Multiplicative and Additive Perturbation Effects on the Recovery of Sparse Signals on the Sphere using Compressed Sensing

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1 Multiplicative and Additive Perturbation Effects on the Recovery of Sparse Signals on the Sphere using Compressed Sensing ibeltal F. Alem, Daniel H. Chae, and Rodney A. Kennedy The Australian National University, Canberra, AUSTRALIA s: Abstract In this paper, we show how perturbation can affect the reconstruction of sparse spherical harmonic (SH) signals, whose domain is the sphere, with compressed sensing (CS) techniques. Our results show that the multiplicative perturbation, which can be generated by the jitter of the sampling locations (co-latitude and longitude) of the signal, can cause significant error in the SH-signal recovery with compressed sensing. To apply compressed sensing practically to the reconstruction of SHsignal, the method of calibration to minimize the multiplicative perturbation is suggested. Furthermore, we showed that the possibility of noise reduction in SH-signal reconstruction with a diversity technique which combines multiple sparse recoveries. I. INTRODUCTION Spherical harmonics (SH) [1 3] are the solution of spherical Laplace s equation, and their linear combination can represent any square integrable complex-valued signal (function) whose domain is the sphere. The SH coefficients (the Fourier coefficients corresponding to the spherical harmonics), however, generally need to be computed from sampled measurements on the sphere, and this computation requires a quite sizeable number of measurement points even for signals with a modest band-limit. The most common way of computing the coefficients is the least squares (LS) method [4, 5] which generally requires the number of measurements to be larger than that of the harmonics of the signal at hand. When the signal of interest is sparse in some domain, compressed sensing (CS) [6 10] can often be employed to determine efficiently the components of the signal. That is, when a relatively few coefficients are non-zero, there is a potential to significantly reduce the number of measurements required to calculate the coefficients, and correspondingly the computational requirements are reduced. In summary, CS can recover the coefficients even when the nominal linear system is quite under-determined because it exploits the sparsity in the signal. Focussing back on our specific applications, since it is possible to represent a spherical signal using SH, we can apply CS to the reconstruction of SH-signals as long as the signal is sufficiently sparse in the SH domain. However, CS is known to be sensitive to perturbations. The presence of additive perturbation [10], which is very common in most systems, will cause CS recovery noise. Furthermore, if there exists a multiplicative perturbation [11] in the sampled measurements, more detrimental problems may occur in the CS recovery. In sampling a SH-signal on the sphere, the noise on the measurements and the jitter on the sphere sampling points will behave as additive and multiplicative perturbations, respectively. In this paper, we are interested in applying CS to the reconstruction of SH-signals. There is already an attempt to apply CS to the reconstruction of SH-signals with preconditioning the sensing matrix method for improvement of stability during reconstruction [1]. However, we focus on the problems in reconstructing SH-signals with CS when there exist both additive and multiplicative perturbations. These perturbations can be incurred during sampling process and can cause severe noise in the CS-recovery. To reduce the noise from additive perturbation, we apply a diversity method [13, 14] to the CS reconstruction of a SH-signal. Furthermore, to reduce the effects from the multiplicative perturbation, we introduce a SH calibration technique. In general, the contributions of this paper are: 1. We raise the problem of perturbations in applying CS to reconstruct SH-signals. This includes the types of perturbations and the reasons for their occurrence.. We reveal the effect of each perturbation in reconstructing SH-signals using the CS scheme. We have undertaken a simulation study to show the adverse effects of both additive and multiplicative perturbations in recovering a sparse SH coefficient vector. 3. We finally suggest a couple of methods, calibration and segmentation, to mitigate the effects of multiplicative and additive perturbations in reconstructing SH-signals using CS. The remainder of this paper is organized as follows. Section II introduces CS into the reconstruction of SH-signals. In Section III, we show how additive and multiplicative perturbations are generated in sampling SH-signals and how these adversely affect the CS recovery of signals on sphere. To remove the adverse effects of these perturbations, SH calibration and SH diversity methods are suggested in Section IV. The adverse effects of these perturbations and their mitigation are presented in Section V. Finally, Section VI finalizes our paper with conclusions and some suggestions for future work /1/$ IEEE

