Research of Bin Han

Research Outlines of Bin Han

My research interests concentrate on roughly four major areas:

Numerical PDEs and computing using high order compact finite difference methods, wavelet-based Galerkin methods, and Fourier-based methods. Currently, I am extensively working on this area on many different topics of numerical PDEs.
Framelets (=wavelet frames) with applications in image processing, data science, and statistical/functional data analysis. Currently, I am working on several projects on directional or quasi-tight framelet applications to image processing, statistical or functional data analysis, as well as their applications to deep learning algorithms.
Wavelet theory including wavelets on bounded intervals for the purposes of wavelet-methods in numerical PDEs and data/image sciences for handling the boundary effect using wavelets and framelets on bounded intervals. Currently, I am working on several projects on adapting wavelets and framelets from the real line to bounded intervals and exploring their applications in numerical PDEs and functional data analysis.
Refinable functions and subdivision schemes including interpolating or Hermite subdivision schemes and multivariate vector/matrix subdivision schemes. Currently, I am working on several projects on multivariate vector subdivision schemes with various properties, special refinable vector functions, as well as explore their applications in computer aided geometric design.

Book: Bin Han, Framelets and Wavelets: Algorithms, Analysis, and Applications, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, Cham, (2017), 724 pages.

I. Numerical PDEs and computing We are interested in developing numerical methods for effectively solving various linear and nonlinear PDEs using finite difference methods, wavelet-base methods, and Fourier-based methods. Currently, we are working on Helmholtz equations, elliptic interface problems, and Burgers' equations etc.

Helmholtz equation: Despite numerous applications of the Helmholtz equation in sciences and industry, numerically solving the Helmholtz equation is very difficult largely due to its highly oscillatory solution. A very fine mesh size is necessary to deal with a large wavenumber, leading to a severely ill-conditioned huge coefficient matrix and pollution effects. We are interested on analysis and numerical solutions of the Helmholtz equations by developing high order compact finite difference methods, wavelet-based Galerkin methods, and Fourier-based schemes.
1. Bin Han and Michelle Michelle, Sharp wavenumber-explicit stability bounds for 2D Helmholtz equations, SIAM Journal on Numerical Analysis, 60 (2022), 1985-2013. [arXiv]
2. Bin Han, Michelle Michelle and Yau Shu Wong, Dirac assisted tree method for 1D Helmholtz equations with arbitrary variable wave numbers, Computers & Mathematics with Applications, 97 (2021), 416-438. [arXiv]
3. Qiwei Feg, Bin Han, and Michelle Michelle, Sixth order compact finite difference method for 2D Helmholtz equations with singular sources and reduced pollution effect, preprint (2021). [arXiv]

Elliptic interface problems: Elliptic interface problems arise in many applications such as modeling of underground waste disposal, oil reservoirs, composite materials, and many other applications in sciences and industry. However, the solution of elliptic interface problems has very low smoothness (even discontinuous) and it is challenging to numerically solving such interface problems. We are interested in developing high order compact finite difference methods to solve such elliptic interface problems.
1. Qiwei Feng, Bin Han and Peter Minev, A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients, Applied Mathematics and Computation, 431 (2022), 127314. [arXiv]
2. Qiwei Feng, Bin Han, and Peter Minev, Sixth order compact finite difference schemes for Poisson interface problems with singular sources, Computers & Mathematics with Applications, 99 (2021), 2-25. [arXiv]
3. Qiwei Feg, Bin Han, and Michelle Michelle, Sixth order compact finite difference method for 2D Helmholtz equations with singular sources and reduced pollution effect, preprint (2021). [arXiv]

