Image denoising using a tight frame - IEEE Xplore

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Image Denoising Using a Tight Frame Lixin Shen, Manos Papadakis, Ioannis A. Kakadiaris, Ioannis Konstantinidis, Donald Kouri, and David Hoffman

Abstract—We present a general mathematical theory for lifting frames that allows us to modify existing filters to construct new ones that form Parseval frames. We apply our theory to design nonseparable Parseval frames from separable (tensor) products of a piecewise linear spline tight frame. These new frame systems incorporate the weighted average operator, the Sobel operator, and the Laplacian operator in directions that are integer multiples of 45 . A new image denoising algorithm is then proposed, tailored to the specific properties of these new frame filters. We demonstrate the performance of our algorithm on a diverse set of images with very encouraging results. Index Terms—Image denoising, tight frame, wavelets.

I. INTRODUCTION

N

OISE REDUCTION has a long history as an area of interest in the image processing community. Digital images are often degraded by noise in the acquisition and/or transmission phase. The goal of image denoising is to recover the true/original image from such a distorted/noisy copy. This is accomplished via a combination of methods involving suitable filtering/transforms and statistical estimation. Typically, the image is transformed onto some domain where the noise component can be identified more easily, and a statistical estimation is performed to identify and remove its influence. This is accomplished by an estimation operator that suppresses noise while preserving the true signal. Finally, the transformation is inverted to obtain the denoised image. In recent years, a wide class of image denoising algorithms have been based on the discrete wavelet transform. The usefulness of the wavelet transform was first demonstrated by Donoho and Johnstone [1]–[3], when they proved that thresholding estimators in a wavelet basis have nearly minimax risk for sets

Manuscript received June 26, 2003; revised February 22, 2005. This work was supported in part by Grants NSF IIS-0431144, NSF IIS-0335578, NSF DMS1041539, NSF CHE-0074311, ARP 003652-0071-2001, NSF DMS-0406748, the Welch Foundation E-0608, and the University of Houston’s TLCC funds. The associate editor coordinating the review of this manuscript and approving it for publication Dr. Ivan W. Selesnick. L. Shen is with the Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 USA. M. Papadakis is with the Department of Mathematics, University of Houston, Houston, TX 77204 USA. I. A. Kakadiaris is with the Computational Biomedicine Laboratory, Department of Computer Science, University of Houston, Houston, TX 77204 USA. I. Konstantinidis is with the Norbert Wiener Center for Harmonic Analysis and Applications, Department of Mathematics, University of Maryland, College Park, MD 20742 USA D. Kouri is with the Departments of Chemistry and Physics, University of Houston, Houston, TX 77204 USA. D. Hoffman is with the Department of Chemistry and Ames Laboratory, Iowa State University, Ames, IA 50011 USA. Digital Object Identifier 10.1109/TIP.2005.864240

of piecewise regular images. For the case of additive Gaussian noise they suggested two thresholding functions

otherwise where is the threshold level and is the wavelet coefficient of the underlying image. The threshold is to be selected using VisuShrink [1] or SureShrink [3]. Coifman and Donoho [4] established that the use of nondecimated transforms minimizes artifacts in the denoised data; the translation invariant denoising scheme they proposed is equivalent to thresholding in the shift-invariant redundant representation implemented by a nonsubsampled filter bank, or frame. In addition, it has been shown [5]–[8] that a redundant representation is substantially superior to a nonredundant representation for image denoising in terms of mean-squared error and signal-to-noise ratio. Several redundant representations have been applied to image denoising. Simoncelli et al. [9] introduced the “steerable pyramid”, a tight frame also used by image denoising algorithms [10]. The dual-tree complex wavelet transform introduced by Kingsbury [11] is also redundant, and it has been used by Sendur and Selesnick [12], [13]. In order to improve the selection criteria for the threshold one needs to depart from the minimax framework, which is optimal when no a priori information about the signal itself is assumed, and move to a Bayesian approach, where both the noise and the true image signal coefficients in the wavelet domain can be modeled using some prior distribution. This approach was first used by Simoncelli and Adelson [5], [14]. This is also the approach used by Chang et al. [6], [15] where BayesShrink, an adaptive, data-driven threshold, is derived in a Bayesian framework using context modeling of the global coefficients histogram. Another approach exploiting the local structure of wavelet coefficients was proposed by Mihçak et al. [16]. Recently, Portilla et al. [10] developed a local Gaussian scale mixture (GSM) model with excellent results, while Pizurica et al. [17] use an empirical model to estimate the prior. Even better results can be obtained by exploiting the fact that wavelet coefficients are statistically dependent [18], [19]. For example, a binary hidden Markov model (HMM) based on wavelet trees for denoising of one-dimensional (1-D) signals was proposed by Crouse et al. [20] and extended to image denoising by Romberg et al. [21]. It is interesting to note that the GSM model [10] can be interpreted as a continuous form of this binary HMM. HMMs were also used by Fan [22]. Sendur and Selesnick [12], [13] propose a new bivariate model by considering dependencies between the wavelet coefficients and their parents.

