Bradley Lucier

Professor Emeritus of Mathematics and Computer Science

• 1 765 49-41979

• MATH G138

lucier@purdue.edu
Personal Website

Personal Website

Research Interest(s):

numerical analysis, wavelets

Research Interests

My research started in numerical methods for partial differential equations, motivated by scalar hyperbolic conservation laws, which can have rough (even discontinuous) solutions. This study led in 1985 to moving-grid numerical methods with rigorous second-order error bounds, the only such methods known. Later I worked with Ron DeVore, then at the University of South Carolina, on applying some ideas from nonlinear approximation theory (specifically, the recently-developed theory of approximation by free-knot splines) to prove new regularity results for conservation laws. These results state that if the nonlinear flux of the conservation law is smooth and convex, and the initial data has smoothness order K for any K>0, then even after discontinuities develop, the solution maintains that smoothness. One must measure smoothness in the space Lq for q=1/(K+1) for this result to hold, however, and these spaces are non-convex.

Around the same time Ron was working with Bjorn Jawerth and Vasil Popov to develop the theory of nonlinear compression of wavelet decompositions of functions; this theory in one dimension was closely related to the results on approximation by free-knot splines. I wrote a series of papers with Ron and Bjorn applying these ideas to image and surface compression, and data compression in general; this included the first paper to apply nonlinear approximation theory to image processing, specifically to analyze the effects of quantization of wavelet coefficients in image compression. This paper introduced a number of other techniques, including integer-to-integer wavelet transforms, nonlinear wavelet transforms of binary images, etc. The paper shows how different strategies of quantizating wavelet coefficients are equivalent to minimizing the error in different Lp spaces.

Later work continued on other areas of applying wavelets to image processing, including noise removal, image reconstruction, etc. This work was in done in collaboration with, among others, Ron, Antonin Chambolle, and Nam-Yong Lee.

I've collaborated loosely, but over a considerable time, with Maria Kallergi and other researchers at the University of South Florida Moffit Cancer Research Center. For one study, I designed a wavelet compression method specifically for mammography, by learning from radiologists which image features are important for diagnosis and tailoring the quantization strategy so that those features are given importance as data is removed. The previous mathematical study was helpful here, in that it interpreted the compression method as, first, determining the relative importance of features at different scales, and second, determining the quantization strategy so that all important feature scales degrade at the same rate. The study concluded that, after compressing the digitized mammograms at an average rate of over 50-1 (individual compression rates ranged from 14-to-1 to over 2000-to-1, depending on the complexity of the individual mammographic image), radiologists interpreted the compressed mammograms more accurately than they did the originals. This study was published in Radiology .

I have a number of other interests of a more amateur nature, mainly in computing, including random number generation, the Scheme programming language, algorithms for computing with large integers, computation of elementary functions, high-performance computing, etc. Scientific computation has motivated nearly all aspects of my research program.

Publications

Certain papers on wavelets and image processing and on numerical methods for partial differential equations are available for download.

Publications listed in Math Reviews as of October 6, 2015

See also "Other Publications" below.

