Direction sensitive analysis of higher order jump discontinuities along circles on the sphere

In recent years, scale-discretized directional wavelets and second-generation curvelets have been introduced on the unit sphere, yielding directional and localized polynomial frames for band-limited signals. In this paper, we show that these functions are able to detect the positions and orientations of all higher order jump discontinuities which lie along circles on the 2-sphere. Specifically, we prove upper and lower estimates for the magnitude of the corresponding inner products when the analysis function is concentrated in the neighborhood of such a singularity. Although similar results already exist in certain two-dimensional settings, this paper is the first one to consider frames and signals that are given on the 2-sphere. As a side product of our investigations, we also develop a new localization bound as well as an explicit formula for the auto-correlation function for second-generation curvelets.


Introduction
In many fields of geosciences, the signals of interest live on a spherical surface and can therefore be represented in terms of the well known spherical harmonics, which form a natural orthonormal basis of L 2 (S 2 ).The corresponding Fourier transform yields a discrete set of harmonic coefficients in the frequency domain which contains all the information about the given signal.Because of the far-reaching benefits of this approach, the harmonic coefficients are often known in practice.However, a big disadvantage of the Fourier transform is the fact that the information is only given in terms of global quantities, since the basis functions are not localized.To deal with this shortcoming, frames consisting of localized analysis functions can be used instead of the spherical harmonics to obtain a position-frequency analysis where local structures can additionally be probed in terms of different orientations whenever the frame is directional.Two such systems, which will lay the foundation of this paper, have recently been proposed by Chan et al (2017) and McEwen et al (2018).In particular, both frames consist of polynomials, i.e., the corresponding analysis coefficients are simply linear combinations of the harmonic coefficients obtained by the Fourier transform and therefore often readily available.The authors in the just mentioned works also gave a variety of examples for applications in geosciences, further indicating their wide applicability.Consequently, it is of great interest to develop a good understanding of these frames.In this paper, we investigate how non-smooth structures like edges or higher order discontinuities are characterized by their analysis coefficients with regards to both systems.
In the setting of univariate functions, several localized algebraic and trigonometric polynomial frames have been shown to be suitable for detecting jump discontinuities of higher order derivatives (see e.g.Khabiboulline and Prestin 2006;Mhaskar andPrestin 1999, 2000a, b).The corresponding results consist of precise asymptotic estimates, in particular upper and lower bounds, for the frame coefficients in the neighborhood of such a singularity of a given signal.In two dimensions the problem is more complex since singularities can lie on curves.Consequently, localized directional analysis functions are needed to identify such features both in terms of their position and orientation.In this context, certain systems of so-called shearlets have been proven to be able to detect singularities along curves while also being directionally sensitive (see e.g.Guo andLabate 2009, 2018;Kutyniok and Petersen 2017;Schober and Prestin 2021;Schober et al 2021).The corresponding results are, again, given in the form of upper and lower estimates for the analysis coefficients.However, in this case, the estimates depend not only on the distance between the singularity and the analysis function, but also on their relative orientation towards each other.
The above mentioned two-dimensional problem also arises when dealing with functions f : S 2 → R which are given on the unit sphere.However, while there exist a variety of localized directional frames which natively live on S 2 and, intuitively speaking, should be suitable for detecting singularities (see e.g.Chan et al 2017;McEwen et al 2018;Iglewska-Nowak 2018), so far no precise statements on the magnitude of the analysis coefficients have been proven.This paper is a first approach at collecting such results.More specifically, we derive upper and lower estimates for the magnitude of the inner products f , D(α, β, γ ) N , where f is a signal with higher order jump discontinuities which lie on circles on the sphere and N ∈ L 2 (S 2 ) corresponds to one of two recently proposed directional polynomial frames.Loosely speaking, our results state that the analysis coefficients are only large when the rotated function D(α, β, γ ) N matches some singularity curve of f locally both in terms of its position and orientation.The precise estimates are given in the Theorems 3.1 and 3.3 and are understood in an asymptotic sense as the dilation parameter N becomes large and, consequently, N becomes more and more localized.The proofs are mainly based on methods used by Mhaskar and Prestin (2000b).It follows from our results that all higher order jump discontinuities of f can be identified, both in terms of their position and orientation, by the asymptotic decay of the corresponding analysis coefficients for large N .
There have been different approaches at constructing localized polynomial frames on the sphere.As in Conrad and Prestin (2002), Mhaskar et al (2000), Narcowich et al (2007), they often consist of isotropic analysis functions.In these situations, the task of detecting singularities along circles reduces to the one-dimensional problem, which was covered by Mhaskar and Prestin (2000b).Consequently, we are mainly interested in anisotropic frames.More specifically, we consider the directional wavelets and second-generation curvelets presented by McEwen et al (2018) and Chan et al (2017).Our main results state that both systems are able to detect the positions and orientations of higher order singularities.Furthermore, our estimates reflect their corresponding directional sensitivities.In particular, directional wavelets are somewhat limited in their ability to distinguish between different orientations, whereas second-generation curvelets are not.
The remainder of this paper is organized as follows.Section 2 is intended to serve as a preparation for our main results.We start with some basic preliminaries in Sect.2.1, followed by an introduction to the concept of directional analysis coefficients in Sect.2.2.Sections 2.3 and 2.4 are then devoted to showcasing the directional wavelets (McEwen et al 2018) and second-generation curvelets (Chan et al 2017).Besides collecting known results, we also derive a new auto-correlation formula as well as a localization bound for second-generation curvelets.In Sect.3, we present our main results, which consist of upper and lower estimates for the magnitude of the corresponding analysis coefficients.Finally, in Sect.4, we illustrate our results in the case where the signal under consideration is the indicator function of a spherical cap.
2 Directional wavelets and curvelets on the sphere

