"Rewiring" Filterbanks for Local Fourier Analysis: Theory and Practice

SUBMITTED MANUSCRIPT 1 “Re wiring” Filterbanks for Local F ourier Analysis: Theory and Practice K eigo Hirakaw a, Member , IEEE , and Patrick J. W olfe, Senior Member , IEEE Abstract — This article describes a series of new results outlin- ing equivalences between certain “rewirings” of filterbank system block diagrams, and the corresponding actions of convolution, modulation, and downsampling operators. This gives rise to a general framework of r everse-order and con volution subband structures in filterbank transforms, which we show to be well suited to the analysis of filterbank coefficients arising from subsampled or multiplexed signals. These results thus provide a means to understand time-localized aliasing and modulation properties of such signals and their subband repr esentations— notions that are notably absent fr om the global viewpoint afforded by Fourier analysis. The utility of filterbank rewirings is demonstrated by the closed-form analysis of signals subject to degradations such as missing data, spatially or temporally multiplexed data acquisition, or signal-dependent noise, such as are often encountered in practical signal pr ocessing applications. Index T erms — Aliasing, likelihood methods, modulation, mul- tiplicative noise, sampling, signal enhancement, time-frequency analysis, wavelets. I . I N T R O D U C T I O N Since the earliest days of signal and wav eform analysis, engineers have recognized the wide utility of parameterized families of filters : conv olution operators that are directly rep- resented by finite-length sequences of real numbers. Parallel banks of such finite-impulse-response filters, including short- time F ourier and wa velet transforms, hav e long been a canoni- cal tool for analyzing signals, images, and other data sets that arise in a variety of applications across scientific fields [1], [2]. The purpose of this article is to further expand filterbank theory and practice by dev eloping a general frame work of rev erse-order and con v olution subband structures in filterbank transforms. It describes a series of ne w results outlining equiv alences between certain “re wirings” of filterbank block diagrams, and the “localized” aliasing and modulation prop- erties of sampled signals and their subband representations, which we describe below . Sampled signals are typically acquired as linear functionals of the underlying data object of interest, which in turn is defined with respect to a continuous variable such as time or space. The actions of the conv olution operators that comprise filterbanks are studied through their Fourier transforms, under the correspondence of element-wise multiplication on the dual Based upon work supported in part by the National Science Foundation under Grant No. DMS-0652743. K. Hirakawa is with the Intelligent Signal Systems Laboratory , Univ ersity of Dayton, College Park, Dayton, OH 45469 (e-mail: k.hirakawa@notes.udayton.edu); and P . J. W olfe is with the Statistics and In- formation Sciences Laboratory , Harvard Uni versity , Oxford Street, Cambridge, MA 02138 (e-mail: patrick@seas.harvard.edu). This work was completed while K. Hirakawa was a Research Associate at Harvard University . group. Since sampling a continuous-time function periodizes its Fourier transform, ho we ver , care must be taken that no information is lost in the process. Indeed, in the absence of additional assumptions, it is not in general possible to recover signals that hav e aliased ; that is, signals whose Fourier trans- forms are supported on interv als so large that this periodization mixes distinct Fourier coefficients. As the bandwidth of any function is directly determined by its global smoothness, Fourier analysis does not lend itself to a meaningful analysis of signals whose smoothness varies and hence are not lo w-pass e verywher e . In contrast, parallel banks of con v olution operators with finite support are fundamentally local in nature. It is well known, for instance, that the flexibility afforded by wav elets to adapt to the local regularity of functions is essential in yielding the sparsity properties necessary for effecti ve signal and image analysis, as well as contemporary signal acquisition techniques such as compressed sensing [3]–[8]. Filterbanks are hence essential engineering tools for data analysis. Howe ver , definitions of aliasing and frequency mod- ulation in the global, Fourier context preclude the closed-form filterbank analysis of signals subject to missing data, spatially or temporally multiplexed acquisition, or signal-dependent noise effects. While it is well kno wn how to apply filterbanks to analyze, modify , and enhance signals that are free from aliasing or modulatory ef fects, the literature presently lacks a unified filterbank theory for these settings. In this work we dev elop a set of results necessary to fully understand and e xploit the local aliasing and modulation prop- erties of sampled signals and their subband representations. Though our moti vation stems from signal processing problems typically encountered in practice (such as those mentioned abov e, to which we return at the end of the article), our re- sults are more general, showing equiv alences between certain “rewirings” of filterbank block diagrams, and the correspond- ing actions of con volution, modulation, and do wnsampling operators. Our primary contributions are the introduction and analysis of two cardinal rewiring mechanisms—reverse-order subband structure (R OSS) and subband conv olution structure (SCS)—by which filterbank subbands are coupled together to describe the relationship between localized aliasing, modula- tion, and conv olution. The frame work we introduce is distinct from work in volving signal recov ery methods [3]–[8] and sampling theorems [9]– [12] in the extant literature. Such work has