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

Compressed SensingYibeltal F. Alem, Daniel H. Chae, and Rodney A. Kennedy

The Australian National University, Canberra, AUSTRALIAEmails: {yibeltal.alem,daniel.chae,rodney.kennedy}@anu.edu.au

Abstract—In this paper, we show how perturbation can affectthe 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 significanterror in the SH-signal recovery with compressed sensing. Toapply compressed sensing practically to the reconstruction of SH-signal, the method of calibration to minimize the multiplicativeperturbation is suggested. Furthermore, we showed that thepossibility of noise reduction in SH-signal reconstruction witha diversity technique which combines multiple sparse recoveries.

I. INTRODUCTION

Spherical harmonics (SH) [1–3] are the solution of sphericalLaplace’s equation, and their linear combination can representany square integrable complex-valued signal (function) whosedomain is the sphere. The SH coefficients (the Fourier coef-ficients corresponding to the spherical harmonics), however,generally need to be computed from sampled measurementson the sphere, and this computation requires a quite sizeablenumber of measurement points even for signals with a modestband-limit. The most common way of computing the coeffi-cients is the least squares (LS) method [4, 5] which generallyrequires the number of measurements to be larger than that ofthe 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 todetermine efficiently the components of the signal. That is,when a relatively few coefficients are non-zero, there is apotential to significantly reduce the number of measurementsrequired to calculate the coefficients, and correspondingly thecomputational requirements are reduced. In summary, CS canrecover the coefficients even when the nominal linear systemis quite under-determined because it exploits the sparsity inthe signal. Focussing back on our specific applications, sinceit is possible to represent a spherical signal using SH, we canapply CS to the reconstruction of SH-signals as long as thesignal is sufficiently sparse in the SH domain.

However, CS is known to be sensitive to perturbations. Thepresence of additive perturbation [10], which is very commonin most systems, will cause CS recovery noise. Furthermore,if there exists a multiplicative perturbation [11] in the sampledmeasurements, more detrimental problems may occur in the

CS recovery. In sampling a SH-signal on the sphere, the noiseon the measurements and the jitter on the sphere samplingpoints will behave as additive and multiplicative perturbations,respectively.

In this paper, we are interested in applying CS to thereconstruction of SH-signals. There is already an attempt toapply CS to the reconstruction of SH-signals with precondi-tioning the sensing matrix method for improvement of stabilityduring reconstruction [12]. However, we focus on the problemsin reconstructing SH-signals with CS when there exist bothadditive and multiplicative perturbations. These perturbationscan be incurred during sampling process and can cause severenoise in the CS-recovery. To reduce the noise from additiveperturbation, we apply a diversity method [13, 14] to theCS reconstruction of a SH-signal. Furthermore, to reduce theeffects from the multiplicative perturbation, we introduce aSH calibration technique. In general, the contributions of thispaper are:

1. We raise the problem of perturbations in applying CS toreconstruct SH-signals. This includes the types of pertur-bations and the reasons for their occurrence.

2. We reveal the effect of each perturbation in reconstructingSH-signals using the CS scheme. We have undertakena simulation study to show the adverse effects of bothadditive and multiplicative perturbations in recovering asparse SH coefficient vector.

3. We finally suggest a couple of methods, calibration andsegmentation, to mitigate the effects of multiplicative andadditive perturbations in reconstructing SH-signals usingCS.

The remainder of this paper is organized as follows. SectionII introduces CS into the reconstruction of SH-signals. In Sec-tion III, we show how additive and multiplicative perturbationsare generated in sampling SH-signals and how these adverselyaffect the CS recovery of signals on sphere. To remove theadverse effects of these perturbations, SH calibration and SHdiversity methods are suggested in Section IV. The adverseeffects of these perturbations and their mitigation are presentedin Section V. Finally, Section VI finalizes our paper withconclusions and some suggestions for future work.

978-1-4673-2393-2/12/$31.00 ©2012 IEEE

II. PRELIMINARIES

A. Spherical Harmonics

Any square-integrable function f(θ, φ) as a function of co-latitude θ and longitude φ, can be expressed as

f(θ, φ) =

∞∑`=0

∑m=−`

xm` Ym` (θ, φ), (1)

where ` and m denote the degree and order, respectively, ofthe SH basis functions represented by Y m` (θ, φ) and definedas

Y m` (θ, φ) =

√2`+ 1

4π

(`−m)!