Preliminaries
Let S 2 = x ∈ R 3 : x 2 = 1 denote the unit sphere in R 3 , where • 2 is the Euclidean norm, induced by the inner product x, y 2 = x y for x, y ∈ R 3 .As visualized in Fig. 1, every point x ∈ S 2 can be identified by its latitude θ ∈ [0, π] and longitude ϕ ∈ [0, 2π ) through In the following, we will often use the shorthand notation (θ, ϕ) to address the corresponding element x(θ, ϕ) in S 2 .Moreover, we will use (1) for arbitrary ϕ ∈ R. The associated longitude contained in the interval [0, 2π ) can then simply be recovered by the 2π -periodicity x(θ, ϕ) = x(θ, ϕ + 2 mπ) for all m ∈ Z.By What follows is a brief discussion on Fourier analysis on the 2-sphere.In particular, we give an explicit orthonormal basis consisting of spherical harmonics as well as a representation of rotations via Wigner D-functions.For more details regarding spherical harmonics we refer to Conrad and Prestin (2002), where a slightly different notation has been used.Further information on the Wigner D-functions can be found in Varshalovich et al (1989).We consider the Hilbert space L 2 (S 2 ) with the inner product denotes the spherical harmonic of degree n and order k, defined as Here we have used the associated Legendre polynomials P k n : [−1, 1] → R, which can be defined by and for positive and negative order, respectively.The Legendre polynomial 2) is given by the Rodrigues formula We note that the associated Legendre polynomials can also be written in terms of Jacobi polynomials as where for α, β > −1.From the above definitions, we can directly deduce the symmetry Since the set of spherical harmonics forms a complete orthonormal system of L 2 (S 2 ), these function play a fundamental role in signal analysis on the unit sphere.In particular, every signal f ∈ L 2 (S 2 ) can be expanded into a Fourier series where f n,k = f , Y k n denotes the Fourier coefficient of f with respect to Y k n .In practice, the above series is replaced by a finite sum and the harmonic coefficients can be computed via suitable sampling theorems which are exact for polynomials up to a certain degree (see e.g.Driscoll and Healy 1994;McEwen and Wiaux 2011).Finally, we want to mention the well known addition theorem which relates the spherical harmonics to the one-dimensional Legendre polynomials.Functions in L 2 (S 2 ) can be arbitrarily rotated by successively rotating around the x 1 , x 2 and x 3 axes.Here, we use the x 3 x 2 x 3 convention with Euler angles α, γ ∈ [0, 2π ) and β ∈ [0, π].Let R x 2 and R x 3 be the rotation matrices which rotate vectors around the x 2 and x 3 axis, respectively.More precisely, We define the rotation operator D(α, β, γ ): where is the composition of the three Euler rotations.An important tool for working with rotations on the unit sphere are the Wigner D-functions , where the sum is performed over all values of j with max(0, k − k) ≤ j ≤ min(n − k, n + k ).As a special case, which will be relevant for us later, we mention that Furthermore, the symmetries and hold for all β ∈ [0, π].The connection between the Wigner D-functions and the rotation operator is given by Consequently, it follows that Finally, we note that the spherical harmonics can be written in terms of the Wigner D-functions as