Other PDEs and computing: We are interested in numerical solutions of the Burgers' equations and hyperbolic telegraph equations, which are nonlinear PDEs. We develop newly modified cubic B-splines with accuracy order 4 and use differential quadrature techniques to solve these nonlinear PDEs. We are also interested in developing wavelet-based methods for numerical solutions of other nonlinear and linear PDEs.
1. Athira Babu, Bin Han, and Noufal Asharaf, Numerical solution of the hyperbolic telegraph equation using cubic B-spline-based differential quadrature of high accuracy, Computational Methods for Differential Equations, published online,
2. Bin Han and Michelle Michelle, Wavelets on intervals derived from arbitrary compactly supported biorthogonal multiwavelets, Applied and Computational Harmonic Analysis, 53 (2021), 270-331. [arXiv]
3. Athira Babu, Bin Han and Noufal Asharaf, Numerical solution of the viscous Burgers' equation using Localized Differential Quadrature method, Partial Differential Equations in Applied Mathematics, 4 (2021), 100044.
4. E. Ashpazzadeh, B. Han, M. Lakestani and M. Razzaghi, Derivative-orthogonal wavelets for discretizing constrained optimal control problems , International Journal of Systems Science, 51 (2020), 786-810.
5. Bin Han and Michelle Michelle, Derivative-orthogonal Riesz wavelets in Sobolev spaces with applications to differential equations, Applied and Computational Harmonic Analysis, 47 (2019), Issue 3, 759-794.
6. Bin Han and Michelle Michelle, Construction of wavelets and framelets on a bounded interval, Analysis and Applications, 16 (2018), No. 06, 807-849.
7. Elmira Ashpazzadeh, Bin Han, and Mehrdad Lakestani, Biorthogonal multiwavelets on the interval for numerical solutions of Burgers' equation, Journal of Computational and Applied Mathematics, 317 (2017), 510-534.
8. Byungil Kim, Hoyoung Jeong, Hyoungkwan Kim, and Bin Han, Exploring wavelet applications in civil engineering, KSCE Journal of Civil Engineering, 21 (2017), no. 4, 1076-1086.
9. Dao-Qing Dai, Bin Han, Rong-Qing Jia, Galerkin analysis for Schroedinger equation by wavelets, Journal of Mathematical Physics, 45 (2004), Issue 3, 855-869..

II. Framelets (=Wavelet frames) with applications Wavelet frames are of particular interest for their applications in signal and image processing. In comparison with classical wavelets, framelets enjoy the desired properties of redundancy for robustness against noise and perturbation, directionality for capturing singularities, flexibility for relatively easy construction, and the near-translation invariance for noise removal. Except image compression and wavelet-based methods for numerical PDEs, one often prefers framelets instead of classical wavelets in many other applications, in particular in image processing and data sciences.

Framelets with directionality for image processing: Directional tight framelets are very attractive for high dimensional data analysis, largely because they have the desired properties of directionality for capturing singularities and of redundancy for robustness against noise. In particular, we developed a particular family of tensor product complex tight framelets with directionality for the purpose of their applications in image processing.
1. Bin Han, Qun Mo, Zhenpeng Zhao, and Xiaosheng Zhuang, Directional compactly supported tensor product complex tight framelets with applications to image denoising and inpainting , SIAM Journal on Imaging Sciences, 12 (2019), Issue 4, 1739-1771.
2. Yi Shen, Bin Han, and Elena Bravermann, Image inpainting from partial noisy data by directional complex tight framelets, ANZIAM (Australia and New Zealand Industrial and Applied Mathematics), 58 (2017), 247-255.
3. Bin Han, Zhenpeng Zhao and Xiaosheng Zhuang, Directional tensor product complex tight framelets with low redundancy, Applied and Computational Harmonic Analysis, 41 (2016), Issue 2, 603-637.
4. Yi Shen, Bin Han and Elena Braverman, Removal of mixed Gaussian and impulse noise using directional tensor product complex tight framelets, Journal of Mathematical Imaging and Vision, 54 (2016), 64-77.
5. Yi Shen, Bin Han and Elena Braverman, Adaptive frame-based color image denoising, Applied and Computational Harmonic Analysis, 41 (2016), Issue 1, 54-74.
6. Bin Han, Qun Mo, and Zhenpeng Zhao, Compactly supported tensor product complex tight framelets with directionality, SIAM Journal on Mathematical Analysis, 47 (2015), Issue 3, 2464-2494.
7. Bin Han and Xiaosheng Zhuang, Smooth affine shear tight frames with MRA structure, Applied and Computational Harmonic Analysis, 39 (2015), 300-338.
8. Bin Han and Zhenpeng Zhao, Tensor product complex tight framelets with increasing directionality, SIAM Journal on Imaging Sciences, 7 (2014), Issue 2, 997-1034. [PDF]
9. Bin Han, Nonhomogeneous wavelet systems in high dimensions, Applied and Computational Harmonic Analysis, 32 (2012), Issue 2, 169-196.

