## Papers on Wavelets and Image Processing

The following papers by Bradley Lucier and his collaborators and students deal with image processing and wavelet theory and applications. All papers are in Adobe Acrobat format (PDF).

Wavelets and Approximation Theory, by Bradley J. Lucier.

Abstract: These notes, a work in progress, explore the relationship between wavelets and approximation theory. I intend to eventually cover Besov spaces, sequence norms of wavelet coefficients, nonlinear approximation with wavelets, connection to image compression and noise removal, etc. The notes, based loosely on a graduate course I taught a number of times in the 1990s, were first produced in Fall 2005 while I was a long-term visitor at the IMA during their emphasis year on Imaging, and they've been revised a number of times since then, most notably in 2012--2013 when I was on sabbatical visiting Stacey Levine at Duquesne University.

Wavelets, by Ronald A. DeVore and Bradley J. Lucier, Acta Numerica 92, A. Iserles, ed., Cambridge University Press, New York, 1992, 1-56.

Review: This is a survey of the fast-growing area of wavelets from an approximation theory point of view. It is good reading for those who have an interest in learning about wavelets for the first time (as an overview and guide to further reading) as well as for those who know some aspect of wavelets and want to see what the approximation-theoretic perspective has to offer. The paper begins with a fairly detailed description of the Haar wavelets to motivate the material that follows. The construction of wavelets section begins with an overview of multiresolution analysis and the basic framework of shift invariant spaces in which context constructions of orthogonal wavelets and prewavelets are discussed. Primary examples here are cardinal spline wavelets and prewavelets in one dimension and box spline wavelets in several dimensions. The authors include a heuristic discussion of the Daubechies construction of compactly supported orthogonal wavelets highlighting the major points. There is a section on the fast wavelet transform, a section that relates the coefficients in a wavelet expansion of a function to its smoothness properties, and a final section on applications. The connections to approximation theory are evident throughout: from the role of shift invariant spaces in the theory, the discussions of smoothness and approximation order in the constructions, and most decidedly in the relation of spaces of coefficients, to smoothness spaces and how this relation together with nonlinear approximation theory techniques can be used in applications. (Review by S. Riemenschneider in Math. Reviews)

Data Compression using Wavelets: Error, Smoothness, and Quantization, by Ronald A. DeVore, Bjorn Jawerth, and Bradley J. Lucier, in DCC-91, Data Compression Conference, J. A. Storer and J. H. Reif, eds., IEEE Computer Society Press, Los Alamitos, CA, 1991, 186-195.

Abstract: Recently, a theory, developed by DeVore, Jawerth, and Popov, of nonlinear approximation by both orthogonal and nonorthogonal wavelets has been applied to problems in surface and image compression by DeVore, Jawerth, and Lucier. This theory relates precisely the norms in which the error is measured, the rate of decay in that error as the compression decreases, and the smoothness of the data. In addition, one can interpret the error incurred by the quantization of wavelet coefficients in terms of this theory. In this talk we give an overview of the previous results, and expand our argument, made earlier for image compression, that frequency-amplitude response curves that arise quite naturally in problems involving human visual and audio perception should be used to decide the quantization strategy for wavelet coefficients and the norm in which to measure the error in compressed data.

Wavelets and Image Compression, by Bradley J. Lucier, in Mathematical Methods in CAGD and Image Processing, Tom Lyche and L. L. Schumaker (eds.), Academic Press, 1992, 391-400.

Abstract: In this paper we present certain results about the com- pression of images using wavelets. We concentrate on the simplest case of the Haar decomposition and compression in L 2 . Further results about compression in L p , p not 2, are mentioned.

Image Compression Through Wavelet Transform Coding, by Ronald A. DeVore, Bjorn Jawerth, and Bradley J. Lucier, IEEE Transactions on Information Theory, 38 (1992), 719-746.

Abstract: A novel theory is introduced for analyzing image compression methods that are based on compression of wavelet decompositions. This theory precisely relates (a) the rate of decay in the error between the original image and the compressed image as the size of the compressed image representation increases (i.e., as the amount of compression decreases) to (b) the smoothness of the image in certain smoothness classes called Besov spaces. Within this theory, the error incurred by the quantization of wavelet transform coefficients is explained. Several compression algorithms based on piecewise constant approximations are analyzed in some detail. It is shown that, if pictures can be characterized by their membership in the smoothness classes considered, then wavelet-based methods are near-optimal within a larger class of stable transform-based, nonlinear methods of image compression. Based on previous experimental research it is argued that in most instances the error incurred in image compression should be measured in the integral sense instead of the mean-square sense.

