Computing Spherical Transform and Convolution on the 2-Sphere

Transcription

1 Computing Spherica Transform and Convoution on the 2-Sphere Boon Thye Thomas Yeo May, 25 Abstract We propose a simpe extension to the Least-Squares method of projecting sampes of an unknown spherica function onto the spherica harmonics. Using Gram-Schmidt orthogonaization, our spherica transform (unike previous agorithms) guarantees that the inverse transform converges to sampe points without any assumption about samping density or grid (such as the atitude-ongitude grid), aowing us to perform spherica convoution with non at-on sampes. Athough we have not derived theoretica proofs, we wi demonstrate that our agorithm achieves ow interpoation errors on the uniformy random grid even when samping beow the theoretica non-aiasing rate of the at-on grid. Whie we wi ony consider rea-vaued spherica functions, our method shoud be readiy extensibe to compexvaued functions. Introduction We define spherica functions, L 2 (S 2 ) to be the Hibert space of square integrabe functions on the two dimensiona sphere, S 2. In spherica coordinates, the usua inner product on the sphere is given by: f, h = π 2π f(θ, φ)h(θ, φ) sin θdφdθ where by convention, θ π (measured down from the z-axis) and φ < 2π measured countercockwise off the x-axis. Spherica functions arise in many fieds, such as shape modeing in computer vision [2] and medica imaging [9], computer graphics [7, 8], astrophysics [] and geophysics [5]. The fitering of spherica functions is therefore an active area of research. However, even simpe inear space-invariant fitering is hard because it is not possibe to discretize the surface of the sphere eveny such that the neighborhood of each point is the same. Therefore a popuar method of performing spherica convoution is to first project the discretized spherica function and fiter onto the span of spherica harmonics and perform the convoution in the fourier domain via simpe mutipications. The samping theorem, convoution theorem and fast spherica transform presented in Drico and Heay s semina paper [4] provides theoretica guarantees for this approach. The samping theorem ensures that the discretization of the spherica functions is reversibe (i.e. no aiasing), and the fast spherica transform aows for fast computation. Since then, there have been many improvements in creating faster spherica transform agorithms [3, ]. However, the samping theorem ony hods for samping on the at-on grid, which suffers from excessive samping near the poes. In practice, the bandwidth of our function might not be known in advance, and so it is conceivaby better if we spread out our measurements over the sphere. In addition to the convoution theorem mentioned above, another reason for representing functions as a inear combination of spherica harmonics is the uniform resoution property of the spherica harmonics (see section 2) [4]. Our own motivation comes from the representation of brain cortica surfaces as functions on spheres. Neurobioogists beieve that most of our higher cognitive abiities originate from the cerebra cortex, and that neuroogica growth or diseases significanty ater the structure of the cortex. Currenty, we are working with MRI-segmented cortica surfaces that are mapped onto the topoogicay-equivaent spherica surfaces. In this case, the samping is ceary of a more uniform nature than the at-on grid and the samping density is outside our contro. In this paper, we present a method of computing the harmonic decomposition of a spherica function, not necessariy samped on the at-on grid. Our transform guarantees that the inverse transform converges to the sampe points. We wi demonstrate empiricay that our agorithm performs better than the usua Least-Squares method and fast spherica transform on the at-on grid. On the uniformy random grid, both our agorithm and Least-Squares achieve ow interpoation errors when samping beow the theoretica non-aiasing rate of the at-on grid. This paper is organized as foows. Section 2 goes through the basic definitions and properties of spherica harmonics, whie section 3 formuates the Least-Squares approximation to the harmonic decomposition probem. This eads to the discussion of