Fast Directional Spatially Localized Spherical Harmonic Transform

Transcription

1 1 Fast Directiona Spatiay Locaized Spherica Harmonic Transform Zubair Khaid, Student Member, IEEE, Rodney A. Kennedy, Feow, IEEE, Saman Durrani, Senior Member, IEEE, Parastoo Sadeghi, Senior Member, IEEE, Yves Wiaux, Member, IEEE, and Jason D. McEwen, Member, IEEE Abstract We propose a transform for signas defined on the sphere that reveas their ocaized directiona content in the spatio-spectra domain when used in conjunction with an asymmetric window function. We ca this transform the directiona spatiay ocaized spherica harmonic transform directiona SLSHT) which extends the SLSHT from the iterature whose usefuness is imited to symmetric windows. We present an inversion reation to synthesize the origina signa from its directiona- SLSHT distribution for an arbitrary window function. As an exampe of an asymmetric window, the most concentrated bandimited eigenfunction in an eiptica region on the sphere is proposed for directiona spatio-spectra anaysis and its effectiveness is iustrated on the Mars topographic data-set. Finay, since such typica data-sets on the sphere are of considerabe size and the directiona SLSHT is intrinsicay computationay demanding depending on the band-imits of the signa and window, a fast agorithm for the efficient computation of the transform is deveoped. The foating point precision numerica accuracy of the fast agorithm is demonstrated and a fu numerica compexity anaysis is presented. Index Terms Signa anaysis, spherica harmonics, -sphere. EDICS Category: MDS-APPL, DSP-TFSR. I. INTRODUCTION Signas that are inherenty defined on the sphere appear in various fieds of science and engineering, such as medica image anaysis [1], geodesy [], computer graphics [3], panetary science [4], eectromagnetic inverse probems [5], cosmoogy [6], 3D beamforming [7] and wireess channe modeing [8]. In order to anayze and process signas on the sphere, many signa processing techniques have been extended from the Eucidean domain to the spherica domain [], [9] [3]. Z. Khaid, R. A. Kennedy, S. Durrani and P. Sadeghi are with the Research Schoo of Engineering, Coege of Engineering and Computer Science, The Austraian Nationa University, Canberra, Austraia. Y. Wiaux is with the Institute of Eectrica Engineering and the Institute of Bioengineering, Ecoe Poytechnique Fédérae de Lausanne EPFL), Lausanne, Switzerand. Y. Wiaux is aso with the Institute of Bioengineering, EPFL, CH-1015 Lausanne, Switzerand, and the Department of Radioogy and Medica Informatics, University of Geneva UniGE), Geneva, Switzerand. J. D. McEwen is with the Department of Physics and Astronomy, University Coege London, London, UK. Z. Khaid, R. A. Kennedy and P. Sadeghi are supported by the Austraian Research Counci s Discovery Projects funding scheme Project No. DP ). Y. Wiaux is supported in part by the Center for Biomedica Imaging CIBM) of the Geneva and Lausanne Universities, EPFL, and the Leenaards and Louis-Jeantet foundations, and in part by the SNSF by Grant PP00P J. D. McEwen is supported by a Newton Internationa Feowship from the Roya Society and the British Academy. E-mai: {zubair.khaid, rodney.kennedy, saman.durrani, parastoo.sadeghi}@anu.edu.au, yves.wiaux@epf.ch, jason.mcewen@uc.ac.uk Due to the abiity of waveets to resove ocaized signa content in both space and scae, waveets have been extensivey investigated for anayzing signas on the sphere [9], [13] [15], [19] [3] and have been utiized in various appications e.g., in astrophysics [4] [9] and geophysics [4], [30], [31]). Some of the waveet techniques on the sphere aso incorporate directiona phenomena in the spatia-scae decomposition of a signa e.g., [1] [3]). As an aternative to spatia-scae decomposition, spatio-spectra spatia-spectra) techniques have aso been deveoped and appied for ocaized spectra anaysis, spectra estimation and spatiay varying spectra fitering of signas [10], [1], [18], [3], [33]. The spectra domain is formed through the spherica harmonic transform which serves as a counterpart of the Fourier transform for signas on the sphere [5], [34] [36]. The ocaized spherica harmonic transform, composed of spatia windowing foowed by spherica harmonic transform, was first devised in [18] for ocaized spectra anaysis. We note that the ocaized spherica harmonic transform was defined in [18] for azimuthay asymmetric i.e., directiona) window functions, however, it was appied and investigated for azimuthay symmetric functions ony. Furthermore, a spectray truncated azimuthay symmetric window function was used for spatia ocaization [18]. Due to spectra truncation, the window used for spatia ocaization may not be concentrated in the region of interest. This issue was resoved in [3], where azimuthay symmetric eigenfunctions obtained from the Sepian concentration probem on the sphere were used as window functions the Sepian concentration probem is studied for arbitrary regions on the sphere in []). Foowing [18], the spatiay ocaized spherica harmonic transform SLSHT) for signas on the sphere has been devised in [10] to obtain the spatio-spectra representation of signas for azimuthay symmetric window functions, where the effect of different window functions on the SLSHT distribution is studied. Subsequenty, the SLSHT has been used to perform spatiay varying spectra fitering [1], again with azimuthay symmetric window functions. In obtaining the SLSHT distribution for spatio-spectra representation of a signa, the use of an azimuthay symmetric window function provides mathematica simpifications, however, such an approach cannot discriminate ocaized directiona features in the spatio-spectra domain. This motivates the use of asymmetric window functions in the spatio-spectra transformation of a signa using the SLSHT. In order to serve this objective, we empoy the definition of the ocaized spherica harmonic transform in [18] and define the SLSHT