Surface Compression, by Ronald A. DeVore, Bjorn Jawerth, and Bradley J. Lucier, Computer Aided Geometric Design, 9 (1992), 219-239.

Abstract: We propose wavelet decompositions as a technique for compressing the number of control parameters of surfaces that arise in Computer-Aided Geometric Design. In addition, we give a specific numerical algorithm for surfact compression based on wavelet decompositions of surfaces into box splines.

Fast Wavelet Techniques for Near-Optimal Image Processing, by Ronald A. DeVore and Bradley J. Lucier, IEEE Military Communications Conference Record, San Diego, October 11-14, 1992, IEEE, Piscataway, NJ, 1992, 1129-1135.

Best approximations in $L_1$ are near best in $L_p$, $p<1$, by Lawrence G. Brown and Bradley J. Lucier, Proceedings of the American Mathematical Society, 120 (1994), 97-100

Abstract: We show that any best $L^1$ polynomial approximation to a function $f$ in $L^p$, $0<p<1$, is near best in $L^p$.

Classifying the Smoothness of Images: Theory and Applications to Wavelet Image Processing, by Ronald A. DeVore and Bradley J. Lucier, in ICIP-94: Proceedings of the 1994 IEEE International Conference on Image Processing, Austin, TX, November 13-16, 1994, IEEE Press, Los Alamitos, CA, Vol. II, 6-10.

Abstract: Devore, Jawerth, and Lucier have previously intro- duced a definition of the smoothness of images that is directly related to the performance of wavelet com- pression schemes. In this paper we survey previous results on the equivalence between smoothness, rate of decay of the wavelet coefficients, and efficiency of wavelet compression techniques applied to images. We report on other applications including deciding how many pixel quantization intervals are needed to pre- serve smoothness, and the fast solution of variational problems that arise naturally in several areas of image processing.

Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage, by Antonin Chambolle, Ronald A. DeVore, Nam-yong Lee, and Bradley J. Lucier, IEEE Transactions on Image Processing, 7 (1998), 319-355. Special Issue on Partial Differential Equations and Geometry-Driven Diffusion in Image Processing and Analysis.

Abstract: This paper examines the relationship between wavelet-based image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem: given an image $F$ defined on a square $I$,minimize over all $g$ in the Besov space $B^1_1(L_1(I))$ the functional $\|F-g\|_{L_2(I)}^2+\lambda\|g\|_{B^1_1(L_1(I))}$. We use the theory of nonlinear wavelet image compression in $L_2(I)$ to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d.,&nbsp;mean zero, Gaussian noise. A new signal-to-noise ratio, which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image $F$: the largest $\alpha$ for which $F\in B^\alpha_q(L_q(I))$, $1/q=\alpha/2+1/2$, and the norm $\|F\|_{B^\alpha_q(L_q(I))}$. Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Donoho and Johnstone's VisuShrink procedure; an example suggests, however, that Donoho and Johnstone's SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves lower error than our procedure.

The version of the paper contained on this web page is longer than the published version, and contains the proofs of some theorems omitted from the published paper, an extra section on biorthogonal wavelets, and a section that claims, through several examples, that the human visual system is quite sensitive to image processes that introduce changes to the Besov smoothness of images.

Program Optimization for Faster Genetic Programming, by Bradley J. Lucier, Sudhakar Mamillapalli, and Jens Palsberg, Proceedings of Genetic Programming 1998: Proceedings of the Third Annual Conference, J. R. Koza et al., eds., 1998, Morgan Kaufmann, San Francisco, 202--207.

Abstract: We have used genetic programming to develop efficient image processing software. The ultimate goal of our work is to detect certain signs of breast cancer that cannot be detected with current segmentation and classification methods. Traditional techniques do a relatively good job of segmenting and classifying small-scale features of mammograms, such as micro-calcification clusters. Our strongly-typed genetic programs work on a multi-resolution representation of the mammogram, and they are aimed at handling features at medium and large scales, such as stellated lesions and architectural distortions. The main problem is efficiency. We employ program optimizations that speed up the evolution process by more than a factor of ten. In this paper we present our genetic programming system, and we describe our optimization techniques.