2 A. Spherical Harmonics II. PRELIMINARIES Any square-integrable function f(, ) as a function of colatitude and longitude, can be expressed as 1X `X f(, )= x m` ` m (, ), (1) `=0 m= ` where ` and m denote the degree and order, respectively, of the SH basis functions represented by ` m (, ) and defined as s ` m ` +1(` m)! (, )= 4 (` + m)! P m` (cos )e im, () and P m` ( ) is the associated Legendre function. The xm` in (1) is the SH coefficient. If the scalar function f(, ) is known, the coefficient of the SH x m` can be calculated by x m` = Z Z 0 0 f(, ) m ` (, ) sin( )d d, (3) where denotes a complex conjugate. However, in this paper, we want to determine x m` with a limited number of signal measurements on the sphere without complete information about the signal f(, ). This is regarded as SH-signal reconstruction, since we have to recover the SH coefficients from samples of the signal corresponding to the measurements. If we choose M measurements on sphere f i = f( i, i) for i =1,...,M, where ( i, i) is the i-th measurement on sphere, the sampled measurement equation can be written f = x. (4) where f R M 1 is the vector of sampled signal f = [f 1...f M ] T, R M N is the matrix of SH, N is the total number of harmonics, and x R N 1 is the coefficient vector corresponding to. If the SH-signal is band-limited in degree to L, the total number of harmonics will be N =(L + 1). The matrix is defined as = y n=1 y n=n where y n is a column vector expressed as y n = mn `n ( 1, 1) mn `n ( M, M ) T, and `n and m n are the degree and order of the harmonic corresponding to the n-th column of, respectively. As we have mentioned briefly in the introduction, the most common way of finding x is the Least squares (LS) method as x? = arg min k fk = f, (5) where x? is the estimation of x, is the pseudo-inverse matrix of defined as =( T ) 1 T. In using the LS method, the number of measurements, M, should as large as or bigger than the number of total harmonics, N (depending on the noise and possible degeneracies). Fig. 1: Perturbations on the sampling of SH-signal : one is deviation from the intended point due to jitter and the other is additive noise. B. Applying CS to SH If the signal is sparse in the basis domain, i.e., most entries of the vector x are zero, it is possible to estimate x even though M is much smaller than N by using compressed sensing (CS) techniques [6 10]. The vector x can be estimated as x? = arg min kk 1 s.t. = f, (6) where kk 1 := P N n=1 n. For successful recovery, there is a requirement on the matrix called the restricted isometry property (RIP) as (1 K) apple kx Kk kx K k apple (1 + K ), (7) where K is the sparsity of x defined as K = {j : x j 6=0}. If K, known as the restricted isometry constant (RIC), is quite smaller than one for any K-sparse vector x K, the recovery of (6) is possible with high probability. For the satisfaction of the RIP, we need the minimum number of measurement, M min given by M C K log N, (8) where C is some positive constant. III. CS SENSITIVIT PROBLEM Sensitivity problems occur when there are perturbations in the compressively sampled measurements. In sampling a SHsignal, there can be two kinds of perturbations. As shown in Fig. 1, one is deviation from intended point due to jitter on the spherical sampling point, and the other is additive noise on the sampled measurements.

3 A. Perturbations of SH-signal If there is any jitter on the spherical sampling, the i-th sampling position at ( i, i) is given by ( p i, p i )=( i + i, i + i), (9) where the superscript p notes perturbation, i and i are the deviations from i and i, respectively. Furthermore, with additive noise, the i-th measurement will be expressed as f p i = f( p i, p i )+e i, (10) where e i is the additive noise on the i-th sample. Then, the perturbed measurements can be expressed in matrix form as f p = p x + e, (11) where p is the SH matrix with jitter, and e =[e 1...e M ] T. Since p can be decomposed into two terms as p = +, (1) where is a deviation matrix from, the sampled signal is re-expressed as where f = x+ e. B. Perturbation Effect f p = x+ f, (13) Since we have no information about the perturbations, blind CS reconstruction should be performed as x? = arg min kk 1, s.t. k f p k apple ", (14) where " is the bound of total noise bound due to the sampling jitter and the additive noise. In CS, the measurement noise e and the deviation matrix are modelled as additive [10] and multiplicative [11] perturbations, respectively. With quantitative modelling method [11], the magnitudes of these perturbations are bounded as kek kfk apple " f, k k (K) k k (K) apple " (K), (15) where " f and " (K) are relative upper bounds on additive and multiplicative perturbation, respectively. k k (K) denotes the spectral norm of all K-column sub-matrices of a matrix. If the K-RIC satisfies p K < 1+" (K) the reconstruction error is known to obey [11] where C is a constant given by C = 1, (16) k xk apple C", (17) 4 p 1+ K (1 + " (K) ) 1 (1 + p ) (1 + K )(1 + " (K) ) 1, (18) and " is the bound of total noise calculated as p 1+ K " = p " (K) 1 + " f kfk. (19) K The noise of CS-recovered signal gets larger as the magnitudes of perturbation increase. Furthermore, the multiplicative perturbation due to sampling jitter is more detrimental to the recovery of the SH-signal since the error