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

Abstract

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.

1 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 $L^2({\mathbb {S}}^2)$. 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 and Prestin 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 and Labate 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 $f :{\mathbb {S}}^2 \rightarrow {\mathbb {R}}$ which are given on the unit sphere. However, while there exist a variety of localized directional frames which natively live on ${\mathbb {S}}^2$ 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 $\left\langle f, {\mathcal {D}}(\alpha , \beta , \gamma )\Psi ^{\scriptscriptstyle N} \right\rangle$, where f is a signal with higher order jump discontinuities which lie on circles on the sphere and $\Psi ^{\scriptscriptstyle N} \in L^2({\mathbb {S}}^2)$ 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 ${\mathcal {D}}(\alpha , \beta , \gamma )\Psi ^{\scriptscriptstyle N}$ 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, $\Psi ^{\scriptscriptstyle N}$ 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

2.1 Preliminaries

Let ${\mathbb {S}}^2 = \left\{ \, {\textbf{x}}\in {\mathbb {R}}^3: \left\| {\textbf{x}} \right\| _2 = 1 \, \right\}$ denote the unit sphere in ${\mathbb {R}}^3$, where $\left\| \cdot \right\| _2$ is the Euclidean norm, induced by the inner product $\langle {\textbf{x}}, {\textbf{y}} \rangle _2 = {\textbf{x}}^\top {\textbf{y}}$ for ${\textbf{x}}, {\textbf{y}} \in {\mathbb {R}}^3$. As visualized in Fig. , every point ${\textbf{x}} \in {\mathbb {S}}^2$ can be identified by its latitude $\theta \in \left[ 0, \pi \right]$ and longitude $\varphi \in \left[ 0, 2 \pi \right)$ through

$$\begin{aligned} {\textbf{x}} (\theta , \varphi ) = \begin{pmatrix} \sin \theta \, \cos \varphi \\ \sin \theta \, \sin \varphi \\ \cos \theta \end{pmatrix}. \end{aligned}$$

(1)

In the following, we will often use the shorthand notation $(\theta , \varphi )$ to address the corresponding element ${\textbf{x}} (\theta , \varphi )$ in ${\mathbb {S}}^2$. Moreover, we will use (1) for arbitrary $\varphi \in {\mathbb {R}}$. The associated longitude contained in the interval $\left[ 0, 2\pi \right)$ can then simply be recovered by the $2\pi$-periodicity ${\textbf{x}} (\theta , \varphi ) = {\textbf{x}} (\theta , \varphi + 2\,m \pi )$ for all $m \in {\mathbb {Z}}$. By

Fig. 1

Representation of a point ${\textbf{x}} \in {\mathbb {S}}^2$ by its spherical coordinates

$$\begin{aligned} C({\textbf{z}}, \phi ) = \left\{ \, {\textbf{x}}\in {\mathbb {S}}^2: \langle {\textbf{x}}, {\textbf{z}} \rangle _2 > \cos \phi \, \right\} \end{aligned}$$

we denote the spherical cap with center ${\textbf{z}} \in {\mathbb {S}}^2$ and opening angle $\phi \in \left( 0, \pi \right)$. Its boundary

$$\begin{aligned} \partial C({\textbf{z}}, \phi ) = \left\{ \, {\textbf{x}}\in {\mathbb {S}}^2: \langle {\textbf{x}}, {\textbf{z}} \rangle _2 = \cos \phi \, \right\} \end{aligned}$$

constitutes a circle on the unit sphere.

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 $L^2({\mathbb {S}}^2)$ with the inner product

$$\begin{aligned} \langle f, g \rangle = \int _0^{2\pi } \int _{0}^\pi f(\theta , \varphi ) \, \overline{g(\theta , \varphi )} \, \sin \theta \, \textrm{d}\theta \, \textrm{d}\varphi \quad \text {for} \; f, g \in L^2({\mathbb {S}}^2). \end{aligned}$$

A common choice for an orthonormal basis of $L^2({\mathbb {S}}^2)$ is the set $\{ \, Y_n^k: k \in {\mathbb {Z}}, \; |k |\le n, \; n \in {\mathbb {N}}_0 \,\}$, where $Y_n^k :{\mathbb {S}}^2 \rightarrow {\mathbb {C}}$ denotes the spherical harmonic of degree n and order k, defined as

$$\begin{aligned} Y_{n}^k(\theta , \varphi ) = (-1)^k \, \sqrt{\frac{2 n + 1}{4 \pi }\frac{(n-k)!}{(n +k)!}}\, P_n^k(\cos \theta )\, \textrm{e}^{\textrm{i} k \varphi }. \end{aligned}$$

Here we have used the associated Legendre polynomials $P_n^k :[-1, 1]\rightarrow {\mathbb {R}}$, which can be defined by

$$\begin{aligned} P_n^k (x) = (1-x^2)^{k/2}\, \frac{\textrm{d}^{ k}}{\textrm{d}x^{k}} P_n(x), \quad k \ge 0, \end{aligned}$$

(2)

and

$$\begin{aligned} P_n^{-k}(x) = (-1)^k \, \frac{(n-k)!}{(n+k)!} \, P_n^k(x), \quad k \ge 0, \end{aligned}$$

for positive and negative order, respectively. The Legendre polynomial $P_n :[-1,1] \rightarrow {\mathbb {R}}$ of degree $n \in {\mathbb {N}}_0$ in (2) is given by the Rodrigues formula

$$\begin{aligned} P_n (x) = \frac{1}{2^n n !} \frac{\textrm{d}^{n}}{\textrm{d}x^{n}} \Big ( (x^2-1)^n \Big ). \end{aligned}$$