Quasi-tight framelets: Due to the energy preservation property, tight (wavelet) framelets are of particular interest in signal and image processing. However, spline tight framelets often have no more than one vanishing moments and their constructions are often not that easy. Quasi-tight framelets are dual wavelet frames but they behaves almost identical to tight framelets. Therefore, we are interested in developing the theory on quasi-tight framelets and in exploring their applications. In particular, quasi-tight framelets can possess the desired property of directionality to capture singularities in high-dimensional data such as image processing.
1. Bin Han and Ran Lu, Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions , Science China Mathematics, 65 (2022), 81-110. [arXiv]
2. Bin Han and Ran Lu, Compactly supported quasi-tight multiframelets with high balancing orders and compact framelet transforms, Applied and Computational Harmonic Analysis, 51 (2021), 295-332. [arXiv]
3. Chenzhe Diao and Bin Han, Generalized matrix spectral factorization and quasi-tight framelets with minimum number of generators, Mathematics of Computation, 89 (2020), 2867-2911. [arXiv]
4. Chenzhe Diao and Bin Han, Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions, Applied and Computational Harmonic Analysis, 49 (2020), Issue 1, 123-151. [arXiv]
5. Bin Han, Tao Li and Xiaosheng Zhuang, Directional compactly supported box spline tight framelets with simple geometric structure, Applied Mathematics Letters, 91 (2019), 213-219. [arXiv]

Theory of framelets: We study the theoretical aspects of univariate and multivariate framelets and investigate their properties such as stability, Gibbs phenomenon, smoothness, approximation orders and vanishing moments.
1. Bin Han, Gibbs phenomenon of framelet expansions and quasi-projection approximation , Journal of Fourier Analysis and Applications, 25 (2019), 2923-2956. [arXiv]
2. Bin Han, Homogeneous wavelets and framelets with the refinable structure, Science China Mathematics, 60 (2017), 2173-2198. [arXiv]
3. Bin Han, The projection method for multidimensional framelet and wavelet analysis, Mathematical Modelling of Natural Phenomena, 9 (2014), No. 5, 83-110. [PDF]
4. Bin Han, Properties of discrete framelet transforms, Mathematical Modelling of Natural Phenomena, 8 (2013), Issue 1, 18-47. [PDF]
5. Say Song Goh, Bin Han and Zuowei Shen, Tight periodic wavelet frames and approximation orders, Applied and Computational Harmonic Analysis, 31 (2011), Issue 2, 228-248.
6. Bin Han, Dual multiwavelet frames with high balancing order and compact fast frame transform, Applied and Computational Harmonic Analysis, 26 (2009), 14-42.
7. Bin Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, Journal of Computational and Applied Mathematics, 155 (2003), Issue 1, 43-67.
8. Ingrid Daubechies and Bin Han, Pairs of dual wavelet frames from any two refinable functions, Constructive Approximation, 20 (2004), No. 3, 325-352.
9. Ingrid Daubechies, Bin Han, Amos Ron, and Zuowei Shen, Framelets: MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), No. 1, 1-46.
10. Ingrid Daubechies and Bin Han, The canonical dual frame of a wavelet frame, Applied and Computational Harmonic Analysis, 12 (2002), No. 3, 269-285.
11. Bin Han and Qun Mo, Multiwavelet frames from refinable function vectors, Advances in Computational Mathematics, 18 (2003), 211-245.
12. Bin Han, On dual wavelet tight frames, Applied and Computational Harmonic Analysis, 4 (1997), no. 4, 380-413.