Wavelet-Vaguelette Decompositions and Homogeneous Equations, by Namyong Lee (Ph.D. Thesis, December 1997).

Abstract: We describe the wavelet-vaguelette decomposition (WVD) for solving a homogeneous equation $Y = Af + Z$, where $A$ satisfies $\widehat{A^{\ast}Af}(\xi) = {\vert{\xi}\vert}^{-2\alpha}\widehat{f}(\xi)$ for some $\alpha\ge0$. We find a sufficient condition on functions to have a WVD. This result generalizes Daubechies's work on the discrete wavelet transform. We examine the relation between the WVD-based method and variational problems for solving a homogeneous equation. Algorithms are derived as exact minimizers of variational problems of the form; given observed function $Y$, minimize over all $g$ in the Besov space $B_{1,1}^{\beta_0}(R^d)$ the functional ${\|Y-Ag\|}_{\mathcal Y}^2+2\gamma{\vert{g}\vert}_{B_{1,1}^{\beta_0}}$, where $\mathcal Y$ is a separable Hilbert space. We use the theory of nonlinear wavelet approximation in $L^2(R^d)$ to derive accurate error bounds for recovering $f$ through wavelet shrinkage applied to observed data $Y$ corrupted with independent and identically distributed mean zero Gaussian noise $Z$. We give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter. We conduct tomographic reconstruction computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about a phantom image $f$: the largest $\beta$ for which $f \in B_{p,p}^{\beta}(R^2)$, $p = {\frac{3}{\beta+3/2}}$, and the seminorm ${\vert{f}\vert}_{B_{p,p}^{\beta}}$. Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Kolaczyk's procedure and classical filtered backprojection method.

The next paper is based on this thesis and some later work.

Wavelet Methods for Inverting the Radon Transform with Noisy Data, by Namyong Lee and Bradley J. Lucier , IEEE Transactions on Image Processing, 10 (2001), 79-94.

Abstract: Because the Radon transform is a smoothing transform, any noise in the Radon data becomes magnified when the inverse Radon transform is applied. Among the methods used to deal with this problem for the Radon transform and other homogeneous equations is the Wavelet-Vaguelette Decomposition (WVD) coupled with Wavelet Shrinkage, as introduced by David Donoho. We extend several results of Donoho and others here. First, we introduce a new sufficient condition on wavelets to generate a WVD, which generalizes a result of Daubechies on the discrete wavelet transform. For a general homogeneous operator $A$, which class includes the Radon transform, we show that a variant of Donoho's method for solving inverse problems can be derived as exact minimizers of variational problems of the form: given the observed data $Y$, minimize over all $g$ in the Besov space $B_{1}^{\beta_0}(L_1(\Bbb R^d))$ the functional ${\|Y-Ag\|}_{\mathcal Y}^2+2\gamma{\vert{g}\vert}_{B_{1}^{\beta_0}(L_1(\Bbb R^d))}$, where $\mathcal Y$ is a separable Hilbert space containing the range of $A$. We use the theory of nonlinear wavelet approximation in $L_2(\Bbb R^d)$ to derive accurate error bounds for recovering $f$ through wavelet shrinkage applied to observed data $Y$ corrupted with independent and identically distributed, mean zero, Gaussian noise $Z$. One intriguing result of this analysis is that there is only one value of $\beta_0$, depending on $\alpha$, the homogeneity index of $A$, and $d$, for which the error remains bounded no matter the number of observations or the value of the regularizing parameter $\gamma$. (For the Radon transform, $\alpha=1/2$, and the optimal value of $\beta_0$ is $d/2-\alpha=1/2$ in two dimensions.) We give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter. We conduct tomographic reconstruction computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image $f$: the largest $\beta$ for which $f \in B_{p}^{\beta}(L_p(\Bbb R^d))$, $p = {{3}/{(\beta+3/2)}}$, and the semi-norm ${\vert{f}\vert}_{B_{p}^{\beta}(L_p(\Bbb R^d))}$. Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Kolaczyk's variant of Donoho's method and the classical filtered backprojection method.