Gram-Schmidt orthogonaization in section 4 and computationa issues in section 5, foowed by experimenta resuts and concusions in section 6 and 7. For competeness, we wi mention that there are other competing methods of spherica fitering. Among these are adhoc techniques designed for particuar fiters, and therefore not extensibe to a fiters. A notabe exampe is Sweden s ifting scheme [], which constructs second generation waveets capabe of handing non-eucidean manifods, such as the sphere [7], [8]. Spherica waveets utiizes the geodesic grid, which is obtained by projecting the icosahedron onto the sphere and then continuay subdividing its faces into smaer trianges. The advantage of

2 (a) (c) (b) (d) The spherica harmonics equations can be overwheming at first, so we dispay a few of the harmonics in figure. The spherica harmonics, Y m (θ, φ) form an orthonorma basis for L 2 (S 2 ). Therefore, we can expand a function, f L 2 (S 2 ) as f = m f(, m)y m (θ, φ) (4) where f(, m) denotes the (, m) harmonic coefficient, equa to f, Y m. Because of eq (4), approximating the harmonics coefficients is equivaent to interpoating our sampes. We state the foowing theorems without proof [4, 3]: Theorem (Convoution Theorem [4]). For functions f, h L 2 (S 2 ), the transform of the convoution is a pointwise product of the transforms: (f 4π h)(, m) = 2π f(, m)ĥ(, ) 2 + Figure : Low-order spherica harmonics. Ony the rea part is potted for compex-vaued harmonics. (a) Y (θ, φ) = 2 (b) Re{Y π 2 (θ, φ)} = 5 4 π (3cos2 θ ) (c) Re{Y2 2 (θ, φ)} = 5 4 2π sin2 θcos2φ (d) Re{Y2 (θ, φ)} = sinθcosθcosφ. The coor of the surface represents 2 5 2π the vaue of the function. For exampe, in (a), Y (θ, φ) is actuay a sphere and hence a surface points are equidistant from the origin, and thus are of the same coor. starting with the icosahedron is that the resuting trianguation has the east imbaance in area between its constituent trianges [8]. Indeed, besides the at-on grid, the geodesic grid is often the spherica grid of choice. It is for exampe used in the iterative soution of partia differentia equations that simuate ocean dynamics [5]. However, as a samping grid, it suffers from the probem that we are unabe to sampe at intermediate number of points, as the number of vertices jump sharpy with each subdivision. At eve 7, there are aready vertices. 2 Spherica Harmonics Preiminaries For any integer and integer m, m, the spherica harmonic of degree and order m, Y m (θ, φ) takes the form for m : Y m (θ, φ) = and for m < : (2 + )( m)! P m (cos θ)e imφ () 4π( + m)! Y m (θ, φ) = ( ) m Y m (θ, φ) (2) where P m (x) are the associated Legendre Poynomias: P m (x) = ( )m 2 ( x 2 m/2 dm+ )! dx m+ (x2 ) (3) 2 Observe the asymmetry of the convoution theorem: ĥ(, ) vs f(, m). This comes from the definition of convoution as a eft-convoution (averaging eft transations). There is a simiar theorem for right convoution (averaging right transations). Another important thing to note about the convoution theorem is that it is independent of the samping. Hence, as ong as we can project our sampes onto the span of spherica harmonics accuratey, we can perform convoution via the fourier domain accuratey, regardess of the samping grid. Theorem 2 (Samping Theorem [3]). Let f L 2 (S 2 ) be a bandimited function of bandwith b, i.e. f(, m) = for b. Then: f(, m) = 2π 2b 2b j= 2b k= a (b) j f(θ j, φ k )Y m(θ j, φ k ) for < b and m. Here θ j = π 2j+ 4B, φ k = πk/b and are deterministic weights designed to compensate for a (b) j the oversamping at the poes. Note that the proof ony showed (2b) 2 sampes to be sufficient (not necessary), athough empiricay our methods cannot hande ess than (2b) 2 sampes on the at-on grid (see section 6.2). Whie the samping theorem ony appies to the at-on grid, for the sake of comparison, we wi sti consider a function of bandwidth b to be undersamped when ess than (2b) 2 sampes are taken. Theorem 3 (Uniform Resoution [4]). Under a rotation, each spherica harmonic of degree is transformed into a inear combination of ony those harmonics of the same degree. Hence, the eve of resoution of bandimited functions obtained through the truncation of the harmonics is uniform a

