MULTI-INPUT single-output deconvolution (MISO-D)

Transcription

1 2752 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 Texas Two-Step: A Framework for Optimal Multi-Input Single-Output Deconvolution Ramesh Neelsh Neelamani, Member, IEEE, Max Deffenbaugh, Member, IEEE, and Richard G. Baraniuk, Fellow, IEEE Abstract Multi-input single-output deconvolution (MISO-D) aims to extract a deblurred estimate of a target signal from several blurred and noisy observations. This paper develops a new two step framework Texas Two-Step to solve MISO-D problems with known blurs. Texas Two-Step first reduces the MISO-D problem to a related single-input single-output deconvolution (SISO-D) problem by invoking the concept of sufficient statistics (SSs) and then solves the simpler SISO-D problem using an appropriate technique. The two-step framework enables new MISO-D techniques (both optimal and suboptimal) based on the rich suite of existing SISO-D techniques. In fact, the properties of SSs imply that a MISO-D algorithm is mean-squared-error optimal if and only if it can be rearranged to conform to the Texas Two-Step framework. Using this insight, we construct new wavelet- and curvelet-based MISO-D algorithms with asymptotically optimal performance. Simulated and real data experiments verify that the framework is indeed effective. Index Terms Curvelets, deblurring, deconvolution, minimax optimal, multichannel, restoration, sufficient statistics, wavelet-vaguelette, wavelets. I. INTRODUCTION MULTI-INPUT single-output deconvolution (MISO-D) is a recurring and important problem in several areas such as motion blur compensation [1], seismic imaging [2], astronomy [3], and medical imaging [4]. For example, in a digital video taken with a trembling hand, each frame ( ) can be modeled as a noisy version of a desired image blurred with linear time-invariant (LTI) filters ( ) [1], [5]. In astronomy, the large binocular telescope (LBT) obtains several different interferometric images ( ), each of which is a blurred version (with ) of the same target [3]. In both the motion blur compensation problem and the LBT problem, MISO-D is necessary to extract a high-resolution image of the target from the multiple measurements ( ). The MISO-D problem setup that we address can be formally described as follows (see Fig. 1). We describe our MISO-D setup using 1-D signals, but all analysis can be directly extended Manuscript received February 18, 2006; revised February 13, The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Thomas S. Denney, Jr. R. N. Neelamani is with the ExxonMobil Upstream Research Company, Houston, TX USA ( ramesh.neelamani@exxonmobil.com). M. Deffenbaugh is with the ExxonMobil Research and Engineering Company, Annandale, NJ USA ( max.deffenbaugh@ exxonmobil.com). R. G. Baraniuk is with the Department of Electrical and Computer Engineering, Rice University, Houston, TX USA ( richb@ece. rice.edu). Digital Object Identifier /TIP to higher dimensions. Each observation,, consists of the unknown desired signal first filtered by a known and then corrupted by additive noise. Using vector formulation (with boldface symbols denoting vectors), the MISO-D setup can be described in the continuous Fourier domain as 1 with denoting frequency; column vectors,, and, and denoting vector transpose. The noise vector comprises wide-sense stationary (WSS) zero-mean jointly Gaussian random processes that are independent of. At each, the noise across the various inputs can be correlated with cross-correlation matrix, with denoting conjugate transpose and denoting the statistical expectation operator. However, across frequencies, will be uncorrelated; that is, for.given,, and,we seek to estimate. Such MISO-D problems where is known are said to be nonblind. In many real life problems, the blurring filters in a MISO-D problem are not known. Such MISO-D problems are said to be blind. Typically, the first step in solving a blind MISO-D problem comprises estimating the blurs from the observations using some a priori information [1], [6], [7]. 2 The second step comprises employing an appropriate nonblind MISO-D technique that uses the blurs from the first step to estimate the target image. Recent papers have made a lot of progress in the difficult problem of blur estimation by specifically exploiting the plurality of observations in a MISO-D setup; [1], [6], and [7] show that the blur estimation step in a blind image MISO-D problem can be performed accurately in theory and practice. Our paper would be useful to perform the second step (described above) in solving practical blind MISO-D problems. See Fig. 4 for an application of our work to a real-life problem (in addition to the sequential two-step blind MISO-D approach adopted by [1], [6], and [7], there exist blind MISO-D techniques [8], [9] that estimate both the blur and the target image jointly; our work s utility in such approaches is currently not known). A special (and simpler) case of nonblind MISO-D, which results when in (1), is the nonblind single-input singleoutput deconvolution (SISO-D) problem. Similar to MISO-D, 1 All discussions can be extended to finite-length discrete-time signals assuming circular convolution. 2 In some cases, such as in the LBT MISO-D problem, accurate blur estimates can also be obtained through a separate calibration process [3]. (1) /$ IEEE