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SHEN et al.: IMAGE DENOISING USING A TIGHT FRAME

The work presented in this paper complements the existing literature. We design nonseparable Parseval frames from separable (tensor) products of a piecewise linear spline tight frame. These new frame systems incorporate the weighted average operator, the Sobel operator, and the Laplacian operator in directions that are integer multiples of 45 . We propose a mixed thresholding strategy that is tailored to the specific operators in this tight frame and takes advantage of the dependencies between them. The remainder of this paper is organized as follows. Section II provides a method of modifying existing frames in order to produce new ones. In Section II, we apply this method to construct a frame-based filtering scheme that is sensitive to detecting singularities of first or second order at certain orientations. This filtering scheme is employed in Section III for the construction of a denoising algorithm. In addition, we describe an iterative scheme to further improve the quality of the denoised image. Our experimental results are presented in Section IV, and our conclusions are given in Section V. II. CONSTRUCTING NEW FRAMELET DIGITAL FILTERS In this section, we present a general mathematical theory for lifting frames that allows us to modify existing frame-based (framelet) filters to construct new ones that form Parseval frames. We apply our theory to design nonseparable Parseval frames from separable (tensor) products of the piecewise linear spline tight frame described by Ron and Shen [23]. These new framelet filters implement the Sobel and discrete Laplace operators at directions which are integer multiples of 45 . As explained in the introduction, redundant representations, such as frames, are preferable for denoising purposes. Designing frames also proves to be more efficient than designing bases, especially when one designs Riesz or orthonormal bases in multi-dimensions arising from scaling functions [24]. Our goal is to construct digital filters using the integer translates of certain finite-length filters. More precisely, the Hilbert space of digital signals we wish to work with in our applications , where , although the results presented in this is section hold true for any natural number . We define the transacting on by , for lation operator . An element of is a digital filter if its every inverse Fourier transform is a bounded function. In order to eliminate any possible ambiguity, we define the Fourier transby form of

This definition imposes a scaled version of the definition of as follows: the usual inner product in , where . A digital filter acts on a digital signal by con). This defines a bounded operator on volution (i.e., . If is a bounded operator mapping the Hilbert space to , then the operator-adjoint of , also denoted by , is de, fined as the operator mapping into via and . It can be proved that is bounded. for every If , then the operator-adjoint of and the conjugate

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transpose are identical. This is the reason for using the same notation for both. of elements of a Hilbert space , A family where is an index set, is a frame for if there exist constants such that for every we have that

We refer to the optimal positive constants as frame bounds. We refer to the frame as tight if , and as Parseval frame . The advantage of PF versus other types of (PF) if frames is that the same set of vectors can be used for decomposition and reconstruction, and signal energy remains constant, just as in the case of orthonormal bases. Exact frames (frames having no redundancy) are Riesz bases and vice versa. A finite of elements in generates a frame set is a frame of if the family of . We begin by stating the following general result. , be a finite set of digProposition 1: Let , is a frame of ital filters. Then if and only if there exist such that for almost the following inequality holds: every

Moreover, this frame is a Parseval frame if and only if . The proof of this proposition is provided in the Appendix. The following corollary immediately follows from Proposition 1, and it describes the idea on which our method of lifting existing framelet filters is based. is a finite Corollary 1: Assume that , with set of digital filters that generates a frame for . For a given -periodic positive integer , let be a matrix-valued function whose entries are continuous (or more generally, measurable and bounded functions). If there such that for almost every exists then the matrix multiplication defines a new family of digital filters which generate . If, in particular, is an isometry, for a frame for , then the resulting and the original almost every frames have the same frame bounds. The subject of three-channel filterbanks based on tight frames that are derived from MRAs has an extensive bibliography [25]–[30]. We use Corollary 1 to lift the frame described by Ron and Shen [23] as being the simplest example of a compactly supported tight spline frame. We select splines of degree one in order to keep the number of wavelets to a minimum. The Parseval frame is generated by the following low-pass , filters: bandpass , and high-pass and Their corresponding inverse Fourier transforms are, respectively and