3 over the sphere, thus providing an avenue of getting around the inabiity to discretize the sphere eveny []. We wi now estabish some properties of rea-vaued spherica functions. Consider a rea-vaued function f and define c m to be the projection coefficient of the function f onto the spherica harmonic of degree and order m. Since c m = f, Y m, we have c m = = π 2π π 2π = ( ) m π f(θ, φ)y m (θ, φ) sin θdφdθ f(θ, φ)[( ) m Y m (θ, φ)] sin θdφdθ 2π f(θ, φ)y m (θ, φ) sin θdφdθ = ( ) m c m (5) where we used eq (2) and f(θ, φ) = f(θ, φ) since f is reavaued. Hence, when trying to find the spherica transform coefficients c m, we ony need to consider m. Therefore, using the above, for m > : c m Y m + c m Y m = ( ) m c m ( ) m Y m + c m Y m = c m Y m + c m Y m = 2Re{c m Y m } = 2(a m R m + b m ( I m )) (6) where c m = a m + ib m and R m, I m are the rea and imaginary parts of Y m. Note that for m =, Y m is rea, and therefore c must aso be rea in order for f to be reavaued. 3 Least-Squares Approximation Whie c m = f, Y m, in practice, we are unabe to evauate the inner product because we ony know the vaues of f at its sampe points. Because the samping theorem ony hods for at-on grid, we are unsure what to do for non at-on sampes. Suppose we sampe f at (θ i, φ i ), i n, not necessariy on the at-on grid, we coud try to discretize the integra [2]: c m 4π n n f(θ i, φ i )Y m (θ i, φ i ) i= Unfortunatey, whie the spherica harmonics are orthonorma, their vaues evauated at some set of parameter pairs (θ i, φ i ) wi generay not form an orthonorma set of vectors. However, what we reay want is a spherica harmonic series that passes near the sampe points [2], based on the assumption that a series that passes near the sampe points wi be cose to the origina function everywhere. Here, we deviate from the formuation of [2], adapting their method to our probem at hand. Once again, etting R m, I m be the rea and imaginary parts of Y m, we define 3 R(θ, φ ) R(θ 2, φ 2 )... R(θ n, φ n ) R(θ, φ ) R(θ 2, φ 2 )... R(θ n, φ n ) 2R(θ, φ ) 2R(θ 2, φ 2 )... 2R(θ n, φ n ) B T = 2I (θ, φ ) 2I (θ 2, φ 2 )... 2I (θ n, φ n ) R2(θ, φ ) R2(θ 2, φ 2 )... R2(θ n, φ n ) I (θ, φ ) 2I (θ 2, φ 2 )... 2I (θ n, φ n ) (7) Hence, B is a n x 2 matrix containing the spherica harmonics up to degree, evauated at our sampe points (with some scaing). Ony the rea part of order harmonics are incuded because order harmonics are rea. The factor 2 and minuses are incuded here due to eq (6). We denote the sampes of f by the n x vector f and define c T = [a a a b a 2... b ], where cm = a m + ib m. Then our objective of finding a harmonic series that passes near the sampe points is equivaent to finding a c that minimizes the error vector e, such that f = B c + e. If we adopt the 2 -norm as our measure of e, then c = (B T B) B T f (8) is the Least-Squares soution that minimizes our error measure. Hence, to project our samped function onto the spherica harmonics up to degree, we form the n x 2 matrix, B and estimate the coefficients using eq (8). In practice, we can increase the degree progressivey, unti a pre-determined accuracy is achieved. The probem with the Least-Squares method is that B T B might not be invertibe even when 2 < n or 2 < b, where b is the bandwidth of our function. Whie B T B seems to be mosty invertibe for uniformy random samping (see section 6), there is no theoretica guarantees. In fact, when samping on the at-on grid, the coumns of B are often not ineary independent, when we have < (2b) 2 sampes. 4 Gram-Schmidt Orthogonaization Notice that in the Least-Squares approximation, as we progressivey increase the degree of the spherica harmonics to improve our approximation, the number of coumns of B increases but not the rows. Hence, to avoid having to worry about inear independence of B s coumns, we can perform Gram-Schmidt orthogonaization. Since Gram-Schmidt orthogonaization is done in an iterative fashion, it is just right for the progressive addition of coumns to B. The coumns of B that are aready in the span of previous coumns are discarded. Formay, we et { v, v 2, v 3,...} denote the coumns of B. Our aim is to produce an orthogona sets of vectors W = { w, w 2, w 3,...} spanning the coumn space of B, giving priorities to the earier coumns (ower harmonics). We do this iterativey: w = v w 2 = v 2 v 2, w w, w w

