15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP
HYPERCOMPLEX ALGEBRAS IN DIGITAL SIGNAL PROCESSING: BENEFITS AND DRAWBACKS
Daniel Alfsmann∗, Heinz G. Göckler∗, Stephen J. Sangwine†, and Todd A. Ell‡
∗Digital Signal Processing Group †Dept. Electron. Syst. Engineering ‡Goodrich Sensor Systems, Ruhr-Universität Bochum University of Essex, Wivenhoe Park Burnsville, MN 55306, USA 44780 Bochum, Germany Colchester CO4 3SQ, UK email: [email protected] email: [email protected] email: [email protected]
ABSTRACT R−,C−,H−algebras and the hyperbolic algebras being investi- This tutorial contribution presents a short historical introduction gated in more detail only most recently [3, 4, 5]. and a survey of hypercomplex algebras in conjunction with some It is a commonplace in mathematics that some or all of the fol- beneficial applications in predominantly time-based digital signal lowing fundamental properties of R and C may vanish in hyper- processing. Potential advantages and shortcomings of hypercom- complex algebras of higher dimension: Commutativity, associativ- plex digital signal processing are discussed. ity and multiplicative inverse. If an algebra is not a division alge- bra, some elements may lack a multiplicative inverse and divisors 1. INTRODUCTION of zero exist with the consequence that the product of two non-zero numbers may vanish . This was thoroughly investigated by WEIER- For a long time, mathematicians have studied various (hypercom- STRASS (1815-97) [6]. plex) algebras as a self-contained discipline. Most recently, physi- cists and engineers have gradually grasped the underlying theories This tutorial presents basic material for the understanding of for a variety of applications in signal and image processing, sig- time-based digital signal processing applying hypercomplex alge- nal and system representation and analysis, computer graphics, etc. bra. To this end, we recall the mainstream use of established CDSP under C (section 2). In section 3, we first give our reasons for tran- This tutorial paper presents an introduction to hypercomplex alge- C bras needed for hypercomplex digital signal processing (HCDSP), scending to higher dimension or other algebras, we present the and discusses the potentials and weaknesses of HCDSP. Note that necessary definitions and representations of hypercomplex algebras pure geometric signal processing is beyond the scope of this paper. for HCDSP and, finally, discuss benefits and drawbacks of HCDSP A basic scientific motivation for the investigation of hyper- in view of a variety of suitable applications. In conclusion, in sec- complex algebras is to extend complex (digital) signal processing tion 4 open issues are presented for future research. (CDSP) to HCDSP. We start our presentation with a short histori- cal survey of the advent of complex numbers and the discovery of 2. ESTABLISHED USE OF COMPLEX ALGEBRA IN hypercomplex algebras referring to [1]. SIGNAL PROCESSING (CDSP) Already during the Renaissance it had been recognised in Italy that the real algebra (R) is algebraically not closed under exponen- Almost all physical signals and (equidistantly) sampled versions tiation, when CARDANO (1501-76) and his competitor FONTANA- thereof are real-valued: s(t),s(k) ∈ R. Nevertheless, a widely used TARTAGLIA (1499-1557) were looking for a general solution of 3rd family of versatile and illustrative integral transforms, LAPLACE-, order equations, nowadays known as Cardanic formulae. To over- z-Transform and some varieties of FOURIER Transform (FT) [7], come this limitation, CARDANO in conjunction with BOMBELLI provide a one-to-one mapping to a physically supported complex (1526-72) first introduced complex numbers (C). GAUSS (1777- frequency domain under C. For instance, the FT discloses the spec- 1855) proposed the illustrative complex plane to represent complex tral content of a signal: ULER numbers in Cartesian and polar coordinates, respectively. E FT (1707-83) contributed a great multitude of widely used complex- s(t), s(k) ∈ R ←→ S( jω), S(e jΩ) ∈ C, (1) valued functional relationships. Moreover, GAUSS proved the fun- where any signal is likewise uniquely represented by its spectrum damental theorem of algebra encompassing the result that the com- jΩ plex algebra (C) is algebraically closed. along the frequency “axis” s := jω = j2π f or z := e , Ω = Another track leads to integral or functional transforms, as in- 2π f / fS, respectively (sampling rate: fS = 1/T ). troduced by LAPLACE (1749-1827), FOURIER (1768-1830) and By interchanging time and frequency in (1), time-frequency du- LAURENT (1813-54), respectively: Real or complex functions of ality [7] suggests the existence of complex signals that possess real one or more independent parameters (time, location, etc.) are (in general complex) spectra. The analytic signal, the most impor- mapped onto a complex variable domain, the frequency or spectral tant class of complex signals, has found a variety of applications domain. It is commonplace that a signal spectrum typically gives predominantly in CDSP: i) Efficient digital baseband processing of much more insight into the nature and properties of a signal than narrow-band bandpass signals with reduced sampling rate [8] (e.g. the original signal. homodyne transceiver and single sideband amplitude modulation), A first step beyond complex algebra (C) was made by HAMIL- ii) spectrally compacted data transmission by combining sequential TON (1805-65) discovering the four-dimensional (4-D) quaternions binary data to higher level complex symbols (Quadrature Ampli- (H) in 1843. Soon after HAMILTON’s publication of the quater- tude Modulation, Orthogonal Frequency Division Multiplex, etc.), nion algebra, his student GRAVES and later CAYLEY (1821-95) iii) most efficient processing of discrete orthogonal transforms, such introduced a first kind of 8-D hypercomplex algebras, the Oc- as DFT and FFT, iv) twofold system parallelisation for sample rate taves/Octonions (O) or Cayley numbers, respectively. Moreover, reduction by two, and v) efficient baseband simulation techniques HAMILTON himself introduced still another 8-D hypercomplex al- based on the complex envelope of the analytic bandpass signal. gebra [2], known as complexified quaternions or biquaternions. A The imaginary part of an analytic signal is given by the kind of generalisation to the n-D case was presented by CLIFFORD HILBERT Transform (HT) of its real part. In general, the HT is (1845-79) emphasising, however, geometrical viewpoints and n- implemented by means of a linear and time-invariant (LTI) digital D rotation. Nevertheless, CLIFFORD algebras include the former system [8], for instance, as an FIR system with (zero-phase) im-
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pulse response hHT(k) = 2/(kπ) for odd k and 0 elsewhere: whereas multiplication is generally distributive over addition. Obvi- ously, a hypercomplex addition (7) consists of n K-valued additions. FT jΩ jΩ 2 sˆ(k) = hHT(k) ∗s(k) ←→ Sˆ(e ) = − jsgn(sinΩ) · S(e ), (2) A hypercomplex multiplication demands n K-valued multiplica- where (∗) denotes convolution, yielding the digital analytic signal: tions. For instance, the multiplication of two complex numbers, z1 = x1 + iy1, z2 = x2 + iy2,(C is considered as a 2-D hypercom- FT jΩ jΩ plex algebra with K = R and i2 = i2 = −1) according to: s+(k) = s(k) + jsˆ(k) ←→ S+(e ) = S(e )[1+ sgn(sinΩ)]. (3) 2 As a result, the overall spectral bandwidth of any analytic signal z1z2 = x1x2 − y1y2 + i(x1y2 + x2y1), (8) s (k) comprises just half the width of the original real signal s(k), + requires n = 2 real additions and n2 = 4 real multiplications. Of allowing for most of the aforementioned applications exploiting the course, these operations have to be considered when evaluating the potential of unconstrained frequency shifting of analytic signals. computational load of HCDSP. Every associative algebra can be represented by an isomorphic 3. HYPERCOMPLEX ALGEBRAS IN DIGITAL SIGNAL K-valued n× n matrix algebra. For instance, the matrix PROCESSING (HCDSP) q1 −q2 −q3 q4 3.1 Motivation q q −q −q Q = 2 1 4 3 ∈ R4×4 (9) Exploiting the potential of CDSP to advantage, as outlined in sec- q3 q4 q1 q2 tion 2, the following questions related to HCDSP are motivated: −q4 q3 −q2 q1 What are the benefits of HCDSP w.r.t. the above collection of ap- is completely equivalent to the quaternion (5), and all operations plications and beyond? What is the impact on HCDSP, if the un- and properties can likewise be validated with both representations derlying hypercomplex algebra lacks commutativity, associativity, (e.g. that Q is not commutative). However, (9) is highly redundant divisors of zero and/or contains ? Hence, at least the following ba- and therefore computationally inefficient compared to the direct cal- sic system properties, well understood for real and complex DSP culation derived from the algebra’s multiplication table (6). [8], call for thorough investigation in the case of HCDSP: i) LTI