Construction of framelets: We construct univariate and multivariate framelets with high vanishing moments, balancing property, and/or symmetry property.
1. Bin Han, Qingtang Jiang, Zuowei Shen, and Xiaosheng Zhuang, Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness, Mathematics of Computation, 87 (2018), 347-379. [PDF][arXiv]
2. Bin Han, Algorithm for constructing symmetric dual framelet filter banks, Mathematics of Computation, 84 (2015), 767-801. [PDF]
3. Bin Han, Symmetric tight framelet filter banks with three high-pass filters, Applied and Computational Harmonic Analysis, 37 (2014), Issue 1, 140-161. [PDF]
4. Bin Han, Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters, Applied and Computational Harmonic Analysis, 35 (2013), Issue 2, 200-227. [PDF]
5. Bin Han and Zuowei Shen, Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames, Israel Journal of Mathematics, 172 (2009), No. 1, 371-398.
6. Bin Han, Construction of wavelets and framelets by the projection method, International Journal of Applied Mathematics and Applications, 1 (2008), Issue 1, 1-40.
7. Martin Ehler and Bin Han, Wavelet bi-frames with few generators from multivariate refinable functions, Applied and Computational Harmonic Analysis, 25 (2008), 407-414.
8. Bin Han and Qun Mo, Symmetric MRA tight wavelet frames with three generators and high vanishing moments, Applied and Computational Harmonic Analysis, 18 (2005), Issue 1, 67-93,
9. Bin Han and Qun Mo, Splitting a matrix of Laurent polynomials with symmetry and its application to symmetric framelet filter banks. SIAM Journal on Matrix Analysis and its Applications, 26 (2004), No. 1, 97-124.
10. Bin Han and Qun Mo, Tight wavelet frames generated by three symmetric B-spline functions with high vanishing moments, Proceedings of the American Mathematical Society, 132 (2004), No. 1, 77-86.
11. Ingrid Daubechies and Bin Han, Pairs of dual wavelet frames from any two refinable functions, Constructive Approximation, 20 (2004), No. 3, 325-352.
12. Bin Han, Recent developments on dual wavelet frames, in Representations, Wavelets, and Frames: A celebration of the Mathematical Work of Lawrence W. Baggett, (P. E. T. Jorgensen, K. Merrill, and J. A. Parker eds.), (2008), 103-130.

III. Wavelet theory Classical wavelet theory deals with orthonormal wavelets, biorthogonal wavelets, and Riesz wavelets. Such wavelets are of particular interest in image/data compression and in wavelet-methods in numerical PDEs. We are interested in studying various properties of wavelet and framelet systems. Such properties includes characterization of function spaces by wavelets, approximation property of shit-invariant spaces, stability and linear independence of refinable vector functions. We are also interested in constructing various types of wavelets including multivariate biorthogonal wavelets and Riesz wavelets.

Wavelets on bounded intervals: To apply wavelets to boundary value problems in numerical PDEs and scientific computing, it is crucial to adapt wavelets from the real line to bounded intervals. We develop particular and general methods to adapt various wavelets from the real line to the unit bounded interval [0,1]. In particular, we develop a direct approach which can adapt any compactly supported biorthogonal (multi)wavelets from the real line to the bounded interval [0,1] with high vanishing moments of the boundary wavelets.
1. Bin Han and Michelle Michelle, Wavelets on intervals derived from arbitrary compactly supported biorthogonal multiwavelets, Applied and Computational Harmonic Analysis, 53 (2021), 270-331. [arXiv]
2. E. Ashpazzadeh, B. Han, M. Lakestani and M. Razzaghi, Derivative-orthogonal wavelets for discretizing constrained optimal control problems , International Journal of Systems Science, 51 (2020), 786-810.
3. Bin Han and Michelle Michelle, Construction of wavelets and framelets on a bounded interval, Analysis and Applications, 16 (2018), No. 06, 807-849.
4. Elmira Ashpazzadeh, Bin Han, and Mehrdad Lakestani, Biorthogonal multiwavelets on the interval for numerical solutions of Burgers' equation, Journal of Computational and Applied Mathematics, 317 (2017), 510-534.
5. Bin Han and Qing-Tang Jiang, Multiwavelets on the interval, Applied and Computational Harmonic Analysis, 12 (2002), No. 1, 100-127.
6. Wolfgang Dahmen, Bin Han, Rong-Qing Jia, and Angela Kunoth, Biorthogonal multiwavelets on the interval: cubic Hermite splines, Constructive Approximation, 16 (2000), 221-259.