We note that the associated Legendre polynomials can also be written in terms of Jacobi polynomials as

$$\begin{aligned} P_n^k (\cos \theta ) = \frac{(n+k)!}{n!} \left( \frac{\sin \theta }{2} \right) ^{\!\!k} P_{n-k}^{(k, k)}(\cos \theta ) \quad \text {for} \; k \ge 0, \end{aligned}$$

(3)

where

$$\begin{aligned} P_{n}^{(\alpha , \beta )}(x) = \frac{(-1)^n}{n! 2^n} \frac{1}{(1-x)^\alpha \, (1+x)^{\beta }} \frac{\textrm{d}^n}{\textrm{d}x^n} \left( (1-x)^{\alpha +n} \, (1+x)^{\beta +n} \right) \end{aligned}$$

for $\alpha , \beta >-1$. From the above definitions, we can directly deduce the symmetry $Y_n^{-k} = (-1)^k \, \overline{Y_n^k}$. Since the set of spherical harmonics forms a complete orthonormal system of $L^2({\mathbb {S}}^2)$, these function play a fundamental role in signal analysis on the unit sphere. In particular, every signal $f \in L^2({\mathbb {S}}^2)$ can be expanded into a Fourier series

$$\begin{aligned} f = \sum _{n=0}^\infty \sum _{k=-n}^n f_{n,k} \, Y_n^k, \end{aligned}$$

where $f_{n, k} = \langle f, Y_n^k\rangle$ denotes the Fourier coefficient of f with respect to $Y_n^k$. 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

$$\begin{aligned} \sum _{k = -n}^n Y_n^k({\textbf{x}}) \, \overline{Y_n^k({\textbf{y}})} = \frac{2 n +1}{4 \pi } \, P_n(\langle {\textbf{x}}, {\textbf{y}}\rangle _2) \quad \quad \text {for all} \; {\textbf{x}}, {\textbf{y}} \in {\mathbb {S}}^2, \end{aligned}$$

(4)

which relates the spherical harmonics to the one-dimensional Legendre polynomials.

Functions in $L^2({\mathbb {S}}^2)$ can be arbitrarily rotated by successively rotating around the $x_1, x_2$ and $x_3$ axes. Here, we use the $x_3x_2x_3$ convention with Euler angles $\alpha , \gamma \in \left[ 0, 2 \pi \right)$ and $\beta \in \left[ 0, \pi \right]$. Let ${\textbf{R}}_{x_2}$ and ${\textbf{R}}_{x_3}$ be the rotation matrices which rotate vectors around the $x_2$ and $x_3$ axis, respectively. More precisely,

$$\begin{aligned} {\textbf{R}}_{x_2}(\beta ) = \begin{pmatrix} \cos \beta &{} \quad 0 &{} \quad \sin \beta \\ 0 &{} \quad 1 &{} \quad 0 \\ - \sin \beta &{} \quad 0 &{} \quad \cos \beta \end{pmatrix}\quad \text {and} \quad {\textbf{R}}_{x_3}(\alpha ) = \begin{pmatrix} \cos \alpha &{}\quad - \sin \alpha &{}\quad 0 \\ \sin \alpha &{} \quad \cos \alpha &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 \end{pmatrix}. \end{aligned}$$

We define the rotation operator $\mathcal {{\mathcal {D}}}(\alpha , \beta , \gamma ):L^2({\mathbb {S}}^2) \rightarrow L^2({\mathbb {S}}^2)$ by

$$\begin{aligned} (\mathcal {{\mathcal {D}}}(\alpha , \beta , \gamma ) f)({\textbf{x}})=f([{\textbf{R}}(\alpha , \beta , \gamma )]^{-1} {\textbf{x}}) \quad \quad \text {for all}\; {\textbf{x}} \in {\mathbb {S}}^2 \;\text {and all}\; f \in L^2({\mathbb {S}}^2), \end{aligned}$$

where ${\textbf{R}}(\alpha , \beta , \gamma ) = {\textbf{R}}_{x_3}(\alpha )\, {\textbf{R}}_{x_2}(\beta ) \, {\textbf{R}}_{x_3}(\gamma ) \in {\mathbb {R}}^{3 \times 3}$ is the composition of the three Euler rotations. An important tool for working with rotations on the unit sphere are the Wigner D-functions $D_{k, k'}^n:[0, 2\pi ) \times [0, \pi ] \times [0, 2 \pi ) \rightarrow {\mathbb {C}}$, defined by

$$\begin{aligned} D_{k, k'}^n (\alpha , \beta , \gamma ) = \textrm{e}^{-\textrm{i} k \alpha } \, d_{k, k'}^n(\beta )\, \textrm{e}^{-\textrm{i} k' \gamma }, \quad \quad \; |k |, |k' |\le n,\; n \in {\mathbb {N}}_0, \end{aligned}$$

for integers k and $k'$. The Wigner d-functions $d_{k, k'}^n :[0, \pi ] \rightarrow {\mathbb {R}}$ are given by

$$\begin{aligned} d_{k, k'}^n(\beta ) = \sum _j&(-1)^{j - k' + k} \frac{\sqrt{(n+k')! (n-k')! (n+k)! (n-k)!}}{(n+k'-j)! j! (n-j-k)! (j-k'+k)! } \nonumber \\&\times \left( \cos \frac{\beta }{2}\right) ^{\!\!2n-2j+k'-k} \!\left( \sin \frac{\beta }{2}\right) ^{\!\!2j-k'+k} , \end{aligned}$$

where the sum is performed over all values of j with $\max (0, k'-k) \le j \le \min (n-k, n+k')$. As a special case, which will be relevant for us later, we mention that

$$\begin{aligned} d_{k, n}^n (\beta ) = \sqrt{\left( {\begin{array}{c}2n\\ n+k\end{array}}\right) } \left( \cos \frac{\beta }{2}\right) ^{\!\!n+k} \!\left( \sin \frac{\beta }{2}\right) ^{\!\!n-k}. \end{aligned}$$