2 NEELAMANI et al.: TEXAS TWO-STEP: A FRAMEWORK FOR OPTIMAL MULTI-INPUT SINGLE-OUTPUT DECONVOLUTION 2753 Fig. 1. Overview of the MISO-D problem and the Texas Two-Step framework. SISO-D is ubiquitous [10]. However, unlike MISO-D, SISO-D has been throughly studied and is well understood [9], [11] [22]. Conventionally, Fourier-based approaches [9], [16] have been employed to tackle SISO-D because the Fourier domain is ideally suited to attenuate the noise that becomes amplified during operator inversion [13]. In fact, the LTI Wiener SISO-D filter [9], [16], [23] provides the best mean-squared-error (MSE) estimate when the signal is a WSS Gaussian random process. However, Fourier-based SISO-D approaches become ineffective for signals containing localized features such as 1-D signals with discontinuities, images with smooth contoured edges or point discontinuities, or higher dimensional signals with hyper-surfaces. 3 The reason is that the Fourier transform is poorly suited to exploit the local structure of such signals [12], [13]. In contrast to the Fourier transform, multiscale transforms such as wavelets [21], [24] and curvelets [25], [26] are better suited to exploit the structure of signals containing spatially localized features. For example, the wavelet transform is particularly suited to exploit the structure of 1-D signals that are piecewise continuous and of images that comprise spike discontinuities such as stellar images [20], [26]. Transforms such as curvelets [25], [26] are tailored to exploit the structure of images, such as seismic images [14], [25], that contain directionally oriented features. By employing wavelet and curvelet representations to exploit s structure, recent SISO-D techniques [11] [15], [17] [20] provide significant improvements over traditional Fourier-based SISO-D estimates. In addition to demonstrating good performances in practice, recent SISO-D approaches also enjoy some desirable optimality properties. In fact, for a wide class of, including piecewise discontinuous ones and for certain convolution operators, no 3 From a statistical perspective, such signals can be characterized as being nonstationary and non-gaussian. SISO-D technique can significantly improve (asymptotically) upon the performance of wavelet-based SISO-D techniques such as wavelet-vaguelette deconvolution (WVD) 4 [12] and Fourier-wavelet regularized deconvolution (ForWaRD) [13]. Even for arbitrary convolution operators, SISO ForWaRD seems to improve upon the performance of Fourier-based SISO-D approaches (experimentally verified) for a wide class of [13]. 5 For cartoon-like images and certain convolution operators, the biorthogonal curvelet decomposition (BCD) is asymptotically optimal [11]. Although a few approaches have been proposed to tackle the MISO-D problem [3], [4], [27] [29], relative to SISO-D, little is known about the optimality of MISO-D techniques. To the best of our knowledge, an optimal MISO-D technique is known only for the case when the signal is a WSS Gaussian random processes. In such a case, the LTI Wiener MISO-D filter [30] provides the best MSE estimate of from the s. Like the Wiener SISO-D filter, the Wiener MISO-D filter is entirely Fourier-based. Therefore, the Wiener MISO-D filter also would become suboptimal when the contains localized features such as discontinuities and edges. The gap between the current understanding of SISO-D and MISO-D problems naturally raises the following question: How can we design a suite of MISO-D techniques (both optimal and suboptimal) that is as rich as the suite of SISO-D techniques? Our contribution is to identify an optimal two-step framework, called the Texas Two-Step, to solve the MISO-D problem (see Fig. 1). i) Texas Two-Step first reduces the MISO-D problem (1) into an easier SISO-D problem 4 This is a slight abuse of nomenclature; the author referred to the general algorithm as wavelet-vaguelette decomposition algorithm. 5 For arbitrary convolution operators, ForWaRD is no longer provably optimal. (2)

3 2754 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 In (2), the scalar,asufficient statistic (SS) for, is constructed by combining all according to function (defined in Section IV-A). The properties of SSs guarantee that the SS in (2) contains all information about that the various measurements in (1) contain. The and the in (2) are the effective blurring filter and the Gaussian noise with spectrum, respectively. ii) Second, Texas Two-Step estimates from (2) by employing an appropriate SISO-D technique. The Texas Two-Step framework enables us to design a rich suite of optimal MISO-D techniques by simply choosing an appropriate SISO-D technique in the second step. In fact, a MISO-D algorithm that cannot be rearranged to conform to the Texas Two-Step framework is guaranteed to be suboptimal with respect to strictly convex performance metrics such as MSE. Consequently, the best way to exploit the structure (say, piecewise smoothness) of the signal is to first reduce the MISO-D problem (1) to (2) and then employ a suitable (say, wavelet-based or curvelet-based) SISO-D technique to solve (2). Consider, for instance, the MISO-D technique termed Texas Two-Step WVD (TX2-WVD) that first reduces (1) to (2) and then employs WVD [12] to solve (2). For a wide class of including piecewise discontinuous ones and for certain collections of s, no MISO-D technique can significantly improve upon TX2-WVD s asymptotic (as the noise level approaches zero) performance. We can make similar optimality claims about TX2-ForWaRD (for the same and as for TX2-WVD) and TX2-BCD (for same as for TX2-WVD, but for different ); these optimality criteria are specified in Section IV-C. Our problem (1) is similar to the well-studied receiver diversity problem in wireless communication [31]. For example, could be the transmitted message, the received message at the th receiver antenna, the receiver channel s fading gain, and the additive noise corrupting the receiver s observation. To leverage the receiver diversity, all the receiver observations are typically combined using maximal ratio combining (MRC) [31] to form an SS. The MRC step is equivalent to Texas Two-Step s first step. However, to the best of our knowledge, the imaging community has not fully exploited such SSs-based reductions to solve the MISO-D problem. Among the MISO-D methods known in the imaging community, our work is closely related from an algorithmic perspective to the independent and insightful work in [28]. Like the Texas Two-Step, [28] also proposes to solve the MISO-D problem by first reducing it to a SISO-D problem, and then using a known SISO-D technique to solve the reduced problem. In fact, the method in [28] can be recast to conform to the Texas Two-Step framework when the noise corrupting are uncorrelated. However, our work differs significantly from [28] from an optimality perspective. The work in [28] is