Award Abstract # 1418772
Collaborative Research: Wavelet Frames for Variational Models in Imaging: Bridging Discrete and Continuum

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF ARIZONA
Initial Amendment Date: July 25, 2014
Latest Amendment Date: August 12, 2016
Award Number: 1418772
Award Instrument: Continuing Grant
Program Manager: Leland Jameson
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: August 1, 2014
End Date: July 31, 2017 (Estimated)
Total Intended Award Amount: $170,832.00
Total Awarded Amount to Date: $170,832.00
Funds Obligated to Date: FY 2014 = $54,454.00
FY 2015 = $62,350.00

FY 2016 = $54,028.00
History of Investigator:
  • Moysey Brio (Principal Investigator)
    brio@math.arizona.edu
  • Leonid Kunyansky (Co-Principal Investigator)
  • Kenneth T-R McLaughlin (Former Principal Investigator)
  • Bin Dong (Former Principal Investigator)
Recipient Sponsored Research Office: University of Arizona
845 N PARK AVE RM 538
TUCSON
AZ  US  85721
(520)626-6000
Sponsor Congressional District: 07
Primary Place of Performance: University of Arizona/Department of Mathematics
617 N. Santa Rita Ave.
Tucson
AZ  US  85721-0089
Primary Place of Performance
Congressional District:
07
Unique Entity Identifier (UEI): ED44Y3W6P7B9
Parent UEI:
NSF Program(s): COMPUTATIONAL MATHEMATICS
Primary Program Source: 01001415DB NSF RESEARCH & RELATED ACTIVIT
01001516DB NSF RESEARCH & RELATED ACTIVIT

01001617DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 9263
Program Element Code(s): 127100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

From the beginning of science, visual observations have been playing important roles. Advances in computer technology have made it possible to apply some of the most sophisticated developments in mathematics and the sciences to the design and implementation of fast algorithms running on a large number of processors to process image data. As a result, image processing and analysis techniques are now applied to virtually all natural sciences and technical disciplines ranging from computer sciences and electronic engineering to biology and medical sciences; and digital images have come into everyone's life. Mathematics has been playing an important role in image and signal processing from the very beginning. There are two major mathematical approaches for image restoration, namely, wavelet tight frame approaches and differential/variational approaches. The main research objective of this project is to investigate geometric aspects of the former approach by connecting it with the latter. It will give rise to new mathematical models and numerical algorithms that benefit researchers in academia, national research laboratories, as well as in industry. The understandings of the geometric aspects of the wavelet frames and the connections with differential operators will contribute to both the community of computational harmonic analysis and the community of variational techniques and numerical PDEs. The education plan will bring undergraduate and graduate students to the frontiers of research in computational mathematics, computer vision and medical imaging; and strengthen the collaborations among mathematicians, engineers, computer scientists and medical doctors.

Wavelet frames are systems of functions that provide linear representations of functions living in certain function spaces such as L2(Rn). In contrast to the classic (bi)orthogonal wavelet bases, such representations are generally redundant which is desirable in many applications. Although most theoretical aspects of wavelet frames have already been well understood in the literature, geometric meanings of the wavelet frame transform are still generally unknown. In fact, the lack of geometric interpretations is one of the major flaws of wavelet frames that prohibits the applications of wavelet frames in some important problems of data analysis that require geometric regularization of the objects-of-interest reside in the data. The main research objective of this proposal is to develop a generic geometric interpretation to the wavelet frame transform, by studying its relations with differential operators within various variational frameworks. Based on the geometric interpretation, we propose new models and algorithms for several important applications such image restoration (deblurring, inpainting, CT/MR imaging, etc.). Through both theoretical analysis and numerical experiments, we will explore the advantages of the proposed wavelet frame based models over the existing variational and differential models for different applications. The proposed research will focus on: (1) the approximation of the differential operators by the wavelet frame transform within general variational frameworks; (2) solving large-scaled ill-posed inverse problems (e.g., image restoration, blind deconvolution) through convex/nonconvex optimizations using wavelet frames; (3) designing and solving wavelet frame based models in real-world applications in imaging such as low-dose CT image reconstruction, removing blurs caused by camera shaking, etc. The study of the geometric meanings of the wavelet frame transform will interpret wavelet frames and their associated optimization models from a whole new angle. Such fundamental study enables us, for the very first time, to fully utilize the unique properties of wavelet frames in geometry-involved data analysis tasks and finding numerical solutions of PDEs. The practical advantages (such as the quality of restoration for inverse problems) of wavelet frame transform over standard finite difference approximations in various applications will become more evident after the proposed studies. Furthermore, this project will also bring new understandings to numerical methods solving variational models; and answers some fundamental and important questions of variational models that are unclear from the literature.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Bin Dong, Zuowei Shen, and Peichu Xie "Image Restoration: A General Wavelet Frame Based Model and Its Asymptotic Analysis" SIAM Journal on Imaging Sciences , 2016
E. Davis and S. Sethhuraman "Consistency and modularity Clustering of Random Graphs" Annals Applied Probability , 2016
L Kunyansky, C P Ingram and R S Witte "Rotational magneto-acousto-electric tomography (MAET): theory andexperimental validation" Physics in Medicine and Biology , v.62 , 2017 , p.3025

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

Image processing and analysis techniques are now applied to virtually all natural sciences and technical disciplines ranging from computer sciences and electronic engineering to biology and medical sciences; and digital images have come into everyone's life. Mathematics has been playing an important role in image and signal processing from the very beginning.

In this project we worked on mathematical algorithms used in image restoration, and on mathematical foundations of inverse problems arising in medical imaging and related areas.

A significant part of the effort went into establishing important theoretical connections between two major mathematical approaches for image restoration, namely, wavelet tight frame approaches and differential/variational approaches. In particular, it was shown that the wavelet based techniques converge to the same result as obtain by using a total variation minimization. This allows researchers to develop highly efficient image restoration algorithms that are expected to benefit researchers in academia, national research laboratories, as well as in industry.

The second important topic was the study and development of efficient numerical algorithms for solving scattering equations and Riemann-Hilbert problems, and development of a detailed asymptotic analysis of meromorphic  Riemann-Hilbert problems associated to discrete orthogonal polynomials. These techniques find applications in the so-called inverse problems that form a foundation of the medical imaging algorithms.

Finally, this grant also partially supported work on experimental and theoretical development of Magneto-Acousto-Electric tomography (MAET). This emerging medical imaging modality is formed by combining electrical measurements with acoustic illumination of an object placed in a strong magnetic field. This emerging technique promises significant advances in early cancer detection. We have built (in collaboration with our colleagues from the Medical Imaging Department a first tomographic MAET scanner, that uses rotation of the test object in order to obtain multiple views of it. We were able to obtain first tomographic MAET images (as opposed to inferior images obtained previously by uni-directional scanning). In addition to the experimental implementation, we developed the mathematical foundation of this technique.

This project also have had a significant education impact, bringing undergraduate and graduate students and postdocs to the frontiers of research in computational mathematics, computer vision and medical imaging. Three graduate students have defended their Ph.D. thesis and two more are expected to do so this year.

 


Last Modified: 08/28/2017
Modified by: Leonid A Kunyansky

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