2 and the SLSHT distributions using azimuthay asymmetric window functions for spatia ocaization. Since the use of an asymmetric window function enabes the transform to revea directiona features in the spatio-spectra domain, we ca the proposed transform the directiona SLSHT. We aso provide a harmonic anaysis of the proposed transform and present an inversion reation to recover the signa from its directiona SLSHT distribution. Since the directiona SLSHT distribution of a signa is required to be computed for each spatia position and for each spectra component, and data-sets on the sphere are of considerabe size e.g., three miion sampes on the sphere for current data-sets [37] and fifty miion sampes for forthcoming data-sets [38]), the evauation of the directiona SLSHT distribution is computationay chaenging. We deveop fast agorithms for this purpose. Through experimenta resuts we show the numerica accuracy and efficient computation of the proposed directiona SLSHT transform. Furthermore, due to the fact that the proposed directiona SLSHT distribution depends on the window function used for spatia ocaization, we anayze the asymmetric band-imited window function with nomina concentration in an eiptica region around the north poe, which is obtained from the Sepian concentration probem on the sphere. We aso iustrate, through an exampe, the capabiity of the proposed directiona SLSHT to revea directiona features in the spatio-spectra domain. The remainder of the paper is structured as foows. In Section II, we review mathematica preiminaries reated to the signas on the sphere, which are required in the seque. We present the formuation of the directiona SLSHT, its harmonic anaysis and signa reconstruction from the SLSHT distribution in Section III. Different agorithms for the evauation of the SLSHT distribution are provided in Section IV and the detaied anaysis of the asymmetric window function is presented in Section V. In Section VI, we show timing and accuracy resuts of our agorithms and an iustration of the transform. Concuding remarks are presented in Section VII. II. MATHEMATICAL BACKGROUND In order to carify the adopted notation, we review some mathematica background for signas defined on the sphere and the rotation group. A. Signas on the Sphere In this work, we consider the square integrabe compex functions fˆx) defined on unit sphere S {u R 3 : u = 1}, where denotes Eucidian norm, ˆx ˆxθ, φ) sin θ cos φ, sin θ sin φ, cos θ) T R 3 is a unit vector and parameterizes a point on the unit sphere with θ [0, π] denoting the co-atitude and φ [0, π) denoting the ongitude. The inner product of two functions f and h on S is defined as [39] f, h fˆx)hˆx) dsˆx), 1) S where ) denotes the compex conjugate, dsˆx) = sin θdθdφ and the integration is carried out over the unit sphere. With the inner product in 1), the space of square integrabe compex vaued functions on the sphere forms a compete Hibert space L S ). Aso, the inner product in 1) induces a norm f f, f 1/. We refer the functions with finite induced norm as signas on the sphere. The Hibert space L S ) is separabe and the spherica harmonics form the archetype compete orthonorma set of basis functions. The spherica harmonics, Y mˆx) = Y m θ, φ), for degree 0 and order m are defined as [5], [36] Y m θ, φ) = N m P m cos θ)e imφ, ) where N m +1 m)! = 4π +m)! denotes the normaization constant and P m are the associated Legendre poynomias [36]. With the above definitions, the spherica harmonics form an orthonorma set of basis functions, i.e., they satisfy Y m, m δ mm, where δ is the Kronecker deta. By competeness and orthonormaity of the spherica harmonics, we can expand any signa f L S ) as where fˆx) = =0 m= f ) m Y m ˆx), 3) ) m f f, Y m = fˆx)y m ˆx) dsˆx) 4) S denotes the spherica harmonic coefficient of degree and order m. The signa f is said to be band-imited with maximum spherica harmonic degree L f if f ) m = 0, > L f. B. Rotations on the Sphere and Wigner-D Functions Rotations on the sphere are often parameterized using Euer anges α, β, γ) SO3), where α [0, π), β [0, π] and γ [0, π) [36]. Using the zyz Euer convention, we define the rotation operator D ρ, for ρ = α, β, γ) SO3), which rotates a function on a sphere in the sequence of γ rotation around z-axis, then β rotation about y-axis foowed by a α rotation around z-axis. The spherica harmonic coefficient of a rotated signa D ρ f is reated to the coefficients of the origina signa by Dρ f ) m = m = D m,m ρ) f ) m, ρ = α, β, γ), 5) where Dm,m ρ) denotes the Wigner-D function [36] of degree and orders m and m and is given by Dm,m ρ) = D m,m α, β, γ) 6) = e imα d m,m β) e imγ, ρ = α, β, γ), where d m,m β) is the Wigner-d function [36]. C. Signas on the Rotation Group SO3) For 0 and m, m Z such that m, m, the Wigner-D functions in 6) form a compete set of orthogona

3 3 functions for the space L SO3)) of functions defined on the rotation group SO3) and foow the orthogonaity reation SO3) Dm,m ρ)dp q,q ρ) dρ = 8π + 1 δ pδ mq δ m q, 7) where dρ = dα sin βdβdγ and the integra is a tripe integra over a rotations α, β, γ) SO3) [36]. Thus, any function f L SO3)) may be expressed as where fρ) = =0 m= m= f ) m,m = + 1 8π SO3) ) f D m,m m,m ρ), 8) fρ)dm,m ρ) dρ. 9) The signa f is said to be band-imited with maximum degree L f if f ) m,m = 0, > L f. D. Discretization of S and SO3) In order to represent functions on S and SO3), it is necessary to adopt appropriate tesseation schemes to discretize both the unit sphere domain and the Euer ange domain of SO3). We consider tesseation schemes that support a samping theorem for band-imited functions, which is equivaent to supporting an exact quadrature. For the unit sphere domain, we adopt the equianguar tesseation scheme [35] defined as S L = {θ nθ = πn θ +1)/L+ 1), φ nφ = πn φ /L + 1) : 0 n θ L, 0 n φ L}, which is a grid of L+1) L+1) sampe points on the sphere incuding repeated sampes of the south poe) that keeps the samping in θ and φ independent. For a band-imited function on the sphere f L S ) with maximum spherica harmonic degree L f, the samping