4 w 3 = v 3 v 3, w w, w w v 3, w 2 w 2, w 2 w 2 w 4 = v 4 v 4, w w, w w v 4, w 2 w 2, w 2 w 2 v 4, w 3 w 3, w 3 w 3.. (9) Note that if v i ies in the span of the previous v j s, then w i = and we shoud discard it. In practice, we discard w i if its norm is beow a certain threshod. It is easy to verify that the w i s are by construction orthogona. The projection of f on the span of W is therefore given by f = i a i w i = i f, w i w i, w i w i. We can summarize our agorithm as foows:. Set W = {} (the empty set), = and residua vector f = f. 2. Evauate f = f, f f, f, the normaized residua error. Normaization by f, f is neccessary to eiminate scaing effects. 3. Whie ( f > threshod), (a) Set = +. (b) Use Gram-Schmidt orthogonaization to check if the samped harmonics of degree is in the span of W (c) Add ony the set of new orthogona vectors, W (if any) to W. (d) Set f = f i W f, w i w i, w i. (e) Evauate f f, w i w i, w i w i and record In our experiments, threshod of f is set to. Observe that our agorithm has ony computed the projection coefficients of f on W, but what we reay want are the corresponding projection coefficients of f on the independent spherica harmonics. In theory, these can be recovered from the projection coefficients of f on W and the projection coefficients of v i s on w i s obtained from the orthogonaization process. However, the inversion process can be quite messy to program and so in practice, we keep track of the independency between the samped spherica harmonics during the orthogonaization process. When we compete the agorithm from above, we assembe the set of independent samped harmonics and perform a east-squares projection of f on that set. The coefficients of the dependent harmonics are set to. Note that the coefficients obtained this way wi be exacty the same as those obtained by the inversion process and wi in fact avoid numerica issues associated with a progressivey tinier residua vector, f. 5 Evauating the Harmonics To evauate the spherica harmonics, we make use of the three-term recurrence for P m (x), the associated Legendre 4 Poynomias []: where, xp m α m = = α m P m + α m +P m + () ( m)( + m) (2 )(2 + ) () and hence to cacuate P m (x), we begin with P m m (x) = and P m m (x) = ( ) m ( x2 ) m 2 2 m m! and use the recurrence eq. () to obtain 2m + (2m)! (2) 2 Pm+ m = αm+ m (xpm m αmp m m ) m (3) and repeat ti we achieve the desired P m (x). Care must be taken to evauate the constant 2m+ 2 m m! 2 (2m)!. If 2 m, m! or (2m)! is evauated directy, the foating-point range can be easiy exceeded even with moderate vaues of m. To avoid overfow, we 2m+ begin with 2, and mutipy a factor from the denominator when the constant is greater than and mutipy a factor from the numerator when the constant is ess than. When a the factors from either the numerator or denominator have been used up, we mutipy the remaining factors []. 6 Experiments In this section, we compare the performance of the Gram- Schmidt projection, the Least-Squares approximation and the Fast Spherica Transform on both the at-on grid and the uniformy random grid. For the fast spherica transform, we actuay utiize the non-speedup version, S2kit [6] because the fast spherica transform package, SpharmonicKit [6] ony handes powers-of-2 bandwidths. However, this is fine since they both use the samping theorem and we are ony testing for accuracy not speed. The uniformy random grid is defined as foows. Given an input N, we sampe (2N) 2 random points uniformy around the sphere. This can be impemented as foows: For each sampe, we generate the coordinate (θ, φ) by samping φ uniformy in the range [, 2π] and θ from the probabiity density distribution 2sin θ, θ π. The sin serves to compensate for the smaer area near the poes and pays an anaogous roe to the sin factor arising from integration on the sphere. We generate two random rea-vaued bandimited functions, f (bandwidth 2) and g (bandwidth 5), by setting the rea and imaginary parts of the spherica harmonic coefficients uniformy in the range [, ], for m. This determines the coefficients of negative order harmonics since c m = c m ( ) m (see eq. 2) for rea-vaued functions.