(5)

Furthermore, the symmetries

$$\begin{aligned} d_{k, k'}^n (\beta ) = (-1)^{k-k'} \, d_{k', k}^n (\beta ) \end{aligned}$$

(6)

and

$$\begin{aligned} d_{k, k'}^n (\pi -\beta ) = (-1)^{n-k} \, d_{k', -k}^n (\beta ) \end{aligned}$$

(7)

hold for all $\beta \in [0, \pi ]$. The connection between the Wigner D-functions and the rotation operator is given by

$$\begin{aligned} \langle \mathcal {{\mathcal {D}}}(\alpha , \beta , \gamma ) f, Y_n^k \rangle = \sum _{k' = -n}^n f_{n,k'} \, D_{k, k'}^n (\alpha , \beta , \gamma ) \quad \text {for all} \; f \in L^2({\mathbb {S}}^2). \end{aligned}$$

Consequently, it follows that

$$\begin{aligned} \mathcal {{\mathcal {D}}}(\alpha , \beta , \gamma ) f = \sum _{n=0}^\infty \sum _{k=-n}^n Y_n^k \, \sum _{k'=-n}^n f_{n, k'} \, D_{k, k'}^n (\alpha , \beta , \gamma )\quad \text {for all} \; f \in L^2({\mathbb {S}}^2). \end{aligned}$$

(8)

Finally, we note that the spherical harmonics can be written in terms of the Wigner D-functions as

$$\begin{aligned} D_{k, 0}^n (\varphi , \theta , 0) = \sqrt{\frac{4 \pi }{2n+1}} \, \overline{Y_n^k(\theta , \varphi )}, \quad |k |\le n, \; n \in {\mathbb {N}}_0. \end{aligned}$$

(9)

2.2 Directional analysis coefficients

For an analysis function $\Psi \in L^2({\mathbb {S}}^2)$ which is localized at the north pole and a signal $f \in L^2({\mathbb {S}}^2)$, we consider the directional analysis coefficients

$$\begin{aligned} \left\langle f, {\mathcal {D}}(\alpha , \beta , \gamma )\Psi \right\rangle , \quad \alpha , \gamma \in [0, 2 \pi ), \; \beta \in [0, \pi ]. \end{aligned}$$

(10)

In this case, the Euler angles $\alpha , \beta$ and $\gamma$ have a concrete meaning, as illustrated in Fig. . While $\gamma$ defines a specific orientation, $\beta$ and $\alpha$ determine the position of the analysis function. Indeed, ${\mathcal {D}}(\alpha , \beta , \gamma )\Psi$ is localized at $(\beta , \alpha ) \in {\mathbb {S}}^2$. We remark that by (8), the above inner product can be written in terms of Wigner D-functions as

Fig. 2

Interpretation of the three Euler rotations in the context of computing the directional analysis coefficients of a signal f. First, the analysis function $\Psi$ gets rotated around the $x_3$ axis by an angle $\gamma$. The resulting function ${\mathcal {D}}(0, 0, \gamma )\Psi$ has a characteristic directional orientation. Subsequently, ${\mathcal {D}}(0, 0, \gamma )\Psi$ gets relocated to $(\beta , \alpha ) \in {\mathbb {S}}^2$ through consecutive rotations around the $x_2$ and $x_3$ axis with angles $\beta$ and $\alpha$, respectively

$$\begin{aligned} \left\langle f, {\mathcal {D}}(\alpha , \beta , \gamma )\Psi \right\rangle = \sum _{n=0}^\infty \sum _{k=-n}^n \sum _{k'=-n}^n f_{n, k} \, \overline{\Psi _{n, k'}} \, \overline{ D_{k, k'}^n (\alpha , \beta , \gamma )}. \end{aligned}$$

(11)

The quality of the analysis coefficients depends strongly on the localization of $\Psi$ 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

$$\begin{aligned} \gamma \mapsto \langle \Psi , {\mathcal {D}}(0, 0, \gamma )\Psi \rangle , \quad \gamma \in [0, 2 \pi ), \end{aligned}$$

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 ${\textbf{z}}\in {\mathbb {S}}^2$ such that f is invariant under rotations around the ${\textbf{z}}$ axis. Hence, all higher order discontinuities of f lie on circles on the sphere. A simple example would be the indicator function ${\textbf{1}}_{C({\textbf{z}}, \phi )}$ of a spherical cap with center ${\textbf{z}}$, as shown in Fig. 2. However, since the surface integral is rotationally invariant, we may assume, without loss of generality, that

$$\begin{aligned} f = \sum _{n=0}^\infty f_{n, 0}\, Y_n^0, \end{aligned}$$

i.e., f is is invariant under rotations around the $x_3$ axis. In this case, it follows that

$$\begin{aligned} \left\langle f, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi \right\rangle = \left\langle f, {\mathcal {D}}(0, \beta , \gamma ) \Psi \right\rangle \quad \text {for all} \; \alpha , \gamma \in [0, 2\pi ), \; \beta \in [0, \pi ] \end{aligned}$$

and therefore we only have to take into account two of the three Euler angles.