An LTI system based on a hypercomplex algebra can always property tightly related to convolution, ii) existence of hypercom- be decomposed either into K-valued basic operations, as in (8), or plex spectral transforms [9], similar to z- and FOURIER Transform, into K-valued subsystems. For the latter, a MIMO (Multiple In- their impact on convolution theorem and the availability of some put Multiple Output) representation is useful, where the compo- kind of analytic signal [10, 11, 12], iii) overall expenditure of sys- nents x k y k K of the hypercomplex input and output signals tem implementation, and potential benefits of divisors of zero for ν ( ), ν ( ) ∈ x k y k A are combined to the respective input and output vec- reduced computational burden [13, 14], and iv) potential advantages ( ), ( ) ∈ tors x k , y k Rn, k Z. To this end, a transfer matrix H z can of multidimensional [11, 15, 16] and multirate HCDSP. ( ) ( ) ∈ ∈ ( ) be derived from the multiplication rule of the underlying algebra Furthermore, there are three basic motivations for the appli- by replacing each real multiplication (·) with the convolution oper- cation of HCDSP: i) The holistic, compact processing of vector- ator (∗); cf. (8) for complex systems. For the quaternion example, valued signals that are a function of one or more independent pa- matrix representation results in the general 4 × 4 MIMO transfer rameters (e.g. time, location, physical quantities). Here, the di- matrix mension of the algebra must be chosen in compliance with the di- mension of the signal vector. This means that each vector-sample Hr(z) −Hi(z) −Hj(z) −Hk(z) H (z) H (z) −H (z) H (z) is treated as a whole rather than treating its components separately. H i r k j (z) = H z H z H z H z (10) Classically, the reason for this is that the sample as a whole conveys j( ) k( ) r( ) − i( ) information (direction in vector space) that is lost if the compo- Hk(z) −Hj(z) Hi(z) Hr(z) nents of the sample are processed independently. ii) Hypercomplex digital processing of real or complex signals, respectively, with ap- that is composed of only four different real subsystems Hr(z), Hi(z), plications, for instance, in digital filtering [17, 18]. iii) Concurrent Hj(z), Hk(z). processing of a couple of independent signals in a hypercomplex system, where the dimension of the algebra must be consistent with 3.3 Classes of hypercomplex algebras and their advantages the number of signals to be processed. For applications in digital 3.3.1 Division algebras filter banks see [19, 20, 21]. Due to the great variety of possible vector spaces and particularly 3.2 Definitions and representations their associated multiplication tables, one has to choose an appro- priate algebra for a specific application. However, the need for We define a general hypercomplex algebra A [22] beneficial properties explicitly narrows the number of alternatives. Firstly, the most important distinction of algebra classes is whether a = a + a i + ... + anin ∈ A, a ,...,an ∈ K (4) 1 2 2 1 or not an algebra A is a division algebra. If this is the case, every as an n-D K-vector space over the field K = R,C with an associated non-zero element a ∈ A has a unique inverse element a−1 ∈ A. In multiplication rule, comprising the unit element 1 (1·a = a·1 = a). 1877, FROBENIUS stated his famous theorem that there exist only The multiplication table defines every relation between the imagi- three associative division algebras A over the real numbers: The real nary units i2,...,in and implies the specific properties of the alge- (R) and complex numbers (C), and the quaternions (H) [23]. Since bra. For instance, the quaternions are a 4-D R-algebra the latter are not commutative regarding multiplication (6), the only commutative and associative division algebras (or fields) are gener- q = q1 + q2i + q3j + q4k ∈ H, q1,q2,q3,q4 ∈ R, (5) ally limited to dimension n ≤ 2 (R, C) [22]. This means that we gen- with the multiplication table: erally have to renounce the familiar properties of R and C if we call 2 2 2 for higher algebra dimensions. For instance, if we lower the require- ij = −ji = k, jk = −kj = i, ki = −ik = j, i = j = k = −1. (6) ments regarding the associativity of multiplication, resulting in the From (6) it becomes clear that quaternions are not commutative: In so-called alternative property (aa)b = a(ab) and a(bb) = (ab)b, general pq 6= qp for p,q ∈ H. Nevertheless, they are associative: respectively, we can utilise a division algebra even for n = 8: The p(qr) = (pq)r. Addition of two hypercomplex numbers a,b ∈ A is octonions O. However, as ZORN has detected in 1931, there is no always performed componentwise: real alternative division algebra other than R, C, H and O [24]. Therefore, these algebras have an exceptional position compared n a + b = a + b + (a + b )i , (7) with any other possible algebra [22]. They have in common that 1 1 ∑ ν ν ν they can be generated with the doubling procedure developed by ν=2