motivated by analyzing the least squares cost function. Such an approach limits the authors from analyzing the optimality of their overall MISO-D algorithm. In contrast, our sufficient statistics-based approach enables us to analyze and guarantee the optimality of our MISO-D solution. Another (relatively minor) difference is that we have also considered cases where the noises corrupting the various observations are correlated. All MISO-D methods [1], [3] published before [28] aim to find an estimate by either optimizing a penalized least squares (PLS) cost function or by setting up convex constraints. As described in the review paper [3], the nonblind LBT MISO-D problem can be solved by optimizing the following Tikhonovregularized [32] least squares cost function: with denoting a regularization constant. [3] also outlines other iterative methods to optimize an PLS cost function with additional positivity constraints on. The authors of [3] point out that many conventional approaches produce ringing artifacts when the target image contains sharp intensity variations. The authors of [1] describe a novel PLS-based method to obtain a high-resolution deblurred image by combining two reallife motion blurred images. The authors first estimate the motion blurs and then use the estimated blurs to find an deblurred image that optimizes with and denoting the partial derivatives of in the and directions, respectively. The authors choose to ensure that large isolated discontinuities such as edges in are not over-penalized. We believe that the MISO-D framework described in [1] and [3] prevents us from easily exploiting the known structure of. Further, the error in the resulting MISO-D estimate cannot be analyzed easily. In fact, our analysis (see Theorem 3) indicates that the MISO-D estimates obtained in [1] and [3] can often be improved. Our views seem to be substantiated by Fig. 2, which illustrates that the TX2-For- WaRD estimate improves upon the estimate in [1] in a simulated MISO-D problem. The rest of the paper is organized as follows. Sections II and III provide an overview of SSs and SISO-D, respectively. Section IV elaborates on the Texas Two-Step framework and identifies some optimal MISO-D algorithms. Section V describes the results obtained by using the Texas Two-Step framework in a simulated and a real-life example. Section VI provides the conclusions and sketches future directions. Appendices A and B contain short, key technical proofs. II. OVERVIEW OF SSS SSs play an central role in statistical inference and estimation. Let be an observation set characterized by the probability density function, with denoting the set of parameters that we desire to estimate; can be deterministic or random. A statistic is sufficient for if the conditional distribution of given is independent of for all [33, p. 35]. In other words, the SS for completely summarizes all the information that contains about. We review some important, well-known properties of SSs below. See [33] for discussions and proofs. (3)

4 NEELAMANI et al.: TEXAS TWO-STEP: A FRAMEWORK FOR OPTIMAL MULTI-INPUT SINGLE-OUTPUT DECONVOLUTION X 2 Y H 2755 Y Fig. 2. (a) Desired ( samples). (b) Observation (30-dB BSNR; blurred with the radially symmetric blur in Fig. 3). (c) Observation (30-dB in Fig. 3). (d) PLS-based MISO-D estimate (19.4-dB SNR)[1]. (e) TX2-ForWaRD estimate (20.7-dB SNR). BSNR; blurred with the radially symmetric blur (f) TX2-ForCuRD estimate (22.6-dB SNR). H Theorem 1: Let be distributed according to be sufficient for. Then, for any estimator let and of, there exists an estimator (possibly randomized) based on that has the same performance as.

5 2756 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 Theorem 1 [33, p. 33] captures the most important property of an SS: the estimation performance obtained by processing the complete data can always be replicated by processing only the SS. In fact, for strictly convex performance metrics, a remarkably strong claim is possible [33]. A function is said to be strictly convex with respect to if for and Theorem 2: Let be distributed according to and let be sufficient for. Further assume that the performance metric is a strictly convex function of the estimate. Then, given an estimator which cannot be expressed as a function of, there exists a better estimator that depends only on Theorem 2 s proof is outlined in [33, p. 47, Rao-Blackwell theorem]. It asserts that an estimator is optimal with respect to a strictly convex performance metric (for example, MSE) only if it can be expressed as condensing into an SS and then operating on. SSs are not unique. A statistic obtained by an invertible transformation of an SS is also sufficient. Note that the entire set of observations is always an SS for, but it is not a particularly useful SS. Typically, an SS is valuable if it is significantly smaller in size than the original data because such a size reduction often implies a reduction in the dimensionality of the problem. For the MISO-D problem, from the infinite set of possible SSs, we will choose an SS that recasts the MISO-D problem into a well-studied SISO-D problem. III. OVERVIEW OF SISO-D SISO-D is an important problem that has been studied extensively [9], [11] [21]. We will be exploiting this well-studied field to solve the MISO-D problem effectively. Given the measurement in (2) and the blurring, and noise spectrum, SISO-D seeks to estimate from. If the is simply inverted, then noise components at frequencies where is small are amplified SISO-D s challenge lies in inverting without noise amplification. A. Fourier-Based SISO-D The Fourier transform is ideally suited to represent WSS Gaussian random processes and smooth -Sobolev space signals 6 in the sense that most of the signal s energy (and, thereby, its structure) is captured by optimally few Fourier components. The Fourier transform is said to provide economical representations for such signals. Further, the Fourier domain also provides the most economical representation for the noise in (4) when is WSS and Gaussian [13]. 