on the grid S Lf ensures that a information of the function is captured in the finite set of sampes and, moreover, that exact quadrature can be performed [35]. Note that this samping theorem was deveoped ony recenty [35] and requires approximatey haf as many sampes on the sphere as required by aternative equianguar samping theorems on the sphere [34]. For the Euer ange representation of the rotation group SO3), we consider the equianguar tesseation scheme E L = {α nα = πn α /L + 1), β nβ = πn β /L + 1), γ nγ = πn γ /L + 1) : 0 n α, n γ L, 0 n β L}. Again for a function f L SO3)) with maximum spectra degree L f, the samping of a function f on E Lf ensures that a information of the function is captured and aso permits exact quadrature which foows from the resuts deveoped on the sphere [35]). III. DIRECTIONAL SLSHT We describe in this section the directiona SLSHT, which is capabe of reveaing directiona features of signas in the spatio-spectra 1 domain. For spatia ocaization, we consider 1 When we refer to spatio-spectra, we consider the SO3) spatia domain, instead of S. This is due to the reason that we are considering a possibe rotations, parameterized using Euer anges which form the SO3) domain. the band-imited azimuthay asymmetric window function which is spatiay concentrated in some asymmetric region around the north poe. Since the rotation around the z-axis does not have any affect on an azimuthay symmetric function, the ocaized spherica harmonic transform using an azimuthay symmetric window function can be parameterized on the sphere. However, if an azimuthay asymmetric window is used to obtain ocaization in the spatia domain, the rotation of the window function is fuy parameterized with the consideration of a three Euer anges α, β, γ) SO3). We refer to the spatiay ocaized transform using an asymmetric window as the directiona SLSHT. Here, we first define the directiona SLSHT distribution which presents the signa in the spatio-spectra domain. Later in this section, we present the harmonic anaysis of SLSHT distribution and provide an inversion reation to obtain the signa from its given directiona SLSHT distribution. A. Forward Directiona SLSHT Definition 1 Directiona SLSHT): For a signa f L S ), define the directiona SLSHT distribution component gρ;, m) L SO3)) of degree and order m as the spherica harmonic transform of a ocaized signa where ocaization is provided by the rotation operator D ρ acting on window function h L S ), i.e., gρ;, m) fˆx) D ρ h ) ˆx) Y m S ˆx) dsˆx) 10) for 0 L g, m, where L g = L f + L h denotes the maximum spherica harmonic degree for which the distribution components gρ;, m) are non-zero, and L f and L h denote the band-imits of the signa f and the window function h, respectivey. Aso, each distribution component gρ;, m) is band-imited in ρ = α, β, γ) SO3) with maximum degree L h, i.e., when expressed in terms of Wigner-D functions. We eaborate on this shorty. Furthermore, we consider unit energy normaized window functions such that h, h = 1. Remark 1: The directiona SLSHT distribution component in 10) can be interpreted as the spherica harmonic transform of the ocaized signa where the window function h provides asymmetric ocaization at spatia position ˆx = ˆxβ, α) S and the first rotation, through γ, determines the orientation of the window function at ˆx. If the window function is azimuthay symmetric, this orientation of the window function by γ becomes invariant and the SLSHT distribution components are defined on L S ) [10]. Since the maximum spectra degree for which the SLSHT distribution is defined is L g = L f +L h, we consider the bandimited window function such that L h L f to avoid extending L g significanty above L f. We discuss the ocaization of the window function in spatia and spectra domains ater in the paper. B. Harmonic Anaysis We now present the formuation of the directiona SLSHT distribution if the signa f and the window function h are represented in the spectra domain. Using the expression of

4 4 the spherica harmonics of a rotated function in 5), we can write the SLSHT distribution component gρ;, m) in 10) as gρ;, m) = L f =0 m = p=0 q= p q = p h ) q where T, m ; p, q;, m) = f ) m 11) p Dp q,q ρ) T, m ; p, q;, m), Y m S ˆx) Y p q ˆx) Y m ˆx) dsˆx) denotes the spherica harmonic tripe product, which can be evauated using Wigner-3j symbos or Cebsch-Gordan coefficients [36], [40]. Remark : By comparing gρ;, m) in 11) with 8), we note that the band-imit of gρ;, m) in ρ is given by L h. Since L f and p L h in 11), our statement that the distribution component gρ;, m) is non-zero for L g = L f + L h foows since the tripe product T, m ; p, q;, m) is nonzero for L f + L h ony. C. Inverse Directiona SLSHT Here, we define the inverse directiona SLSHT to reconstruct a signa from its SLSHT distribution. The origina signa can be reconstructed from its directiona SLSHT distribution through the spectra domain margina, that is, by integrating the SLSHT distribution components over the spatia domain SO3) [18]. Using our harmonic formuation in 11), define m ˆf) as the integra of the SLSHT distribution component gρ;, m) over SO3) giving m ˆf) = gρ;, m)dρ, 0 L f = SO3) =0 m = f ) m T, m ; p, q;, m) p=0 q= p q = p SO3) h ) q p D p q,q ρ) dρ = 16π 3 h ) 0 0 f ) m, 1) where we have used the orthogonaity reation of Wigner-D functions see 7)). Using the expression in 1), we can find the spherica harmonic coefficient f ) m of the signa f as m ) m ˆf) f = 16π 3 h ) 0, 13) 0 which indicates that we ony need to know the DC component of the window function h ) 0 in order to obtain the signa 0 from its directiona SLSHT distribution. It further imposes the condition that the DC component of the window function must be non-zero. Athough the distribution components in 11) are defined up to degree L g = L f + L h, we ony require the components up to L f for signa reconstruction. Computing the forward and inverse directiona SLSHT is computationay demanding. Since the directiona SLSHT distribution