constant C is increasing highly even with small value of " (K). IV. MINIMIZING PERTURBATION EFFECTS A. Calibration For Multiplicative Perturbation If we have the information on the perturbations of ( i, i) for i =1,...,M, we can minimize the adverse effect of multiplicative perturbation. We can use a reference signal to perform a calibration or to estimate the multiplicative perturbation by transmitting and receiving the reference signal before transmitting actual signal. The calibration method can be varying depending on applications. With the assumption of the calibration, the recovery can be performed with the calibrated SH matrix of p, Ỹ p as x? = arg min kk 1, s.t. kỹ p f p k apple " add, (0) instead of (14), where " add = " f kfk. The noise bound is reduced from " to " add since the basis mismatch term is removed by calibration. Then, the recovery noise can be reduced as k xk apple C add " add, (1) where C add is a constant given by In (), 4 p 1+ p K C add = 1 (1 + p ) p. () K p K is the RIC of p. B. Segmentation for Additive Perturbation If the total measurements can be segmented with D multiple sets of M min measurements, we can extract diversity gain by performing and combining of multiple CS recoveries. We can have multiple CS recoveries as x? d = arg min kk 1 s.t. kỹ p d f p d k apple " d, (3) for d =1,,...,D, where " d is the upper bound on k f d k. Then, we can combine multiple versions of x? d to achieve diversity as DX x? = w d x? d, (4) where w d is the combining weight for x? d. If we set the weights of parallel branches w d =1/D for d =1,,...,D, the noise vector of the combined recovery is given by x = 1 D DX x d, (5)

4 Fig. : Signal on sphere consists of spherical harmonics whose coefficients are sparse. Fig. 3: Sampling points for CS reconstruction. Only M = 100 points are selected for CS. where x d R N 1 is the noise vector of the d-th CS recovery. Since the multiplicative perturbation is removed by the calibration, x d,d = 1,,...,D can be regarded as uncorrelated as long as the additive noise is not correlated. If the noise of multiple recoveries are at the same level, the noise bound of the combined CS recovery is inversely proportional to p D as k xk / C add" p add, (6) D shown in Section VII. A. Simulation Set-ups V. NUMERICAL RESULTS We want to reconstruct a spherical signal with CS which is sparse in basis domain as shown in Fig.. The signal is assumed to be band-limited to L = 50 and for simplicity, only positive order spherical harmonics are computed. As a result, we will have N = 136 harmonics for a maximum bandlimit of L = 50. The sparsity of the coefficients is assumed as K = 10 out of N = 136. However, these parameters can be extended in case of more complex signal. Furthermore, Gaussian noise whose variance is is intentionally added to the signal for the generation of the additive perturbation. Fig. 3 shows the intended sampling points on the sphere for CS. Only M = 100 measurements are selected on the sphere for CS reconstruction. For the emulation of multiplicative perturbation, the variances of the deviations on (, ) of the intended points are set to 0.01 rad. B. CS Reconstruction 1) Under Both Perturbations: Fig. 4(a) is the CS-recovered signal under both additive and multiplicative perturbations, and Fig. 4(b) compares the pure coefficients and the CS-recovered coefficients of the SH. These results show that there is severe noise in the recovered signal. It indicates that we cannot reconstruct the original SH-signal with CS if there are both perturbations. ) Removing Multiplicative Perturbation: Fig. 5(a) and Fig. 5(b) show the CS recovered SH-signal with the assumption that the multiplicative perturbation is removed. The recovery is tolerable if there are only additive perturbations, even though there is still noise. This result implies that the multiplicative perturbation is more detrimental to the recovery than the additive perturbation. This, therefore, suggests minimizing the multiplicative perturbation is important in the reconstruction of SH-signal with CS. 3) Reducing Additive Perturbation: CS diversity technique is applied to reduce the additive noise further. Even though the recovery noise is reduced significantly by removing the multiplicative perturbation, the recovery still suffers from the noise due to the additive perturbation. To extract the diversity gain by combining multiple recoveries, we selected 4 sets of M = 100 CS samples on the sphere. Fig. 6 is the combined SH coefficients with the multiple recoveries. Comparing Fig. 5(b), the additive noise is reduced significantly with the application of diversity gain technique. VI. CONCLUSIONS In this paper, we showed the perturbation effects and problems in reconstructing SH-signals with CS. Our results show that the multiplicative perturbation, which is generated by the jitter of sampling position on SH-signal, can cause serious error on the SH-signal recovery with CS. To apply CS practically to the reconstruction of SH-signal, the method of calibration to minimize the multiplicative perturbation is suggested. Furthermore, we showed that the possibility of noise reduction in SH-signal reconstruction with the diversity technique of combining multiple recoveries.