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It is easy to note that (1)

to entwine the two new copies while leaving the originals unfrom changed. In summary, we apply the isometry matrix into , where

and, therefore, as Proposition 1 implies, , , and form a . Parseval frame for and , The Riesz scaling function and the wavelets , , and , respectively, are given by associated with Next, by considering the decompositions and , we define the matrix-valued function and

The interesting properties of these functions are the following: are symmetric while is 1) is interpolatory; 2) and and are supported on the interval anti-symmetric; 3) , ; and 4) the frame-wavelets and generate a frame . The anti-symmetric function can be used as a firstof can order singularity detector while the symmetric function be used as a second-order singularity detector. This is the reason for selecting these framelets for the proposed denoising scheme. The tensor product of the 1-D PF of Ron and Shen with itself is another PF arising from nine separable (tensor product) filters

where

. The following equality holds:

which is also unitary, since and, thus, are isometries , for every . Thus, Corollary 1 applies to gives rise to an 11-band filter bank, with the integer and translates of the filters defined by the coordinate functions of forming a PF of . We call this the UHF11 filter bank. In summary, the UHF11 filter bank is implemented by the is the low pass filter; the highfollowing filters: pass filters are , , , , , , , , , and . We note that the second and third coordinates of still define ) in the horizontal and verthe Sobel operators (scaled by tical directions, respectively. In addition, the impulse responses of the fourth and fifth coordinates of are given by

(2) Thus, we view as a low-pass filter, and the remaining eight filters as band-pass and high-pass. We refer to these filters collectively as the UHF9 filter bank. The filter taps corresponding . to it are given by the nine 3 3 matrices and in the UHF9 filter bank We note that the filters are the Sobel operators detecting vertical and horizontal edges. This motivates us to augment bank UHF9 with two diagonal first order singularity detectors. The UHF11 filter bank: (2) implies that for every in , the vector defined by

is a unit vector in . To construct the new filters, we first into the quadruplet “clone” the pair of filters . To achieve this isometrically, we , where define the mapping

Since the columns of form an orthonormal subset of , we defines an isometry mapping from into . obtain that We then use the rotation matrix , where

respectively. These two filters act as derivatives parallel to the and or, equivalently, as Sobel operators in directions these directions. The UHF14 filter bank: To further illustrate the power of Corollary 1, we now augment UHF11 with filters capable of detecting second order singularities in more directions. First, , , and are the horizontal, we note that the filters vertical and isotropic Laplacians. Again, as a first step we use

to “clone” these three filters, and then apply the rotation

to the newly defined copies. The end result is the isometry where

SHEN et al.: IMAGE DENOISING USING A TIGHT FRAME

TABLE I PSNR AFTER ITERATIVE APPLICATION OF UHDA1 WITH VARIOUS WEIGHTS !

Using Corollary 1, gives rise to a PF . The special characteristic of this new 14-band filter of bank is that the linear combinations

and

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TABLE II COMPARISON OF PSNR VALUES OF RESULTS FOR DIFFERENT TEST IMAGES AND NOISE LEVELS

defined by the ninth and tenth coordinates of are the Laplace and . We call operators parallel to the directions of this new 14-band filter bank UHF14. The impulse responses of the filters of this new 14-band filter bank are the following: , , , , , , , , , , , , and . The technique employed to construct both the UHF11 and the UHF14 filter bank relies on Corollary 1 in two ways. First, to construct isometries that increase the redundancy of an existing filter-based PF by producing scaled duplicates of a subset of those filters, and, second, to apply unitary operators on selected collections of those filters to produce new filters with desirable characteristics (e.g., geometry). In both cases, the resulting frame is a Parseval frame. This is a versatile technique that can be employed in a much more general setting than the two examples discussed above.

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Fig. 1.

Detail of the denoising the “Barbara” image with  = 20, using (a) wiener2, (b) bishrink, (c) GSM, and (d) UHDA2.