Interpreting Translation-Invariant Wavelet Shrinkage as A New Image Smoothing Scale Space, by Antonin Chambolle and Bradley J. Lucier, IEEE Transactions on Image Processing, 10 (2001), 993-1000.

Abstract: Ronald Coifman and David Donoho suggested translation-invariant wavelet shrinkage as a means of removing noise from images. Basically, this applies wavelet shrinkage to a two-dimensional version of the semi-discrete wavelet representation of Mallat and Zhong. Coifman and Donoho also showed how the method could be implemented in $O(N\log N)$ operations, where there are $N$ pixels, which compares to $O(N)$ operations for ordinary wavelet shrinkage, and $O(N\log N)$ operations for the Fast Fourier Transform. In this paper, we provide a mathematical framework for iterated translation-invariant wavelet shrinkage, and show, using a theorem of Kato and Masuda, that with orthogonal wavelets it is equivalent to gradient descent in $L_2(I)$ along the semi-norm for the Besov space $B^1_1(L_1(I))$, which, in turn, can be interpreted as a new nonlinear wavelet-based image smoothing scale space.

Nonlinear Wavelet Approximation in Anisotropic Besov Spaces, by Christopher Leisner.

Abstract: We introduce new anisotropic wavelet decompositions associated with the smoothness $\boldsymbol\beta$, $\boldsymbol\beta=(\beta_1,\dots,\beta_d)$, $\beta_1,\dots,\beta_d>0$ of multivariate functions as measured in anisotropic Besov spaces $B^{\boldsymbol\beta}$. We give the rate of nonlinear approximation of functions $f\in B^{\boldsymbol\beta}$ by these wavelets. Finally, we prove that, among a general class of anisotropic wavelet decompositions of a function $f\in B^{\boldsymbol\beta}$, the anisotropic wavelet decomposition associated with $\boldsymbol\beta$ gives the optimal rate of compression of the wavelet decomposition of $f$.

Wavelet Smoothing of Functional Magnetic Resonance Images: A Preliminary Report, by Bradley J. Lucier, Proceedings of the SPIE-Volume 5207, Wavelets: Applications in Signal and Image Processing X, M. A. Unser, A. Aldroubi, and A. F. Laine, eds., November 2003, 134-146.

Abstract: Functional (time-dependent) Magnetic Resonance Imaging can be used to determine which parts of the brain are active during various limited activities; these parts of the brain are called activation regions. In this preliminary study we describe some experiments that are suggested from the following questions: Does one get improved results by analyzing the complex image data rather than just the real magnitude image data? Does wavelet shrinkage smoothing improve images? Should one smooth in time as well as within and between slices? If so, how should one model the relationship between time smoothness (or correlations) and spatial smoothness (or correlations). The measured data is really the Fourier coefficients of the complex image---should we remove noise in the Fourier domain before computing the complex images? In this preliminary study we describe some experiments related to these questions.

High-performance Wavelet Compression for Mammography: Localization Response Operating Characteristic Evaluation, by Maria Kallergi, Bradley J. Lucier, Claudia G. Berman, Marla R. Hersh, Jihai J. Kim, Margaret S. Szabunio, and Robert A. Clark, Radiology, 238 (2006), 62-73.

Purpose: To evaluate the accuracy of a visually lossless, image-adaptive, wavelet-based compression method for achievement of high compression rates at mammography. Materials and Methods: The study was approved by the institutional review board of the University of South Florida as a research study with existing medical records and was exempt from individual patient consent requirements. Patient identifiers were obliterated from all images. The study was HIPAA compliant. An algorithm based on scale-specific quantization of biorthogonal wavelet coefficients was developed for the compression of digitized mammograms with high spatial and dynamic resolution. The method was applied to 500 normal and abnormal mammograms from 278 patients who were 32-85 years old, 85 of whom had biopsy-proved cancer. Film images were digitized with a charge-coupled device-based digitizer. The original and compressed reconstructed images were evaluated in a localization response operating characteristic experiment involving three radiologists with 2-10 years of experience in reading mammograms. Results: Compression rates in the range of 14:1 to 2051:1 were achieved, and the rates were dependent on the degree of parenchymal density and the type of breast structure. Ranges of the area under the receiver operating characteristic curve were 0.70-0.83 and 0.72-0.86 for original and compressed reconstructed mammograms, respectively. Ranges of the area under the localization response operating characteristic curve were 0.39-0.65 and 0.43-0.71 for original and compressed reconstructed mammograms, respectively. The localization accuracy increased an average of 6% (0.04 of 0.67) with the compressed mammograms. Localization performance differences were statistically significant with P = 0.05 and favored interpretation with the wavelet-compressed reconstructed images. Conclusion: The tested wavelet-based compression method proved to be an accurate approach for digitized mammography and yielded visually lossless high-rate compression and improved tumor localization.

