Publication List and Research Statement

Referred Journal Publications

  1. Wavelet series expansion in Hardy spaces with approximate duals (with H. Lim), Analysis Mathematica, Vol. 50, No. 2, 2024, pp. 563-595. (preprint in arXiv)
  2. Invertibility of circulant matrices of arbitrary size (with J. Choi), Linear and Multilinear Algebra, Vol. 70, No. 21, 2022, pp. 7057-7074. (preprint in arXiv)
  3. Multivariate tight wavelet frames with few generators and high vanishing moments (with Z. Lubberts and K. A.Okoudjou), International Journal of Wavelets, Multiresolution and Information Processing, Vol. 20, Issue 5, September 2022, Article No. 225009 (27 pages). (preprint (of an earlier version) in arXiv)
  4. Tight wavelet filter banks with prescribed directions, International Journal of Wavelets, Multiresolution and Information Processing, Vol. 24, Issue 4, July 2022, Article No. 225008 (20 pages). (preprint in arXiv)
  5. Interpolatory tight wavelet frames with prime dilation (with Z. Lubberts), Applied and Computational Harmonic Analysis, Vol. 49, Issue 3, Nov. 2020, pp. 897-915. (preprint)
  6. New constructions of nonseparable tight wavelet frames (with Z. Lubberts), Linear Algebra and its Applications, Vol. 534, Dec. 2017, pp. 13-35. (preprint)
  7. Prime Coset Sum: A systematic method for designing multi-D wavelet filter banks with fast algorithms (with F. Zheng), IEEE Transactions on Information Theory, Vol. 62, No. 11, Nov. 2016, pp. 6580-6593. (preprint in arXiv)
  8. Identification of cancer-driver genes in focal genomic alterations from whole genome sequencing data (with H. Jang and H. Lee), Scientific Reports, May 2016, 6:25582. (preprint)
  9. Monitoring nonlinear profiles adaptively with a wavelet-based distribution-free CUSUM chart (with H. Wang, S.-H. Kim, X. Huo, and J. R. Wilson), International Journal of Production Research, Vol. 53, No. 15, 2015, pp. 4648-4667. (preprint)
  10. Scaling Laplacian pyramids (with K. A. Okoudjou), SIAM Journal on Matrix Analysis and Applications, Vol. 36, No. 1, 2015, pp. 348-365. (preprint in arXiv)
  11. Multi-D wavelet filter bank design using Quillen-Suslin theorem for Laurent polynomials (with H. Park and F. Zheng), IEEE Transactions on Signal Processing, Vol. 62, No. 20, Oct. 2014, pp. 5348-5358. (preprint)
  12. Coset Sum: An alternative to the tensor product in wavelet construction (with F. Zheng), IEEE Transactions on Information Theory, Vol. 59, No. 6, June 2013, pp. 3554-3571. (preprint in arXiv)
  13. Monitoring nonlinear profiles using a wavelet-based distribution-free CUSUM chart (with J. Lee, S.-H. Kim, and J. R. Wilson), International Journal of Production Research, Vol. 50, No. 22, Nov. 2012, pp. 6574-6594. (preprint)
  14. High-performance very local Riesz wavelet bases of L2(Rn) (with A. Ron), SIAM Journal on Mathematical Analysis, Vol. 44, No. 4, July. 2012, pp. 2237-2265.
  15. Wavelet-based identification of DNA focal genomic aberrations from single nucleotide polymorphism arrays (with H. Lee), BMC Bioinformatics, Vol. 12, No. 146, May 2011.
  16. Effortless critical representation of Laplacian pyramid, IEEE Transactions on Signal Processing, Vol. 58, No. 11, Nov. 2010, pp. 5584-5596.
  17. L-CAMP: Extremely local high-performance wavelet representations in high spatial dimension (with A. Ron), IEEE Transactions on Information Theory, Vol. 54, No. 5, May 2008, pp. 2196-2209.
  18. New constructions of piecewise-constant wavelets (with A. Ron), ETNA, Special Volume on Constructive Function Theory 25, 2006, pp. 138-157.

Book Chapter

  1. Use of Quillen-Suslin theorem for Laurent polynomials in wavelet filter bank design, Book chapter of Excursions in Harmonic Analysis, Volume 5 (R. Balan, J. J. Benedetto, W. Czaja, M. Dellatorre, and K. A. Okoudjou, Eds.), Birkhauser (published in June 2017), pp. 303-313. (preprint)