2.3 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 band-limit. Very similar to the construction in McEwen et al (2018, 2013), we define the directional wavelets $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ in harmonic space by

$$\begin{aligned} \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} = \sum _{n=0}^\infty \sum _{k=-n}^n \sqrt{\frac{2n+1}{8\pi ^2}} \, \kappa \!\left( \frac{n}{N} \right) \zeta _{n, k}^{\scriptscriptstyle K} \, Y_n^k, \end{aligned}$$

(12)

where the localization function $\kappa$ satisfies $\kappa \in C^{q+1}([0, \infty ))$ for some $q \in {\mathbb {N}}_0$ and $\emptyset \ne \text {supp}(\kappa )\subset [t_1, t_2]$ with $0< t_1 <t_2$ and $t_2 \in {\mathbb {N}}$. To impose uniqueness on $t_1$ and $t_2$, we demand that the interval $[t_1, t_2]$ is minimal. The parameter $N \in {\mathbb {N}}$ 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

$$\begin{aligned} \zeta _{n, k}^{\scriptscriptstyle K} = {\left\{ \begin{array}{ll} \displaystyle \eta \, \nu \, \sqrt{\frac{1}{2^p} \left( {\begin{array}{c}p\\ \frac{p-k}{2}\end{array}}\right) }, \quad &{} \text {if}\; |k |\le K-1, \\ 0, &{} \text {if}\;|k |\ge K, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \eta = {\left\{ \begin{array}{ll} 1, \quad &{} \text {if}\; K -1 \; \text {even},\\ \textrm{i}, &{} \text {if}\; K-1 \; \text {odd}, \end{array}\right. } \quad \quad \quad \nu = {\left\{ \begin{array}{ll} 1, \quad &{} \text {if}\; K+k \; \text {odd},\\ 0, &{} \text {if}\; K+k \; \text {even} \end{array}\right. } \quad \end{aligned}$$

and

$$\begin{aligned} p ={\left\{ \begin{array}{ll} \min (K-1, n-1), \quad &{} \text {if}\; K+n \; \text {even},\\ \min (K-1, n), &{} \text {if}\; K+n \; \text {odd}. \end{array}\right. } \end{aligned}$$

Here, $K \in {\mathbb {N}}$ is the parameter controlling the directionality. In particular, it holds that $(\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N})_{n, k} = 0$ if $|k |\ge K$. Hence, the directionality component is imposing an azimuthal band-limit, which yields steerable wavelets (Wiaux et al 2008). Obviously, $\zeta _{n, k}^{\scriptscriptstyle K}$ does not depend on n for $n \ge K$. In this case, we also write $\zeta _{k}^{\scriptscriptstyle K}$ instead of $\zeta _{n, k}^{\scriptscriptstyle K}$. For $K=1$, we obtain

$$\begin{aligned} \Psi _{\text {W}, \scriptscriptstyle 1}^{\scriptscriptstyle N}(\theta , \varphi ) = \frac{1}{\sqrt{2\pi }}\sum _{n=0}^\infty \frac{2n+1}{4 \pi } \, \kappa \!\left( \frac{n}{N} \right) P_n(\cos \theta ) \end{aligned}$$

and thus axisymmetric wavelets. Larger values of K result in a stronger directionality, as measured by the auto-correlation, since

$$\begin{aligned} \sum _{k=-\min (K-1, n)}^{\min (K-1, n)} \left|\zeta _{n, k}^{\scriptscriptstyle K} \right|^2 \, \textrm{e}^{\textrm{i} k \gamma } = \cos ^p \gamma \end{aligned}$$

and therefore

$$\begin{aligned} \langle \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}, {\mathcal {D}}(0, 0, \gamma )\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle = \sum _{n=0}^\infty \frac{2n+1}{8 \pi ^2} \, \kappa ^2\!\left( \frac{n}{N} \right) \cos ^p \gamma . \end{aligned}$$

(13)

For later use, we state the following proposition.

Proposition 2.1

Let $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ be defined as in (12) with $q \ge 4$ and $\text {supp}(\kappa ) \subset [1/2, 2]$. Then

$$\begin{aligned} \left\Vert \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \right\Vert _{L^1({\mathbb {S}}^2)} = \int _{0}^{2 \pi } \int _0^\pi |\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} (\theta , \varphi ) |\, \sin \theta \, \textrm{d}\theta \, \textrm{d}\varphi \le c, \end{aligned}$$

where $c>0$ is a constant that depends only on $\kappa$ and K.

Proof

For $n \ge N/2 \ge K$, the directionality component $\zeta _{ k}^{\scriptscriptstyle K}=\zeta _{n, k}^{\scriptscriptstyle K}$ does not depend on n. By changing the order of summation in (11), straightforward calculations yield

$$\begin{aligned} \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}(\theta , \varphi ) = \sum _{k=0}^{K-1} w_{k}(\gamma ) \sum _{n=0}^{2 N} \kappa \!\left( \frac{n}{N} \right) (2n+1) \, \sqrt{\frac{(n-k)!}{(n+k)!}} \, P_n^k (\cos \theta ), \end{aligned}$$

where $w_k$, $k=0, 1,..., K-1$, are continuous functions. Now let $k\in \{ \, 0, 1,..., K-1\,\}$ be fixed. It is easy to see that there are constants $c_0$ and $c_1$ such that

$$\begin{aligned} \sqrt{\frac{(n-k)!}{(n+k)!}} = n^{-k} \left( c_0 + c_1 n^{-1} + {\mathcal {O}}(n^{-2}) \right) \quad \text {as} \; n \rightarrow \infty . \end{aligned}$$