[1]	Antonin Chambolle, Stacey E. Levine, and Bradley J. Lucier. An upwind finite-difference method for total variation-based image smoothing. SIAM J. Imaging Sci. , 4(1):277-299, 2011. [ bib | DOI | http ]
[2]	Jingyue Wang and Bradley J. Lucier. Error bounds for finite-difference methods for Rudin-Osher-Fatemi image smoothing. SIAM J. Numer. Anal. , 49(2):845-868, 2011. [ bib | DOI | http ]
[3]	Antonin Chambolle and Bradley J. Lucier. Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space. IEEE Trans. Image Process. , 10(7):993-1000, 2001. [ bib | DOI | http ]
[4]	Nam-Yong Lee and Bradley J. Lucier. Wavelet methods for inverting the Radon transform with noisy data. IEEE Trans. Image Process. , 10(1):79-94, 2001. [ bib | DOI | http ]
[5]	Antonin Chambolle, Ronald A. DeVore, Nam-yong Lee, and Bradley J. Lucier. Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. , 7(3):319-335, 1998. [ bib | DOI | http ]
[6]	Antonin Chambolle and Bradley J. Lucier. Un principe du maximum pour des opérateurs monotones. C. R. Acad. Sci. Paris Sér. I Math. , 326(7):823-827, 1998. [ bib | DOI | http ]
[7]	Ronald A. DeVore and Bradley J. Lucier. On the size and smoothness of solutions to nonlinear hyperbolic conservation laws. SIAM J. Math. Anal. , 27(3):684-707, 1996. [ bib | DOI | http ]
[8]	C.-c. Hsiao, B. Jawerth, B. J. Lucier, and X. M. Yu. Near optimal compression of orthonormal wavelet expansions. In Wavelets: mathematics and applications , Stud. Adv. Math., pages 425-446. CRC, Boca Raton, FL, 1994. [ bib ]
[9]	Lawrence G. Brown and Bradley J. Lucier. Best approximations in L 1 are near best in L p , p <1. Proc. Amer. Math. Soc. , 120(1):97-100, 1994. [ bib | DOI | http ]
[10]	Ronald A. DeVore, Björn Jawerth, and Bradley J. Lucier. Surface compression. Comput. Aided Geom. Design , 9(3):219-239, 1992. [ bib | DOI | http ]
[11]	Bradley J. Lucier. Wavelets and image compression. In Mathematical methods in computer aided geometric design, II (Biri, 1991) , pages 391-400. Academic Press, Boston, MA, 1992. [ bib ]
[12]	Ronald A. DeVore and Bradley J. Lucier. Wavelets. In Acta numerica, 1992 , Acta Numer., pages 1-56. Cambridge Univ. Press, Cambridge, 1992. [ bib ]
[13]	Ronald A. DeVore, Björn Jawerth, and Bradley J. Lucier. Image compression through wavelet transform coding. IEEE Trans. Inform. Theory , 38(2, part 2):719-746, 1992. [ bib | DOI | http ]
[14]	Ronald A. DeVore and Bradley J. Lucier. High order regularity for conservation laws. Indiana Univ. Math. J. , 39(2):413-430, 1990. [ bib | DOI | http ]
[15]	Ronald A. DeVore and Bradley J. Lucier. High order regularity for solutions of the inviscid Burgers equation. In Nonlinear hyperbolic problems (Bordeaux, 1988) , volume 1402 of Lecture Notes in Math. , pages 147-154. Springer, Berlin, 1989. [ bib | DOI | http ]
[16]	Bradley J. Lucier. Regularity through approximation for scalar conservation laws. SIAM J. Math. Anal. , 19(4):763-773, 1988. [ bib | DOI | http ]
[17]	David Hoff and Bradley J. Lucier. Numerical methods with interface estimates for the porous medium equation. RAIRO Modél. Math. Anal. Numér. , 21(3):465-485, 1987. [ bib ]
[18]	Bradley J. Lucier and Ross Overbeek. A parallel adaptive numerical scheme for hyperbolic systems of conservation laws. SIAM J. Sci. Statist. Comput. , 8(2):S203-S219, 1987. Parallel processing for scientific computing (Norfolk, Va., 1985). [ bib | DOI | http ]
[19]	Bradley J. Lucier. On nonlocal monotone difference schemes for scalar conservation laws. Math. Comp. , 47(175):19-36, 1986. [ bib | DOI | http ]
[20]	Bradley J. Lucier. A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. , 46(173):59-69, 1986. [ bib | DOI | http ]
[21]	Bradley J. Lucier. Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal. , 22(6):1074-1081, 1985. [ bib | DOI | http ]
[22]	Bradley J. Lucier. On Sobolev regularizations of hyperbolic conservation laws. Comm. Partial Differential Equations , 10(1):1-28, 1985. [ bib | DOI | http ]
[23]	Bradley J. Lucier. A stable adaptive numerical scheme for hyperbolic conservation laws. SIAM J. Numer. Anal. , 22(1):180-203, 1985. [ bib | DOI | http ]
[24]	Thomas Nagylaki and Bradley Lucier. Numerical analysis of random drift in a cline. Genetics , 94(2):497-517, 1980. [ bib ]

Other Publications

See also "Publications listed in Math Reviews" above.