In addition, several key properties of the Ron and Shen frame are also lifted to the filter banks constructed in our and have vanishing examples. The two frame wavelets moments of order 1 and 2, respectively (i.e., and for ). Thus, constant signals , which corresponds cannot pass through the high-pass filter to , while neither constant nor linear signals can pass through . Due to the linearity of integration the high-pass filter these properties impose similar characteristics to the UHF9 filter-bank which are subsequently inherited by the UHF11 (and the UHF14) filter bank constructed via Corollary 1.

is a good choice for large values The threshold when a unitary wavelet transform is used [1]. However, of are only isothe transforms induced by convolution with metric, and not unitary. This results in an overall reduction of the energy contribution of the noise in the transformed image [31]. Therefore, the threshold needs to be scaled by a factor , , which is selected experimentally. The critewhere rion for selecting it is maximization of the peak signal-to-noise ratio (PSNR). is not known, it is estimated by the robust median estiIf mator

III. DENOISING ALGORITHMS Let be a noisy image. We filter using the UHF11 filter bank constructed in the previous section. We stress that the output of each channel in this filter bank is undecimated. Let be the output of the image through the th band (i.e., ). We separate the ten high-pass subband and , outputs into two groups, respectively. Note that filters in the first group can be used to detect first order singularities, while filters in the second group can be used to detect second order singularities. Accordingly, we choose different thresholding strategies for each of the two groups. , For the first group, we modify the coefficients in the using the hard threshold operator , where , is a thresholding factor, is the number of is the noise variance. pixels in , and

where is the output of using 1-level Haar high-pass filtering. and to obOur proposed algorithm jointly thresholds , . It should be noted that the proposed shrinkage of tain the wavelet coefficients is not the same as the classical wavelet shrinkage. For if or (3) otherwise where and . The scaled are obtained by computing the maximum thresholds and , respecmagnitude of the response of the filters tively. Similarly if or (4) otherwise

SHEN et al.: IMAGE DENOISING USING A TIGHT FRAME

Fig. 2.

for directly

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Detail of the result of denoising the “House” image with  = 20, using (a) wiener2, (b) bishrink, (c) GSM, and (d) UHDA2.

. For

, we use the hard thresholding operator (5)

Outputs in the second group are denoised by applying locally adaptive window-based denoising using maximum a posteriori estimates (LAWMAP), a method proposed by Mihçak et al. are independent [16]. We assume that the coefficients . zero-mean Gaussian variables with unknown variance is formed based on a local neighborhood An estimate of which is a square window of size centered at . . Then, the We postulate an exponential prior , using the maximum a posteriori (MAP) estimator for exponential prior, is given by

and denoise the We can further decompose the output wavelet outputs using the above described process. Our algorithm can be summarized as follows:

Denoising Algorithm 1 (UHDA1) 1)

2)

3) 4) where the sum is over all in . We impose a positivity condition by setting all negative estimates equal to zero, as suggested by Mihçak et al. [16], since it is possible to obtain negis too small. ative values from the actual MAP estimate if . With the estimated Thus, we use and , we apply a Wiener (least-squares fit) filter to all , (6)

Initialization: Input the noisy image , a threshold factor , and the number of decomposition levels . up to level Decompose the image using the UHF11 filter bank to , obtain Compute , . using Equations (3)-(6). from and Reconstruct image , by using the UHF11 filter bank.

We can improve UHDA1 as follows: Let be a weight between 0 and 1. The linear combination will be considered as a new noisy image. Using UHDA1, we obtain a new denoised image. We vary the weight and use UHDA1 iteratively as follows:

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Fig. 3. Detail of the result of denoising the “Boat” image with  = 20, using (a) wiener2, (b) bishrink, (c) GSM, and (d) UHDA2.