Observer Evaluations of Wavelet Methods for the Enhancement and Compression of Digitized Mammograms, by Maria Kallergi, John J. Heine, and Bradley J. Lucier, in IWDM 2006, Lecture Notes in Computer Science 4016 (2006), 482-489.

Abstract: Two observer experiments were performed to evaluate the performance of wavelet enhancement and compression methodologies for digitized mammography. One experiment was based on the localization response operating characteristic (LROC) model. The other estimated detection and localization accuracy rates. The results of both studies showed that the two algorithms consistently improved radiologists' performance although not always in a statistically significant way. An important outcome of this work was that lossy wavelet compression was as successful in improving the quality of digitized mammograms as the wavelet enhancement technique. The compression algorithm not only did not degrade the readers' performance but it improved it consistently while achieving compression rates in the range of 14 to 2051:1. The proposed wavelet algorithms yielded superior results for digitized mammography relative to conventional processing methodologies. Wavelets are valuable and diverse tools that could make digitized screen/film mammography equivalent to its direct digital counterpart leading to a filmless mammography clinic with full inter- and intra-system integration and real-time telemammography.

An Upwind Finite-Difference Method for Total Variation--Based Image Smoothing, by Antonin Chambolle, Stacey E. Levine, and Bradley J. Lucier, SIAM Journal on Imaging Sciences, 4 (2011), 277-299.

Abstract: In this paper we study finite-difference approximations to the variational problem using the BV smoothness penalty that was introduced in an image smoothing context by Rudin, Osher, and Fatemi. We give a dual formulation for an upwind finite-difference approximation for the BV seminorm; this formulation is in the same spirit as one popularized by the first author for a simpler, less isotropic, finite-difference approximation to the (isotropic) BV seminorm. We introduce a multiscale method for speeding the approximation of both Chambolle's original method and of the new formulation of the upwind scheme. We demonstrate numerically that the multiscale method is effective, and we provide numerical examples that illustrate both the qualitative and quantitative behavior of the solutions of the numerical formulations.

Error Bound for Numerical Methods for the ROF Image Smoothing Model, by Jingyue Wang (Ph.D. Thesis, August 2008).

Abstract: The Rudin-Osher-Fatemi variational model has been extensively studied and used in image analysis. There have been several very successful numerical algorithms developed to compute the minimizer of the discrete version of the ROF energy. We study the convergence of numerical solutions of discrete total variation models to the solution of the continuous model. We use the discrete ROF energy with a symmetric discrete TV operator and obtain an error bound between the minimizer for the discrete ROF model with a symmetric TV operator and the minimizer for the continuous ROF model. Partial results are also obtained on error bounds of some non-symmetric discrete TV minimizers.

The next paper is based on this thesis and some later work.

Error Bounds for Finite-Difference Methods for Rudin-Osher-Fatemi Image Smoothing, by Jingyue Wang and Bradley J. Lucier, SIAM Journal on Numerical Analysis, 49 (2011), 845-868.

Abstract: We bound the difference between the solution to the continuous Rudin-Osher-Fatemi image smoothing model and the solutions to various finite-difference approximations to this model. These bounds apply to "typical" images, i.e., images with edges or with fractal structure. These are the first bounds on the error in numerical methods for ROF smoothing.

Photon Level Chemical Classification using Digital Compressive Detection, by David S. Wilcox, Gregery T. Buzzard, Bradley J. Lucier, Ping Wang, and Dor Ben-Amotz, Analytica Chimica Acta, 755 (2012), 17-27.