components gρ;, m) in 10) are defined for L g, the number of distribution components are of the order L g, whie the samping of ρ is of the order L 3 h ; thus, the direct evauation of the directiona SLSHT distribution is prohibitivey computationay expensive. Therefore efficient agorithms need to be deveoped which reduce the computationa compexity. We address this probem in the next section. IV. EFFICIENT COMPUTATION OF DIRECTIONAL SLSHT DISTRIBUTION Here, we present efficient agorithms for the computation of the directiona SLSHT distribution of a signa and the signa reconstruction from its directiona SLSHT distribution. First, we discuss the computationa compexities if the SLSHT distribution components are computed using direct quadrature as given in 10) or using the harmonic formuation in 11). Later, we deveop an aternative harmonic formuation which reduces the computationa burden. Finay, we present an efficient agorithm that incorporates a factoring of rotations [41] and expoits the FFT. First we need to parameterize the required tesseation schemes for S for the representation of the signa f and the window h and for SO3) which forms the spatia domain of the directiona SLSHT distribution. Since the maximum spectra degree of the signa f is L f, we therefore consider the equianguar tesseation S Lf to represent f. Since the maximum degree for a SLSHT distribution components gρ;, m) in ρ is L h, we therefore consider the tesseation E Lh to represent the SLSHT distribution components on L SO3)). A. Direct Quadrature and Harmonic Formuation We define the forward spatio-spectra transform as evauation of each SLSHT distribution component gρ;, m). Evauation of the forward spatio-spectra transform using exact quadrature in 10) requires the computation of two dimensiona summation over the tesseation of S for each 3-tupe α, β, γ). Since there are OL 3 h ) such 3-tupes in the tesseation scheme E Lh and the SLSHT distribution components are of the order OL f ), the computationa compexity to compute a distribution components using direct quadrature is OL 4 f L3 h ). Using the harmonic formuation in 11), the compexity to compute each SLSHT distribution component is OL f L6 h ) and to compute a SLSHT distribution components is OL 4 f L6 h ). Athough the harmonic formuation in 11) is usefu to estabish that the signa can be reconstructed from the directiona SLSHT distribution, it is much more computationay demanding than direct quadrature. We deveop efficient agorithms in the next subsection which improve the computationa compexity of the harmonic formuation and make it more efficient than direct quadrature. For the inverse directiona SLSHT distribution, we ony need to integrate over SO3) to obtain the signa in the spherica harmonic domain as proposed in 1). Since the integra can be evauated by a summation over a Euer anges using quadrature weights, an efficient way to recover the signa from its SLSHT distribution is through direct quadrature, with compexity of OL 3 h ) for each distribution component and ) for a components. OL f L3 h

5 5 In order to evauate the integra in 1) exacty, we need to define quadrature weights aong Euer ange β in the tesseation E Lh. We evauate the integra in 1) by the foowing summation ˆf) m = 1 L h + 1) 3 n α=0 n β =0 n γ=0 gα nα, β nβ, γ nγ ;, m) qβ nβ ), 14) where the quadrature weights qβ nβ ) foow from [35], with 4π L h 1, + ) 1 βnβ = 0 qβ nβ ) = 8π w m) cos mβ nβ, otherwise m= L h where wm) is defined as [35] ±iπ, m = ±1, wm) = 0, m odd, m 1, m even. 1 m, B. Fast Agorithm for Forward Directiona SLSHT 15) 16) Here, we deveop a fast agorithm to reduce the computationa compexity of the forward SLSHT. We first consider an aternative harmonic formuation of the forward SLSHT and then empoy the factoring of rotations approach which was first proposed in [41] and has been used in the impementations of the fast spherica convoution [4] and the directiona spherica waveet transform [15]. We may write the directiona SLSHT distribution component gρ;, m) in 10) as a spherica convoution [15] of h and the spherica harmonic moduated signa f Y m, giving gρ;, m) = p=0 q= p q = p ) fy m q ) q p h p Dp q,q α, β, γ), 17) which can be expressed, using the definition of the Wigner-D function in 6), as ) gρ;, m) = fy m q p p=0 q= p q = p h ) q p dp q,q β)e iqγ e iqα. 18) The band-imit of the spherica harmonic moduated signa f Y m is L f +. Since the maximum for which g is nonzero is L f + L h, we must compute up to f YL m f +L h, which is band-imited to L f +L h. However, we ony need to compute the spherica harmonic coefficients ) fy m q of the moduated p signa up to degree p L h. Therefore, the computation of the spherica harmonic transform of f Y m is an interesting subprobem. We show in Appendix A that the spherica harmonic In the evauation of 14) we have computed the summation over L h + 1 sampe points in both α and γ. This is due to the tesseation E Lh required to capture a information content of gα, β, γ;, m). However, if one were considered in recovering f ony, then given the quadrature rue in [35] m ˆf) in 14) coud be computed exacty with ony L h + 1 sampe points in α and γ. coefficients ) fy m q p for 0 p L h, q p of the signa f Y m can be computed in OL 3 f L h ) time for a and m. By factoring the singe rotation by α, β, γ) into two rotations [15], [41], [4] D ρ = D ρ1 D ρ, ρ = α, β, γ), ρ 1 = α π/, π/, β), ρ = 0, π/, γ + π/), 19) and noting the effect of rotation on spherica harmonic coefficients in 5), we can write the Wigner-D function in 6) as D p q,q α, β, γ) = iq q q = p p q q p q q e iqα iq β iq γ, 0) where p qq = dp q,q π/) and we have used the foowing symmetry properties of Wigner-d functions [40] d p q,q β) = 1)q q d p q,q β) = 1)q q d p q, q β) = 1) q q d p q,q β) = dp q, qβ). 