6 The L -Sobolev space comprises signals with rapidly decaying Fourier coefficients. (4) SISO-D techniques such as the Wiener SISO-D filter [23] and Tikhonov-regularized SISO-D [32] leverage the economy of Fourier domain signal and noise representations to effectively suppress the amplified noise in (4). See the excellent articles [34], [35] and the references therein for further insights to the importance of economical representations in estimation. Conceptually, Fourier-based SISO-D comprises the following two steps. 1) Invert as in (4). 2) Shrink each Fourier component of independently to attenuate the amplified noise. For example, the Wiener SISO-D estimate is given by with the power spectral density (PSD) of. 7 The second bracketed term in (5), which takes values between 0 and 1, shrinks the noisy Fourier components of (4) based on the signal-to-noise ratio (SNR) at each frequency. Wiener SISO-D provides the best MSE estimate of when is a WSS Gaussian random process. For cases where belongs to the -Sobolev space and the noise is white (that is, is constant with ), Tikhonov SISO-D is asymptotically optimal. Its estimate s MSE decay rate as the noise variance tends to zero (denoted by henceforth) cannot be improved [12], [13]. Unfortunately, the Fourier domain does not provide economical representations for signals with singularities such as images with edges. Consequently, for such signals, even with the best scalar Fourier shrinkage, Fourier-based SISO-D provides unsatisfactory estimates. For instance, these limitations manifest themselves as ringing around edges in images [13]. B. Wavelet-Based SISO-D Wavelets represent a signal in terms of shifted versions of a low-pass scaling function and shifted and dilated versions of a prototype band-pass wavelet function [21]. Wavelets are well-suited to exploit the structure of piecewise continuous 1-D signals and of images that comprise spike discontinuities such as stellar images [20], [26]. Formally, wavelets provide optimally economical representations for signals in smoothness spaces such as Besov spaces [35]; the wavelet coefficients of Besov space signals decay exponentially fast. Roughly speaking, a Besov space contains functions with derivatives in, with measuring finer smoothness distinctions [36]. Besov spaces with different,, and characterize many classes of signals in addition to -Sobolev space signals; for example, in 1-D, contains piecewise polynomial signals [21]. WVD is a relatively new algorithm that leverages wavelets economical signal representation to obtain improved deconvolution estimates [12], [13]. Conceptually, WVD comprises operator inversion as in (4) followed by noise suppression via shrinkage of s wavelet coefficients. However, WVD is not applicable when contains Fourier domain nulls. ForWaRD is an extension of WVD that can be applied 7 For deterministic X, an estimate of jx(f )j is used instead of P (f ). (5)

6 NEELAMANI et al.: TEXAS TWO-STEP: A FRAMEWORK FOR OPTIMAL MULTI-INPUT SINGLE-OUTPUT DECONVOLUTION 2757 to all types of SISO-D problems. ForWaRD comprises (4) followed by noise suppression via a balanced combination of Fourier-based and wavelet-based shrinkage [13]. WVD and ForWaRD provide near-optimal rates of MSE decay for certain classes of SISO-D problems. Let be a -dimensional signal that belongs to the Besov function space and let with (6) Above, denotes the -dimensional frequency vector and denotes its magnitude. By, we mean that there exist constants and such that. Let the in (2) be WSS Gaussian noise such that (that is, is nearly white) for some scalar. Then, both WVD and ForWaRD (with appropriate Fourier shrinkage [13]) provide estimates whose MSE decays with as for some constant and all. Furthermore, for this problem, this rate of error decay is the best achievable rate among all SISO-D estimators [12], [13]. In practice, ForWaRD also provides good estimates for a wide variety of including those containing Fourier domain nulls [13], [37]. While wavelets are ideally suited to represent a wide class of 1-D signals [26], they do not provide economical representations for some types of high-dimensional signals. For example, the wavelet domain does not economically represent images containing localized directional features such as smooth edges (e.g., cartoon images) or oriented textures (e.g., seismic and fingerprint images), and 3-D volumes containing smooth surface discontinuities [26]. For such higher dimensional signals, wavelet-based SISO-D techniques, such as WVD and ForWaRD, are expected to provide suboptimal performance [11]. C. Curvelet-Based SISO-D Curvelets are new multiscale transforms that represent an image in terms of shifted versions of a low-pass scaling function and shifted, dilated, and rotated versions of a prototype band-pass curvelet function. Unlike wavelet basis functions, each bandpass curvelet basis function has an elongated envelope with the envelope s length scaling as its width squared; this is referred to as the parabolic scaling law [26]. Curvelets are tailored to exploit the structure of images such as seismic images [14], [25] that contain localized directional features and oriented textures. Formally, curvelets provide optimally economical representations for images in so-called spaces. An image comprises twice differentiable regions separated by piecewise twice differentiable boundary curves. In other words, an image is a piecewise smooth image with piecewise smooth boundary edges [11], [26]. For example, many cartoon images would belong to the space. (7) BCD is a recently proposed approach that employs curveletbased thresholding to suppress the amplified noise in (4) [11]. Conceptually, BCD comprises operator inversion as in (4) followed by noise suppression via shrinkage of s curvelet coefficients. Consider a SISO-D problem where belongs to the space and the 2-D filter, with and the 2-D frequency vector. Let in (2) be nearly white Gaussian noise (WGN); that is,, for some frequency independent. Then, the BCD estimate s MSE decays with as for some constant and all. This asymptotic rate of error decay is the best achievable rate among all SISO-D estimators [11]. For the same SISO-D problem, Fourier-based SISO-D