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CAYLEY and DICKSON: A 2n-D algebra B is constructed of two 3.3.3 Commutative algebras A n-D subalgebras by (possibly multiple) application of If we forgo both the call for a division algebra and a geometric b = a1 + a2in+1 ∈ B, a1,a2 ∈ A (11) meaning according to sec. 3.3.2, we can utilise commutative and 2 associative non-division algebras, which always exhibit zero divi- and a new imaginary unit in+1 = −1 for each iteration step, which sors for n > 2. Mostly, commutative algebras are constructed by a anticommutes (iα iβ = −iβ iα ) with the already existing imaginary modified CAYLEY-DICKSON rule (11) with commuting imaginary units i2,...,in. For instance, with (11) the quaternions B = H (n = units. The first attempt to apply these algebras to signal processing 2 · 2) can be generated from complex numbers (n = 2) with A = C, was made by SCHÜTTE [17]: He proposed the Reduced Biquater- 2 i3 = −1 and i2i3 = −i3i2. To comply with (5) and (6), the imaginary nions defined by (12) with q3 ≡ q4 ≡ 0. Alternatively, they can be units have to be relabeled only: i = i2, j = i3, k = i4. represented as bicomplex numbers [37]: Due to the exceptional position of the quaternions within hy- ′ ′′ ′ ′′ C C C C percomplex algebras (the only associative real division algebra with a = a1 + ja2 ∈ ⊗ , a1 = a1 + ia1 ∈ , a2 = a2 + ia2 ∈ , (13) n > 2), they have been investigated most thoroughly and most com- where i2 = j2 = −1, which belong to the family of multicomplex monly been used in HCDSP. In particular, their usefulness in the numbers [38]. Hypercomplex Commutative Algebras (HCA) [11, n-D signal (image) processing has been shown [12, 15, 25], espe- 39] (with CLIFFORD-like generation scheme) and Tessarines [14, cially in connection with an intrisically quaternionic FT (QFT). 40, 41] are isomorphic to (13). The latter are based on the 2-D There are many possible definitions of a quaternionic FT, all hyperbolic numbers [3, 4] D (double numbers [22], split-complex stemming from the non-commutative nature of the algebra. Besides numbers). They can be defined like complex numbers the possibility of placing the transform’s exponential kernel on the ′ ′′ ′ ′′ left or right of the input function, the kernel itself can take multiple a = a + ja ∈ D, a ,a ∈ R, j2 = +1, j ∈/ R, (14) forms. Multiple kernel definitions are a result of the fact that, in q+r q r H but with a different definition of the imaginary unit j [40, 4]. As a general, e 6= e e , where q,r ∈ . consequence, zero divisors exist already for 2-D numbers (14): The first definition of a quaternionic FT was that of Ell [9, 26]. Sangwine’s discrete version of this transform [27] has shown appli- (1− j)(1+ j) = 1− j2 = 1− 1 = 0. (15) cability by allowing generalizations of many image processing tech- Hyperbolic numbers can be extended to any power-of-two n-D hy- niques dependent on FT of vector data. In particular, using a more perbolic numbers [5]: e.g. D ⊗ D for n = 4. Moreover, these com- recent hypercomplex FT definition [25], the validity of hypercom- mutative algebras have the benefit that they retain their properties plex auto- and cross-correlation, and vector phase correlation have throughout all dimensions n. been demonstrated [28, 29, 30]. In [12] a hypercomplex FT was The existence of zero divisors may cause difficulties (cf. sec. applied in the analysis of greyscale images because of its symme- 3.4.2), but it also confers a major advantage: The potential of try properties (which are similar to the symmetry properties of the orthogonal decomposition, allowing for the representation of any complex FT of real signals). It was not until 2007 that suitable hy- number in terms of its orthogonal components, e.g. percomplex spectral convolution operator formulas were provided 1 ′ ′′ 1 ′ ′′ for use in vector image processing [31]. a˜ = a + a ∈ R, a˜ = a − a ∈ R (16) The QFT suffers from the same lacuna as the complex FT; the 1 2 2 2 class of useful functions do not all have a QFT (e.g. unit step func- in case of 2-D