1. Owen G. Rehrauer, Bharat R. Mankani, Gregery T. Buzzard, Bradley J. Lucier, and Dor Ben-Amotz. Fluorescence Modeling for Optimized-Binary Compressive Detection Raman Spectroscopy. Optics Express , 23 (2015), 23935-23951
2. David S. Wilcox, Gregery T. Buzzard, Bradley J. Lucier, Owen G. Rehrauer, Ping Wang, and Dor Ben-Amotz. Digital compressive quantitation and hyperspectral imaging. Analyst , 138:4982-4990, 2013
3. Gregery T. Buzzard and Bradley J. Lucier. Optimal filters for high-speed compressive detection in spectroscopy. In Proceedings of SPIE Volume 8657, Computational Imaging XI, 865707 (February 14, 2013).
4. David S. Wilcox, Gregery T. Buzzard, Bradley J. Lucier, Ping Wang, and Dor Ben-Amotz. Photon level chemical classification using digital compressive detection. Analytica Chimica Acta , 755:17-27, 2012
5. Maria Kallergi, John J. Heine, and Bradley J. Lucier. Observer evaluations of wavelet methods for the enhancement and compression of digitized mammograms. In Digital Mammography , volume 4042 of Lecture Notes in Computer Science , pages 482-489. Springer, Berlin, 2006.
6. Maria Kallergi, Bradley J. Lucier, Claudia G. Berman, Maria R. Hersh, J. Kim Jihai, Margaret S. Szabunio, and Robert A. Clark. High-performance wavelet compression for mammography: localization response operating characteristic evaluation. Radiology , 238(1):62-73, 2006.
7. Bradley J. Lucier. Wavelet smoothing of functional magnetic resonance images: a preliminary report. In Wavelets: Applications in Signal and Image Processing X , Michael A. Unser, Akram Aldroubi, and Andrew F. Laine, eds., volume 5207 of Proceedings of SPIE, pages 134-146. SPIE, 2003.
8. Bradley J. Lucier. Numerical partial differential equations in Scheme. In Proceedings of the Workshop on Scheme and Functional Programming 2000 .
9. Dhiraj Kacker, A. Ufuk Agar, Jan P. Allebach, and Bradley J. Lucier. Wavelet decomposition based representation of nonlinear color transformations and comparison with sequential linear interpolation. In Proceedings of the Fifth IEEE International Conference on Image Processing , vol. 1, pages 186-190. IEEE, Piscataway, NJ, 1998.
10. Bradley J. Lucier, Sudhakar Mamillapalli, and Jens Palsberg. Program optimization for faster genetic programming. In Genetic Programming 1998: Proceedings of the Third Annual Conference , J. R. Koza et al., eds, pages 202-207. Morgan Kaufmann, San Francisco, 1998.
11. Ronald A. DeVore, Bradley J. Lucier, and Zeshang Yang. Feature extraction in digital mammography. In Wavelets in Medicine and Biology , A. Aldroubi and M. Unser, eds., pages 145-161. CRC Press, Boca Raton, 1996.
12. Zeshang Yang, Maria Kallergi, Ronald A. DeVore, Bradley J. Lucier, Wei Qian, Robert A. Clark, Lawrence P. Clarke. Effect of wavelet bases on compressing digital mammograms. Engineering in Medicine and Biology Magazine , 14(5):570-577, 1995.
13. Ronald A. DeVore and Bradley J. Lucier. Classifying the smoothness of images: theory and applications to wavelet image processing. In ICIP-94: Proceedings of the 1994 IEEE International Conference on Image Processing , vol. 2, pages 6-10. IEEE Press, Piscataway, NJ, 1994.
14. Bradley J. Lucier, Maria Kallergi, Wei Qian, Ronald A. DeVore, Robert A. Clark, Edward B. Saff, and Lawrence P. Clarke. Wavelet compression and segmentation of mammographic images. Journal of Digital Imaging: The Official Journal of the Society for Computer Applications in Radiology , 7:27-38, 1994.
15. Ronald A. DeVore and Bradley J. Lucier. Smoothness spaces and wavelet decompositions. In Curves and Surfaces in Computer Vision and Graphics III , J. D. Warren, ed., volume 1830 of the Proceedings of SPIE, pages 2-12. SPIE, 1992.
16. Ronald A. DeVore and Bradley J. Lucier. Fast wavelet techniques for near-optimal image processing. In Military Communications Conference, 1992. MILCOM '92, Conference Record , pages 1129-1135. IEEE, Piscataway, NJ, 1992.
17. Ronald A. DeVore, Björn Jawerth, and Bradley J. Lucier. Data compression using wavelets: error, smoothness, and quantization. In DCC '91, Data Compression Conference, 1991 , J. A. Storer and J. H. Reif, eds., pages 186-195. IEEE Computer Society, Los Alamitos, CA, 1991.
18. Bradley J. Lucier. Performance evaluation for multiprocessors programmed using monitors. In Proceedings of the 1988 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems , Special Issue of the SIGMETRICS Performance Evaluation Review, 16(1):22-29, 1988.