1) Using the Wigner-D expansion given in 0), we can write the aternative harmonic formuation of the SLSHT distribution component gρ;, m) in 17) as gρ;, m) = p=0 q= p q = p q = p ) fy m q ) q p h p iq q p q q p q q e iqα iq β q γ, ) where ρ = α, β, γ). By reordering the summations we can write where gρ;, m) = L C q,q,q, m) = iq q q= L h q = L h q = L h C q,q,q, m) e iqα iq β q γ, ρ = α, β, γ), 3) h p=max q, q, q ) p q q p q q fy m ) q p h ) q Comparativey, the computation of the SLSHT distribution components using the expression given by 3) is not more efficient than the initia expression 18). However, the presence of compex exponentias can be expoited by empoying FFTs to evauate the invoved summations. The objective of factoring the rotations is to carry out the β rotation aong the y-axis as a rotation aong the z-axis. The rotations aong the z-axis are expressed using compex exponentias and thus these rotations can be appied with much ess computationa burden, by expoiting the power of an FFT, reative to a rotation about the y-axis. A the three rotations which characterize the spatia domain of the SLSHT distribution components appear in compex exponentias in 3) and thus we can use FFTs to evauate the summation of C q,q,q, m) over q, q and q. First we need to compute C q,q,q, m) for each and for each m which requires p.

6 6 the one-dimensiona summation over three dimensiona grid formed by q, q and q and thus can be computed in OL 4 h ). Using C q,q,q, m), the summation over the compex exponentias in 3) can be carried out in OL 3 h og L h ) using FFTs. The overa compexity of this approach is dominated by the computation of C q,q,q, m), that is, OL4 h ) for each SLSHT distribution component and OL f L4 h ) for the compete SLSHT distribution. We note that the evauation of C q,q,q, m) requires the computation of p qq which can be evauated over the q, q ) pane for each p using the recursion formua of [43] with a compexity of OL h ). Since p is of the order L h, the matrices can be evauated in OL 3 h ), which does not change the overa compexity of our proposed agorithm. The overa asymptotic compexity of our fast agorithm is thus OL 3 f L h + L f L4 h ). Remark 3: In order to evauate 18), we note that the separation of variabes approach [1] can be used as an aternative to the factoring of rotation approach to deveop a fast agorithm. This is due to the factorized form of Wigner-D function and the consideration of equianguar tesseation scheme for SO3), which keeps the independence between the sampes aong different Euer anges. In terms of the computationa compexity, the separation of variabe approach has the same computationa compexity as the factoring of rotation approach. However, the separation of variabe approach needs to compute Wigner-d functions for a vaues of β but ony requires a two dimensiona FFT, whereas the factoring of rotation approach ony requires the evauation of Wigner-d function for π/ but requires a three dimensiona FFT. Since both approaches have the same compexity, we use the factoring of rotation in our impementation of the fast agorithm. Remark 4: If we want to anayze the signa f with mutipe window functions, then we do not need to recacuate the spherica harmonic transform of the moduated signa f Y m, which accounts for the OL 3 f L h ) factor in the overa compexity. Once it is computed, the SLSHT distribution can be computed in OL f L4 h ) time for each window function of the same band-imit using the proposed efficient impementation. Our proposed formuation and efficient impementation can be further optimized in the case of a steerabe window function. Steerabe functions have an azimutha harmonic bandimit in m that is ess than the band-imit in see [0], [1] for further detais about steerabiity on the sphere). In this case, the L f L4 h factor contributing to the overa asymptotic compexity of the fast agorithm is reduced to L f L3 h. Furthermore, we may then compute the directiona SLSHT for any continuous γ [0, π) from a sma number of basis orientations due to the inearity of the SLSHT). If the signa and window function are rea, the computationa time can be further reduced by considering the conjugate symmetry reation of the spherica harmonic coefficients. Furthermore, in this setting, the SLSHT distribution components aso satisfy the conjugate symmetry property gρ;, m) = 1) m gρ;, m) 4) and we do not need to compute the SLSHT distribution components of negative orders. V. WINDOW LOCALIZATION IN SPATIAL AND SPECTRAL DOMAINS The directiona SLSHT distribution is the spherica harmonic transform of the product of two functions, the signa f and the rotated window function h and we must be carefu in interpreting the directiona SLSHT distribution in the sense that we do not mistake using the signa to study the window because there is no distinction mathematicay. The window function shoud be chosen such that it provides spatia ocaization in some spatia region around the north poe origin). Since we have considered a band-imited window function, the window function cannot be perfecty ocaized in the spatia domain due to the uncertainty principe on the sphere [44]. However, it can be optimay ocaized by maximizing the energy concentration of the window function in the desired directiona region []. The interpretation and the effectiveness of the directiona SLSHT distribution depends on the chosen window function. The window function with maximum ocaization in some defined asymmetric region provides directiona ocaization and thus reveas directiona features in the spatio-spectra domain. The more directiona the window function, the more directiona features it can revea in the spatio-spectra domain but this tends to increase the maximum spherica harmonic degree L h. Reca that the maximum degree of the directiona SLSHT distribution components is given by L g = L f + L h. Thus, when the signa is expressed in the spatio-spectra domain its spectra domain is extended by L h, which resuts in spectra eakage. Therefore, we want the window function to be simutaneousy maximay ocaized in some spatia region R S and have the minimum possibe band-imit which achieves the desired eve of energy concentration in the spatia region R. Here, we propose