hyperbolic numbers (14). Any operation can be car- tion). The Laplace transform broadens the class of transformable ried out componentwise on the orthogonal components (16) and, functions [9]. Moreover, the concept of the analytic signal (3) hence, the computational load of multiplication/convolution is dra- depending on a single (time) parameter has been extended to the matically reduced. (Regular 4-D hypercomplex multiplication re- quaternionic analytic signal, which is a function of two independent quires 42 = 16 real multiplications, whereas multiplication employ- variables, by means of a QFT [11, 12, 32, 33]. ing orthogonal decomposition requires only 4 real multiplications!) The operation saving effect increases with algebra dimension n. In 3.3.2 Geometric algebras contrast to n-D hyperbolic algebras referring to (16), commutative Geometric algebra is a branch of mathematics and computer science algebras containing complex subalgebras can only be decomposed (especially robotics) concerned with the representation and manip- to n/2 orthogonal components [14]. Mathematically, the orthogo- ulation of algebraic quantities representing points, lines and planes nal decomposition is based on the fact that any commutative and in 3-D space [34, 35]. Arbitrary rotation in 3-D space is efficiently associative algebra over R is isomorphic to a direct sum of R [6]. described and performed using quaternions H. Moreover, most ad- vantageously, repeated rotation does not accumulate errors caused 3.4 Problems by angle quantisation. As a result, “integer” rotation is feasible. 3.4.1 Non-commutativity and non-associativity Most common geometric algebras are examples of the CLIF- FORD algebra class. They utilise a formulation law different from Many hypercomplex algebras lack either or both of commutativity 2 and associativity. This is fundamental to the algebras and it fol- (11) and permit the so-called generator units both with in+1 = −1 2 lows from their geometric meaning distinguishing them from the (a) and in+1 = +1 (b). The number p of type-a generator units, and commutative algebra classes. Resulting from this meaning, it fol- the number q of type-b generator units completely determine the lows that the ordering of cascaded sections becomes crucial and K C K particular CLIFFORD algebra over the field with ℓ(p,q)( ). the convolution theorem, linking the time and spectral domains, Generally, every CLIFFORD algebra is associative, but is not neces- does not hold in the common form [11, 41]. Apart from that, non- C H O sarily a division algebra. They include and , but not , since commutativity does not pose fundamental problems in HCDSP. In the latter is not associative. In [11], a CLIFFORD FT is presented. the case of quaternions, as an example, re-ordering of products is Differing from CLIFFORD’s biquaternions [1], HAMILTON was expressed by the generalized conjugate rule the first to introduce a biquaternion algebra by replacing the real quantities of quaternions H according to (5) with complex parame- pqr = rqp, p,q,r ∈ H, (17) ters: where the overbar represents a quaternion conjugate [22]. If we q = q1 + q2i + q3j + q4k, q1,q2,q3,q4 ∈ C, (12) have two terms in an algebraic expression which we wish to re- The biquaternion algebra is not a division algebra, but it is asso- order, we must apply the rule: pq = qp. This makes algebraic ciative. The biquaternion FT has been defined [36] and some of derivations more difficult. A good example of such a derivation is its properties elucidated. Furthermore, biquaternions have been ap- given in [42], where it is proved that the QFT is uniquely invertible. plied to vector-sensor array processing [15], but much work remains Associativity is a more complex problem. An associative bi- to be done on the application of biquaternions to signal processing. nary operation ◦ is one for which (a ◦ b) ◦ c = a ◦ (b ◦ c) for all