successfully char- acterized sufficient conditions for exact reconstruction when filterbank theory is used to restrict the class of signals under consideration, or to specify the fundamental compressibility of SUBMITTED MANUSCRIPT 2 (a) x (b) x m (c) x s (d) ˆ x (e) ˆ x m (f) ˆ x s Fig. 1. Pictorial illustrations of modulation and sampling in both the time (a-c) and frequency (d-f) domains. its members. In contrast, this article employs filterbank theory to describe the data acquisition and sampling process itself, rather than an y properties of a given signal class. Notions of localized aliasing and localized modulation are intimately connected with the R OSS and SCS analyses that we introduce below , and are also complementary to other well-understood concepts in filterbank analysis. The article is organized as follo ws. In the remainder of Section I we introduce ke y definitions and filterbank nota- tion, and provide a simple example of local aliasing and local modulation that motiv ates our subsequent analysis. In Section II we introduce our first “rewiring” notion—that of rev erse-order subband structure—and deriv e corresponding expressions for the filterbank coefficients corresponding to a subsampled signal. In Section III we build on this work to introduce the notion of subband con volution structure— our second means of filterbank rewiring—and show how it leads to a conv olution theorem particularly suited to the local modularity of the filterbank transform. W e conclude with a discussion in Section IV where we consider the practical use of these two notions in problems inv olving missing data, multiplex ed signal acquisition, and signal-dependent noise. A. Ke y Definitions and Filterbank Notation Throughout, let x ∈ ` 2 ( Z ) be a real-valued sequence index ed by n ∈ Z . Subsampling is the operation of replacing ev ery odd-numbered element of x by zero, and hence the subsampled sequence x s is defined element-wise as x s [ n ] := ( x [ n ] if n even, 0 if n odd. (Note that this is distinct from downsampling , a dilation of the index set of x to yield x [2 n ] , such that odd-numbered sam- ples are “dropped” and only even-numbered ones retained.) Equiv alently , x s [ n ] is an arithmetic av erage of x [ n ] and its frequency- modulated version x m [ n ] := ( − 1) n x [ n ] .: x s [ n ] = 1 2 ( x [ n ] + x m [ n ]) . (a) One-lev el filterbank structure (b) One-lev el complementary filterbank structure Fig. 2. One-lev el filterbank (a) and its complementary structure (b). Diagram (a) represents analysis and synthesis filters { g 0 , g 1 } and { h 0 , h 1 } , respectiv ely , as well as filterbank coefficient sets { v x 0 , v x 1 } resulting from the action of the analysis con volution operator and subsequent downsampling on a sequence x . Rev ersing the downsampling procedure and applying the synthesis filters yields a “reconstructed” sequence x r . Diagram (b) represents filters ˜ g i , ˜ h i “complementary” to (a), obtained by swapping the roles of g and h , respectively , and then applying an affine transformation. Figures 1(a)-(c) serve as a reminder to illustrate how samples in x and x m with opposite signs cancel out to yield x s ; we shall frequently refer back to them later . Let ˆ x denote the discrete-time Fourier transform of x , with ω ∈ R / 2 π its corresponding normalized angular frequency . Then it follows that ˆ x m ( ω ) = ˆ x ( ω − π ) and ˆ x s ( ω ) = 1 2 h ˆ x ( ω ) + ˆ x ( ω − π ) i . (1) Here we see that when the bandwidth of x —i.e., the support of ˆ x —is suf ficiently large, ˆ x ( ω ) and ˆ x m ( ω ) are indistinguishable in ˆ x s ( ω ) ; as shown in Figures 1(d)-(g), their supports overlap in the Fourier domain. This phenomenon is called aliasing ; in the absence of additional information, aliased portions of x cannot be recovered from ˆ x s ( ω ) alone. The Fourier transform is a fundamentally global operation; modulation and aliasing mix non-local information from the sequence x . In contrast, a filterbank maps a sequence x to some alternativ e representation by way of localized fil- terbank coef ficients (“analysis”), and subsequently yields a linear reconstruction x r (“synthesis”); well-known examples include short-time Fourier and wav elet representations. As the analysis operator acts linearly , we write its action as an inner product, calling it a filter when it is a con volution operator parameterized by translation along a sublattice of Z , as is the case considered here. If this con volution operator is represented by the actions of a finite-length, real-valued sequences in ` 2 ( Z ) , then we refer to a real-v alued, finite- impulse-response analysis filter g and corresponding synthesis filter h . Figure 2(a) illustrates a basic filterbank structure, with two analysis filters { g 0 , g 1 } and tw o synthesis filters { h 0 , h 1 } . W e denote by { v x 0 , v x 1 } the corresponding filterbank coefficient sequences, defined as follows. Definition 1.1 (F ilterbank Coefficient Sequence): W e call v x i ∈ ` 2 ( Z ) a one-le vel filterbank coef ficient sequence corresponding to x if v x i [ n ] :=  g i [ m ] ? m x [ m ]  [2 n ] , (2) SUBMITTED MANUSCRIPT 3 where the summation in the discrete con volution ? m is per - formed over the index m , and the subsequent notion of downsampling by two is reflected by the index set { 2 n : n ∈ Z } . This composition of con volution and dilation implies in turn that ˆ v x i ( ω ) = 1 2 h ˆ g i  ω 2  ˆ x  ω 2  + ˆ g i  ω 2 − π  ˆ x  ω 2 − π i , (3) and the set of transform coefficients { v x i [ n ] } n ∈ Z is collectiv ely referred to as the i th filterbank subband. T ypically g 0 and h 0 are smooth (i.e., low-pass) filters, while g 1 and h 1 hav e zero av erage. Thus, v x 0 provides a measure of local lo w- frequency energy concentration, while v x 1 captures local high- frequency energy , with temporal localization provided by the finite support of { g i , h i } i ∈ Z 2 . A filterbank’ s joint time- frequency resolution can be fine-tuned by recursi vely nesting copies of the basic one-lev el transform structure illustrated in Figure 2(a), yielding the multi-lev el filterbank structures that we consider later in Section II-C. Note that the Fourier representation of (3) implies the superposition of shifted copies of the resultant filtered spectra, which in general will gi ve rise to aliasing of the type illustrated in Figure 1(f). It is thus natural to ask for conditions under which this aliasing will cancel—a prerequisite for the exact reconstruction of any input sequence x ∈ ` 2 ( Z ) from its filterbank coefficients, such that x r = x in the diagram of Figure 2(a). T o this end we arriv e at the following well-known definition, which stems from global properties of the Fourier transform. Definition 1.2 (P erfect Reconstruction F ilterbank): A per- fect reconstruction filterbank { g i , h i } i ∈ Z 2 admits for all x ∈ ` 2 ( Z ) the relation ˆ x ( ω ) := ˆ h 0 ( ω ) ˆ v x 0 (2 ω ) + ˆ h 1 ( ω ) ˆ v x 1 (2 ω ) = ˆ x ( ω ) . (4) Equiv alently , as shown in Figure 2(a), we hav e for all n ∈ Z that x r [2 n ] =  h 0 [2 m ] ? m v x 0 [ m ]  [ n ] +  h 1 [2 m ] ? m v x 1 [ m ]  [ n ] = x [2 n ] x r [2 n + 1] =  h 0 [2 m + 1] ? m v x 0 [ m ]  [ n ] +  h 1 [2 m + 1] ? m v x 1 [ m ]  [ n ] = x [2 n + 1] . As described earlier, the sequences of operations correspond- ing to the forward transform step in (2) and the reconstruction step in (4) are commonly referred to as the analysis and synthesis filterbanks , respectively . Remark 1.1 (Haar F ilterbank T ransform): Perhaps the most well-known example of a perfect reconstruction filterbank is given by the so-called Haar transform, which may be defined in terms of its z -transform as: X n g i [ n ] z − n =1 + ( − 1) i z , X n h i [ n ] z − n = 1 2 h ( − 1) i + z − 1 i . (5) Note that g 0 and g 1 in (5) amount to the sum and difference of neighboring samples, respectiv ely , and it is clear that the original sequence x [ n ] is easily recoverable from the corresponding sequences of filterbank coefficients. Important and well-known results associated with Definition 1.2 established the following (see, e.g., [1]) . Pr operty 1.1 (Alias Cancellation Equivalence): The set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank if and only if 2 = ˆ g 0 ( ω ) ˆ h 0 ( ω ) + ˆ g 1 ( ω ) ˆ h 1 ( ω ) (6) 0 = ˆ g 0 ( ω ) ˆ h 0 ( ω − π ) + ˆ g 1 ( ω ) ˆ h 1 ( ω − π ) (7) Pr operty 1.2 (Analysis-Synthesis Symmetry): If finite- impulse-response filters { g i , h i } i ∈ Z 2 comprise a perfect reconstruction filterbank, then there exist a ∈ R \ { 0 } and b ∈ Z such that ˆ g i ( ω ) = ( − 1) 1 − i ae j (2 b +1) ω ˆ h 1 − i ( ω − π ) . Remark 1.2: The condition of (7) in Property 1.1 guaran- tees that the aliased components in v x 0 and v x 1 cancel, so the impulse response of the overall filterbank structure in Figure 2(a) is equal to one-half of the expression of (6); i.e., ev erywhere constant and equal to unity . Property 1.2 makes explicit the fact that an analysis filterbank uniquely defines its corresponding synthesis filterbank. These properties serve as the foundation for the reverse-order subband filterbank structure that we introduce in Section II below . B. Motivating Example for F ilterbank “Rewiring” The preceding section has served to introduce the basic notions of filterbank theory that we shall employ here. Before continuing, it is instructiv e to consider a simple moti vating example based on the simplest case of the Haar filterbank transform. In essence, we will see that “rewiring” filterbank diagrams such as those in Figure 3 can be related to the actions of con volution, modulation, and downsampling operators. In subsequent sections we develop these properties formally , and show ho w the y yield ne w insights into important practical problems. T o begin, consider the symmetric Hadamard matrix Φ =  1 1 1 − 1  which maps two-dimensional v ectors to the corresponding sums and difference of their components, and thus serves to define the (unnormalized) one-lev el Haar filterbank. Note that Φ − 1 = 1 2 Φ , and consider two systems of linear equations in Φ that will serve to illustrate the concepts of reverse-order and conv olution subband structure:  q r  = Φ  a b  ,  s t  = Φ  c d  . Example 1.1 (Reverse-Or dering and Subsampling): Consider the first system of equations above, and suppose that we replace b with − b , yielding a “modulated” version of [ a, b ] T . W e then observe that the Haar transform of [ a, − b ] T results in a re verse-or dering of q and r , which play the roles SUBMITTED MANUSCRIPT 4 of low-pass and high-pass components, respectiv ely:  1 1 1 − 1   a − b  = 1 2  1 1 1 − 1   1 1 − 1 1   q r  =  r q  . Since summing [ a, b ] T and its modulated version corresponds to subsampling, we next compute the Haar transform of [ a, 0] T , and observe that this results in an arithmetic averaging of q and r :  1 1 1 − 1   a 0  = 1 2  1 1 1 − 1   a b  +  a − b  = 1 2  q + r q + r  . W e see from this simple example that the “swapping” and the “combining” of low-pass and high-pass components are reminiscent of modulation and aliasing in the traditional Fourier sense, as illustrated respectiv ely in Figures 1(e) and 1(f). Example 1.2 (Convolution and P ointwise Multiplication): Now consider the element-wise product of the vectors [ a, b ] T and [ c, d ] T . The Haar transform of this product [ ac, bd ] T is:  1 1 1 − 1   ac bd  = 1 4  1 1 1 − 1   ( q + r )( s + t ) ( q − r )( s − t )  = 1 4  ( q + r )( s + t ) + ( q − r )( s − t ) ( q + r )( s + t ) − ( q − r )( s − t )  = 1 2  q s + r t q t + sr  . The symmetry of q s + rt and q t + sr suggests a kind of cyclic con volution of [ q , r ] T and [ s, t ] T . In fact, we will see in Section III that our filterbank rewiring techniques reco ver precisely this notion of group structure, in direct analogy to global