Construction of wavelets: To construct various wavelets, two key issues are (1) how to construct suitable refinable (vector) functions and their associated low-pass filters with desired properties; (2) how to derive the associated wavelet functions and their associated high-pass filters from given refinable (vector) functions and the low-pass filters. We provide general methods for both topics. In particular, we introduce a CBC (coset-by-coset) algorithm to construct all biorthogonal (multi)wavelets.
1. Bin Han and Xiaosehgn Zhuang, Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields, Mathematics of Computation. 82 (2013), 459-490.
2. Charles K. Chui, Bin Han and Xiaosheng Zhuang, A dual-chain approach for bottom-up construction of wavelet filters with any integer dilation, Applied and Computational Harmonic Analysis, 33 (2012), 204-225.
3. Bin Han and Xiaosheng Zhuang, Matrix extension with symmetry and its application to symmetric orthonormal multiwavelets, SIAM Journal on Mathematical Analysis, 42 (2010), Issue 5, 2297-2317.
4. Bin Han and Hui Ji, Compactly supported orthonormal complex wavelets with dilation 4 and symmetry, Applied and Computational Harmonic Analysis, 26 (2009), 422-431.
5. Ning Bi, Bin Han, and Zuowei Shen Componentwise polynomial solutions and distribution solutions of refinement masks, Applied and Computational Harmonic Analysis, 27 (2009), Issue 1, 117-123.
6. Bin Han, Symmetry property and construction of wavelets with a general dilation matrix, Linear Algebra and its Applications, 353 (2002), 207-225.
7. Bin Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, Journal of Approximation Theory, 110 (2001), No. 1, 18-53.
8. Di-Rong Chen, Bin Han and Sherman D. Riemenschneider, Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments, Advances in Computational Mathematics, 13 (2000), 131-165.
9. Bin Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM Journal on Mathematical Analysis, 31 (1999/2000), 274-304.
10. Bin Han, Construction of multivariate biorthogonal wavelets by CBC algorithm, Wavelet analysis and multiresolution methods (Urbana-Champaign, IL, 1999), 105-143, Lecture Notes in Pure and Appl. Math., 212, Dekker, New York, 2000.
11. Bin Han, Symmetric orthonormal scaling functions and wavelets with dilation factor 4, Advances in Computational Mathematics, 8 (1998), 221-247.

Riesz wavelets: Riesz wavelets are of particular interest in numerical PDEs. In particular, wavelets used in numerical PDEs are built on Riesz wavelets in Sobolev spaces. Therefore, it is important to analyze and construct various Riesz wavelets for their applications in computational mathematics and scientific computing.
1. Bin Han and Michelle Michelle, Derivative-orthogonal Riesz wavelets in Sobolev spaces with applications to differential equations, Applied and Computational Harmonic Analysis, 47 (2019), Issue 3, 759-794.
2. Bin Han, Qun Mo, and Zuowei Shen, Small support spline Riesz wavelets in low dimensions, Journal of Fourier Analysis and Applications, 17 (2011), 535-566.
3. Bin Han and Zuowei Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constructive Approximation, 29 (2009), Issue 3, 369-406.
4. Bin Han, Refinable functions and cascade algorithms in weighted spaces with Holder continuous masks, SIAM Journal on Mathematical Analysis, 40 (2008), Issue 1, 70-102.
5. Bin Han and Rong-Qing Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Applied and Computational Harmonic Analysis, 23 (2007), Issue 3, 321-345.
6. Bin Han, On a conjecture about MRA Riesz wavelet bases, Proceedings of American Mathematical Society, 134 (2006), 1973-1983.
7. Bin Han, Soon-Geol Kwon and Sang Soo Park, Riesz multiwavelet bases, Applied and Computational Harmonic Analysis, 20 (2006), 161-183.
8. Bin Han and Zuowei Shen, Wavelets with short support, SIAM Journal on Mathematical Analysis, 38 (2006), Issue 2, 530-556.
9. Bin Han and Zuowei Shen, Wavelets from the Loop scheme, Journal of Fourier Analysis and its Applications, 11 (2005), No. 6, 615-637.