Furthermore, by using the addition theorem (4) for ${\textbf{x}} = {\textbf{y}}$, it is straightforward to show that $|P_n^k(\cos \theta ) |\le c_2 n^k$, where $c_2>0$ depends only on k. Consequently, we obtain

$$\begin{aligned} \sum _{n=0}^{2 N} \kappa \! \left( \frac{n}{N} \right)&(2n+1) \, \sqrt{\frac{(n-k)!}{(n+k)!}} \, P_n^k (\cos \theta )\\&= \sum _{j=0}^1 c_j N^{-k-j} \sum _{n=0}^{2 N} \eta _{k, j}\!\left( \frac{n}{N} \right) (2n+1) \, P_n^k (\cos \theta ) + {\mathcal {O}}(1), \end{aligned}$$

in which $\eta _{k, j} (t) = \kappa (t) t^{-k-j}$. It now follows from a simple substitution and from (2) that

$$\begin{aligned} \int _0^\pi \bigg |\sum _{n=0}^{2 N} \eta _{k, j}\!\left( \frac{n}{N} \right)&(2n+1) \, P_n^k (\cos \theta ) \bigg |\sin \theta \, \textrm{d}\theta \\&= \int _{-1}^1 \bigg |\sum _{n=0}^{2 N}\eta _{k, j}\!\left( \frac{n}{N} \right) (2n+1) \, P_n^k (x) \bigg |\, \textrm{d}x\\&= \int _{-1}^1 \bigg |(1-x^2)^{k/2} \sum _{n=0}^{2 N} \eta _{k, j}\!\left( \frac{n}{N} \right) (2n+1) \, P_n^{(k)} (x) \bigg |\, \textrm{d}x. \end{aligned}$$

Furthermore, the classical Bernstein inequality for algebraic polynomials (see Daugavet and Rafal’son 1972) yields

$$\begin{aligned} \int _{-1}^1 \bigg |(1-x^2)^{k/2}&\sum _{n=0}^{2 N} \eta _{k, j}\!\left( \frac{n}{N} \right) (2n+1) \, P_n^{(k)} (x) \bigg |\, \textrm{d}x \\&\le c_3 N^k \int _{-1}^1 \bigg |\sum _{n=0}^{2 N}\eta _{k, j}\!\left( \frac{n}{N} \right) (2n+1)\, P_n (x) \bigg |\, \textrm{d}x, \end{aligned}$$

where $c_3 >0$ only depends on k. Since $q\ge 4$, it holds that $\eta _{k, j} \in C^{5}([0, \infty ))$ and thus, according to (Petrushev and Xu (2005), Proposition 1), the latter integral is bounded independent of N. This completes the proof. $\square$

2.4 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 ${\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}$ in harmonic space by

$$\begin{aligned} {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N} = \frac{1}{\sqrt{2}} \sum _{n=0}^\infty \sqrt{\frac{2n+1}{8 \pi ^2}} \, \kappa \!\left( \frac{n}{N} \right) \left( Y_n^{-n} + (-1)^n Y_n^n \right) , \end{aligned}$$

(14)

where $\kappa$ and N have the same meaning as in (12). That is, $\kappa \in C^{q+1}([0, \infty ))$ for some $q \in {\mathbb {N}}_0$ and $\emptyset \ne \text {supp}(\kappa )\subset [t_1, t_2]$ with $0< t_1 <t_2$ and $t_2 \in {\mathbb {N}}$. Again, we assume that $[t_1, t_2]$ is the smallest possible interval with these properties. We note that ${\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}$ possesses the following useful representation.

Proposition 2.2

The curvelet ${\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}$ defined in (14) can be written as

$$\begin{aligned} {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}(\theta , \varphi ) = \frac{1}{2 \pi ^{3/2}} \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) \frac{2n+1}{2} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) }\, \sin ^n \theta \, \cos n \varphi . \end{aligned}$$

Proof

By using the symmetry $Y_n^{-n} = (-1)^n \overline{Y_n^n}$ of the spherical harmonics as well as their connection to the Wigner d-functions in (9) and the symmetry (6), we obtain

$$\begin{aligned} {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}(\theta , \varphi ) = \sqrt{2} \sum _{n=0}^\infty \sqrt{\frac{2n+1}{8\pi ^2}} \,\kappa \! \left( \frac{n}{N} \right) \sqrt{\frac{2n+1}{4\pi }} \,\, d_{0,n}^n (\theta ) \, \cos n \varphi . \end{aligned}$$

In addition, it follows directly from (5) that

$$\begin{aligned} d_{0,n}^n (\theta ) = \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) } \left( \frac{\sin \theta }{2} \right) ^{\!\!n}. \end{aligned}$$

$\square$

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 ${\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}$ is localized at $(\pi /2, 0) \in {\mathbb {S}}^2$. The latter will be discussed in greater detail in Proposition 2.4. In the following, however, we will mostly refer to the curvelet $\Psi _{\text {C}}^{\scriptscriptstyle N} = {\mathcal {D}}(0, \pi /2, \pi ){\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}$, 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

$$\begin{aligned} \Psi _{\text {C}}^{\scriptscriptstyle N} = \frac{1}{\sqrt{2}} \sum _{n=0}^\infty \sum _{k=-n}^n \sqrt{\frac{2n+1}{8 \pi ^2}} \, \kappa \!\left( \frac{n}{N} \right) d_{k, n}^n(\pi /2) \left( 1 +(-1)^k \right) Y_n^k, \end{aligned}$$

(15)

where

$$\begin{aligned} d_{k, n}^n(\pi /2) = \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n+k\end{array}}\right) }. \end{aligned}$$

The following proposition provides an explicit formula for the directional auto-correlation 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 $\Psi _{\text {C}}^{\scriptscriptstyle N}$ be the curvelet defined in (15). It holds that

$$\begin{aligned} \langle \Psi _{\text {C}}^{\scriptscriptstyle N}, {\mathcal {D}}(0, 0, \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N} \rangle = \sum _{n=0}^\infty \frac{2n+1}{8 \pi ^2} \, \kappa ^2\!\left( \frac{n}{N} \right) \left( \cos ^{2n} \frac{\gamma }{2} + \sin ^{2n} \frac{\gamma }{2} \right) \end{aligned}$$

for all $\gamma \in [0, 2\pi )$.