(`+m)!Pm` (cos θ)eimφ, (2)

and Pm` (·) is the associated Legendre function. The xm` in (1)is the SH coefficient. If the scalar function f(θ, φ) is known,the coefficient of the SH xm` can be calculated by

xm` =

∫ 2π

0

∫ π

0

f(θ, φ)Y m` (θ, φ)∗ sin(θ)dθdφ, (3)

where ∗ denotes a complex conjugate.However, in this paper, we want to determine xm` with a

limited number of signal measurements on the sphere withoutcomplete information about the signal f(θ, φ). This is regardedas SH-signal reconstruction, since we have to recover theSH coefficients from samples of the signal corresponding tothe measurements. If we choose M measurements on spherefi = f(θi, φi) for i = 1, . . . ,M , where (θi, φi) is the i-thmeasurement on sphere, the sampled measurement equationcan be written

f = Y x. (4)

where f ∈ RM×1 is the vector of sampled signal f =[f1 . . . fM ]T , Y ∈ RM×N is the matrix of SH, N is the totalnumber of harmonics, and x ∈ RN×1 is the coefficient vectorcorresponding to Y . If the SH-signal is band-limited in degreeto L, the total number of harmonics will be N = (L + 1)2.The matrix Y is defined as

Y =[yn=1 · · · yn=N

]where yn is a column vector expressed as

yn =[Y mn

`n(θ1, φ1) · · · Y mn

`n(θM , φM )

]T,

and `n and mn are the degree and order of the harmoniccorresponding to the n-th column of Y , respectively.

As we have mentioned briefly in the introduction, the mostcommon way of finding x is the Least squares (LS) methodas

x? = arg minx‖Y x− f‖2

= Y †f , (5)

where x? is the estimation of x, Y † is the pseudo-inversematrix of Y defined as Y † = (Y TY )−1Y T . In using the LSmethod, the number of measurements, M , should as large asor bigger than the number of total harmonics, N (dependingon the noise and possible degeneracies).

Fig. 1: Perturbations on the sampling of SH-signal : one isdeviation from the intended point due to jitter and the otheris additive noise.

B. Applying CS to SH

If the signal is sparse in the basis domain, i.e., most entriesof the vector x are zero, it is possible to estimate x eventhough M is much smaller than N by using compressedsensing (CS) techniques [6–10]. The vector x can be estimatedas

x? = arg minx‖x‖1 s.t. Y x = f , (6)

where ‖x‖1 :=∑Nn=1 |xn|. For successful recovery, there is

a requirement on the matrix Y called the restricted isometryproperty (RIP) as

(1− δ2K) ≤ ‖Y x2K‖22‖x2K‖22

≤ (1 + δ2K), (7)

where K is the sparsity of x defined as K = |{j : xj 6= 0}|. Ifδ2K , known as the restricted isometry constant (RIC), is quitesmaller than one for any 2K-sparse vector x2K , the recoveryof (6) is possible with high probability. For the satisfactionof the RIP, we need the minimum number of measurement,Mmin given by

M ≥ C ·K · logN, (8)

where C is some positive constant.

III. CS SENSITIVITY PROBLEM

Sensitivity problems occur when there are perturbations inthe compressively sampled measurements. In sampling a SH-signal, there can be two kinds of perturbations. As shown inFig. 1, one is deviation from intended point due to jitter onthe spherical sampling point, and the other is additive noiseon the sampled measurements.

A. Perturbations of SH-signal

If there is any jitter on the spherical sampling, the i-thsampling position at (θi, φi) is given by

(θpi , φpi ) = (θi + ∆θi, φi + ∆φi), (9)

where the superscript p notes perturbation, ∆θi and ∆φi arethe deviations from θi and φi, respectively. Furthermore, withadditive noise, the i-th measurement will be expressed as

fpi = f(θpi , φpi ) + ei, (10)

where ei is the additive noise on the i-th sample. Then, theperturbed measurements can be expressed in matrix form as

fp = Y px + e, (11)

where Y p is the SH matrix with jitter, and e = [e1 . . . eM ]T .Since Y p can be decomposed into two terms as

Y p = Y + ∆Y , (12)

where ∆Y is a deviation matrix from Y , the sampled signalis re-expressed as

fp = Y x + ∆f , (13)

where ∆f = ∆Y x + e.

B. Perturbation Effect

Since we have no information about the perturbations, blindCS reconstruction should be performed as

x? = arg minx‖x‖1, s.t. ‖Y x− fp‖2 ≤ ε, (14)

where ε is the bound of total noise bound due to the samplingjitter and the additive noise.