5 6. Toy Exampe As a sanity check, we sampe f on the at-on grid using (2 2) 2 = 6 sampes, and appy Least-Squares, Gram-Schmidt and S2kit transforms to the sampes. We then compare the coefficients obtained by the transforms with the origina known coefficients. The resuts are summarized in the foowing tabe: Least-Squares Gram-Schmidt S2Kit Max Absoute Difference 4.523e e e-4 We repeat this with 6 uniformy random sampes, but without S2kit since it ony handes at-on sampes: Normaized Difference (Test Error) Least Square Residua (Training Error) Gram Schmidt Residua(Training Error) S2kit Residua (Training Error) Least Square Normaized Difference (Test Error) Gram Schmidt Normaized Difference (Test Error) S2kit Normaized Difference (Test Error) Least-Squares Gram-Schmidt Max Absoute Difference e e-5 As expected, our agorithms work we with (2b) 2 sampes. Note that in the second tabe, Least-Squares and Gram-Schmidt shoud have the same errors except that we generate the 6 randomy uniform sampes on the fy, and so the agorithms were running on different set of sampes. 6.2 Undersamping on the Lat-Lon Grid Next, we undersampe f on the at-on grid with varying number of sampes, and appy Least-Squares, Gram- Schmidt and S2kit transforms. This time, the transform coefficients are definitey different from the origina coefficients, and so to measure interpoation errors between sampe points, we reconstruct our function, f from our transform coefficients using eq (4). We then compare the vaues of f and f at uniformy random points. Denoting f and f as the sampe vectors, we define the normaized difference between f and f to be f f / f, where the norm is taken to be the usua inner product norm,,. We can interpret the normaized difference to be the test error because it measures how we the coefficients we found generaize to unseen sampe points. We summarize the resuts in figure 2. From the figure, notice that the normaized difference (test error) of the Gram-Schmidt is in genera ower than Least-Squares and S2kit. We can aso concude from the figure that Gram-Schmidt suffers from overfitting, since test error is a ot higher than training error (amost ). Indeed, in the case of sampes (Gram-Schmidt suffers from the worst test error), setting the threshod for f to be.5 instead of increases the training error to.3393, whie decreasing the test error to.2284 (not shown in figure). An important observation is that athough Gram-Schmidt manages to beat Least-Squares and S2kit, in genera, the resuts are sti consideraby weaker than the amost-zero errors obtained in the previous section, when there s sufficient samping. We sha see in the next section that on the uniformy random grid, we can achieve ow errors despite undersamping Number of Sampes Figure 2: Performing Spherica Transform on the Undersamped Lat-Lon Grid: Training (Normaized Residua) Error and Test (Normaized Difference) Error of Least- Squares, Gram-Schmidt and S2kit. 6.3 Undersamping on the Uniformy Random Grid In this section, we undersampe f using the uniformy random grid with varying number of sampes. Once again we use normaized difference to measure our generaization error. The resuts are shown in figure 3. From the figure, we find that as ong as we have at east 4 sampes, both Least-Squares and Gram-Schmidt yied amost error. Interestingy, once we have ess than 4 sampes, the errors simpy bow up. Indeed, when we compare the experimenta errors between 399 and 4 sampes, we notice a big jump in error. This is summarized in the tabe beow: Number of Sampes Least-Squares f f / f 2.257e Gram-Schmidt f f / f.7346e To ensure that our resuts are not dependent on the error measure, we try a different error measure. We define the maximum normaized absoute difference between f and f to be max( abs( f f) abs( f) ). It basicay measures the argest percentage error in the approximation. The foowing tabe summarizes the resut: Number of Sampes Least-Squares.269e e+4 Gram-Schmidt.5e e+5 We do not think that it is a coincidence that 4 = 2 2, where 2 is the bandwidth of our samped function, f. It might be that for a bandimited spherica function of bandwidth b that is uniformy samped, the Nyquist rate might be b 2, rather than (2b) 2 on the at-on grid. Aso observe