Directional analysis coefficients
For an analysis function ∈ L 2 (S 2 ) which is localized at the north pole and a signal f ∈ L 2 (S 2 ), we consider the directional analysis coefficients In this case, the Euler angles α, β and γ have a concrete meaning, as illustrated in Fig. 2. While γ defines a specific orientation, β and α determine the position of the analysis function.Indeed, D(α, β, γ ) is localized at (β, α) ∈ S 2 .We remark that by (8), the above inner product can be written in terms of Wigner D-functions as The quality of the analysis coefficients depends strongly on the localization of in real space, as well as its directionality.The latter can be measured in terms of the directional auto-correlation which is defined as the function where a greater peakedness corresponds to a greater directional sensitivity.
In this paper, we will assume the signal f to be isotropic.By that we mean that there exists a direction z ∈ S 2 such that f is invariant under rotations around the z axis.Hence, all higher order discontinuities of f lie on circles on the sphere.A simple example would be the indicator function 1 C(z,φ) of a spherical cap with center z, as shown in Fig. 2.However, since the surface integral is rotationally invariant, we may assume, without loss of generality, that i.e., f is is invariant under rotations around the x 3 axis.In this case, it follows that and therefore we only have to take into account two of the three Euler angles.

Directional wavelets
Scale-discretized directional wavelets on the sphere were first introduced by Wiaux et al ( 2008) and later revisited in McEwen et al (2018), McEwen et al (2013).They are designed to be well localized in real space as well as to show an optimal directionality with respect to the auto-correlation function, while also having an azimuthal bandlimit.Very similar to the construction in McEwen et al (2018McEwen et al ( , 2013)), we define the directional wavelets N W,K in harmonic space by where the localization function κ satisfies κ ∈ C q+1 ([0, ∞)) for some q ∈ N 0 and ∅ = supp(κ) ⊂ [t 1 , t 2 ] with 0 < t 1 < t 2 and t 2 ∈ N. To impose uniqueness on t 1 and t 2 , we demand that the interval [t 1 , t 2 ] is minimal.The parameter N ∈ N is controlling the dilation, in which larger values correspond to a better localization in real space.
As in McEwen et al (2018), we define the directionality component by Here, K ∈ N is the parameter controlling the directionality.In particular, it holds that ( N W,K ) n,k = 0 if |k| ≥ K .Hence, the directionality component is imposing an azimuthal band-limit, which yields steerable wavelets (Wiaux et al 2008).Obviously, ζ K n,k does not depend on n for n ≥ K .In this case, we also write and thus axisymmetric wavelets.Larger values of K result in a stronger directionality, as measured by the auto-correlation, since For later use, we state the following proposition.

123
Proposition 2.1 Let N W,K be defined as in (12) with q ≥ 4 and supp(κ) where c > 0 is a constant that depends only on κ and K .
does not depend on n.By changing the order of summation in (11), straightforward calculations yield where w k , k = 0, 1, ..., K −1, are continuous functions.Now let k ∈ { 0, 1, ..., K −1 } be fixed.It is easy to see that there are constants c 0 and c 1 such that Furthermore, by using the addition theorem (4) for x = y, it is straightforward to show that |P k n (cos θ)| ≤ c 2 n k , where c 2 > 0 depends only on k.Consequently, we obtain It now follows from a simple substitution and from (2) that Furthermore, the classical Bernstein inequality for algebraic polynomials (see Daugavet and Rafal'son 1972) where and thus, according to (Petrushev and Xu (2005), Proposition 1), the latter integral is bounded independent of N .This completes the proof.