Proof

It is easy to see that

$$\begin{aligned} \langle \Psi _{\text {C}}^{\scriptscriptstyle N}&, {\mathcal {D}}(0, 0, \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N} \rangle \\&\quad = \sum _{n=0}^\infty \sum _{k=-n}^n \frac{2n+1}{8\pi ^2} \, \kappa ^2\! \left( \frac{n}{N}\right) \left( \textrm{e}^{\textrm{i}\pi k} +1 \right) \left( \textrm{d}_{k,n}^n (\pi /2)\right) ^2 \,\textrm{e}^{\textrm{i}k \gamma }. \end{aligned}$$

Furthermore, by applying the addition theorem for Wigner d-functions (see Varshalovich et al.1989, p. 87), we obtain

$$\begin{aligned} \sum _{k=-n}^n \textrm{d}_{k,n}^n (\pi /2) \, \textrm{d}_{k,n}^n (\pi /2) \, \textrm{e}^{\textrm{ik}(\gamma + \pi )}&= (-1)^n \sum _{k=-n}^n \textrm{d}_{n,k}^n (\pi /2) \, \textrm{d}_{k,n}^n (\pi /2) \, \textrm{e}^{\textrm{i}k\gamma }\\&= \sin ^{2n}\frac{\gamma }{2}. \end{aligned}$$

Consequently, we also have

$$\begin{aligned} \sum _{k=-n}^n \textrm{d}_{k,n}^n (\pi /2) \, \textrm{d}_{k,n}^n (\pi /2) \, \textrm{e}^{\textrm{i}k\gamma } = \cos ^{2n}\frac{\gamma }{2}. \end{aligned}$$

$\square$

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 ${\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}$ be the curvelet defined in (14), where we additionally assume that there is a value $z \in (t_1, t_2]$ such that $\kappa ^{(q+1)}(t) \ne 0$ for all $t \in (t_1, z)$. Then there exists a constant $c_q >0$, which depends only on $\kappa$ and q, such that

$$\begin{aligned} |{\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}(\theta , \varphi ) |\le \frac{c_q \, (\sin \theta )^{t_1 N} \, N^{7/4}}{(1 + N |\varphi |)^{q+1}}, \quad \theta \in [0, \pi ], \; \varphi \in [-\pi , \pi ]. \end{aligned}$$

Proof

According to Proposition 2.2,

$$\begin{aligned} {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}(\theta , \varphi ) = \frac{1}{2 \pi ^{3/2}} \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) \frac{2n+1}{2} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) }\, \sin ^n \theta \, \cos n \varphi . \end{aligned}$$

Since $\text {supp}(\kappa ) \subset [t_1, t_2]$, all non-zero terms of this series fulfill $t_1 N \le n \le t_2 N-1$. Furthermore, it can be derived from Stirling’s formula that

$$\begin{aligned} \frac{1}{2 \pi ^{3/2}} \frac{2n+1}{2} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) } = \sum _{\ell = -1}^{q+1} d_\ell n^{-\ell -1/4} + {\mathcal {O}}(n^{-q-9/4}) \end{aligned}$$

for some constants $d_{-1}, d_0,..., d_{q+1}$ and consequently

$$\begin{aligned} {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N} (\theta , \varphi )&= \sum _{\ell = -1}^{q+1} d_\ell \sum _{n=0}^{t_2N-1} \kappa \!\left( \frac{n}{N}\right) n^{-\ell -1/4} \, \sin ^{n} \theta \, \cos n \varphi \nonumber \\&\quad + (\sin \theta )^{t_1 N}\, {\mathcal {O}}( N^{-q-5/4} ). \end{aligned}$$

(16)

We now want to derive an upper bound for the inner sum. For

$$\begin{aligned} h_{\scriptscriptstyle N,\ell , \theta }(t)=\kappa \!\left( \frac{t_2}{2\pi } \cdot t\right) t^{-\ell - 1/4} \,(\sin \theta )^{\frac{t_2 N }{2\pi } t}, \end{aligned}$$

we get

$$\begin{aligned} \sum _{n=0}^{t_2N-1} \kappa \!\left( \frac{n}{N}\right) n^{-\ell -1/4} \, \sin ^n \theta \, \cos n \varphi = \left( \frac{2 \pi }{t_2 N}\right) ^{\! \! \ell + 1/4} \, \sum _{n=0}^{t_2 N-1} h_{\scriptscriptstyle N,\ell , \theta }\!\left( \frac{2\pi n}{t_2 N} \right) \cos n \varphi .\nonumber \\ \end{aligned}$$

(17)

Now let $\theta \in (0, \pi )$ and

$$\begin{aligned} s = (\sin \theta )^\frac{t_2 N }{2\pi }. \end{aligned}$$

From Leibniz’s rule, it follows that

$$\begin{aligned} h_{\scriptscriptstyle N,\ell , \theta }^{(p)}(t)&= \sum _{j=0}^p \left( {\begin{array}{c}p\\ j\end{array}}\right) \left( \frac{t_2}{2\pi }\right) ^{\!\!p-j} \kappa ^{(p-j)}\!\left( \frac{t_2}{2\pi } \cdot t\right) \; \frac{\textrm{d}^{j}}{\textrm{d}t^{j}} \! \left( t^{-\ell - 1/4} s^t \right) \\&= \sum _{j=0}^p\sum _{k=0}^j \left( {\begin{array}{c}p\\ j\end{array}}\right) \left( {\begin{array}{c}j\\ k\end{array}}\right) \left( \frac{t_2}{2\pi }\right) ^{\!\!p-j} \; \kappa ^{(p-j)}\!\left( \frac{t_2}{2\pi } \cdot t\right) (\ln s)^k s^t \, \frac{\textrm{d}^{j-k}}{\textrm{d}t^{j-k}} \! \left( t^{-\ell - 1/4} \right) \end{aligned}$$