s MSE decays as, whereas WVD s MSE decays as. The rate (8) is conjectured to hold for arbitrary [11]. Similar to WVD, BCD cannot be employed when contains Fourier domain nulls. However, simply replacing the wavelet-based shrinkage steps in ForWaRD with curvelet-based shrinkage would effectively deblur images (and images containing oriented features) even when contains Fourier domain nulls. We will employ the resulting curvelet-based SISO-D, which we naturally call Fourier-curvelet regularized deconvolution (ForCuRD), in the Section V. Similar to For- WaRD and WVD, we expect that ForCuRD and BCD would also enjoy the same optimality properties. Rigorous analysis of ForCuRD s optimality properties is left as a topic for future investigation. D. Miscellaneous SISO-D In addition to the transform domain noniterative SISO-D techniques described in Sections III-A C, there exist a vast literature on iterative deconvolution techniques [9], [14], [17] [20], [38]. Most iterative SISO-D techniques seek an estimate that optimizes a cost function comprising two terms (typically) The first term measures the given estimate s fit to whereas the second term (called the penalty term) measures the estimate s conformance to prior information about the structure (say, piecewise smoothness) of the true. Note that the noisy estimate (4) minimizes the first term in (9). The term is included to ensure that the solution minimizing (9) is not too noisy. For example, [22] obtains good deblurring results by setting to be the total variation of. In [17], the authors provides excellent estimates by choosing a penalty term that measures the sparsity of the estimate s wavelet representation. To optimize cost functions comprising a variety of sparsity-based penalty terms, [38] identifies an elegant thresholding-based iterative SISO-D approach (similar to the one in [17]) with guaranteed convergence. Recently, [14] demonstrated promising results using an iterative curvelet-based SISO-D technique based on [38]. Similar to ForCuRD, the (8) (9)

7 2758 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 method in [14] can also handle SISO-D problems where the contains Fourier domain nulls. IV. TEXAS TWO-STEP MISO-D FRAMEWORK We are now poised to describe the paper s key contribution the Texas Two-Step framework. Sections IV-A and B describe Texas Two-Step s steps and Section IV-C describes Texas Two-Step s optimality. A. Step i): From MISO-D to SISO-D Texas Two-Step first reduces the MISO-D problem (1) to a related SISO-D problem (2) by identifying an appropriate SS. In the following lemma and henceforth, denotes the Moore- Penrose pseudo-inverse of and denotes an identity matrix. Lemma 1: Consider the MISO-D problem setup (1). Then,,defined in (10) (12), shown at the bottom of the page, forms an SS for. See Appendix A for the proof. In typical MISO-D problems, would lie in the column space of [hence, in (12)]. For example, s column space when comprises WSS, zero-mean, independent, white, Gaussian random processes; in this case, is diagonal with all diagonal elements nonzero. We can easily recast (11) into the SISO-D setup (2) by setting and (13) (14) Thanks to the properties of SSs, instead of estimating from various measurements, we can just focus on estimating from the single effective measurement. Hence, the MISO-D setup (1) is transformed into a SISO-D setup (2) with the same solution. B. Step ii): Apply Appropriate SISO-D After transforming the MISO-D problem (1) into (2), the Texas Two-Step invokes an appropriate SISO-D technique to solve (2). The choice of the SISO-D technique is based on the structure of the desired signal and the effective blurring operator and noise in (13) and (14) respectively. See Section III for some popular SISO-D techniques and their properties. Note that [see (14)] in the reduced SISO-D problem (2) is a colored Gaussian random process with spectrum (15) Hence, before applying a SISO-D technique that assumes white, an additional noise prewhitening step [39] would be necessary. The prewhitening can be performed by simply dividing by. In practice, for the sake of stability, the noise prewhitening should be combined with the inversion step of the chosen SISO-D technique. C. Optimality and New MISO-D Techniques The properties of SSs guarantee that Texas Two-Step with an appropriate SISO-D in the second step can match the performance of any MISO-D algorithm. In fact, a MISO-D algorithm can be MSE-optimal only if it can be rearranged to conform to the Texas Two-Step framework. Theorem 3: Let denote an arbitrary performance metric for an arbitrary MISO-D estimator. Then, there exists a SISO-D estimator (possibly randomized) such that the Texas Two-Step estimator with employed to solve (2) satisfies (16) Further assume that is a strictly convex function of.if cannot be recast into the Texas Two-Step framework, then it is suboptimal with respect to. Theorem 3 immediately follows from Theorem 1, Theorem 2 and Lemma 1. We can infer from Theorem 3 that if an optimal SISO-D technique is chosen in the second step, then the Texas Two-Step MISO-D technique is also optimal. Corollary 1: Let, with an arbitrary function space. Let denote the SISO-D estimator of from (2) that is optimal with respect to an arbitrary performance metric. Let denote the two-step MISO-D estimator that employs in the second step. Then, among all MISO-D techniques, is optimal with respect to. Corollary 1 enables us to construct several new optimal MISO-D techniques. The following subsections describes the optimality of existing and new optimal MISO-D techniques (see Table I for a list both optimal and suboptimal approaches). 1) Fourier-Based MISO-D: This section specifies the optimality of the Texas Two-Step algorithm when a Fourier-based SISO-D technique is employed in the second step to solve (2). Corollary 2 specifies Texas Two-Step s optimality when the Wiener SISO-D filter is employed in the second step; the resulting algorithm is termed TX2-Wiener. Corollary 2: Assume that the in (1) is a WSS Gaussian random process with spectrum. The TX2-Wiener estimate is given by (17), shown at the bottom of the next page, if otherwise column space of (10) (11) (12)