Referred Conference Proceedings

  1. Simplifying formal proof-generating models with ChatGPT and basic searching techniques (with S. Han, T. Hur, K. S. Lee, M. Lee, and H. Lim), Computing Conference 2025. (accepted)
  2. Laplacian pyramid-like autoencoder (with S. Han and T. Hur), Intelligent Computing: Proceedings of the 2022 Computing Conference, Volume 2, pp. 59-78. Part of the Lecture Notes in Networks and Systems book series (LNNS,volume 507). Springer International Publishing. (preprint in arXiv and codes)
  3. Deep scattering network with max-pooling (with T. Ki), Proceedings of the 2021 Data Compression Conference, 2021, pp. 348-348. (preprint; based on the work in the preprint in arXiv and codes)
  4. Understanding the scattering transform using univariate signals (with H. Lim), Proceedings of the 11th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), 2018, pp. 1-7. (preprint)
  5. Scalable filter banks (with K. A. Okoudjou), Proceedings of SPIE Vol. 9597, Wavelets and Sparsity XVI, 95970Q, 2015, pp. 1-6. (preprint)
  6. The design of non-redundant directional wavelet filter bank using 1-D Neville filters (with F. Zheng), Proceedings of the 10th International Conference on Sampling Theory and Applications, 2013, pp. 216-219. (preprint)
  7. Committee algorithm: An easy way to construct wavelet filter banks, Proceedings of the 37th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2012, pp. 3485-3488. (preprint in arXiv)
  8. Designing thin wavelet filters (with F. Zheng), Proceedings of the 45th Asilomar Conference on Signals, Systems and Computers, 2011, pp. 2019-2024.

Works Submitted

  1. Design of wavelet filter banks for any dilation using extended Laplacian pyramid matrices (with S. Kim)
  2. New tight wavelet frame constructions sharing responsibility (with H. Lim) (preprint in arXiv)

Works in Progress

  1. Applications of wavelets in the quantum field theory for the strong interaction (with H. Lee)

Research Statement

My research interests are In Applied and Computational Harmonic Analysis, I am mostly interested in constructing multi-dimensional (multi-D) wavelets and their extensions and variants. While 1-D wavelets have been extensively studied and their constructions are well understood, there are still many remaining problems for multi-D wavelets. A typical method for constructing multi-D wavelets is to use the tensor product of 1-D wavelets along coordinate directions. This tensor product method provides an easy and straightforward way, but the resulting multi-D wavelets present many limitations. The main tools that I have used and plan to use to solve these problems are ranging from more traditional means (cf. Referred Journal Publications [1,2,5]) to less traditional ones (cf. Referred Journal Publications [3,7,8,9,12,13,14,15,16,18]) from areas such as Algebra and Algebraic Geometry. Research in this direction also requires understanding the interplay between many essential properties of multi-D wavelets, including directionality, vanishing moments, smoothness, and fast algorithms. Understanding this interplay is especially important in wavelet applications.

Among the most successful wavelet applications up to date have been in Signal and Image Processing. Applications to other areas tend to be more challenging. A critical problem in such cases is to find out which wavelet-based signal processing tools would be best suited for the application and the data at hand. Through collaboration with experts in related fields, I have developed new wavelet-based signal processing methods for biological/manufacturing data (cf. Referred Journal Publications [4,6,10,11]), which present many advantages over other existing methods. As the technology for obtaining datasets is fast-growing, the datasets in Science and Engineering are abundant nowadays. Furthermore, recent advances in Deep Learning have changed a lot of research around it, and the research in Signal and Image Processing is no exception. I plan to employ Deep Learning algorithms to solve wavelet application problems, including both traditionally studied and less explored ones.

Currently, I am exploring the possibility of learning wavelet filters using Deep Learning techniques. It can be called a data-driven wavelet filter construction problem. For a given dataset, one tries to find the best wavelet filters or their variants that can represent the dataset. It can provide a way to find a function space that approximates the given dataset well or even help to bypass function spaces altogether and directly connect the dataset with wavelet filters. Using currently available tools in wavelets, this is very hard to attain, if possible at all. After this process, one can replace some of the filters appearing in Deep Learning algorithms with these wavelet filters, making the algorithms more transparent and controllable. It will complement the current Deep Learning algorithms well as they present surprisingly high performance in various applications, but theoretical understanding about them is still very low. In this approach, it is crucial to understand how one can add some desirable properties of wavelet filters (e.g. directionality), or weaken some essential properties of wavelet filters (e.g. perfect reconstruction) in such a way the modification is theoretically sound and practically implementable.

Another direction of my research is to describe the quantum field theory using wavelets, especially the quark and gluon fields confined in hadrons such as protons and neutrons. The quantum field theory was developed using Fourier expansion, as it was a great success in the first quantum field theory, QED, dealing with electrons and photons. The general picture is that an incoming wave from a far distance approaches a target and has a small interaction, and goes away. This picture does not work very efficiently in describing the quark and gluon waves confined in a small volume, though. Noting that wavelets can provide much more localized expansion than Fourier expansion, I am currently trying to use this localization property of the wavelets to develop the wavelet-based QCD to describe the quark and gluon fields in the hadron in collaboration with a theoretical particle physicist.