for $0 \le p \le q+1$. Additionally, it holds for all $t \in [2\pi t_1/t_2, 2 \pi ]$ that

$$\begin{aligned} \left|\left( {\begin{array}{c}p\\ j\end{array}}\right) \left( {\begin{array}{c}j\\ k\end{array}}\right) \left( \frac{t_2}{2\pi }\right) ^{\!\!p-j} \frac{\textrm{d}^{j-k}}{\textrm{d}t^{j-k}} \! \left( t^{-\ell - 1/4} \right) \right|\le c_{j, k}, \end{aligned}$$

where $c_{j, k}\ge 0$ depends only on j, k, p and $\ell$. Consequently,

$$\begin{aligned} \int _{\frac{2 \pi t_1}{t_2}}^{2 \pi } |h_{\scriptscriptstyle N,\ell , \theta }^{(p)}(t) |\, \textrm{d}t \le \sum _{j=0}^p \sum _{k=0}^j c_{j, k} \,|\ln s |^k \int _{\frac{2 \pi t_1}{t_2}}^{2 \pi } s^t \left|\kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \right|\textrm{d}t. \end{aligned}$$

We note that $\kappa ^{(m)}(t_1) =\kappa ^{(m)}(t_2) = 0$ for $0 \le m \le q+1$. Furthermore, by assumption, there exists a $z \in (t_1, t_2]$ such that $\kappa ^{(q+1)}(t) \ne 0$ for all $t \in (t_1, z)$. Hence, by the mean value theorem, it follows that $\kappa ^{(p-j)}(t)\ne 0$ for all $t \in (t_1, z)$. We obtain

$$\begin{aligned} \int _{\frac{2 \pi t_1}{t_2}}^{2 \pi } s^t&\left|\kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \right|\textrm{d}t \\&= \left|\int _{\frac{2 \pi t_1}{t_2}}^{\frac{2\pi z}{t_2}} s^t \, \kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \textrm{d}t \right|+ \int _{\frac{2 \pi z }{t_2} }^{2 \pi } s^t \left|\kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \right|\textrm{d}t \\&\le \left|\int _{\frac{2 \pi t_1}{t_2}}^{\frac{2\pi z}{t_2} } s^t \, \kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \textrm{d}t \right|+ s ^{\frac{2\pi z}{t_2}} \int _{\frac{2 \pi z }{t_2}}^{2 \pi } \, \left|\kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \right|\textrm{d}t. \end{aligned}$$

Furthermore, if $s\ne 1$, repeated integration by parts yields

$$\begin{aligned}&\left|\int _{\frac{2 \pi t_1}{t_2}}^{\frac{2\pi z}{t_2} } s^t \, \kappa ^{(p-j)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \textrm{d}t \right|\\&\quad \le s^{\frac{2\pi z}{t_2}} \sum _{m=0}^{k-1} \frac{\left( \frac{t_2}{2\pi }\right) ^{\! m}|\kappa ^{(p-j+m)}(z) |}{|\ln s|^{m+1}} + \frac{\left( \frac{t_2}{2\pi }\right) ^{\! k}}{|\ln s|^{k}} \left|\int _{\frac{2 \pi t_1}{t_2}}^{\frac{2\pi z}{t_2} } s^t \, \kappa ^{(p-j+k)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \textrm{d}t \right|\\&\quad \le s^{\frac{2\pi z}{t_2}} \sum _{m=0}^{k-1} \frac{\left( \frac{t_2}{2\pi }\right) ^{\! m}|\kappa ^{(p-j+m)}(z) |}{|\ln s|^{m+1}} + \frac{s^{\frac{2\pi t_1}{t_2}}\left( \frac{t_2}{2\pi }\right) ^{\!k}}{|\ln s|^{k}} \int _{\frac{2 \pi t_1}{t_2}}^{\frac{2\pi z}{t_2} } \left|\kappa ^{(p-j+k)}\!\! \left( \frac{t_2 }{2 \pi } \cdot t \right) \right|\textrm{d}t. \end{aligned}$$

Since $s^\delta \ln (s)^m \rightarrow 0$ as $s \rightarrow 0^+$ for every $\delta >0$ and for all $m \in {\mathbb {N}}_0$, there exists a constant $c_p>0$, which depends only on $\kappa , p$ and $\ell$, such that

$$\begin{aligned} \int _{\frac{2 \pi t_1}{t_2}}^{2 \pi } |h_{\scriptscriptstyle N,\ell , \theta }^{(p)}(t) |\, \textrm{d}t \le c_p\, (\sin \theta )^{t_1 N} \quad \text {for all} \; \theta \in [0, \pi ]. \end{aligned}$$

In particular, there exists a $c_q>0$ such that

$$\begin{aligned} \int _{0}^{2 \pi } |h_{\scriptscriptstyle N,\ell , \theta }^{(q+1)} (t) |\, \textrm{d}t =\int _{\frac{2\pi t_1}{t_2}}^{2 \pi } |h_{\scriptscriptstyle N,\ell , \theta }^{(q+1)} (t) |\, \textrm{d}t \le c_q \, (\sin \theta )^{t_1 N}, \end{aligned}$$