8 NEELAMANI et al.: TEXAS TWO-STEP: A FRAMEWORK FOR OPTIMAL MULTI-INPUT SINGLE-OUTPUT DECONVOLUTION 2759 TABLE I OPTIMAL AND SUBOPTIMAL MEMBERS OF THE TEXAS TWO-STEP FAMILY with given by (12). The estimate (17) enjoys the best MSE among all MISO-D estimators. Expression (17) is obtained by substituting (13), (14), and (15) into (5). The optimality follows from Corollary 1 and from the optimality of the Wiener SISO-D technique (see Section III-A). It should come as no surprise to the reader that the TX2-Wiener estimator (17) is indeed the well-known Wiener MISO-D filter [30]. Consider the case where and where the noises and that corrupt the observations and are independent of each other. The noise cross-correlation matrix in this case simplifies to 2) Wavelet-Based MISO-D: For Besov space and certain collections of operators, Texas Two-Step with either WVD or ForWaRD employed in the second step (termed TX2-WVD and TX2-ForWaRD, respectively) are indeed asymptotically optimal. Corollary 3: Assume that in (1) is an arbitrary -dimensional signal that belongs to the Besov function space. In (1), let the noise correlation matrix, with a frequency-independent scaling factor and an arbitrary matrix such that the column space of contains and with and Then (17) can be rearranged into with (19) (18) with and. The estimator (18) is identical to the Wiener MISO-D filter expression (16) in [30]. Thus, the TX2-Wiener estimator (17) is indeed the well-known Wiener MISO-D filter [30]. If belongs to the -Sobolev space, then we can infer that Texas Two-Step with Tikhonov SISO-D employed in the second step would be asymptotically optimal (Tikhonov SISO-D should be applied after noise prewhitening). Then, for every, both TX2-WVD and TX2-For- WaRD (with appropriate Fourier shrinkage [13]) provide estimates whose MSE decays with (that is, with decreasing noise level) as - and - (20) for some constant and all. Furthermore, this rate of error decay is the best achievable rate among all MISO-D estimators. The proof, which is outlined in Appendix B, follows from Corollary 1 and from WVD s and ForWaRD s optimality (see (17)

9 2760 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 Section III-B). Note that (19) is less stringent that the corresponding condition (6) for the SISO-D case. For example, (19) would be satisfied even if just one in the MISO-D setup satisfies (6). Hence, we expect that TX2-WVD and TX2-ForWaRD would be optimal for a broad range of MISO-D problems (especially as the number of observations increases). 3) Curvelet-Based MISO-D: As the reader might expect, Texas Two-Step with BCD employed in the second step (termed TX2-BCD) is an asymptotically optimal estimator for images and certain collections of operators. Corollary 4: Assume that in (1) is an arbitrary image. In (1), let the noise correlation matrix, with a frequency-independent scaling factor and an arbitrary matrix such that s column space contains and with Then, for every image, TX2-BCD provides an estimate whose MSE decays with as - (21) for some constant and all. Furthermore, this rate of error decay is the best achievable rate among all MISO-D estimators. The proof is omitted since it is virtually identical to Corollary 3 s proof. TX2-BCD would be optimal for arbitrary if SISO BCD s rate (8) holds for arbitrary (as conjectured in [11]). (Since ForCuRD s optimality is not yet understood, TX2- ForCuRD s theoretical properties of are also unknown). V. RESULTS This section presents results from a simulated MISO-D experiment and a real-life motion deblurring problem. Our results substantiate that Texas Two-Step is indeed an effective framework to tackle nonblind MISO-D problems. We illustrate the performance of two new MISO-D techniques, TX2-ForWaRD and TX2-ForCuRD, in this section and compare them to the PLS-based (refer to (3)) nonblind MISO-D method employed in [1]. In the first experiment, we choose a pixel piece of the Barbara image illustrated in Fig. 2(a) for because it contains an illustrative collection of complex features. The observations [see Fig. 2(b)] and [see Fig. 2(c)] are constructed by blurring with radially symmetric filters and, respectively. Fig. 3 plots s and s radial frequency responses. The analytic expressions for the two filters are and (22) with and,, where and are the spatial frequency sample numbers of the 2-D discrete Fourier transform (DFT). The additive WGN levels in both the observations corresponds to a blurred signal-to-noise ratio (BSNR) of 30 db; BSNR is defined as [13] blurred signal's variance noise variance Assuming and denote the variances of the noises corrupting and, respectively, the filter is given by (23) Note that while and both contain Fourier domain nulls (at 7.75 and 10.95, respectively), does not go to zero at any frequency. Fig. 3 illustrates the radial frequency response of. Fig. 2(d) illustrates the estimate obtained using the PLS-based nonblind MISO-D method employed by [1]. As prescribed in [1], we chose in (3) and adjusted to maximize the SNR of the estimate db. The SNR is measured as Notice that the PLS estimate does not fully capture the stripes on the woman s scarf and pants. Further, the estimate is also slightly noisy. TX2-ForWaRD exploits the strength of wavelet representations to returns a cleaner estimate [see Fig. 2(e); db] with well-preserved scarf textures. The wavelet-domain Wiener filter (WWF) [13], [40] is employed to perform the wavelet domain denoising step in TX2-ForWaRD. However, notice that the long stripes on the pants are partially washed out. TX2-ForCuRD [see Fig. 2(f); db] further improves upon the PLS-based estimate. Empirical Bayes-based thresholding [41] is used to perform the curvelet domain denoising step in TX2-ForCuRD. Since curvelets are well suited to represent elongated directional features, TX2-ForCuRD preserves the long stripes on the woman s pants particularly well. However, the face region in the TX2-ForCuRD estimate is arguably more blurred than the TX2-ForWaRD estimate. Further, the TX2-ForCuRD estimate also contains several linear artifacts that result from