Analysis of wavelets: We analyze various properties of wavelet and framelet systems, in particular, wavelets in Sobolev function spaces, approximation property of shit-invariant spaces, and stability and linear independence of refinable vector functions.
1. Bin Han, On linear independence of integer shifts of compactly supported distributions, Journal of Approximation Theory, 201 (2016), 1-6. [PDF]
2. Bin Han, Nonhomogeneous wavelet systems in high dimensions, Applied and Computational Harmonic Analysis, 32 (2012), Issue 2, 169-196.
3. Bin Han, Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space, Applied and Computational Harmonic Analysis, 29 (2010), no. 3, 330-353.
4. Bin Han, Wavelets and framelets within the framework of nonhomogeneous wavelet systems, in Approximation Theory XIII: San Antonio 2010 (Springer Proceedings in Mathematics), (2012), 121-161.
5. Bin Han, Gitta Kutyniok, and Zuowei Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM Journal on Numerical Analysis, 49 (2011), 1921-1946.
6. Bin Han, The projection method in wavelet analysis, in Splines and Wavelets: Athens 2005, G. Chen and M.J. Lai eds., (2006), 202-225.
7. Bin Han, Symmetric multivariate orthogonal refinable functions, Applied and Computational Harmnoic Analysis, 17 (2004), 277-292.
8. Bin Han, Projectable multivariate refinable functions and biorthogonal wavelets, Applied and Computational Harmonic Analysis, 13 (2002), 89--102.
9. Bin Han, Some applications of projection operators in wavelets, Acta Mathematica Sinica, 11 (1995), 105-112.

Wavelets with linear-phase moments or balanced property: Linear-phase moments of refinable functions are closely related to vanishing moments of wavelets and polynomial-interpolating property. They also consist of a special family of wavelets that are of particular interest in developing wavelet methods for solving nonlinear PDEs. Though multiwavelets may possess high vanishing moments in the function setting, their associated discrete multiwavelet transform may lose the sparsity and have only low discrete polynomial reproduction property. It is necessary for multiwavelets to have the balanced property so that they are properly balanced in order to have the high discrete vanishing moments for sparsity in their discrete multiwavelet transform for image processing and data sciences.
1. Bin Han, Symmetric orthogonal filters and wavelets with linear-phase moments, Journal of Computational and Applied Mathematics, 236 (2011), Issue 4, 482-503.
2. Bin Han, Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules, Advances in Computational Mathematics, 32 (2010), No. 2, 209-237.
3. B. Han, Matrix extension with symmetry and applications to symmetric orthonormal complex M-wavelets, Journal of Fourier Analysis and its Applications 15 (2009), 684-705.
4. Bin Han, Dual multiwavelet frames with high balancing order and compact fast frame transform, Applied and Computational Harmonic Analysis, 26 (2009), 14-42.
5. Bin Han, The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets, Mathematics of Computation, 79 (2010), 917-951.