Abstract: A key bottleneck to high-speed chemical analysis, including hyperspectral imaging and monitoring of dynamic chemical processes, is the time required to collect and analyze hyperspectral data. Here we describe, both theoretically and experimentally, a means of greatly speeding up the collection of such data using a new digital compressive detection strategy. Our results demonstrate that detecting as few as ~10 Raman scattered photons (in as little time as ~30 microseconds) can be sufficient to positively distinguish chemical species. This is achieved by measuring the Raman scattered light intensity transmitted through programmable binary optical filters designed to minimize the error in the chemical classification (or concentration) variables of interest. The theoretical results are implemented and validated using a digital compressive detection instrument that incorporates a 785 nm diode excitation laser, digital micromirror spatial light modulator, and photon counting photodiode detector. Samples consisting of pairs of liquids with different degrees of spectral overlap (including benzene/acetone and n-heptane/n-octane) are used to illustrate how the accuracy of the present digital compressive detection method depends on the correlation coefficients of the corresponding spectra. Comparisons of measured and predicted chemical classification score plots, as well as linear and non-linear discriminant analyses, demonstrate that this digital compressive detection strategy is Poisson photon noise limited and outperforms total least squares--based compressive detection with analog filters.

Optimal Filters for High-Speed Compressive Detection in Spectroscopy, by Gregery T. Buzzard and Bradley J. Lucier, Proceedings of SPIE Volume 8657, Computational Imaging XI, 865707 (February 14, 2013); doi:10.1117/12.2012700.

Abstract: Recent advances allow for the construction of filters with precisely defined frequency response for use in Raman chemical spectroscopy. In this paper we give a probabilistic interpretation of the output of such filters and use this to give an algorithm to design optimal filters to minimize the mean squared error in the estimated photon emission rates for multiple spectra. Experiments using these filters demonstrate that detecting as few as ~10 Raman scattered photons in as little time as ~30 microseconds can be sufficient to positively distinguish chemical species. This speed should allow "chemical imaging" of samples.

Digital Compressive Quantitation and Hyperspectral Imaging, by David S. Wilcox, Gregery T. Buzzard, Bradley J. Lucier, Owen G. Rehrauer, Ping Wang, and Dor Ben-Amotz, Analyst, 138 (2013), 4982-4990; doi:10.1039/C3AN00309D.

Abstract: Digital compressive detection, implemented using optimized binary (OB) filters, is shown to greatly increase the speed at which Raman spectroscopy can be used to quantify the composition of liquid mixtures and to chemically image mixed solid powders. We further demonstrate that OB filters can be produced using multivariate curve resolution (MCR) to pre-process mixture training spectra, thus facilitating the quantitation of mixtures even when no pure chemical component samples are available for training.

Figures: The journal Analyst embedded the images in the paper using the lossy compression algorithm DCTEncode rather than the lossless compression algorithm FlateEncode. This introduced certain visual distortions into the images. We offer the interested reader the original figures as sent to the journal.

Fluorescence Modeling for Optimized-Binary Compressive Detection Raman Spectroscopy, by Owen G. Rehrauer, Bharat R. Mankani, Gregery T. Buzzard, Bradley J. Lucier, and Dor Ben-Amotz, Optics Express, 23 (2015), 23935-23951; DOI:10.1364/OE.23.023935.

Abstract: The recently-developed optimized binary compressive de- tection (OB-CD) strategy has been shown to be capable of using Raman spectral signatures to rapidly classify and quantify liquid samples and to image solid samples. Here we demonstrate that OB-CD can also be used to quantitatively separate Raman and fluorescence features, and thus facilitate Raman-based chemical analyses in the presence of fluorescence background. More specifically, we describe a general strategy for fitting and suppressing fluorescence background using OB-CD filters trained on third-degree Bernstein polynomials. We present results that demonstrate the utility of this strategy by comparing classification and quantitation results obtained from liquids and powdered mixtures, both with and without fluorescence. Our results demonstrate high-speed Raman-based quantitation in the presence of moderate fluorescence. Moreover, we show that this OB-CD based method is effective in suppressing fluorescence of variable shape, as well as fluorescence that changes during the measurement process, as a result of photobleaching.

Preconditioned Conjugate Gradient Method for Boundary Artifact-Free Image Deblurring, by Nam-Yong Lee and Bradley J. Lucier, Inverse Problems and Imaging, 10 (2016), 195-225.