Fourier analysis. In Section IV, these ideas will reappear in the context of analysis of signals subject to multiplicativ e noise corruption. I I . R E V E R S E - O R D E R S U B B A N D S T RU C T U R E A N D L O C A L I Z E D A L I A S I N G Having introduced the necessary definitions and giv en two brief examples, we now begin our technical dev elopment of filterbank “re wiring. ” Bearing in mind the e xamples considered abov e, we introduce in Section II-A belo w the notion of complementary filterbanks , and then employ them to obtain the following results in Section II-B: the r everse-or dering of subband structur e that results from modulation, and local- ized aliasing that stems from a veraging the low- and high- frequency filterbank coefficients. In Section II-C we extend these results to the setting of multi-le vel filterbanks. A. Complementary F ilterbanks Definition 2.1: (Complementary F ilterbanks and F ilterbank Coefficients): Let { g i , h i } i ∈ Z 2 be a perfect reconstruction filterbank. Then we define the complementary filterbank { ˜ g i , ˜ h i } i ∈ Z 2 as follows: ˜ g i [ n ] := ah i [ n + (2 b + 1)] ˜ h i [ n ] := a − 1 g i [ n − (2 b + 1)] , where a and b are chosen to satisfy Property 1.2 of perfect reconstruction filterbank, and we call w x i [ n ] a one-level com- plementary filterbank coefficient corresponding to a sequence x if w x i [ n ] :=  ˜ g i [ m ] ? m x [ m ]  [2 n ] . The following important property of complementary filter- banks follows directly from Properties 1.1 and 1.2 of perfect reconstruction filterbanks. Pr oposition 2.1 (Complementarity & P erfect Reconstruction): If the set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank, then so is { ˜ g i , ˜ h i } i ∈ Z 2 . Pr oof: Appealing to Property 1.2, we see that Fourier transforms of ˜ g i and ˜ h i respectiv ely yield ˆ ˜ g i ( ω ) = ae j (2 b +1) ω ˆ h i ( ω ) ˆ ˜ h i ( ω ) = a − 1 e − j (2 b +1) ω ˆ g i ( ω ) . (8) By substitution, we verify that (6) and (7) hold for { ˜ g i , ˜ h i } i ∈ Z 2 : ˆ ˜ g 0 ( ω ) ˆ ˜ h 0 ( ω ) + ˆ ˜ g 1 ( ω ) ˆ ˜ h 1 ( ω ) =  ae j (2 b +1) ω ˆ h 0 ( ω )  a − 1 e − j (2 b +1) ˆ g 0 ( ω )  +  ae j (2 b +1) ω ˆ h 1 ( ω )  a − 1 e − j (2 b +1) ˆ g 1 ( ω )  = 2 ˆ ˜ g 0 ( ω ) ˆ ˜ h 0 ( ω − π ) + ˆ ˜ g 1 ( ω ) ˆ ˜ h 1 ( ω − π ) =  ae j (2 b +1) ω ˆ h 0 ( ω )  a − 1 e − j (2 b +1) ˆ g 0 ( ω − π )  +  ae j (2 b +1) ω ˆ h 1 ( ω )  a − 1 e − j (2 b +1) ˆ g 1 ( ω − π )  = 0 . Hence by Property 1.1, the set { ˜ g i , ˜ h i } i ∈ Z 2 comprises a perfect reconstruction filterbank. Figure 2(b) illustrates this complementary filterbank structure, along with the corresponding complementary coefficients w x i . It is natural to ask if a filterbank can be its own complement, and to this end we hav e the following. Definition 2.2 (Self-Complementary F ilterbank): W e call a filterbank { g i , h i } i ∈ Z 2 self-complementary if v x i [ n ] = ( − 1) 1 − i w x i [ n ] . (9) Returning now to Remark 1.1, we note the following. Pr oposition 2.2 (Self-Complementarity of Haar F ilterbank): The Haar filterbank is self-complementary . Pr oof: It follows from (5) that the Haar filterbank satisfies the following symmetry: ˆ g i ( ω ) = ( − 1) i e j (2 b +1) ω ˆ g ∗ i ( ω ) , with a = 1 2 and b = − 1 . Applying Property 1.2 of perfect reconstruction filterbanks in turn yields ( − 1) i e j (2 b +1) ω ˆ g ∗ i ( ω ) =( − 1) 1 − i ae j (2 b +1) ω ˆ h 1 − i ( ω − π ) ˆ g ∗ i ( ω ) = − a ˆ h 1 − i ( ω − π ) . (10) The well-known identity ˆ g i ( ω ) = ( − 1) i e j (2 b +1) ω ˆ g ∗ 1 − i ( ω − π ) of Smith and Barnwell [13] applies; and upon substituting this into (10), we obtain the desired result: ˆ g i ( ω ) =( − 1) i e j (2 b +1) ω ( − a ˆ h i ( ω )) =( − 1) 1 − i ˆ ˜ g i ( ω ) . SUBMITTED MANUSCRIPT 5 Fig. 3. Illustration of Theorem 1, with the left side showing a modulated signal x m [ n ] and the right side showing reverse-ordering of the comple- mentary filterbank. The lo w- and high-frequency filterbank subbands for the modulated signal behave like the high- and low-frequenc y complementary filterbank subbands for the original signal, respectively . As we sho w below , complementary filterbanks play a ke y role in the reverse-ordering of subband structure induced by modulation. B. Reverse-Or der Subband Structur e Figure 3 illustrates the re versal of subband or dering that results when x is modulated by π to yield x m : the low- frequency filterbank coefficient for the modulated signal ( v x m 0 [ n ] ) behav es like the high-frequency complementary fil- terbank coef ficient for the original signal ( w x 1 [ n ] ), and vice- versa. As may be seen by comparing Figure 3 with Figure 1, this filterbank subband “role-rev ersal” is consistent with the Fourier interpretation of modulation by π ; in both cases, the low- and high-frequency components are swapped, modulo- 2 π . W e formalize this notion as follows: Theor em 1 (Reverse-Or der Subband Structur e (ROSS)): If the set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank, then v x m i [ n ] = ( − 1) i w x 1 − i [ n ] . (11) Pr oof: Modulation of x by π implies that we hav e that ˆ v x m i ( ω ) = 1 2 h ˆ g i  ω 2  ˆ x m  ω 2  + ˆ g i  ω 2 − π  ˆ x m  ω 2 − π i = 1 2 h ˆ g i  ω 2  ˆ x  ω 2 − π  + ˆ g i  ω 2 − π  ˆ x  ω 2 i . By Property 1.2 of perfect reconstruction filterbanks and Definition 2.1, ˆ g i ( ω ) = ( − 1) 1 − i ae j (2 b +1) ω ˆ h 1 − i ( ω − π ) = ( − 1) i ˆ ˜ g 1 − i ( ω − π ) ˆ v x m i ( ω ) = ( − 1) i 2 h ˆ ˜ g 1 − i  ω 2  ˆ x  ω 2  + ˆ ˜ g 1 − i  ω 2 − π  ˆ x  ω 2 − π i = ( − 1) i ˆ w x 1 − i ( ω ) . (12) Applying Property 1.1 of perfect reconstruction filterbanks to (12) immediately yields the following important corollary . Cor ollary 2.1 (Modulation induced by ROSS): Suppose the set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank. Then 0 = ˆ ˜ g 1 ( ω ) ˆ h 0 ( ω ) − ˆ ˜ g 0 ( ω ) ˆ h 1 ( ω ) , 2 = ˆ ˜ g 1 ( ω ) ˆ h 0 ( ω − π ) − ˆ ˜ g 0 ( ω ) ˆ h 1 ( ω − π ) . Remark 2.1 (F ilterbank Interpr etation of R OSS Modulation): An intuitiv e interpretation of Corollary 2.1 is that exchanging the low- and high-frequency filterbank subbands results in modulation. T o see this, consider reconstruction of the Fig. 4. Illustration of Corollary 2.1: exchange of low- and high-frequency filterbank subbands results in modulation (compare to the standard comple- mentary filterbank structure of Figure 2(b)). Fig. 5. Illustration of Corollary 2.2, with the left side showing x [ n ] subject to subsampling and the right side showing the corresponding aliasing structure. Filterbank coefficients corresponding to the subsampled signal are arithmetic av erages of complementary low- and high-frequency coef ficients. complementary filterbank coefficients with r everse-or der subbands, as illustrated in Figure 4: ˆ x r ( ω ) = ˆ h 0 ( ω ) ˆ w x 1 (2 ω ) − ˆ h 1 ( ω ) ˆ w x 0 (2 ω ) = 1 2 ˆ h 0 ( ω ) h ˆ ˜ g 1 ( ω ) ˆ x ( ω ) + ˆ ˜ g 1 ( ω − π ) ˆ x ( ω − π ) i − 1 2 ˆ h 1 ( ω ) h ˆ ˜ g 0 ( ω ) ˆ x ( ω ) + ˆ ˜ g 0 ( ω − π ) ˆ x ( ω − π ) i = 1 2 ˆ x ( ω ) h ˆ h 0 ( ω ) ˆ ˜ g 1 ( ω ) − ˆ h 1 ( ω ) ˆ ˜ g 0 ( ω ) i + 1 2 ˆ x ( ω − π ) h ˆ h 0 ( ω ) ˆ ˜ g 1 ( ω − π ) − ˆ h 1 ( ω ) ˆ ˜ g 0 ( ω − π ) i = ˆ x ( ω − π ) . W e also obtain a filterbank interpretation of the aliasing induced by subsampling , in analogy to the Fourier decom- position of (1). As shown in Figure 5, filterbank coefficients corresponding to the subsampled signal x s [ n ] are arithmetic av erages of low- and high-frequency coefficients correspond- ing to x [ n ] , in analogy to the symmetry about π / 2 visible in the Figure 1(f). Cor ollary 2.2 (Localized Aliasing): If the set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank, then by linearity and Theorem 1, v x s i [ n ] = 1 2  v x i [ n ] + ( − 1) i w x 1 − i [ n ]  . Furthermore, this modulation implies the filterbank subband symmetry v x s i [ n ] = ( − 1) i w x s 1 − i [ n ] . As can be seen in Figure 5, localized aliasing occurs when v x i [ n ] and w x 1 − i [ n ] are both simultaneously nonzero and hence indistinguishable in v x s i [ n ] . Unlike the global Fourier aliasing illustrated in Figure 1, howe ver , this aliasing is confined to a temporally localized re gion that depends on the local re gularity of x . SUBMITTED MANUSCRIPT 6 C. Extension to Multi-Level Setting Multi-lev el filterbank analysis corresponds to a recursi ve application of con volution and downsampling operators to successiv e sets of filterbank coefficients. T o inde x the cor- responding subbands, we adopt binary vector notation for indices as follo ws. Let i = ( i I - 1 , . . . , i 1 , i 0 ) T ∈ Z 2 I and i 0 = ( i I - 1 , . . . , i 1 , 1 − i 0 ) T , and recalling v x i [ n ] and w x i [ n ] from Definitions 1.1 and 2.1, define the corresponding I -lev el recursions: v x i [ n ] :=  g i I - 1 ? m I - 1  . . .  g i 1 [ m 1 ] ? m 1  g i 0 [ m 0 ] ? m 0 x [ m 0 ]  [2 m 1 ]  [2 m 2 ] . . .  [2 m I - 1 ]  [2 n ] (13) w x i [ n ] :=  g i I - 1 ? m I - 1  . . .  g i 1 [ m 1 ] ? m 1  ˜ g i 0 [ m 0 ] ? m 0 x [ m 0 ]  [2 m 1 ]  [2 m 2 ] . . .  [2 m I - 1 ]  [2 n ] . (14) Here i k ∈ Z 2 index es the analysis filters used in the k th-lev el decomposition (i.e. g 0 or g 1 ), and i 0 corresponds to a high (low) frequency subband when i is a low (high) frequency subband. Note here that the complementary filters ˜ g 0 and ˜ g 1 are used only in the 0 th-le vel decomposition in (14). The corresponding perfect reconstruction extension of (4) to the case of an I -le vel filterbank is ˆ x r ( ω ) := X i ˆ v x i (2 I ω ) I − 1 Y k =0 ˆ h i k (2 k ω ) . (15) Then, in parallel to our earlier development, the results of Theorem 1 and Corollary 2.2 extend to the multi-lev el setting as follows. Theor em 2 (Multi-Level ROSS): Suppose the set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank, and let i ∈ Z 2 I . Then, v x m i =( − 1) i 0 w x i 0 [ n ]; v x s i =( − 1) i 0 w x s i 0 [ n ] = 1 2  v x i [ n ] + ( − 1) i 0 w x i 0 [ n ]  . Using the same e xample sho wn in Figure 1, Figure 6 illustrates this localized aliasing in the multi-le vel filterbank setting. Note that although the subsampled e xample signal is subject to aliasing in a global sense (Figure 1(f)), the corresponding v x i [ n ] may be recovered from v x s i [ n ] whenever w x i 0 [ n ] = 0 . Theorem 2 simplifies when self-complimentarity is taken into account, illustrated also in Figure 7. Cor ollary 2.3 (Multi-Level Self-Complementary R OSS): If the set { g i , h i } i ∈ Z 2 is a perfect reconstruction filterbank that is also self-complementary , then v x i [ n ] =( − 1) 1 − i 0 w x i [ n ] , v x m i [ n ] = v x i 0 [ n ] ; v x s i [ n ] = 1 2  v x i [ n ] + v x m i [ n ]  = 1 2  v x i [ n ] + v x i 0 [ n ]  . Remark 2.2 (Extension to Discr ete W avelet T ransform): The above results can easily be adapted to discrete wav elet transforms, which employ the same fundamental building blocks as perfect reconstruction filterbanks [1], [2]. As an example, the filterbank re wiring associated with the wav elet transform of a subsampled signal x s [ n ] is illustrated in Figure 8; we leave the details as an easy exercise for the reader . (a) v x i (b) v x m i (c) v x s i (d) w x i Fig. 6. Pictorial illustration of localized aliasing in the 2-lev el filterbank domain, as indicated by Theorem 2. Parts (a-c) show filterbank coeffi- cients sequences corresponding to Fig. 1(a-c), respectiv ely , with (d) the complementary sequence corresponding to Fig. 1(a). From top to bottom, the ordering of the four subbands represented in each subfigure is i = (0 , 0) , (1 , 0) , (1 , 1) , (0 , 1) . “Re wiring” is evident in comparing (b) with (d), and v x i [ n ] is exactly recov erable from v x s i [ n ] whene ver w x i 0 [ n ] = 0 . I I I . S U B B A N D C O N V O L U T I O N S T RU C T U R E A N D L O C A L I Z E D M O D U L A T I O N In the previous section, we introduced and studied