6 Normaized Difference (Test Error) Least Square Residua (Training Error) Gram Schmidt Residua(Training Error) Least Square Normaized Difference (Test Error) Gram Schmidt Normaized Difference (Test Error) Normaized Difference (Test Error) Least Square Normaized Difference (Test Error) Gram Schmidt Normaized Difference (Test Error) S2kit Normaized Difference (Test Error) Number of Sampes Number of Sampes Figure 3: Performing Spherica Transform on Undersamped Uniformy Random Grid: Training (Normaized Residua) Error and Test (Normaized Difference) Error of Least-Squares and Gram-Schmidt. that 2 2 corresponds to the degree of freedom we have in determining our harmonic coefficients (where we consider a singe compex coefficient to contain 2 degrees of freedom). One might wonder why Least-Squares has simiar performance to Gram-Schmidt on the randomy uniform grid, but not on the at-on grid. The probem is that for the aton grid, B T B (see eq. 8) is often non-invertibe even when there are fewer coumns than there are sampes. 6.4 Convoution: Undersamping on the Lat- Lon Grid In this experiment, we convove our random bandimited function, f with our random fiter, g of bandwidth 5 to obtain h, i.e. f g = h. We assume that we have sufficient resources to represent the fiter, so we wi use its true harmonic coefficients. The approximation comes from estimating f s coefficients, using Least-Squares, Gram-Schmidt and S2kit with varying number of sampes. We then use the convoution theorem to obtain h, our approximation of h. Once again, we measure our interpoation error by comparing h and h at uniformy random points using normaized difference as our error measure. Note that we can cacuate h exacty since we generated f and g and so can appy the convoution theorem directy. We summarize our resuts in figure 4. Both S2kit and Gram-Schmidt have amost zero errors up ti 576 ((2(2)) 2 ) sampes. Whie it is not shown in the figure, the accuracy deteriorates rapidy when we use < 576 sampes. On the other hand, Least Squares perform bady the moment we undersampe f (ess than 8 sampes). However, notice that despite this, a three methods have significanty ower errors in figure 4 compared with figure 2. A possibe 6 Figure 4: Performing Convoution on the Undersamped Lat-Lon Grid: Training (Normaized Residua) Error and Test (Normaized Difference) Error of Least-Squares and Gram-Schmidt. Note that the back ine ies exacty on the bue ine, so you can t see the bue ine. expanation is this: because our fiter is of bandwidth 5, it acts as a ow pass fiter and remove the higher harmonics of f by mutipying them with zero. This tends to reduce the errors because the higher frequencies are ess reiabe due to undersamping. What is surprising however, is that whie a three methods have approximatey the same errors when estimating the spherica harmonics coefficients on the undersamped aton grid (see figure 2), S2kit and Gram-Schmidt perform so much better than Least-Squares when performing convoution. 6.5 Convoution: Undersamping on the Uniformy Random Grid We repeat the convoution experiment from the previous section, but this time using the uniformy random grid. From section 6.3, we expect our convoution to be accurate when we have at east 4 sampes, since Least-Squares and Gram-Schmidt Spherica Transforms are accurate over that range. Figure 5 shows the resut of our convoution experiment, and our predictions are indeed correct. However, unike the at-on grid, we do not get any boost in accuracy due to the ow-pass fiter. Despite this, we note that the uniformy random grid sti performs better, requiring ony 4 sampes compared with 576 sampes on the at-on grid. 6.6 Fina Toy Exampe We wi end our experimenta section with a fina toy exampe. Figure 6a shows a spherica image with a noisy protrusion (whose harmonic coefficients we know), and figure 6b shows a spherica fiter with a smooth symmetric bump on