Abstract: Several methods have been proposed to reduce boundary artifacts in image deblurring. Some of those methods impose certain assumptions on image pixels outside the field-of-view; the most important of these assume reflective or anti-reflective boundary conditions. Boundary condition methods, including reflective and anti-reflective ones, however, often fail to reduce boundary artifacts, and, in some cases, generate their own artifacts, especially when the image to be deblurred does not accurately satisfy the imposed condition. To overcome these difficulties ,we suggest using free boundary conditions, which do not impose any restrictions on image pixels outside the field-of-view, and preconditioned conjugate gradient methods, where preconditioners are designed to compensate for the non-uniformity in contributions from image pixels to the observation. Our simulation studies show that the proposed method outperforms reflective and anti-reflective boundary condition methods in removing boundary artifacts. The simulation studies also show that the proposed method can be applicable to arbitrarily shaped images and has the benefit of recovering damaged parts in blurred images.

Binary Complementary Filters for Compressive Raman Spectroscopy, by Owen G. Rehrauer, Vu C. Dinh, Bharat R. Mankani, Gregery T. Buzzard, Bradley J. Lucier, and Dor Ben-Amotz, Applied Spectroscopy, 72 (2018) 69-78. DOI:10.1177/0003702817732324.

Abstract: The previously described optimized binary compressive detection (OB-CD) strategy enables fast hyperspectral Raman (and fluorescence) spectroscopic analysis of systems containing two or more chemical components. However, each OB- CD filter collects only a fraction of the scattered photons and the remainder of the photons are lost. Here, we present a refinement of OB-CD, the OB-CD2 strategy, in which all of the collected Raman photons are detected using a pair of complementary binary optical filters that direct photons of different colors to two photon counting detectors. The OB-CD2 filters are generated using a new optimization algorithm described in this work and implemented using a holographic volume diffraction grating and a digital micromirror device (DMD) whose mirrors are programed to selectively direct photons of different colors either to one or the other photon-counting detector. When applied to pairs of pure liquids or two-component solid powder mixtures, the resulting OB-CD2 strategy is shown to more accurately estimate Raman scattering rates of each chemical component, when compared to the original OB-CD, thus facilitating chemical classification at speeds as fast as 3 ms per measurement and the collection of Raman images in under a second.

Pointwise Besov Space Smoothing of Images, by Gregery T. Buzzard, Antonin Chambolle, Jonathan D. Cohen, Stacey E. Levine, and Bradley J. Lucier, J Math Imaging Vis, 61 (2019), 1--20. DOI:10.1007/s10851-018-0821-1.

Abstract: We formulate various variational problems in which the smoothness of functions is measured using Besov space semi-norms. Equivalent Besov space semi-norms can be defined in terms of moduli of smoothness or sequence norms of coefficients in appropriate wavelet expansions. Wavelet-based semi-norms have been used before in variational problems, but existing algorithms do not preserve edges, and many result in blocky artifacts. Here, we devise algorithms using moduli of smoothness for the $B^1_\infty(L_1(I))$ Besov space semi-norm. We choose that particular space because it is closely related both to the space of functions of bounded variation, BV(I), that is used in Rudin-Osher-Fatemi image smoothing, and to the $B^1_1(L_1(I))$ Besov space, which is associated with wavelet shrinkage algorithms. It contains all functions in BV(I), which include functions with discontinuities along smooth curves, as well as "fractal-like" rough regions; examples are given in an appendix. Furthermore, it prefers affine regions to staircases, potentially making it a desirable regularizer for recovering piecewise affine data. While our motivations and computational examples come from image processing, we make no claim that our methods "beat" the best current algorithms. The novelty in this work is a new algorithm that incorporates a translation-invariant Besov regularizer that does not depend on wavelets, thus improving on earlier results. Furthermore, the algorithm naturally exposes a range of scales that depends on the image data, noise level, and the smoothing parameter. We also analyze the norms of smooth, textured, and random Gaussian noise data in $B^1_\infty(L_1(I))$, $B^1_1(L_1(I))$, $\text{BV}(I)$ and $L_2(I)$ and their dual spaces. Numerical results demonstrate properties of solutions obtained from this moduli of smoothness-based regularizer.

Note: The journal modified (re-dimensioned, and hence smoothed) the original images in publication (which one of my co-authors called "ironic" in an image smoothing paper; I'm trying to be very polite here). The version I'm including above is a corrected preprint from which the original images can be extracted using, e.g., the Linux program "pdfimages".