In CS, the measurement noise e and the deviation ma-trix ∆Y are modelled as additive [10] and multiplica-tive [11] perturbations, respectively. With quantitative mod-elling method [11], the magnitudes of these perturbations arebounded as

‖e‖2‖f‖2

≤ εf ,‖∆Y ‖(K)

2

‖Y ‖(K)2

≤ ε(K)Y , (15)

where εf and ε(K)Y are relative upper bounds on additive and

multiplicative perturbation, respectively. ‖ · ‖(K)2 denotes the

spectral norm of all K-column sub-matrices of a matrix. Ifthe 2K-RIC satisfies

δ2K <

√2(

1 + ε(2K)Y

)2 − 1, (16)

the reconstruction error is known to obey [11]

‖∆x‖2 ≤ Cε, (17)

where C is a constant given by

C =4√

1 + δ2K(1 + ε(2K)Y )

1− (1 +√

2)((1 + δ2K)(1 + ε

(2K)Y )2 − 1

) , (18)

and ε is the bound of total noise calculated as

ε =(√1 + δK√

1− δKε(K)Y + εf

)‖f‖2. (19)

The noise of CS-recovered signal gets larger as the magni-tudes of perturbation increase. Furthermore, the multiplicativeperturbation due to sampling jitter is more detrimental tothe recovery of the SH-signal since the error constant C isincreasing highly even with small value of ε(2K)

Y .

IV. MINIMIZING PERTURBATION EFFECTS

A. Calibration For Multiplicative Perturbation

If we have the information on the perturbations of(∆θi,∆φi) for i = 1, . . . ,M , we can minimize the adverseeffect of multiplicative perturbation. We can use a referencesignal to perform a calibration or to estimate the multiplicativeperturbation by transmitting and receiving the reference signalbefore transmitting actual signal. The calibration method canbe varying depending on applications. With the assumptionof the calibration, the recovery can be performed with thecalibrated SH matrix of Y p, Y

pas

x? = arg minx‖x‖1, s.t. ‖Y p

x− fp‖2 ≤ εadd, (20)

instead of (14), where εadd = εf‖f‖2. The noise bound isreduced from ε to εadd since the basis mismatch term ∆Yis removed by calibration. Then, the recovery noise can bereduced as

‖∆x‖2 ≤ Caddεadd, (21)

where Cadd is a constant given by

Cadd =4√

1 + δp2K1− (1 +

√2)δp2K

. (22)

In (22), δp2K is the RIC of Y p.

B. Segmentation for Additive Perturbation

If the total measurements can be segmented with D multiplesets of Mmin measurements, we can extract diversity gain byperforming and combining of multiple CS recoveries. We canhave multiple CS recoveries as

x?d = arg minx‖x‖1 s.t. ‖Y p

dx− fpd‖2 ≤ εd, (23)

for d = 1, 2, . . . , D, where εd is the upper bound on ‖∆fd‖2.Then, we can combine multiple versions of x?d to achievediversity as

x? =

D∑d=1

wdx?d, (24)

where wd is the combining weight for x?d. If we set the weightsof parallel branches wd = 1/D for d = 1, 2, . . . , D, the noisevector of the combined recovery is given by

∆x =1

D

D∑d=1

∆xd, (25)

Fig. 2: Signal on sphere consists of spherical harmonics whosecoefficients are sparse.

where ∆xd ∈ RN×1 is the noise vector of the d-th CSrecovery. Since the multiplicative perturbation is removed bythe calibration, ∆xd, d = 1, 2, . . . , D can be regarded asuncorrelated as long as the additive noise is not correlated. Ifthe noise of multiple recoveries are at the same level, the noisebound of the combined CS recovery is inversely proportionalto√D as

‖∆x‖2 /Caddεadd√

D, (26)

shown in Section VII.

V. NUMERICAL RESULTS

A. Simulation Set-ups

We want to reconstruct a spherical signal with CS whichis sparse in basis domain as shown in Fig. 2. The signal isassumed to be band-limited to L = 50 and for simplicity, onlypositive order spherical harmonics are computed. As a result,we will have N = 1326 harmonics for a maximum band-limit of L = 50. The sparsity of the coefficients is assumedas K = 10 out of N = 1326. However, these parameterscan be extended in case of more complex signal. Furthermore,Gaussian noise whose variance is 0.0025 is intentionally addedto the signal for the generation of the additive perturbation.