Denoising Algorithm 2 (UHDA2) 1) Initialization: Input the noisy image , a threshold factor , and the number of decomposition levels . 2) Set the weight vector . 3) For do: Apply UHDA1 to obtain ; Replace by to ; 4) Output . IV. RESULTS AND DISCUSSION We have tested our proposed algorithms on a number of images. In this section, we present selected results for four 512 grayscale images (i.e., Barbara, Boat, Lena, and 512 256 images (i.e., House and Peppers). Yogi) and two 256 White Gaussian noise with zero-mean and standard deviation was added to the six images, and denoising performance was evaluated using the PSNR in dB as a performance metric. The PSNR in decibels is given by

where is the root mean-square error between the original and the denoised images. For the Barbara, Boat, and Peppers , while for the House, Lena, and images, we chose . Yogi images, we chose

Performance results for the two proposed algorithms are preare produced by sented in Table I. The results with UHDA1. We observe that the iterative application of UHDA1 with varying weights does improve the PSNR. It is reasonable to expect that this process cannot be used indefinitely and will eventually lead to deteriorated performance, as can already be seen in the Lena image for small . Thus, the choice of weights and the number of iterations necessary need to be carefully calibrated, with more iterations being admissible for larger noise levels. However, the particular choice of weights proposed in UHDA2 is shown to be quite adequate in finding a good balance, with gains over UHDA1 of up to 2 dB in some cases (e.g., for the Yogi image). We also benchmarked UHDA2 against various other methods reported in the literature. Here, we present results comparing it to three other methods. The first is implemented by the wiener2 function in the software package MATLAB [32]. We also tested an implementation of the bishrink algorithm described by Sendur and Selesnick [13], and the GSM algorithm described by Portilla et al. [10]. The comparison results are summarized in Table II. We observe that our method outperforms the others in terms of PSNR when applied to the Peppers and Yogi images, produces similar results for the Boat and House images, and compares favorably in the cases of Lena and Barbara. Figs. 1–4 are presented for visual inspection of these results. Fig. 1 depicts a detail of the Barbara image containing the area around the face. It is important to note that the visual quality exhibited in that detail is comparable to that of the other two

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Fig. 4. Detail of the result of denoising the “Yogi” image with  = 20, using (a) wiener2, (b) bishrink, (c) GSM, and (d) UHDA2.

state-of-the-art methods. The texture of the scarf and the back of the chair are both faithfully reconstructed. Note that our method produces denoised images that exhibit noticeably less ringing artifacts around edges compared to the other methods. This is particularly pronounced in the House image (Fig. 2), and can also be seen in the Boats image. Finally, we also included the Yogi image (Fig. 4) to provide a comparison of our results outside the realm of natural images. A final word about the speed of our algorithm. Implemented as unoptimized MATLAB code on a 2.4-GHz Pentium 4 computer with 1 GB memory, UHDA1 with one level of decomposition can complete processing a 512 512 image in 4 s. Processing time scales linearly with the number of levels chosen for the decomposition. UHDA2 runs about five times slower than UHDA1, as its intermediate steps consist of five iterative executions of UHDA1. Each denoised image reported in this paper was obtained in 2 min or less.

V. CONCLUSION We have presented a general theory of constructing new frames from existing ones. Starting from a piecewise linear spline tight frame in 1-D, we designed two new Parseval frames lifted using our theory. These nonseparable framelets are capable of detecting first and second order singularities in directions that are integer multiples of 45 . To illustrate the appeal of these framelets, two image denoising algorithms

were proposed, tailored to their specific properties. Our results indicate that our algorithms produce denoised images with less ringing artifacts and similar, or better, PSNR when compared to other state of the art algorithms.

APPENDIX THE PROOF OF PROPOSITION 1 We begin with an interesting lemma (Lemma 2.5, [33]). be Hilbert spaces, be a Lemma 1: Let complete family of vectors in (i.e., the closed linear span of is equal to ). Assume that 1) the dimension of is equal to the cardinality of the index set ; 2) is an orthonormal basis of ; , defined by , for every , is a 3) bounded linear operator. (here, the denotes Also, let be the closure of the range of the conjugate transpose (adjoint) of ). Then, the following are equivalent. is invertible and its inverse is bounded. 1) is a frame of . 2) 3) The range of is closed. Finally, is an orthogonal projection if and only if is a PF of ; and is a Riesz basis of if equals . and only if the range of

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Proof: [The Proof of Proposition 1]: Let , , and be the standard . Consider the following operator: orthonormal basis of

where , for every , and , and . The operator is obviously for every is bounded. If linear. It is also bounded because each is the orthogonal projection of onto the , then the , where now the adjoint is not composite operator the conjugate transpose of , but its operator-adjoint, satisfies the following: for every Thus, every commutes with all integer modulations (i.e., multiplications by )