7 Normaized Difference (Test Error) Least Square Normaized Difference (Test Error) Gram Schmidt Normaized Difference (Test Error) Number of Sampes Figure 5: Performing Convoution on the Undersamped Uniformy Random Grid: Training (Normaized Residua) Error and Test (Normaized Difference) Error of Least- Squares and Gram-Schmidt. the North Poe (whose harmonic coefficients we aso know). Both the image and fiter have bandwidths of 64. Convoving our image with the fiter shoud ideay yied an output that has a maximum at the bump (see figure 6c). In our experiment, we sampe our fiter and image at 64 2 uniformy random points (instead of the required (2(64)) 2 points on the at-on grid). The resut is shown in figure 6d. The maximum normaized absoute difference between the idea output and our resut over 4 uniformy random points is.946e 8. 7 Concusions and Future Work In this paper, we proposed and experimented with an extension to the Least-Squares method of projecting sampes of an unknown spherica function onto the span of spherica harmonics. The use of Gram-Schmidt orthogonaization is motivated by the fact that whie Least-Square aims to find a spherica harmonic decomposition that passes near the sampe points, there s no guarantee that the coumns of B (refer to eq. 7) wi be independent even when there are more rows than coumns in B (i.e. more sampes than the degree of freedom we have in the harmonic coefficients). Hence B T B coud become non-invertibe before we coud attain a good fit. Indeed, as shown in section 6.2, the coumns of B quicky became dependent when samping on the at-on grid, resuting in poor training and test error. On the other hand, Gram-Schmidt was abe to fit the sampes competey, athough there were obvious signs of overfitting. We ike to comment on the discarding of dependent spherica harmonics in the Gram-Schmidt orthogonaization process. Since these (samped) spherica harmonics aready ie in the span of previousy samped harmonics, adding them wi not yied a better approximation to the 7 samped points. By throwing them away, we are in fact favoring the harmonics we process earier, i.e. the ower-order harmonics. Hence, our agorithm has an impicit preference for smooth (ower-order) harmonics that expain our sampe points. We aso demonstrated that we needed ess sampes for the randomy uniform grid compared with the at-on grid. Empiricay, we found that a function of bandwidth b required b 2 sampes on the uniformy random grid compared with (2b) 2 sampes on the at-on grid. Finay, our experiments on convoution (section 6.4 and 6.5) showed that if we fitered a function of bandwidth b with a fiter of ower bandwidth, we might require ess than (2b) 2 sampes on the at-on grid in order to get accurate resuts. This was not true for the uniformy random grid, athough overa, the uniformy random grid sti performed better. Such a property is ess surprising if one considers a -D signa. Remember that samping a -D signa is equivaent to repicating the fourier transform of the signa periodicay in the fourier domain. As samping becomes more sparse, these repicas come coser and coser together. When the repicas overap, we have aiasing. Since overaps first occur in the higher frequencies, it means that aiasing effects tend to first show up in the higher frequency bands. Appying a ow-pass fiter wi eiminate the probematic higher frequency bands. The advantage of Least-Squares and Gram-Schmidt is that they hande the case of non at-on samping. However the fast spherica transform is simpy much faster (O(nog 2 n)), where n is the number of sampes. In comparison, our agorithm is at east O(n 3 ) (due to matrix inversion) assuming we use just enough sampes to prevent aiasing. The ess samping we require on the uniformy random grid ony comes up as an agorithmic constant. A possibe aternative might be to perform a one-time decomposition of our image using Least-Squares or Gram-Schmidt. Then for any fiter or cascade of fiters we woud ike to appy to our image, we can perform fast spherica transform to obtain the fiters coefficients assuming we know the anaytic expression of the fiters. There are a ot more work to be done:. We woud ike to test our agorithm on other grids and investigate their samping threshod. 