misestimated curvelet coefficients. TX2-Wiener (estimate not included in the visual comparison) returned an estimate db with substantial ringing around the discontinuities because the Fourier domain is a poor choice to exploit the structure of discontinuous signals such as the Barbara image. In the second experiment, we address a real-life MISO-D problem that the authors of [1] kindly provided us. We seek to combine the motion-blurred images of a hotel s front wall, illustrated in Fig. 4(a) and (b), to estimate a high-resolution deblurred image. The horizontal and vertical blurring in Fig. 4(a) and (b), respectively, were caused by the fast pan and tilt of the camera. Using the software kindly provided by the authors in [1], we estimated the impulse responses (with filter length 17) of the blurring filters from the observations; Fig. 4(c) and (d) displays the estimated blurring filter impulse responses. The impulse responses are 1-D because the blurring is assumed by [1] to be unidirectional. We have verified using simulations that the novel estimation of the blurring filter impulse responses in [1] (which is the paper s focus and main

10 NEELAMANI et al.: TEXAS TWO-STEP: A FRAMEWORK FOR OPTIMAL MULTI-INPUT SINGLE-OUTPUT DECONVOLUTION 2761 Fig. 3. Radial frequency responses of the filters used to obtain Y and Y in Fig. 2. contribution) returns accurate estimates. Fig. 4(e) depicts the impressive deblurred estimate in [1], which is obtained by employing an PLS-based nonblind MISO-D method [refer to (3)] with the estimated blurs. Fig. 4(f) exhibits that replacing the PLS-based nonblind MISO-D method with TX2-ForWaRD provides additional improvement. Hard thresholding [21] is employed to perform the wavelet domain denoising step in TX2-ForWaRD. The TX2-ForWaRD estimate s improved resolution is particularly noticeable in the brick-laid regions. VI. CONCLUSION We have proposed a simple, new framework Texas Two-Step to address nonblind MISO-D problems. Texas Two-Step s key step is to reduce the MISO-D problem into a well-understood SISO-D problem using SSs. The reduction enables Texas Two-Step to invoke a rich suite of algorithms from the well-studied SISO-D field to solve the MISO-D problem. A Texas Two-Step technique inherits the optimality of the SISO-D technique employed in the second step. Indeed, a MISO-D technique can be optimal with respect to a strictly convex metric (such as MSE) only if it can be recast into the Texas Two-Step framework. Using this new insight, we have constructed new wavelet-based (TX2-WVD and TX2-For- WaRD) and curvelet-based (TX2-BCD and TX2-ForCuRD) MISO-D algorithms. Among these new MISO-D algorithms, TX2-WVD, TX2-ForWaRD, and TX2-BCD are guaranteed to enjoy asymptotic optimality properties; the properties of TX2-ForCuRD are still open for further study. Using simulations, we demonstrated that TX2-ForWaRD and TX2-ForCuRD provide improved estimates compared to the conventional Wiener MISO-D filter and the more recent PLS-based MISO-D method [1]. These results substantiate the efficacy of the Texas Two-Step framework in solving nonblind MISO-D problems. Our results on a real-life motion deblurring problem indicate that the Texas Two-Step framework can be also used to effectively solve real-life blind MISO-D problems where the blurs can be accurately estimated from the observations. We have not yet studied the effect of erroneous blur estimation on the Texas Two-Step s performance. This remains a topic for future study. The concepts developed in this paper could be useful to design optimal imaging experiments that conform to the MISO-D setup. For example, given an imaging apparatus with limited aperture (such as a telescope, a medical imaging system, a digital camera, etc.) and a limited quota of measurements, how should we design our imaging experiment? We believe that the sequence of experiments should be designed to minimize the ill-posedness of the underlying dual SISO-D problem (assuming the blurs are known or can be estimated accurately). The Gaussianity of the noise corrupting the observation plays a critical role in ensuring that the MISO-D problem transforms into a simpler SISO-D problem when the SS is constructed. For non-gaussian noises, constructing the SS would in general transform the MISO-D problem into a smaller but possibly less structured (typically, nonlinear) inverse problem. It would be useful to better understand cases when it is easier to obtain an optimal estimate (such as the maximum likelihood estimate) from the smaller, transformed problem instead of solving the larger, original MISO-D problem. Another related avenue for further research is to identify cases where the smaller problem continues to be a linear inverse problem. The approach adopted in this paper might be useful to solve more general classes of linear inverse problems. Consider the linear inverse problem (24)

11 2762 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 Fig. 4. (a) Horizontally blurred observation Y. (b) Vertically blurred observation Y. (c), (d) Impulse responses (with length 17) of the horizontal and vertical blurring filters H and H estimated simultaneously from Y and Y using [1]. (e) PLS-based MISO-D estimate [1]. (f) TX2-ForWaRD estimate. with the observation vector, the target vector, a known matrix operator, and a Gaussian noise vector with an invertible correlation matrix. The statistic forms an SS for. In problems such as super-resolution (see [42] for the formulation) the super-resolved target s size can be smaller than the total sizes of the low-resolution observations. In such cases, we could design faster and better estimators for without any loss of information or optimality by analyzing the smaller dimensional SS,. The application of such ideas to specific linear inverse problems appears promising and remains a topic for future investigation. APPENDIX A Proof of Lemma 1 is an SS. Our proof will be almost identical to the machinery employed to prove the following well-known fact: given several realizations of a Gaussian random variable, the average of the realizations forms an SS for the random variable s mean.