Curvelets
Second-generation curvelets on the sphere have been introduced by Chan et al ( 2017), where they have been shown to be efficient in representing anisotropic signal content.The construction is very similar to that of the directional wavelets.In particular, dilations are performed through a kernel function as before.In contrast, however, second-generation curvelets do not possess an azimuthal band-limit, which causes them to be more directionally sensitive than the directional wavelets.Similar to Chan et al (2017), we define the curvelets ˜ N C in harmonic space by where κ and N have the same meaning as in (12).That is, κ ∈ C q+1 ([0, ∞)) for some q ∈ N 0 and ∅ = supp(κ) ⊂ [t 1 , t 2 ] with 0 < t 1 < t 2 and t 2 ∈ N. Again, we assume that [t 1 , t 2 ] is the smallest possible interval with these properties.We note that ˜ N C possesses the following useful representation.

Proposition 2.2
The curvelet ˜ N C defined in (14) can be written as Proof By using the symmetry Y −n n = (−1) n Y n n of the spherical harmonics as well as their connection to the Wigner d-functions in (9) and the symmetry (6), we obtain In addition, it follows directly from (5) that The definition in ( 14) has a clear spectral interpretation, since we only use spherical harmonics of the highest and lowest order at any given degree.Proposition 2.2, on the other hand, provides an easy to read explicit formula, which reveals, in particular, that ˜ N C is localized at (π/2, 0) ∈ S 2 .The latter will be discussed in greater detail in Proposition 2.4.In the following, however, we will mostly refer to the curvelet C , which is localized at the north pole and therefore a suitable analysis function in the sense of Sect.2.2.This kind of repositioning was also utilized in Chan et al (2017).Straightforward calculations yield where The following proposition provides an explicit formula for the directional autocorrelation of the curvelets.This allows us to measure their directionality and compare it to the corresponding formula for directional wavelets given in (13).
Proposition 2.3 Let N C be the curvelet defined in (15).
Furthermore, by applying the addition theorem for Wigner d-functions (see Varshalovich et al.1989, p. 87), we obtain Consequently, we also have Proposition 2.3 states that, in contrast to the directional wavelets, the second-generation curvelets become more directional as the parameter N increases.Finally, we want to prove a localization bound similar to that of the directional wavelets in McEwen et al (2018).For this, we make use of the representation given by Proposition 2.2.

Proposition 2.4 Let ˜ N
C be the curvelet defined in ( 14), where we additionally assume that there is a value z ∈ (t 1 , t 2 ] such that κ (q+1) (t) = 0 for all t ∈ (t 1 , z).Then there exists a constant c q > 0, which depends only on κ and q, such that Proof According to Proposition 2.2, Furthermore, it can be derived from Stirling's formula that for some constants d −1 , d 0 , ..., d q+1 and consequently We now want to derive an upper bound for the inner sum.For we get Now let θ ∈ (0, π) and From Leibniz's rule, it follows that where c j,k ≥ 0 depends only on j, k, p and .Consequently, We note that κ (m) (t 1 ) = κ (m) (t 2 ) = 0 for 0 ≤ m ≤ q +1.Furthermore, by assumption, there exists a z ∈ (t 1 , t 2 ] such that κ (q+1) (t) = 0 for all t ∈ (t 1 , z).Hence, by the mean value theorem, it follows that κ ( p− j) (t) = 0 for all t ∈ (t 1 , z).We obtain Furthermore, if s = 1, repeated integration by parts yields Since s δ ln(s) m → 0 as s → 0 + for every δ > 0 and for all m ∈ N 0 , there exists a constant c p > 0, which depends only on κ, p and , such that In particular, there exists a c q > 0 such that which means that h (q) N , ,θ has a bounded total variation on [0, 2π ].Thus, by (Mhaskar and Prestin (2000b), Lemma 5), it holds for all N ∈ N with N ≥ 6/t 2 that for all |ϕ| ≤ π , where c q > 0 might differ from the constant in (18).Hence, we get Together with ( 16) and ( 17), this implies that Finally, repeated integration by parts yields for every ϕ ∈ [0, π].Again, c q > 0 is a constant which depends only on κ, q and .Since ˜ N C (θ, −ϕ) = ˜ N C (θ, ϕ), the proof is complete.