using a band-imited eigenfunction obtained from the soution of the Sepian concentration probem [] as a window function, concentrated in a spatiay ocaized eiptica region around the north poe. We first parameterize the eiptica region and ater anayze the resuting eigenfunctions from the perspective of the uncertainty principe on the sphere [44]. We note that the choice of the asymmetric region as an eiptica region is ony one possibiity and the anaysis can be extended to other asymmetric regions such as strip regions around the north poe [45]. A. Parametrization of Window Function We consider a band-imited window function of maximum spherica harmonic degree L h, which is spatiay concentrated in the eiptica region on the sphere with major axis aong the x-axis, and thus is orientated aong the x-axis. The eiptica region can be parameterized using the focus coatitude θ c of the eipse aong the positive x-axis and the arc ength a of the semi-major axis: R θc,a) { θ, φ) : s θ, φ), θc, 0) ) + s θ, φ), θc, π) ) a }, 5) where 0 θ c a π/. Here s θ, φ), θ, φ ) ) = arccos sin θ sin θ cosφ φ ) + cos θ cos θ ) denotes the

7 7 a) b) c) Fig. 1: Band-imited eigenfunction windows h on the sphere with 90% spatia concentration in an eiptica region R θc,a) of focus θ c = π/6 and major axis: a) a = π/6 + π/80, b) a = π/6 + π/10 and c) a = π/6 + π/40. λ max a = π/6 + π/80 a = π/6 + π/10 a = π/6 + π/ degree, L Fig. : The maximum eigenvaue λ max of the eigenfunction h with spherica harmonic band-imit L for spatia concentration in an eiptica region of focus θ c = π/6 and major axis a as indicated. Back vertica ines indicate the bound on the bandimit B h given by 6) and the gray vertica ines indicate the actua L h which ensures 90% concentration in an eiptica region. anguar distance between two points θ, φ) and θ, φ ) on the sphere. Remark 5: For a given focus θ c, the region becomes more directiona as the arc ength a approaches θ c from π/. For a = π/, the region becomes azimuthay symmetric, i.e., we recover the poar cap of centra ange π/. Aso, when θ c = 0, the region becomes azimuthay symmetric poar cap) of centra ange a. B. Sepian Concentration Probem As a resut of the Sepian concentration probem [], [45] to find the band-imited function with bandwidth L h and maxima spatia concentration in an eiptica region R θc,a), we obtain L h + 1) eigenfunctions. Due to the symmetry of the eiptica region about x-y pane, the eigenfunctions are rea vaued [45]. The eigenvaue associated with each eigenfunction serves as a measure of the energy concentration in the spatia region. Here we consider the use of the bandimited eigenfunction with maximum energy concentration in the eiptica region for given band-imit L h and refer to such an eigenfunction as the eigenfunction window. If A denotes the area of the eiptica region, it is shown [], [45] that most of the eigenvaues ie either near zero or unity for both symmetric and asymmetric regions and the sum of a eigenvaues, referred as an equivaent of the Shannon number [], is equa to N 0 = AL h + 1) /4π). Aso, it is shown empiricay in [3] that there exist ess than or equa to N 0 1 spatiay concentrated eigenfunctions with non-insignificant energy concentration. By noting these deveopments and empirica resuts, and considering that L h must be chosen that we obtain at east one eigenfunction which is spatiay concentrated in the eiptica region, we recover the foowing empirica ower bound for L h : π L h B h = A 1, A = R θc,a) dsˆx), 6) where ) denotes the integer ceiing function this bound foows directy from N 0 ). We anayze this bound ater in this section. Let λ max denote the eigenvaue associated with the most spatiay concentrated eigenfunction. By finding the minimum vaue of L h which ensures that λ max is greater than or equa to the desired energy concentration, an eigenfunction window hˆx) with desired energy concentration in the eiptica region and minimum possibe band-imit L h can be obtained. Thus, the focus of an eiptica region θ c, arc ength of semi-major axis a and maximum spectra degree L h fuy parameterize the eigenfunction window. As an iustration, the three eigenfunction windows having 90% spatia concentration in the eiptica regions R θc,a) with focus θ c = π/6 and a {π/6+π/40, π/6+π/10, π/6+π/80} and respective maximum spherica harmonic degree L h {18, 14, 11} are shown in Fig. 1. For these eiptica regions, in Fig. we show λ max versus spherica harmonic band-imit L denoting the band-imit of the most concentrated eigenfunction window, where we note the difference in L h and the bound B h for

8 8 C h π/6+π/80 π/6 + π/10 π/6 + π/40 0 σ S ) or uncertainty product uncertainty product 0σ S σ L σ L ς Fig. 3: Auto-correation function C h ς) as described in 7) for eigenfunction windows h with spatia concentration in an eiptica region of focus θ c = π/6 and major axis a as indicated. the desired concentration eve. For 90% desired concentration eve, we observe that the spherica harmonic degree L h which ensures λ max 0.9 deviates further from the bound B h given by 6) as the concentration region becomes more directiona as expected since the bound does not incorporate the eve of directionaity of the region). C. Anaysis of Eigenfunction Window Concentrated in Eiptica Region 1) Directionaity: For the use of the eigenfunction window in obtaining the directiona SLSHT distribution, the directionaity of the eigenfunction window must be a key criterion in seecting the eigenfunction window for the identification of ocaized features of a signa in the spatio-spectra domain. We use the definition of the auto-correation function on the sphere as a measure of the directionaity of the eigenfunction window [0], []. The auto-correation function is defined as the inner product of eigenfunction window with its version rotated around the z-axis: C h ς) = D ρ1 h, h, ρ 1 = 0, 0, ς), ς [0, π ], 7) which can be expressed in the harmonic domain as C h ς) = =0 m= e imς h m. 8) Due to the symmetry of the eiptica region and the