2. We woud ike to investigate a more theoretica basis of samping imits on a non at-on grid, besides the intuitive reason that the at-on grid wastes too many sampes near the poes. A possibe source of inspiration coud come from non-uniform samping of -D signas. It is known that if non-uniform samping is done correcty, the nyquist rate can be vioated for -D signas. 3. Whie projecting our image onto the span of spherica harmonics provides a convenient way of performing convoution, we are sti unabe to hande non-inear fiters. There are many usefu (even simpe) non-inear fiters in image processing (such as the median fiter) that cannot be impemented via convoution. For such

8 (a) (b) (c) (d) Figure 6: (a) Spherica function, Bandwidth = 64, with a noisy protrusion. (b) Spherica Fiter, Bandwidth = 64, with smooth symmetric bump on the North Poe. (c) Idea Output. (d) Output found by estimating harmonic coefficients using Gram-Schmidt over 642 sampes over the uniformy random grid. fiters, we probaby have to empoy brute force methods, such as projecting our spherica function onto the pane and doing image processing there. Since the sphere and pane are geometricay and topoogicay different, we wi need to warp our image and fiter appropriatey. We wi aso need to interpoate the pixes of the 2D-grid, introducing possiby interpoation errors. It wi be interesting to compare the accuracy of the brute force method with our agorithm appied to inear fitering. [3] D.N. Rockmore P.J. Kosteec D.M. Heay, Jr. and S. Moore. Ffts for 2-sphere-improvements and variations. The Journa of Fourier Anaysis and Appications, pages Vo. 9, No. 4, , 23. [4] James R. Drisco and JR Dennis M. Heay. Computing fourier transforms and convoutions on the 2-sphere. Advances in Appied Mathematics, pages Vo. 5, 22 25, 994. [5] Francis X. Girado. Lagrange-gaerkin methods on spherica geodesic grids. Journa of Computationa Physics, pages 36: 97 23, We have not considered the case when our sampes are noisy. For exampe, we woud not expect our segmented cortica surfaces to be free of noise. Hence we woud probaby not want to fit our data sampes competey. This is reated to the overfitting probem that Gram-Schmidt suffers from. However, for rea data, we actuay do not know the underying function, so it wi be hard to test for overfitting. Perhaps we can use techniques such as eave-one-out cross-vaidation, athough that wi sow down our agorithms even further. [6] Peter J. Kosteec and Danie N. Rockmore. S2kit: A ite version of spharmonic kit. geeong/sphere/. [7] Peter Schroder and Wim Swedens. Spherica waveets: Efficienty representing functions on the sphere. In Computer Graphics Proceedings (SIGGRAPH), pages 6 72, 995. [8] Peter Schroder and Wim Swedens. Spherica waveets: Texture processing. Rendering Techniques, pages , Aug 995. [9] Lawrence H. Staib and James S. Duncan. Mode-based deformabe surface finding for medica images. IEEE Transactions on Medica Imaging, pages Vo. 5, No. 5, 72:73, Oct 996. References [] Martin Bohme. A fast agorithm for fitering and waveet decomposition on the sphere. Thesis (Dipoma), June 22. [2] G. Gerig CH. Brechbuher and O. Kuber. Parametrization of cosed surfaces for 3-d shape description. Computer Vision and Image Understanding, pages Vo. 6, No.2, 54 7, Mar, 995. [] Wim Swedens. The ifting scheme: A construction of second generation waveets. Siam Journa of Mathematica Anaysis, pages Vo 29, No. 2, pp , Mar 998. [] L. Jacques Y. Wiaux and P. Vandergheynst. Corresponding principe between spherica and eucidean waveets. Submitted to Astrophysics, 25. 8