Main results
For r ∈ N 0 and φ ∈ (0, π), we consider the isotropic function f r ,φ : S 2 → R, where As illustrated in Fig. 3, the parameter r controls the smoothness of f r ,φ .More precisely, it follows from Leibniz's rule that Hence, f r ,φ is infinitely often differentiable except for a jump discontinuity of order r at the latitude θ = φ.As discussed in Mhaskar Prestin (2000b), most signals of practical interest which are invariant under rotations around the x 3 axis can be written in the form where g is R times continuously differentiable and also axisymmetric.Furthermore, the smoothness of g causes the corresponding analysis coefficients g, D(α, β, γ ) N to decay rapidly and uniformly as N becomes large.A precise statement of this fact is given by the following remark, where we adopt the notation of Dai and Xu (2013).
In particular, since P, D(α, β, γ ) N = 0 for every 123 Now let g ∈ C r (S 2 ) and ω r be the modulus of smoothness defined in (Dai and Xu (2013), Definition 4.2.1).By (Dai and Xu (2013), Theorems 4.4.2,4.5.5)we obtain inf where From Proposition 2.4, it follows that In the same way, lim follows from Proposition 2.1.
In the following, we will prove upper and lower estimates for the magnitude of the inner products f r ,φ , D(α, β, γ ) N .Since f r ,φ is invariant under rotations around the x 3 axis, we can neglect the Euler angle α.Furthermore, as illustrated by Fig. 4 , |φ − β| is the geodesic distance between the center of the analysis function and the singularity of f r ,φ .Intuitively speaking, since N is localized and has a zero mean, we expect all analysis coefficients with large absolute values to correspond to Euler angles β that are close to φ.In addition, it is obvious that the inner products depend strongly on the orientation of D(α, β, γ ) N , which is determined by γ .The extreme cases γ = 0, where the orientation of the analysis function is opposite to the edge, and γ = π/2, where the orientations match, are visualized in Fig. 4. Before stating our main results, we note that, by using the substitution t = cos θ and applying (Mhaskar and Prestin 2000b, Lemma 4), the harmonic coefficients of f r ,φ can be written as 2n + 1 4π where y = cos φ, provided that n ≥ r + 1.In particular, ( f r ,φ ) n,k = 0 for k = 0.