the rev erse-ordering of subband structure induced by subsampling, for the case of general perfect reconstruction filterbanks. Furthermore, we sa w that this structure simplified considerably when self-complimentarity was taken into account. In this section, we sho w that the symmetry of the Haar filterbank transform also af fords a characterization of Fourier group duality and con volution. As a special case, we recov er the multi-lev el R OSS structure of Corollary 2.3, thereby linking the ROSS results of Section II and the subband con volution structure (SCS) we introduce below . A. Subband Con volution Structure Recall the multi-le vel filterbank decomposition of (13), in which i k ∈ Z 2 index es pairs of analysis filters used for the k th lev el decomposition. Below , we prove the duality of time- domain multiplication and subband con volution of filterbank coefficients in this context. Theor em 3 (Subband Con volution): Let v x i , v y i be I -le vel Haar filterbank coefficient sequences corresponding to x and y , respectively , with v xy i [ n ] that of the element-wise product xy . Then, letting ~ denote cyclic conv olution, we have the relation  v x j ~ j v y j  i [ n ] := 1 2 I X j v x i + j [ n ] v y j [ n ] = v xy i [ n ] . SUBMITTED MANUSCRIPT 7 Fig. 7. Illustration of Corollary 2.3. Filterbank coefficients corresponding to the subsampled signal are arithmetic av erages of reverse-order coefficients. Localized aliasing thus occurs when v x i [ n ] and w x i 0 [ n ] are both supported simultaneously and hence indistinguishable in v x s i [ n ] . Fig. 8. An example of ROSS-based wavelet analysis, with the left side showing x [ n ] subject to subsampling, and the right side showing the corresponding aliasing structure. Pr oof: By the F ourier representation of (15), the product x [ n ] · y [ n ] can be analyzed as: \ ( x · y )( ω ) = ˆ x r ( ω ) ? ω ˆ y r ( ω ) = " X i ˆ v x i (2 I ω ) I − 1 Y k =0 ˆ h i k (2 k ω ) # ? ω   X j ˆ v x j (2 I ω ) I − 1 Y k =0 ˆ h j k (2 k ω )   = X i X j Z π − π ˆ v x i (2 I ν ) ˆ v x j  2 I ( ω − ν )  · " I − 1 Y k =0 ˆ h i k (2 k ν ) ˆ h j k  2 k ( ω − ν )  # dν . (16) It follo ws from the definition of the Haar filterbank transform in (5) that ˆ h i k (2 k ν ) ˆ h j k  2 k ( ω − ν )  = 1 4 h ( − 1) i k + e − j 2 k ν i h ( − 1) j k + e − j 2 k ( ω − ν ) i = 1 2 h ˆ h i k + j k (2 k ω ) + ( − 1) i k e − j 2 k ν ˆ h i k + j k  2 k ( ω − 2 ν )  i . Substituting this expression into (16) for k = 0 , \ ( x · y )( ω ) = 1 2 X i X j Z π − π ˆ v x i (2 I ν ) ˆ v x j ·  2 I ( ω − ν )  " I − 1 Y k =1 ˆ h i k (2 k ν ) ˆ h j k  2 k ( ω − ν )  # · h ˆ h i 0 + j 0 ( ω ) + ( − 1) i 0 e − j ν ˆ h i 0 + j 0 ( ω − 2 ν ) i dν = 1 2 X i X j ˆ h i 0 + j 0 ( ω ) Z π − π ˆ v x i (2 I ν ) ˆ v x j  2 I ( ω − ν )  · " I − 1 Y k =1 ˆ h i k (2 k ν ) ˆ h j k  2 k ( ω − ν )  # dν , where we hav e used the fact that R π − π e − j 2 k ν ˆ f ( ν ) dν = 0 for all ˆ f ( ν ) ∈ L ( R / 2 − k π ) whenev er k ≥ 0 , as e − j (2 k ν − π ) ˆ f ( ν − 2 − k π ) = − e − j 2 k ν ˆ f ( ν ) . By recursion ov er k , the above reduces to \ ( x · y )( ω ) = 1 2 K X i X j " K − 1 Y k =0 ˆ h i k + j k (2 k ω ) # · Z π − π ˆ v x i (2 I ν ) ˆ v x j  2 I ( ω − ν )  · " I − 1 Y k = K ˆ h i k (2 k ν ) ˆ h j k  2 k ( ω − ν )  # dν = 1 2 K +1 X i X j " K Y k =0 ˆ h i k + j k (2 k ω ) # · Z π − π ˆ v x i (2 I ν ) ˆ v x j  2 I ( ω − ν )  · " I − 1 Y k = K +1 ˆ h i k (2 k ν ) ˆ h j k  2 k ( ω − ν )  # dν = · · · = 1 2 I X i X j " I − 1 Y k =0 ˆ h i k + j k (2 k ω ) # · h ˆ v x i (2 I ω ) ? ω ˆ v y j (2 I ω ) i . (17) Note that (17) tak es the form of a multi-le vel in verse filterbank transform, as per (15). As the Haar filterbank transform is one- to-one and onto, v xy i is thus uniquely defined by ˆ v xy i ( ω ) = 1 2 I X j h ˆ v x i + j ( ω ) ? ω ˆ v y j ( ω ) i , which agrees with the claim of the theorem. Figure 9 illustrates the corresponding “rewiring” of filter- bank subbands, in which v x i [ n ] and v y i [ n ] are coupled together to yield v xy 0 [ n ] , and v x i 0 [ n ] and v y i [ n ] are combined to produce SUBMITTED MANUSCRIPT 8 Fig. 9. Illustration of Theorem 3, sho wing the correspondence between time-domain multiplication and “logical con volution” of filterbank subbands. v xy 1 . Remark 3.1 (Logical Con volution): By restricting x and y be finite-dimensional, we recover the so-called logical con- volution theorem [14] as a special case of Theorem 3. This is easily seen by considering an order - 2 I W alsh sequence ~ φ i ∈ R 2 I and its Abelian structure: diag ( ~ φ i ) ~ φ j = ~ φ i + j , i , j ∈ { Z 2 } I . (18) Orthogonality of the W alsh basis sets implies that any ~ x ∈ R 2 I can be expanded in terms of its W alsh-Hadamard coefficients D ~ φ i , ~ x E = ~ φ T i ~ x as ~ x = 2 − I P j D ~ φ j , ~ x E ~ φ j , and hence the group homomorphism of (18) yields the desired relation for all ~ x, ~ y ∈ R 2 I : D ~ φ i , diag ( ~ x ) ~ y E = 1 2 2 I X j , j 0 D ~ φ j , ~ x ED ~ φ j 0 , ~ y ED ~ φ i , diag ( ~ φ j ) ~ φ j 0 E = 1 2 I X j D ~ φ j , ~ x ED ~ φ i + j , ~ y E . B. Localized Modulation and Connection to Reverse-Or dered Subband Structure W e conclude this section by interpreting the result of Theorem 3 in terms of amplitude modulation and sampling. Remark 3.2 (Multi-Level R OSS for Haar F ilterbank T ransform): Suppose that we set v y (0 ,..., 0 , 0) [ n ] = v y (0 ,..., 0 , 1) [ n ] = 2 I − 1 for all n , and v y i [ n ] = 0 otherwise, thus yielding a Dirac comb . Then Theorem 3 agrees precisely with Corollary 2.3:  v x j ~ j v y j  i [ n ] = 1 2  v x i [ n ] + v x i 0 [ n ]  . Remark 3.3 (Generalized Subsampling): More generally , suppose y [ n ] ∈ { 0 , 1 } is a sampling mask of any kind. Then it may be seen from Figure 9 that the subsampled signal x [ n ] y [ n ] is aliased if v x 0 [ n ] v y 0 [ n ] and v x 1 [ n ] v y 1 [ n ] (or , v x 0 [ n ] v y 1 [ n ] and v x 0 [ n ] v y 1 [ n ] ) are simultaneously supported. Remark 3.4 (Localized Modulation): Consider Figure 9 again and suppose at time n = 0 , we have that v y 0 [ n ] = 0 and v y 1 [ n ] = 1 . Then v x 0 [ n ] is “modulated” to v xy 1 [ n ] . This is similar to Fourier amplitude modulation, in which case the energy of the modulated signal is concentrated around a chosen carrier frequency; howe ver , a representation based on Theorem 3 is amenable to temporally local processing. For example, suppose at time n = 1 , we take v y 0 [ n ] = 1 and v y 1 [ n ] = 0 ; then v x 0 [ n ] is mapped to v xy 0 [ n ] instead of v xy 1 [ n ] . In other words, this filterbank interpretation of localized modulation —illustrated in Figure 10—is ideal for tracking modulation when the “carrier” y [ n ] is allo wed to change over time. (a) v x i (b) v y i (c) v xy i Fig. 10. Pictorial illustration of localized modulation in 2-lev el filterbank domain. Time-domain multiplication of x and y , represented by their filter- bank coefficients in (a) and (b), results in subband convolution, shown in (c); “rewiring” is evident in comparing (a) and (c). Exact recovery of v x i [ n ] from v xy i [ n ] is possible when the supports of v x i + j [ n ] are mutually exclusi ve for all i and j corresponding to nonzero v x i [ n ] and v y j [ n ] . Formally , we are concerned with characterizing a sum z [ n ] = P k x k [ n ] y k [ n ] of modulated sequences x k . When the “en velope function” y k [ n ] is chosen carefully , then it follo ws from Theorem 3 that these signals are recoverable. Pr oposition 3.1 (Localized Amplitude Modulation): Suppose y k [ n ] is defined by a sequence of index values j k ∈ { Z 2 } I and Haar filterbank coefficients: v y k i [ n ] = δ ( i , j k [ n ]) . Then x k [ n ] is recov erable from z [ n ] = P k x k [ n ] y k [ n ] when the supports of v x k i + j k [ n ] [ n ] are mutually exclusiv e for all i and k . Pr oof: By the subband con volution result of Theorem 3, v z i [ n ] = X k v x k y k i [ n ] = X k  v x k j ~ j v y k j  i [ n ] = X k v x k i + j k [ n ] [ n ] Howe ver , by the assumed mutual exclusivity of the supports of v x k i + j k [ n ] [ n ] , it follows that v z i [ n ] =  v x k i + j k [ n ] [ n ] if there exists k with nonzero v x k i + j k [ n ] 0 otherwise, and thus we conclude that v x k i [ n ] = v z i + j k [ n ] [ n ] whenever v v k i [ n ] is nonzero. I V . D I S C U S S I O N : I M P L I C A T I O N S F O R S I G NA L A N A LY S I S The preceding two sections have explored properties of rev erse-order and conv olution subband structure (R OSS and SCS) in filterbanks, and shown their relation to the concepts of localized aliasing and modulation. W e now discuss the practical implications of these results for signal analysis, and provide two brief demonstrations that “re wiring” filterbank diagrams in the manner of ROSS and SCS can enable new solutions to problems in volving subsampled data corrupted by additiv e or multiplicati ve noise. As many scientific and engi- neering applications give rise to in verse problems in volving subsampled and/or noisy data, and since filterbanks are the tool of choice for many signal and image processing tasks, it SUBMITTED MANUSCRIPT 9 is natural to analyze the data likelihoods resulting from these problems directly in the filterbank coefficient domain. A. F ilterbank-Domain Likelihoods via R OSS and SCS While data likelihoods often do not admit straightforward closed-form e xpressions through traditional filterbank analysis, the R OSS and SCS concepts provide a new way to characterize signals subject to aliasing or signal-dependent noise effects directly in the filterbank coefficient domain, by virtue of the associated filterbank rewiring techniques. The expression of filterbank data likelihoods is key to solving signal recon- struction and enhancement problems in this context; as signal acquisition models, these likelihoods may be coupled with regularization terms that encourages parsimony as a means of signal modeling, reflected through prior probability densities on filterbank coefficients, or equiv alently through terms that explicitly penalize complexity . T o this end, the following two corollaries sho w how explicit likelihood formulations follo w directly from application of the R OSS and SCS concepts introduced earlier . Cor ollary 4.1 (Noisy , Subsampled Data Likelihood): Fix x ∈ ` 2 ( Z ) as the signal of interest, and let ξ comprise samples of white Gaussian noise of variance σ 2 . Suppose then that we observe subsampled, noisy data y = x s + ξ s and subsequently apply a unitary filterbank transform; then it follo ws from the localized aliasing relation of Theorem 2 that the analysis filterbank coefficients of y satisfy v y i [ n ] = v x i [ n ] + ( − 1) i 0 w x i 0 [ n ] 2 + v ξ i [ n ] + ( − 1) i 0 w ξ i 0 [ n ] 2 , and hence each admits a Normal likelihood with mean  v x i [ n ] + ( − 1) i 0 w x i 0 [ n ]  / 2 and variance σ 2 / 2 . Cor ollary 4.2 (Multiplicative Noise Data Likelihood): Suppose instead that the observation model y [ n ] = x [ n ] + x [ n ] ξ [ n ] is in force. Then the subband con volution structure of Theorem 3 implies the filterbank coefficient relation v y i [ n ] = v x i [ n ] +  v x j [ n ] ~ j v ξ j [ n ]  i , and hence the likelihood form of v y i [ n ] is multiv ariate Normal with mean v x i [ n ] , where the covariance of v y i [ n ] and v y j [ n ] is giv en by σ 2 ( v x k ~ k v x k ) i + j . B. Pr oof of Concept: Application to Image Interpolation and Denoising The likelihood e xpressions of Corollaries 4.1 and 4.2 abov e extend straightforwardly to the case of separable two- dimensional filterbank transforms, which in turn are typically employed in imaging applications. Thus, to illustrate the practical applicability of these results, we no w undertake two proof-of-concept experiments that are representativ e of prob- lems frequently encountered in digital imaging, and for which our R OSS and SCS characterizations—in