eigenfunction window about the major axis x-axis) and minor axis yaxis), we have considered ς in the range of 0 to π/. For the eigenfunction windows presented in the previous subsection, the auto-correation function for each window is shown in Fig. 3, which indicates that C h ς) decays more rapidy from unity at ς = 0 for more directiona window and therefore, the peakedness of C h ς) quantifies the abiity of the window function to revea the ocaized directiona features of a signa in the spatio-spectra domain major axis, a Fig. 4: Spatia variance 0σS ), spectra variance σ L ) and uncertainty product, as described in 9), 30) and 31), are shown for eigenfunctions with spatia concentration in an eiptica region of focus θ c = π/6 and major axis a as indicated. ) Spatia and Spectra Locaization: Here we study the spatia and spectra ocaization of eigenfunction windows from the perspective of the uncertainty principe on the sphere [44], [46], according to which, the function cannot be simutaneousy ocaized in both the spatia and spectra domains. The foowing inequaity specifies the uncertainty principe for unit energy functions defined on the sphere, which reates the trade-off between the spatia and spectra ocaization of a function: σ S 1 σ S σ L 1, 9) where σ S and σ L denote the variance of the band-imited unit energy eigenfunction window in the spatia domain and spectra domain respectivey and are defined as [46] and σ S = 1 1 sinθ) ) hθ, φ) dθdφ 30) S σl = + 1) =0 m= h m. 31) Due to the consideration of unit energy functions, 0 σ S 1. We note that smaer variance indicates better ocaization of the window function. The variance in the spatia and spectra domains and the uncertainty product for eigenfunctions concentrated in an eiptica region of focus π/6 and majoraxis of different vaues are shown in Fig. 4. As expected, the variance in the spatia domain decreases as the region becomes more directiona, whereas the variance in the spectra domain increases because L h increases. We aso note that the uncertainty product increases as the region becomes more directiona.

9 τ τ L f L f a) Computation time to evauate fy m)q p b) Computation time to evauate a SLSHT distribution components given fy m)q p τ ε L f L f c) Inverse transform computation time d) Numerica vaidation Fig. 5: Numerica vaidation and computation time of the proposed agorithms. The computation time in seconds: a) τ 1 b) τ and c) τ 3. For fixed L h, τ 1 evoves as OL 3 f ) and both τ and τ 3 scae as OL f ) as shown by the soid red ines without markers). d) The maximum error ɛ, which empiricay appears to scae as OL), as shown by the soid red ine. VI. RESULTS In this section, we first demonstrate the numerica vaidation and computation time of our agorithms to evauate the directiona SLSHT components. Later, we provide an exampe to iustrate the capabiity of the directiona SLSHT, showing that it reveas the directiona features of signas in the spatiospectra domain. The impementation of the our agorithms is carried out in MATLAB, using the MATLAB interface of the SSHT 3 package the core agorithms of which are written in C and which aso uses the FFTW 4 package to compute Fourier transforms) to efficienty compute forward and inverse spherica harmonic transforms [35]. A. Numerica Vaidation and Computation Time In order to evauate the numerica accuracy and the computation time, we carry out the foowing numerica experiment We use the band-imited function h for spatia ocaization with band-imit L h = 18 and spatia ocaization in the region R π/6,π/6+π/40). We generate band-imited test signas with band-imits 18 L f 130 by generating spherica harmonic coefficients with rea and imaginary parts uniformy distributed in the interva [0, 1]. For the given test signa, we measure the computation time τ 1 to evauate spherica harmonic transform of the moduated signa, i.e., fy m)q p for p L h, q p and for a L f + L h, m, using the method presented in Appendix A. Given the spherica harmonic transform of the moduated signa, we then measure the computation time τ to compute a directiona SLSHT distribution components gρ;, m) for L f + L h and m using our fast agorithm presented in Section IV-B, where we compute the Wigner-d functions on-the-fy for the argument π/ by using the recursion of Trapani [43]. We aso record the computation time τ 3 to recover a signa from its SLSHT distribution components. A numerica experiments are performed using

10 10 a) Fig. 6: Mars signa in the spatia domain. The grand canyon Vaes Marineris and the mountainous regions of Tharsis Montes and Oympus Montes are indicated. MATLAB running on a.4 GHz Inte Xeon processor with 64 GB of RAM and the resuts are averaged over ten test signas. The computation time τ1 and τ are potted against the band-imit Lf of the test signa in Fig. 5a and Fig. 5b, which respectivey evove as OL3f ) and OLf ) for fixed Lh and thus corroborate the theoretica compexity. The computation time τ3 for the inverse directiona SLSHT is potted in Fig. 5c, which scaes as OLf ) for fixed Lh, again supporting the theoretica compexity. We reconstruct the origina signa from its SLSHT distribution components using 14) and 1), in order to assess the numerica accuracy of our agorithms by measuring the maximum absoute error between the origina spherica harmonic coefficients of the test signa and the reconstructed vaues. The maximum absoute error is potted in Fig. 5d for different band-imits Lf, which iustrates that our agorithms achieve very good numerica accuracy with numerica errors at the eve of foating point precision. B. Directiona SLSHT Iustration In order to iustrate the capabiity of the proposed transform to revea the ocaized contribution of spectra contents and probe the directiona features in the spatio-spectra domain, we anayze the Mars topographic map height above geoid) as a signa on the sphere, which is obtained by using the spherica harmonic mode of the topography of Mars5. The Mars topographic map is shown in Fig. 6 in the spatia domain, where the