Singularity detection with directional wavelets
We will now discuss the analysis of higher order jump discontinuities with directional wavelets.Here, the function where s K = (1 − (−1) K )/2 and ζ K k is the directionality component ζ K n,k defined in Sect.2.3 for n ≥ K , plays an important role since it characterizes the directional sensitivity of N W,K .
123 Theorem 3.1 Let N W,K be the directional wavelet defined in ( 12).Furthermore, let s there exists an interval (i 1 , i 2 ) ⊂ R and a constant c 1 > 0, which both depend only on κ, s K , δ and r , such that provided that N is large enough.On the other hand, there exists a constant c 2 > 0, which depends only on κ, q, K , δ and r , such that Proof As discussed before, the directionality component of N W,K becomes independent of n for n ≥ t 1 N ≥ K .By using (11) as well as the fact that ( f r ,φ ) n,k = 0 for k = 0, we get The symmetries ( 6) and ( 7) yield In addition, for k ≥ 0, we have which follows from ( 9) and (3).Now let s k = (1 − (−1) k )/2.According to (Szegö 1975, Theorem 8.21.8), the asymptotic expression holds uniformly for all β ∈ [δ, π − δ].Furthermore, it is easy to see that for any fixed k .Consequently, it follows from the above considerations that holds uniformly for all β ∈ [δ, π − δ] and is valid for all k with |k | ≤ K − 1.We note that in (21) we only need to sum over indices k wich have the same parity as K − 1, since by definition ζ K k = 0 if s k = s K .Thus, we can exchange s k with 1 − s K .By (20), we have where we can plug in the asymptotic expressions 2n + 1 2 which, again, is easy to verify, and which holds uniformly for all φ ∈ [δ, π − δ].By inserting the right hand sides of ( 23) and ( 24) into (21), while also using ( 25) and ( 26), we get uniformly for all β, φ ∈ [δ, π − δ].By using the addition theorem cos x cos y = (cos(x − y) + cos(x + y))/2, we can split the foregoing expression into two sums.Similar to the proof of Proposition 2.4, the first sum can be written as where h(t) = κ(t • t 2 /(2π)) t −r −1 .Applying (Mhaskar and Prestin 2000b, Lemma 5), we obtain The same argument followed by repeated integration by parts yields Hence, the first statement is proven.
The lower bound of Theorem 3.1 now follows directly, since is an entire function and therefore we can find an interval (i 1 , i 2 ) ⊂ R which is free of zeroes.
In combination with Remark 1, Theorem 3.1 states that directional wavelets are suitable for detecting higher order singularities.Furthermore, the function χ K , which is visualized in Fig. 5, is contained as a factor in the dominant part of f r ,φ , D(α, β, γ ) N W,K .Therefore, the directional sensitivity, with regards to detecting discontinuities, is, almost entirely, characterized by χ K and, in particular, independent of N .Consequently, we can view χ K as kind of a directionality measure, where a greater peakedness corresponds to a greater directional sensitivity.Remark 2 So far, we have assumed the same definition of the directionality component as in McEwen et al (2018).However, for the proof of the above theorem we only need that for large values of n Hence, as long as the directionality component possesses these properties, Theorem 3.1 holds.Furthermore, the third condition is not necessary for the upper bound.

Singularity detection with curvelets
Before we state the main theorem regarding the singularity detection with curvelets, let us first prove the following auxiliary lemma.Lemma 3.2 For all β, γ ∈ (0, π) it holds that and, in particular, Proof By using ( 8) and ( 15), we get Now, (29) follows from the addition theorem for Wigner d-functions (Varshalovich et al 1989, p. 87).Finally, simple calculations yield (30).
Proof By using Lemma 3.2 together with (20), straightforward calculations yield where y = cos φ.As in the proof of Theorem 3.1, we use the asymptotic formulas ( 25) and ( 26) as well as uniformly for all φ ∈ [δ, π − δ].By the addition formula for the cosine function, we have where ψ = (φ − r π)/2 − 3π/4.Again, we can apply (Mhaskar and Prestin 2000b, Lemma 5), which yields Thus, repeated integration by parts proves the first statement.Furthermore, the lower bound follows from the same arguments as in Theorem 3.1.
Together with Remark 1, we conclude that second-generation curvelets are able to detect higher order singularities.However, in contrast to the directional wavelets, Theorem 3.3 states that the curvelets are, asymptotically speaking, only sensitive to discontinuities that match their orientation perfectly.Indeed, as implied by the upper bound, the inner products f r ,φ , D(α, β, γ ) N  C decay rapidly for large values of N whenever γ = π/2.