12 NEELAMANI et al.: TEXAS TWO-STEP: A FRAMEWORK FOR OPTIMAL MULTI-INPUT SINGLE-OUTPUT DECONVOLUTION 2763 Our model (1) assumes that the noise comprises WSS zero-mean jointly Gaussian random processes that are independent of. Consequently, the different noise frequency components are independent of each other and uncorrelated across frequencies. This key property enables us to treat each frequency component separately. In the rest of the Appendix A, every symbol (such as and ) is a function of frequency. Henceforth, every symbol s frequency dependence is dropped for the sake of brevity; for example, is referred to as. Case 1: Invertible. Let denote the real part of a complex number. Then Note that is a column vector comprising uncorrelated (and, hence, independent) zero-mean Gaussian random variables such that (29) with denoting the Dirac delta function. Since, there exists a column vector such that (30) (25) From (30), since for, we can infer that the respective s are also 0. Therefore (26) Given, can be factorized into two parts a term (25) that depends only on, and the rest (26) that depends only on. Thus, given, is independent of. Hence, by definition, is an SS for. Case 2: Noninvertible. In this case, we will consider the two scenarios identified by (12) separately. Before analyzing the two scenarios, we define a few necessary terms. Let (27) denote the singular value decomposition of with an unitary matrix and a diagonal matrix with non-negative diagonal entries 8,, such that (31) refers to the th row of the matrix. A column vector that is premultiplied by is denoted using a caret symbol. The th row of a transformed column vector such as is referred to as. Case 2a:. As we shall soon see, this case is nearly identical to Case 1. Multiplying (1) by, we have 8 Note that R is guaranteed to be non-negative definite. (28) (32) Given, is independent of because (31) depends only on and (32) depends only on. Hence, is an SS for. Case 2b: Column Space of. In this case, too, we can verify that given, can be factorized into two parts one depending only on and the other only on.however, for the sake of brevity and clarity, we choose to just highlight the engine that drives the formal proof.

13 2764 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 11, NOVEMBER 2007 Using (10), (12), and (27), we can rewrite would like to thank the reviewers for their insightful and constructive comments. (33) (34) The variance of the noise corrupting is zero. Further, since column space of, for some. Therefore,. Therefore, given, we can exactly determine (with probability 1) as Since captures all information that contains about, forms an SS for. APPENDIX B Proof for Corollary 3: TX2-WVD s and TX2-ForWaRD s Rates. Since lies in the column space of, [from (12)]. Hence, Texas Two-Step s first step yields a SISO-D problem (2) with Gaussian noise with variance and (35) (36) To estimate from such a SISO-D problem, both WVD and ForWaRD first invert and then appropriately attenuate the amplified noise [see (4)]. Note that is white noise and colored noise with variance. 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Baraniuk, Improved wavelet denoising via empirical Wiener filtering, Proc. SPIE Wavelet Applications in Signal and Image Processing V, vol. 3169, pp , Oct [41] M. Figueiredo and R. D. Nowak, Wavelet-based image estimation: An empirical Bayes approach using Jeffrey s noninformative prior, IEEE Trans. Image Process., vol. 10, no. 9, pp , Sep [42] M. Elad and A. Feuer, Restoration of single super-resolution image from several blurred, noisy and down-sampled measured images, IEEE Trans. Image Process., vol. 6, no. 12, pp , Dec Max Deffenbaugh (S 93 M 02) received the B.S.E. degree in electrical engineering from Princeton University, Princeton, NJ, 1991, and the Sc.D. degree from the Massachusetts Institute of Technology/Woods Hole Oceanographic Institution Joint Program, Woods Hole, MA, in He is with the ExxonMobil Research and Engineering Company, Annandale, NJ. His interests include applications of estimation theory and signal processing in exploration seismology, numerical modeling of seismic wave propagation and geological processes, and telemetry of neural and EMG signals from freely swimming fish. Richard G. Baraniuk (S 85 M 93 SM 98 F 01) received the B.Sc. degree from the University of Manitoba, Manitoba, BC, Canada, in 1987, the M.Sc. degree from the University of Wisconsin, Madison, in 1988, and the Ph.D. degree from the University of Illinois at Urbana-Champaign, Urbana, all in electrical engineering, in After spending 1992 and 1993 with the Ecole Normale Supérieure, Lyon, France, he joined Rice University, Houston, TX, where he is currently the Victor E. Cameron Professor of Electrical and Computer Engineerng and Director of the Connexions Project. He spent sabbaticals at Ecole Nationale Supérieure de Télécommunications, Paris, France, in 2001, and the Ecole Fédérale Polytechnique de Lausanne, Switzerland, in His research interests in signal and image processing include wavelets and multiscale analysis, statistical modeling, and sensor networks. Dr. Baraniuk received a NATO postdoctoral fellowship from NSERC in 1992, the National Young Investigator award from National Science Foundation in 1994, a Young Investigator Award from the Office of Naval Research in 1995, the Rosenbaum Fellowship from the Isaac Newton Institute of Cambridge University in 1998, the C. Holmes MacDonald National Outstanding Teaching Award from Eta Kappa Nu in 1999, the Charles Duncan Junior Faculty Achievement Award from Rice University in 2000, the ECE Young Alumni Achievement Award from the University of Illinois in 2000, and the George R. Brown Award for Superior Teaching at Rice University in 2001 and He was elected a Plus Member of AAA in Ramesh Neelsh Neelamani (S 97 M 04) received the B.Tech. degree in 1997 from the Indian Institute of Technology, Bombay, and the M.S. and Ph.D. degrees in 1999 and 2003, respectively, from Rice University, Houston, TX, all in electrical engineering. He joined ExxonMobil in 2003 and is currently a Research Scientist at the ExxonMobil Upstream Research Company, Houston. His research passions include signal and image processing topics such as multiscale signal representations and algorithms, statistical signal processing, color processing, distributed algorithms, and pattern recognition. He is a member of the SEG and SIAM.