This implies that for

, ,

, where Therefore, is defined by matrix multiplication with the . This matrix-valued function matrix has rank one. According to Lemma 1 the restriction of on the closure of the range of must be a bounded and invertible operator for to be a frame of and vice-versa. If the set of such that has positive measure, then the range cannot be dense in , and equivalently, of cannot span . , for almost every Therefore, we conclude that . such that The existence of a constant , follows from the boundedness hypothesis for . The fact that the inverse of the restriction of on the closure of the range of is bounded implies such that , for that there exists . Since is defined by matrix every in the range of multiplication with the matrix-valued function , the is defined orthogonal projection onto the range space of to be the multiplication with the matrix-valued projection defined on whose range, for every , is the 1-D subspace spanned by the vector . The boundedness of the on the closure of the range inverse of the restriction of , implies that the restriction of on the range of is invertible, so, for almost every and of the projection for all vectors we have that

This inequality readily implies . The converse follows by reversing certain of the previous aris a Parseval guments. If , then is an orthogonal projection, imframe of is an plying that for almost every the matrix

orthogonal projection of happen if

, and vice-versa. This can only

ACKNOWLEDGMENT The authors would like to thank J. Portilla and E. Simoncelli for providing them with an early version of their code in order to compare our results and for their helpful comments. They would also like to thank the referees and the Associate Editor I. Selesnick for providing them with valuable comments and insightful suggestions which have brought improvements to several aspects of this paper. REFERENCES [1] D. Donoho and I. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika, vol. 81, pp. 425–455, 1994. [2] D. Donoho, “De-noising by soft thresholding,” IEEE Trans. Inf. Theory, vol. 41, no. 3, pp. 613–627, May 1995. [3] D. Donoho and I. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Amer. Stat. Assoc., vol. 90, no. 432, pp. 1200–1224, 1995. [4] R. Coifman and D. Donoho, “Translation-invariant de-noising,” in Wavelets and Statistics, A. Antoniadis and G. Oppenheim, Eds. Berlin, Germany: Springer-Verlag, 1995, vol. 103, pp. 125–150. [5] E. Simoncelli, “Bayesian denoising of visual images in the wavelet domain,” in Bayesian Inference in Wavelet Based Models, P. Müller and B. Vidakovic, Eds. New York: Springer-Verlag, 1999, vol. 141, ch. 18, pp. 291–308. [6] S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process., vol. 9, no. 9, pp. 1522–1531, Sep. 2000. [7] X. Li and M. Orchard, “Spatially adaptive denoising under overcomplete expansion,” in Proc. Int. Conf. Image Processing, vol. 3, Vancouver, BC, Canada, Sep. 2000, pp. 300–303. [8] H. Zhang, A. Nosratinia, and R. Wells, “Modeling the autocorrelation of wavelet coefficients for image denoising,” in Proc. Int. Conf. Image Processing, vol. 3, Vancouver, BC, Canada, Sep. 2000, pp. 304–307. [9] E. Simoncelli, W. Freeman, E. Adelson, and D. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 587–607, Mar. 1992. [10] J. Portilla, V. Strela, M. Wainwright, and E. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process., vol. 12, no. 11, pp. 1338–1351, Nov. 2003. [11] N. Kingsbury, “Complex wavelets for shift invariant analysis and filtering of signals,” Appl. Comput. Harmon. Anal., pp. 234–253, 2001. [12] L. Sendur and I. Selesnick, “Bivariate shrinkage functions for waveletbased denoising exploiting interscale dependency,” IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2744–2756, Nov. 2002. , “Bivariate shrinkage with local variance estimation,” IEEE Signal [13] Process. Lett., vol. 9, no. 12, pp. 438–441, Dec. 2002. [14] E. Simoncelli and E. Adelson, “Noise removal via Bayesian wavelet coring,” in Proc. Int. Conf. Image Processing, vol. I, Lausanne, Switzerland, Sep. 1996, pp. 379–382. [15] S. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. Image Process., vol. 9, no. 9, pp. 1532–1546, Sep. 2000. [16] M. Mihçak, I. Kozintsev, and K. Ramchandran, “Low-complexity image modeling based on statistical modeling of wavelet coefficients,” IEEE Signal Process. Lett., vol. 6, no. 2, pp. 300–303, Feb. 1999. [17] A. Pizurica, W. Philips, I. Lemahieu, and M. Acheroy, “A joint interand intrascale statistical model for Bayesian wavelet based image denoising,” IEEE Trans. Image Process., vol. 11, no. 5, pp. 545–557, May 2002. [18] J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Acoust., Speech, Signal Process., vol. 41, no. 12, pp. 3445–3462, Dec. 1993.