5 (a) CS-reconstructed SH-signal. (a) CS-reconstructed SH-signal. (b) Comparison of harmonic coefficients between the pure and the CS-reconstructed. Fig. 4: CS Recovering of SH-signal under both additive and multiplicative perturbations. For the emulation of the multiplicative perturbation, the jitter whose variance amounts to 0.01 rad is added on the sampling point (, ). Furthermore, Gaussian noise whose variance is is added to the signal for the generation of the additive perturbation. The CSrecovered SH-signal is totally different with the original under both perturbations. (b) Comparison of harmonic coefficients between the pure and the CS-reconstructed. Fig. 5: CS-reconstructed signal with the assumption of removing multiplicative perturbation. Without the multiplicative perturbation, the CS-recovered SH-signal is tolerable except some noise. where d is the variance of the elements on the noise vector, the noise variance of the d-th recovery can be bounded as Interesting problems for future work include the practical method of minimizing the multiplicative perturbation in SHsignal processing with CS. VII. APPENDIX Since the magnitude of the CS recovery noise with the d-th measurement set can be expressed as k x d k ' p N d, (7) d / C d" d N, (8) where C d is the error bound constant of the d-th CS recovery. If x d,d = 1,,...,D are uncorrelated each other, as a result, the variance of the elements of x is bounded as / 1 D X C d " d D N. (9)

6 Fig. 6: Comparison of harmonic coefficients between the original and the CS-reconstructed with additive noise reduction. The residual noise is further reduced. Finally, the magnitude of the noise bound of the combined vector can be expressed as v k xk / 1 ux t D C d " d. (30) D For intuitive understanding of this noise bound, let us assume that C d C add and " d " add for d =1,,...,D. Then, the noise bound of the combined CS recovery is inversely proportional to p D as [7] E. J. Candès, Compressive sampling, in Proc. of Int. Congress of Mathematicians, vol. 3, Madrid, Spain, Aug. 006, pp [8] E. J. Candès and M. Wakin, An introduction to compressive sampling, IEEE Signal Process. Mag., vol. 5, pp. 1 30, Mar [9] R. G. Baraniuk, Compressive sensing, IEEE Signal Process. Mag., vol. 4, pp , Jul [10] E. J. Candès, J. K. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics, vol. 59, pp , 006. [11] M. A. Herman and T. Strohmer, General deviants : An analysis of perturbations in compressed sensing, IEEE J. Select. Topics Signal Processing, vol. 4, pp , 010. [1] H. Rauhut and R. Ward, Sparse recovery for spherical harmonic expansions, ArXiv, 011. [13] D. G. Brennan, Linear diversity combining techniques, in Proc. IRE, vol. 47, no. 6, 1959, pp [14] A. Goldsmith, Wireless Communication. Cambridge University Press, 005. k xk / C add" add p D. (31) Therefore, we can reduce recovery noise by increasing the number of branches, D. ACKNOWLEDGMENT This work was supported by the Australian Research Council Discovery Grant DP REFERENCES [1] R. A. Kennedy and P. Sadeghi, Hilbert Space Methods in Signal Processing. New ork, N: Cambridge University Press, 01. [] G. Arfken, Mathematical Methods for Physicists. Academic Press, [3] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, [4] Å. Björck, Numerical Methods for Least Squares Problems. SIAM, [5] G. H. Golub and C. F. V. Loan, Matrix Computations. The Johns Hopkins University Press, [6] D. L. Donoho, Compressed sensing, IEEE Trans. Information Theory, vol. 5, pp , Apr. 006.