IV. Refinable functions and subdivision schemes Wavelets and framelets are often derived from refinable functions and most properties of wavelets and framelets are determined that of a refinable function. The properties of a refinable function are often studied through a subdivision scheme and its associated cascade algorithm in the function setting. A subdivision scheme is a fast local averaging algorithm to generate smooth curves and surface. In particular, a subdivision scheme can be employed in wavelet analysis to compute and analyze refinable functions and in the fast wavelet/framelet transform to reconstruct the data. We are interested in the convergence and smoothness of various subdivision schemes

Smoothness of refinable functions: Smoothness of a refinable (vector) function is probably the most important property in wavelet theory because it determines the smoothness of its associated wavelets and framelets. Therefore, it is fundamental to analyze refinable (vector) functions and its properties such as approximation orders, stability, smoothness, symmetry and other properties. We analyze the smoothness property of refinable (vector) functions and how to compute their smoothness exponents.
1. Bin Han, Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix, Advances in Computational Mathematics, 24 (2006), No. 1-4, 375-403.
2. Bin Han, Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM Journal on Matrix Analysis and its Applications, 24 (2003), No. 3, 693-714.
3. Bin Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, Journal of Approximation Theory, 124 (2003), Issue 1, 44-88.

Hermite subdivision schemes:
1. Bin Han, Analysis and convergence of Hermite subdivision schemes, Foundations of Computational Mathematics, published online, [arXiv]
2. Bin Han and Xiaosheng Zhuang Analysis and construction of multivariate interpolating refinable function vectors, Acta Applicandae Mathematicae, 107 (2009), No. 1-3, 143-171.
3. Bin Han and Qun Mo, Analysis of optimal bivariate refinable Hermite interpolants, Communications in Pure and Applied Analysis, 6 (2007), No. 3, 689-718.
4. Bin Han and Thomas Yu, Face-based Hermite subdivision schemes, Journal of Concrete and Applicable Mathematics, 4 (2006), No. 4, 435-450.
5. Bin Han, Thomas Yu, Yonggang Xue, Non-Interpolatory Hermite subdivision schemes, Mathematics of Computation, 74 (2005), 1345-1367.
6. Serge Dubuc, Bin Han, Jean-Louis Merrien and Qun Mo, Dyadic C2 Hermite interpolation on a square mesh, Computer Aided Geometric Design, 22 (2005), Issue 8, 727-752.
7. Bin Han, Thomas P.-Y. Yu, and Bruce Piper, Multivariate refinable Hermite interpolants, Mathematics of Computations, 73 (2004), 1913-1935.
8. Bin Han, Michael L. Overton, and Thomas P.-Y. Yu, Design of Hermite subdivision schemes aided by spectral radius optimization, SIAM Journal on Scientific Computing, 25 (2003), No. 2, 643-656.
9. Bin Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, Journal of Approximation Theory, 110 (2001), No. 1, 18-53.
10. Bin Han, Hermite interpolants and biorthogonal multiwavelets with arbitrary order of vanishing moments SPIE Proc. 3813 (1999), 147-161.

Interpolating subdivision schemes:
1. Bin Han and Xiaosheng Zhuang Analysis and construction of multivariate interpolating refinable function vectors, Acta Applicandae Mathematicae, 107 (2009), No. 1-3, 143-171.
2. Bin Han, Son-Geol Kwon and Xiaosheng Zhuang, Generalized interpolating refinable function vectors, Journal of Computational and Applied Mathematics, 227 (2009), 254-270.
3. Bin Han and Rong-Qing Jia, Optimal C^2 two-dimensional interpolatory ternary subdivision schemes with two-ring stencils, Mathematics of Computation, 75 (2006), 1287-1308.
4. Bin Han and Rong-Qing Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Mathematics of Computation, 71 (2002), No. 237, 165-196.
5. Bin Han, Symmetry property and construction of wavelets with a general dilation matrix, Linear Algebra and its Applications, 353 (2002), 207-225.
6. Bin Han and Sherman D. Riemenschneider, Interpolatory biorthogonal wavelets and CBC algorithm, Wavelet analysis and applications (Guangzhou, 1999), 119-138, AMS/IP Studies in Advanced Mathematics, 25, Amer. Math. Soc., Providence, RI, (2002).
7. Bin Han and Rong-Qing Jia, Optimal interpolatory subdivision schemes in multidimensional spaces, SIAM Journal on Numerical Analysis, 36 (1998),105-124.

