null space of a linear imaging operator

Joseph Kuo, Jason Granstedt, Umberto Villa, Mark A. Anastasio

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Image reconstruction algorithms seek to reconstruct a sought-after object from a collection of measurements. However, complete measurements such that an object can be uniquely reconstructed are seldom available. Analysis of the null components of the imaging system can guide both physical design of the imaging system and algorithmic design of reconstruction algorithms to more closely reconstruct the true object. Characterizing the null space of an imaging operator is a computationally demanding task. While computationally efficient methods have been proposed to iteratively estimate the null space components of a single or a small number of images, full characterization of the null space remains intractable for large images using existing methods. This work proposes a novel learning-based framework for constructing a null space projection operator of linear imaging operators utilizing an artificial neural network autoencoder. To illustrate the approach, a stylized 2D accelerated MRI reconstruction problem (for which an analytical representation of the null space is known) was considered. The proposed method was compared to state-of-the-art randomized linear algebra techniques in terms of accuracy, computational cost, and memory requirements. Numerical results show that the proposed framework achieves comparable or better accuracy than randomized singular value decomposition. It also has lower computational cost and memory requirements in many practical scenarios, such as when the dimension of the null space is small compared to the dimension of the object.

Original languageEnglish (US)
Title of host publicationMedical Imaging 2021
Subtitle of host publicationPhysics of Medical Imaging
EditorsHilde Bosmans, Wei Zhao, Lifeng Yu
ISBN (Electronic)9781510640191
StatePublished - 2021
EventMedical Imaging 2021: Physics of Medical Imaging - Virtual, Online, United States
Duration: Feb 15 2021Feb 19 2021

Publication series

NameProgress in Biomedical Optics and Imaging - Proceedings of SPIE
ISSN (Print)1605-7422


ConferenceMedical Imaging 2021: Physics of Medical Imaging
Country/TerritoryUnited States
CityVirtual, Online


  • Autoencoder
  • Inverse problem
  • Medical image reconstruction
  • Null space

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Atomic and Molecular Physics, and Optics
  • Radiology Nuclear Medicine and imaging
  • Biomaterials


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