grand canyon Vaes Marineris and the mountainous regions of Tharsis Montes and Oympus Montes are shown, eading to the high frequency contents. We note that the mountainous regions are non-directiona features of the Mars map, whereas the grand canyon serves as a directiona feature with direction orientated aong a ine of approximate constant 5 wieczor/sh/ b) Fig. 7: Magnitude of the components of the directiona SLSHT distribution of the Mars signa obtained using the eigenfunction window concentrated in an eiptica region of focus θc = π/1 and major axis a = 7π/1. For fixed orientation γ, the distribution components gρ; `, m) are mapped on the sphere using ρ = φ, θ, γ) for order m = 15 and degree 0 ` 5. The components are shown for orientation a) γ = 0 and b) γ π/ of the window function around the z-axis. Top eft: gρ; 0, 15), top right: gρ;, 15). atitude. The spherica harmonic coefficients provide detais about the presence of higher degree spherica harmonics in the signa, but do not revea any information about the ocaized contribution of higher degree spherica harmonics. If we anayze the signa by empoying the SLSHT using an azimuthay symmetric window function, the presence of ocaized contributions of higher degree spectra contents can be determined in the spatio-spectra domain [10], however, the presence of directiona features cannot be extracted. Here, we iustrate that the use of the directiona SLSHT enabes the identification of directiona features in the spatio-spectra domain, which is due to the consideration of an asymmetric window function for spatia ocaization. We obtain the directiona SLSHT distribution components gρ; `, m) of the Mars map f using the band-imited eigenfunction window h with Lh = 60 and 90% concentration in

11 11 the spatia domain in an eiptica region Rπ/1,7π/1). The magnitude of the SLSHT distribution components gρ; `, m) for order m = 15 and degrees 0 ` 5 and 50 ` 55 are shown in Fig. 7 and Fig. 8 respectivey, where the components in the panes are for Euer ange a) γ = 0 and b) γ = 100π/01 π/. Since the eiptica region is oriented aong the x-axis, the window with orientation γ = 0 provides ocaization aong coatitude and the window with orientation γ π/ provides ocaization aong ongitude. It is evident that using orientation of the window γ π/ probes the information about the grand canyon Vaes Marineris directiona feature) aong ongitude in the spatio-spectra domain. The ocaized contribution of higher degree spherica harmonics towards the mountainous region can aso be observed in Fig. 7 for degree 0 ` 5 and both γ = 0 and γ = π/. However, there is not significant contribution of spherica harmonics of degree 50 ` 55 towards mountainous region as indicated in Fig. 8a where γ = 0, but the ocaization of the directiona features aong the orientation γ π/ are reveaed in the spatio-spectra domain as shown in Fig. 8b. Due to the abiity of the directiona SLSHT to revea the ocaized contribution of spectra contents and the directiona or oriented features in the spatio-spectra domain, it can be usefu in many appications where the signa on the sphere is ocaized in position and orientation. a) VII. C ONCLUSIONS We have presented the directiona SLSHT to project a signa on the sphere onto its joint spatio-spectra domain as a directiona SLSHT distribution. In spirit, the directiona SLSHT is composed of SO3) spatia ocaization foowed by the spherica harmonic transform. Here, we have proposed the use of an azimuthay asymmetric window function to obtain spatia ocaization, which enabes the transform to resove directiona features in the spatio-spectra domain. We have aso presented an inversion reation to synthesize the origina signa from its directiona SLSHT distribution. Since data-sets on the sphere are of considerabe size, we have deveoped a fast agorithm for the efficient computation of the directiona SLSHT distribution of a signa. The computationa compexity of computing the directiona SLSHT is reduced by providing an aternative harmonic formuation of the transform and then expoiting the factoring of rotation approach [41] and the fast Fourier transform. The computationa compexity of the proposed fast agorithm to evauate SLSHT distribution of a signa with band-imit Lf using window function with bandimit Lh is OL3f Lh +Lf L4h ) as compared to the compexity of direct evauation, which is OL4f L3h ). The numerica accuracy and the speed of our fast agorithm has aso been studied. The directiona SLSHT distribution reies on a window function for spatia ocaization; we have anayzed the band-imited window function obtained from the Sepian concentration probem on the sphere, with nomina concentration in an eiptica region around the north poe. We provided an iustration which highighted the capabiity of the directiona SLSHT to revea directiona features in the spatio-spectra domain, which is ikey to be of use in many appications. b) Fig. 8: Magnitude of the components of the directiona SLSHT distribution of the Mars signa obtained using the eigenfunction window concentrated in an eiptica region of focus θc = π/1 and major axis a = 7π/1. For fixed orientation γ, the distribution components gρ; `, m) are mapped on the sphere using ρ = φ, θ, γ) for order m = 15 and degree 50 ` 55. The components are shown for orientation a) γ = 0 and b) γ π/ of the window function around z-axis. Top eft: gρ; 50, 15), top right: gρ; 5, 15). A PPENDIX A S PHERICAL H ARMONIC T RANSFORM OF M ODULATED S IGNAL Our objective is to compute the spherica harmonic transform of the moduated signa f Y`m, up to degree Lh, for a ` and m. In order to serve the purpose, we use a separation variabe technique given by Z π q f Y`m p = N`m Npq P`m cos θ)ppq cos θ) 0 Z π f θ, φ)eim q)φ dφ sin θdθ. 3) 0 {z } Iθ,m q) Since 0 ` Lf + Lh and 0 p Lh, we need to consider the signa f Y`m samped on the grid SLf +Lh for the expicit evauation of exact quadrature note that samping in φ coud be optimized given m q Lf + Lh but this

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