Analysis of subdivision schemes and cascade algorithms: The properties of a refinable function are often studied through a subdivision scheme and its associated cascade algorithm in the function setting. We are interested in studying various properties of refinable (vector) functions, including smoothness and convergence of refinable functions and their associated subdivision schemes (or the cascade algorithms in the function setting).
1. Lincong Fang, Bin Han, and Yi Shen, Quasi-interpolating bivariate dual sqrt{2}-subdivision using 1D stencils, Computer Aided Geometric Design, 98 (2022) 102139. [PDF]
2. Bin Han, Refinable functions and cascade algorithms in weighted spaces with Holder continuous masks, SIAM Journal on Mathematical Analysis, 40 (2008), Issue 1, 70-102.
3. Bin Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, Journal of Approximation Theory, 124 (2003), Issue 1, 44-88.
4. Bin Han, Classification and construction of bivariate subdivision schemes, Proceedings on Curves and Surfaces Fitting: Saint-Malo 2002, A. Cohen, J.-L. Merrien, and L. L. Schumaker eds., (2003), 187-197.
5. Bin Han, The initial functions in a cascade algorithm, Wavelet Analysis: Twenty Years' Developments, Proceedings of International Conference on Computational Harmonic Analysis, D.X. Zhou ed., 154-178, (2002).
6. Bin Han and Rong-Qing Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM Journal on Mathematical Analysis, 29 (1998), 1177-1199.
7. Bin Han and Thomas A. Hogan, How is a vector pyramid scheme affected by perturbation in the mask? Approximation theory IX, Vol. 2 (Nashville, TN, 1998), 97--104, Innov. Appl. Math., Vanderbilt University Press, Nashville, TN, 1998.

V. Other research topics We also worked on compressed sensing, shearlets, sampling theory etc,

Bin Han, Swaraj Paul, and Niraj K. Shukla, Microlocal analysis and characterization of Sobolev wavefront sets using shearlets, Constructive Approximation, 55 (2022), 661-704. [arXiv]
Bin Han and Zhiqiang Xu, Robustness properties of dimensionality reduction with Gaussian random matrices, Science China Mathematics, 60 (2017) 1753-1778 [arXiv]
Yi Shen, Bin Han and Elena Braverman, Stability of the elastic net estimator, Journal of Complexity, 32 (2016), 20-39.
Yi Shen, Bin Han and Elena Braverman, Stable recovery of analysis based approaches, Applied and Computational Harmonic Analysis, 39 (2015), 161-172.
Bin Han, Gitta Kutyniok, and Zuowei Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM Journal on Numerical Analysis, 49 (2011), 1921-1946.
Bin Han, Bivariate (two-dimensional) wavelets, in Encyclopedia of Complexity and System Science. R. A. Meyers ed., (2009), 589-599.
Ning Bi, Bin Han and Zuowei Shen, Examples of refinable componentwise polynomials, Applied and Computational Harmonic Analysis, 22 (2007), Issue 3, 368-373.
Wen Chen, Bin Han and Rong-Qing Jia, Estimate of aliasing error for non-smooth signals prefiltered by quasi-projections into shift invariant spaces, IEEE Transactions on Signal Processing, 53 (2005), 1927-1933.
Wen Chen, Bin Han and Rong-Qing Jia, A simple oversampled A/D conversion in shift invariant spaces, IEEE Transactions on Information Theory, 51 (2005), no. 2, 648-657.
Wen Chen, Bin Han and Rong-Qing Jia, Maximal gap of a sampling set for the exact iterative reconstruction algorithm in shift invariant spaces, IEEE Signal Processing Letters, 11 (2004), No. 8, 655-658.
Paul Shelley, Xiaobo Li and Bin Han, A hybrid quantization scheme for image compression,, Image and Vision Computing, 22 (2004), Issue 3, 203-213.
Jason Knipe, Xiaobo Li and Bin Han, An improved lattice vector quantization scheme for wavelet compression, IEEE Transactions on Signal Processing, 46 (1998), 239-243.