Illustrations
We will now illustrate our results from the previous section.As a test signal we choose the indicator function of a spherical cap.More precisely, let f = 1 C(z,φ) with center z = ((5π − 2)/10, 0) and opening angle φ = π/5, as visualized on the left side of Fig. 6 .By choosing a fixed angle γ in (10), we determine the orientation to which our further analysis will be most sensitive.Here, we set γ = π/4.As discussed in Sect.2.2, this means that the original analysis function is rotated around the x 3 axis by π/4 before being placed at each point on the sphere where we wish to evaluate the corresponding inner product.This process of relocating the pre-rotated analysis function is illustrated on the right side of Fig. 6, where we have also included the boundary of the spherical cap for reference.Furthermore, we highlighted the parts of the boundary where close by analysis functions exhibit approximately the same orientation as the local edge itself.The wavelets N W,K and curvelets N C defined in ( 12) and ( 15) are uniquely determined by the localization function κ and by the parameters K and N .Here, we choose κ to be the kernel constructed in McEwen et al (2018) with supp(κ) = [1/2, 2], as shown in Fig. 7.The resulting pre-rotated analysis functions D(0, 0, π/4) N W,K and D(0, 0, π/4) N C are visualized in Fig. 8 for the parameters K = 1, 8, 16 and N = 16, 32, 64, 128.Furthermore, Fig. 9 shows the corresponding analysis coefficients W K ,N  f : S 2 → R and C N f : S 2 → R, given by W K ,N f (β, α) = f , D(α, β, π/4) N W,K All values W K ,N f and C N f are computed using the expansion (11) in terms of the Wigner D-functions.Since the test signal is a spherical cap, there exist well-known explicit formulas for the harmonic coefficients f n,k .In addition, the expansion (11) reduces to a finite sum due to the fact that the analysis functions are band-limited.For the sake Fig. 9 Visualization of the analysis coefficients corresponding to the signal f in Fig. 6 and to the directional wavelets and curvelets in Fig. 8 of clarity, the images have each been rescaled, such that all values lie between −1 and 1.The results in Fig. 9 can be interpreted as follows: Reading each row from left to right, we see that the analysis coefficients decay rapidly for increasing N , as long as they are not too close to the boundary of the spherical cap.This behavior is reflected in the upper bounds of Theorems 3.1 and 3.3, respectively.Furthermore, the peak of the analysis coefficients moves closer to the edge and does not vanish for large N .This characteristic is described by the lower bounds in the theorems just mentioned.However, the directional sensitivity vastly differs among the considered wavelets and curvelets.The wavelet N W,1 , for K = 1, is axisymmetric and thus possesses no directionality.The resulting image of analysis coefficients must therefore be isotropic.In other words, all sections of the edge are detected equally.This is also in accordance with Fig. 5, since the directionality measure χ 1 is constant for all angles.In contrast, when we consider directional wavelets with K = 8 or K = 16, only the parts of the boundary which exhibit approximately the same orientation as the wavelet itself are visibly detected.It is also clear from the images that the wavelet N W,16 is more directionally sensitive than N W,8 , meaning that only an even smaller portion of the edge remains visible by the analysis coefficients.Indeed, this behavior is accurately described by the function χ K visualized in Fig. 5, since it appears as a factor in the dominant term of the analysis coefficients.In particular, the directional sensitivity of N W,K stays the same as N gets large.For curvelets the latter concept does not hold.As visualized in the last row of Fig. 9, curvelets become more directionally sensitive as N increases.In fact, Theorem 3.3 states that only edges that have exactly the same orientation as the analysis function will remain visible for large N .

Fig. 1
Fig. 1 Representation of a point x ∈ S 2 by its spherical coordinates

Fig. 2
Fig.2Interpretation of the three Euler rotations in the context of computing the directional analysis coefficients of a signal f .First, the analysis function gets rotated around the x 3 axis by an angle γ .The resulting function D(0, 0, γ ) has a characteristic directional orientation.Subsequently, D(0, 0, γ ) gets relocated to (β, α) ∈ S 2 through consecutive rotations around the x 2 and x 3 axis with angles β and α, respectively

Fig. 4
Fig. 4 Directional analysis of an axisymmetric signal f = f r ,φ for r = 0.The Euler angle β controls the latitudinal position of the analysis function and, consequently, its distance |φ − β| to the edge.The orientation of D(0, β, γ ) N is determined by γ and matches the singularity curve when γ = π/2

Fig. 6 Fig. 7
Fig.6Left: Indicator function f = 1 C(z,φ) of a spherical cap.Right: Visualization of an initial orientation on the north pole as well as its relocated versions, that arise from a rotation around the x 2 axis followed by another rotation around the x 3 axis

Fig. 8
Fig. 8 Pre-rotated directional wavelets and curvelets for different parameters