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Lixin Shen received the B.Sc. and M.Sc. degrees from Peking University, China, in 1987 and 1990, respectively, and the Ph.D. degree from Zhongshan University, China, in 1996, all in mathematics. From September 1996 to July 2001, he was a Research Fellow with the Center for Wavelets, Approximation, and Information Processing, National University of Singapore. From August 2001 to July 2002, he was a Postdoctoral Fellow with the Computational Biomedicine Laboratory (formerly Visual Computing Laboratory), University of Houston, Houston, TX. From August 2002 to July 2004, he was with Department of Mathematics, West Virginia University, Morgantown. He is currently an Assistant Professor with the Department of Mathematics, Western Michigan University, Kalamazoo. His current research interests are wavelets and their applications to image processing.

Manos Papadakis received the B.Sc. and Ph.D. degrees in mathematics from the University of Athens, Athens, Greece, in 1985 and 1993, respectively. He is currently an Associate Professor with the Department of Mathematics, University of Houston, Houston, TX, and Associate Director of the University of Houston’s Institute of Digital Informatics and Analysis. Previously held positions include Visiting Lecturer with the Department of Mathematics, University of Ioannina, Greece, and the Military Academy of Greece. His research interests include wavelets, frames, and isotropic multiresolution analysis, and their applications to biomedical and seismic imaging. Dr. Papadakis received the Raph E. Powe Junior Faculty Research Award on Computer Science and Mathematics from the Oak Ridge Associated Universities in 2000.

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Ioannis A. Kakadiaris received the Ptychion (B.Sc.) degree in physics from the University of Athens, Athens, Greece, in 1989, and the M.Sc. degree in computer science from Northeastern University, Boston, MA, in 2001. He is currently pursuing the Ph.D. degree in computer science at the University of Pennsylvania, Philadelphia, PA. He joined the University of Houston (UH), Houston, TX, in August 1997, after completing a Postdoctoral Fellowship at the University of Pennsylvania. He is the Director of the Division of Bio-Imaging and Bio-Computation at the UH Institute for Digital Informatics and Analysis and is the Founder and Co-Director of UH’s Computational Biomedicine Laboratory. His research interests include biomedical image analysis, modeling and simulation, computational biomedicine, biometrics, computer vision, and pattern recognition. Dr. Kakadiaris is the recipient of the 2000 NSF Early Career Development Award, the UH Computer Science Research Excellence Award, the UH Enron Teaching Excellence Award, the James Muller VP Young Investigator Prize, and the Schlumberger Technical Foundation Award.

Ioannis Konstantinidis received the B.Sc. degree in mathematics from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996, and the Ph.D. degree in mathematics from the University of Maryland, College Park, in 2001. He is currently an Assistant Research Scientist with the Norbert Wiener Center for Harmonic Analysis and Applications, University of Maryland. Previously, he was Assistant Director of the Computational Biomedicine Laboratory (formerly Visual Computing Laboratory) at the University of Houston, Houston, TX. His research interests include waveform design, digital geometry processing, nonorthogonal systems (wavelets, frames, etc.), and multiresolution analysis.

Donald Kouri, photograph and biography not available at the time of publication.

David Hoffman received the B.S. degree from the University of Illinois, Urbana, in 1960, and the Ph.D. degree from the University of Wisconsin in 1964, both in chemistry. He was a Postdoctoral Associate with the Mathematical Physics Department, University of Adelaide, Adelaide, Australia, in 1965 and 1966, and in chemical engineering with the University of Minnesota in 1966 and 1967. He joined the Department of Chemistry, Iowa State University, Ames, in the fall of 1967 as an Assistant Professor, where, except for when on academic leave, he has spent his entire career, having been promoted to Associate Professor in 1972 and to Full Professor in 1976. In 2001, he was named University Professor. He returned to the University of Adelaide on leave during the 1971 to 1972 academic year, and he also spent a year at the National Science Foundation during the 1980 to 1981 academic year. His principle research has been in nonequilibrium statistical mechanics and in quantum scattering theory. Recently, he has developed an interest in signal processing.