contrast to standard approaches—yield closed-form likelihood expressions for the corresponding filterbank coefficients. The first of these—image interpolation in the presence of noise—is made dif ficult by the fact that lo w- and high- frequency filterbank subbands interact with one another as well as with the noise itself; the “re wiring” e xpression of Corollary 4.1 in turn provides a closed-form likelihood expression for the filterbank coefficients. In the second experiment, we consider the similarly dif ficult problem of mitigating multiplicative noise; in this case, Corollary 4.2 yields the corresponding likelihood. As filterbank coef ficients of images typically exhibit sparsity [15], one natural approach to utilize these likelihoods through a Bayesian framew ork, in which transform coefficients of the underlying image x are modeled as random variables v X i [ n ] taking zero-mean, symmetric, and unimodal “hea vy-tailed” distributions that exhibit super -Gaussian tail beha vior [16], [17]. W e assume such a prior distribution here, and ev aluate posterior means numerically via Monte Carlo av erages. W e follow typical practice in approximating the o verall joint posterior distrib ution of all filterbank coefficients by a product of marginal distrib utions, where each marginal posterior is associated with a particular subband. In turn, the ` 2 -optimal estimator of filterbank coefficients v X i [ n ] corresponding to the i th subband giv en the corresponding data coefficients v y i [ n ] is giv en by E [ v X i | v y i ] = R v x i p ( v y i | v x i ) p ( v x i ) dv x i p ( v y i ) ; (19) the corresponding synthesis filterbank in turn allows recon- struction of the estimated image. W e first consider a well-kno wn 8-bit test image that has been artificially downsampled and degraded with additiv e white Gaussian noise of variance 400 to yield a signal-to-noise ratio (SNR) of 16.77 dB relativ e to x s (or 1.22 dB relativ e to x ), as shown in Figures 11(a) and 11(b). Figure 11(c) sho ws the corresponding image reconstruction, which retains much of sharpness of the image edges and textures while suppressing noise, resulting in an SNR gain of 3.81 dB. For purposes of comparison, Figure 11(d) shows the result of the recently proposed simultaneous interpolation and denoising method of [18], which yields an SNR gain of 3.55 dB; this reconstruction exhibits edges that are more strongly preserved, but at the expense of greater smoothing of image textures. In our second experiment, we consider a synthetic aperture radar (SAR) image, av ailable at www .sandia.gov/radar; as may be seen from Figure 12(a), such images suffer from the ef fects of multiplicati ve noise [19]. Figure 12(b) shows the enhanced image resulting from a “rewiring” approach, which exhibits reduced noise in smooth and textured regions, and av oids the introduction of artif acts. In contrast, Figure 12(c) illustrates the standard approach: application of a logarithmic transformation to the data, followed by an additiv e denoising technique (here the well-known method of [16]) and subsequent exponentia- tion of the result. Not only are Bayes optimality properties of [16] lost in the exponentiation transformation back to the pixel domain, but also substantial artifacts are seen to result from this standard approach. SUBMITTED MANUSCRIPT 10 (a) Original test image (b) Subsampled, noisy test image (c) “Rewiring” reconstruction (d) Reconstruction via the method of [18] Fig. 11. Example of noisy image interpolation via the R OSS technique of Section II: The first row shows a full-resolution 8-bit test image (a), along with a subsampled version that has been degraded with additiv e white Gaussian noise of variance 400 (b). The bottom row shows a posterior mean reconstruction based on the filterbank-domain likelihood of Corollary 4.1 and a heavy-tailed prior distribution on filterbank coefficients (c), along with a reconstruction according to the recently proposed method of [18], sho wn for comparison (d). C. Concluding Remarks In conclusion, we hav e shown in this article ho w filterbank “rewirings, ” corresponding to compositions of con v olution, modulation, and downsampling operators, admit expressions of localized aliasing and modulation, in directly analogy to the global setting of Fourier analysis. In addition to establishing a number of results that formalize re verse-order and con v olution subband structures in filterbank transforms in Sections II and III, respecti vely , we ha ve demonstrated in this section how these concepts in turn enable the establishment of closed- form likelihood functions for the direct filterbank analysis of signals subject to de gradations such as missing data, spa- tially or temporally multiplex ed data acquisition, or signal- dependent noise, such as are often encountered in practical signal processing applications. A C K N OW L E D G M E N T The authors would like to thank Sandia National Laborato- ries for generously providing access to the synthetic aperture radar data; and Drs. Lei Zhang, Xin Li, and Javier Portilla for SUBMITTED MANUSCRIPT 11 (a) SAR image showing “speckled noise” (b) “Rewiring” enhancement (c) Enhancement via the method of [16], applied after logarithmic transformation Fig. 12. Example of multiplicative noise mitigation via the SCS technique of Section III: Panel (a) shows a portion of SAR imagery data in which noise is visibly present. Panel (b) shows an enhancement based on the filterbank-domain lik elihood of Corollary 4.2 and a hea vy-tailed prior distribution on filterbank coef ficients, with panel (c) showing for comparison an enhancement according to the method of [16], designed for additi ve noise and applied after a v ariance-stabilizing logarithmic transformation. kindly providing their code for the purpose of comparativ e ev aluation. R E F E R E N C E S [1] S. 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