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a, b and c. I.e. the order in which the operations are carried out One approach to overcome these issues is to study quaternion- is significant (not the ordering of the operands). The quaternions, valued functions of one or more reduced quaternions. Ell [9] studied biquaternions and CLIFFORD algebras are associative, so the prob- quaternion-valued functions of ‘ortho-complex’ variables. This al- lem does not arise there, but many other hypercomplex algebras of lowed for the definition of bi-analytic and bi-regular functions and dimension n > 4 lack associativity. This was first discovered when functional compositions, which retained these properties. Likewise, HAMILTON’s ideas were extended to the 8-D octonions, where mul- quaternion-valued functions of a single pure quaternion variable tiplication is not associative. Nevertheless, a digital system based ix + jy + kz, x,y,z ∈ R, were investigated in [51]. on an (associative or non-associative) hypercomplex algebra always An alternate approach is to limit the regions for the definition of remains linear and time-invariant (LTI), if it is realised exclusively the analytic extension to local subfields within the quaternion field. with (ideal) adders, constant multipliers and delays. De LEO [52] generalized analyticity by defining a ‘local’ derivative operator that depends upon the four-dimensional point at which the 3.4.2 Existence of zero divisors derivative is to be made. This approach has the added benefit of being directly generalizable to octonionic functions of octonionic Although the existence of zero divisors allows for orthogonal de- variables. composition (highly reducing computational burden), it also im- n 2 plies that the Euclidean norm |a| = q∑ν=1 aν is not multiplicative: 4. CONCLUSION |a|· |b| 6= |ab|. This results from the underlying non-Euclidean geo- We have presented an overview of the main ideas of hypercomplex metric meaning (sec. 3.3.2), and severely complicates the definition digital signal processing as an extension of the ideas of complex of signal energy and the definition of lossless systems [20]. signal processing. The many possible hypercomplex algebras have been shown to have common features and in some cases common 3.4.3 Non-availability of hypercomplex transforms drawbacks (lack of commutativity, associativity, existence of divi- While useful hypercomplex spectral transforms are known for sig- sors of zero) which, however, can even be used to advantage in cer- nals depending on two variables [9], e. g. spatial variables in case tain cases. Nevertheless, the use of hypercomplex algebras in sig- of image processing, expedient intrinsically hypercomplex spectral nal processing has developed great promise over the last ten years, transforms are claimed to be not available for signals depending on and we believe that the actual difficulties of using hypercomplex only one variable, e. g. on time in case of communication or au- algebras are not insurmountable. The next 5–10 years will show, dio signals. In [11, 32], the argument is based on the number of whether this optimism is justified. symmetries inherent in the signal, forming a hierarchy of spectral Future research in HCDSP will urge us to study the entire transforms. In fact, the QTFM toolbox [43] includes computation knowledge offered by the mathematics and computer science com- of one-dimensional quaternion FOURIER transforms, but so far the munity more thoroughly, and to investigate in how far this knowl- argument of [11, 32] has not been examined or refuted in print. edge is applicable to the classical signal processing tasks predom- inantly considered in this survey paper. This may lead us via the 3.4.4 Non-availability of hypercomplex analytic signals work of FELSBERG [10], BÜLOW [12, 32] and SOMMER [11] and beyond to the monogenic signal as a kind of generalised analytic In the theory of complex analysis there is a four-way equivalence signal, and to the RIESZ transform as a generalised HILBERT trans- between a complex function being analytic, regular, continuously form. differentiable and conjugate harmonic. This four-way equivalence is exploited in the simplification of many theorems and the analysis of mathematical models used in signal processing. For example, the REFERENCES HT used to construct an analytic signal (3) is the boundary value of a [1] B. L. van der Waerden, A History of Algebra, Springer-Verlag, conjugate harmonic function along a closed contour about the upper Berlin, 1985. R2 half plane of . Further, causality is the FOURIER dual to the one- [2] W. R. Hamilton, “On the geometrical interpretation of some sidedness we find in the frequency domain of analytic signals. But results obtained by calculation with biquaternions,” Proceed- in the z-domain causal functions are free of poles outside the unit ings of the Royal Irish Academy, vol. 5, pp. 388–90, 1853. circle, hence they are ‘analytic’ there. It is therefore natural to look for the same expressive power in hypercomplex analytic signals. It [3] G. 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