Fig. 3 shows the intended sampling points on the sphere forCS. Only M = 100 measurements are selected on the spherefor CS reconstruction. For the emulation of multiplicativeperturbation, the variances of the deviations on (θ, φ) of theintended points are set to 0.01 rad.

B. CS Reconstruction

1) Under Both Perturbations: Fig. 4(a) is the CS-recoveredsignal under both additive and multiplicative perturbations, andFig. 4(b) compares the pure coefficients and the CS-recoveredcoefficients of the SH. These results show that there is severe

Fig. 3: Sampling points for CS reconstruction. Only M = 100points are selected for CS.

noise in the recovered signal. It indicates that we cannotreconstruct the original SH-signal with CS if there are bothperturbations.

2) Removing Multiplicative Perturbation: Fig. 5(a) andFig. 5(b) show the CS recovered SH-signal with the assump-tion that the multiplicative perturbation is removed. The re-covery is tolerable if there are only additive perturbations,even though there is still noise. This result implies that themultiplicative perturbation is more detrimental to the recov-ery than the additive perturbation. This, therefore, suggestsminimizing the multiplicative perturbation is important in thereconstruction of SH-signal with CS.

3) Reducing Additive Perturbation: CS diversity techniqueis applied to reduce the additive noise further. Even thoughthe recovery noise is reduced significantly by removing themultiplicative perturbation, the recovery still suffers from thenoise due to the additive perturbation. To extract the diversitygain by combining multiple recoveries, we selected 4 sets ofM = 100 CS samples on the sphere. Fig. 6 is the combined SHcoefficients with the multiple recoveries. Comparing Fig. 5(b),the additive noise is reduced significantly with the applicationof diversity gain technique.

VI. CONCLUSIONS

In this paper, we showed the perturbation effects andproblems in reconstructing SH-signals with CS. Our resultsshow that the multiplicative perturbation, which is generatedby the jitter of sampling position on SH-signal, can causeserious error on the SH-signal recovery with CS. To applyCS practically to the reconstruction of SH-signal, the methodof calibration to minimize the multiplicative perturbation issuggested. Furthermore, we showed that the possibility ofnoise reduction in SH-signal reconstruction with the diversitytechnique of combining multiple recoveries.

(a) CS-reconstructed SH-signal.

(b) Comparison of harmonic coefficients between the pure and theCS-reconstructed.

Fig. 4: CS Recovering of SH-signal under both additiveand multiplicative perturbations. For the emulation of themultiplicative perturbation, the jitter whose variance amountsto 0.01 rad is added on the sampling point (θ, φ). Furthermore,Gaussian noise whose variance is 0.0025 is added to thesignal for the generation of the additive perturbation. The CS-recovered SH-signal is totally different with the original underboth perturbations.

Interesting problems for future work include the practicalmethod of minimizing the multiplicative perturbation in SH-signal processing with CS.

VII. APPENDIX

Since the magnitude of the CS recovery noise with the d-thmeasurement set can be expressed as

‖∆xd‖2 '√Nσd2, (27)

(a) CS-reconstructed SH-signal.

(b) Comparison of harmonic coefficients between the pure and theCS-reconstructed.

Fig. 5: CS-reconstructed signal with the assumption of re-moving multiplicative perturbation. Without the multiplicativeperturbation, the CS-recovered SH-signal is tolerable exceptsome noise.

where σd2 is the variance of the elements on the noise vector,the noise variance of the d-th recovery can be bounded as

σd2 /

(Cdεd

)2N

, (28)

where Cd is the error bound constant of the d-th CS recovery.If ∆xd, d = 1, 2, . . . , D are uncorrelated each other, as aresult, the variance of the elements of ∆x is bounded as

σ2 /1

D2

D∑d=1

(Cdεd

)2N

. (29)

Fig. 6: Comparison of harmonic coefficients between the orig-inal and the CS-reconstructed with additive noise reduction.The residual noise is further reduced.

Finally, the magnitude of the noise bound of the combinedvector can be expressed as

‖∆x‖2 /1

D

√√√√ D∑d=1

(Cdεd

)2. (30)

For intuitive understanding of this noise bound, let us assumethat Cd ≈ Cadd and εd ≈ εadd for d = 1, 2, . . . , D. Then,the noise bound of the combined CS recovery is inverselyproportional to

√D as

‖∆x‖2 /Caddεadd√

D. (31)

Therefore, we can reduce recovery noise by increasing thenumber of branches, D.

ACKNOWLEDGMENT

This work was supported by the Australian Research Coun-cil Discovery Grant DP1094350.

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