(18)

which means that $h_{\scriptscriptstyle N,\ell , \theta }^{(q)}$ has a bounded total variation on $[0, 2\pi ]$. Thus, by (Mhaskar and Prestin (2000b), Lemma 5), it holds for all $N \in {\mathbb {N}}$ with $N \ge 6/t_2$ that

$$\begin{aligned} \bigg |\frac{1}{t_2 N} \sum _{n=0}^{t_2 N-1} h_{\scriptscriptstyle N,\ell , \theta }\!\left( \frac{2 \pi n}{t_2 N}\right) \cos n \varphi&- \frac{1}{2 \pi } \int _{\frac{2 \pi t_1}{t_2}}^{2\pi } h_{\scriptscriptstyle N,\ell , \theta }(t)\, \cos \!\left( \frac{t_2 N \varphi }{2\pi } \cdot t \right) \textrm{d}t \bigg |\\&\le \frac{c_q \, (\sin \theta )^{t_1 N}}{N^{q+1}} \end{aligned}$$

for all $|\varphi |\le \pi$, where $c_q >0$ might differ from the constant in (18). Hence, we get

$$\begin{aligned} \frac{2\pi }{t_2 N} \sum _{n=0}^{t_2 N-1} h_{\scriptscriptstyle N,\ell , \theta }\!\left( \frac{2 \pi n}{t_2 N}\right) \cos n \varphi&= \int _{\frac{2\pi t_1}{t_2}}^{2 \pi } h_{\scriptscriptstyle N,\ell , \theta }(t) \, \cos \!\left( \frac{t_2 N\varphi }{2 \pi } \cdot t \right) \textrm{d}t \\&\quad + (\sin \theta )^{t_1 N} \, {\mathcal {O}}(N^{-q-1}). \end{aligned}$$

Together with (16) and (17), this implies that

$$\begin{aligned} {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N} (\theta , \varphi )&= \sum _{\ell = -1}^{q+1} d_\ell \left( \frac{2 \pi }{t_2 N}\right) ^{\! \! \ell - 3/4} \int _{\frac{2\pi t_1}{t_2}}^{2 \pi } h_{\scriptscriptstyle N,\ell , \theta }(t) \, \cos \!\left( \frac{t_2 N\varphi }{2 \pi } \cdot t \right) \textrm{d}t\\&\quad + (\sin \theta )^{t_1 N} \, {\mathcal {O}}(N^{-q+3/4}). \end{aligned}$$

Finally, repeated integration by parts yields

$$\begin{aligned} \bigg |\int _{\frac{2\pi t_1}{t_2}}^{2 \pi } h_{\scriptscriptstyle N,\ell , \theta }(t) \, \cos \! \left( \frac{t_2 N\varphi }{2\pi } \cdot t \right) \textrm{d}t \bigg |&= \bigg |\Re \bigg \{ \int _{\frac{2\pi t_1}{t_2}}^{2 \pi }h_{\scriptscriptstyle N,\ell , \theta }(t)\, \textrm{e}^{- \textrm{i}\frac{t_2}{2\pi } t} \,\textrm{e}^{\textrm{i}\frac{t_2(1+N\varphi ) }{2\pi } t}\, \textrm{d}t \bigg \}\bigg |\\&\le \bigg |\int _{\frac{2\pi t_1}{t_2}}^{2 \pi } h_{\scriptscriptstyle N,\ell , \theta }(t)\, \textrm{e}^{- \textrm{i}\frac{t_2}{2\pi } t}\, \textrm{e}^{\textrm{i}\frac{t_2(1+N\varphi ) }{2\pi } t}\, \textrm{d}t \bigg |\\&\le \frac{c_q \, (\sin \theta )^{t_1 N}}{(1+ N \varphi )^{q+1}} \end{aligned}$$

for every $\varphi \in [0, \pi ]$. Again, $c_q>0$ is a constant which depends only on $\kappa , q$ and $\ell$. Since ${\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}(\theta , - \varphi ) = {\tilde{\Psi }}_{\text {C}}^{\scriptscriptstyle N}(\theta , \varphi )$, the proof is complete. $\square$

3 Main results

For $r \in {\mathbb {N}}_0$ and $\phi \in (0, \pi )$, we consider the isotropic function $f_{r, \phi } :{\mathbb {S}}^2 \rightarrow {\mathbb {R}}$, where

$$\begin{aligned} f_{r, \phi }(\theta , \varphi ) = {\left\{ \begin{array}{ll} \frac{(\cos \theta - \cos \phi )^r}{(1- \cos \phi )^r}, \quad &{} \theta \in [0, \phi ],\\ 0, &{}\theta \in (\phi , \pi ]. \end{array}\right. } \end{aligned}$$

As illustrated in Fig. , the parameter r controls the smoothness of $f_{r, \phi }$. More precisely, it follows from Leibniz’s rule that

Fig. 3

The graph $\left\{ \, \left( \theta ,f_{r, \phi }(\theta , \varphi )\right) : \theta \in [0, \pi ] \, \right\}$ for different values of r

$$\begin{aligned} \frac{\textrm{d}^r}{\textrm{d}\theta ^r}&(\cos \theta -\cos \phi )^r \bigg \vert _{\theta =\phi } \\&= \sum _{j=0}^r \left( {\begin{array}{c}r\\ j\end{array}}\right) \frac{\textrm{d}^j}{\textrm{d}\theta ^j}(\cos \theta -\cos \phi )^{r-1} \bigg \vert _{\theta = \phi } \, \frac{\textrm{d}^{r-j}}{\textrm{d}\theta ^{r-j}}(\cos \theta -\cos \phi ) \bigg \vert _{\theta = \phi } \\&= -r \sin \phi \, \frac{\textrm{d}^{r-1}}{\textrm{d}\theta ^{r-1}}(\cos \theta -\cos \phi )^{r-1} \bigg \vert _{\theta =\phi } \end{aligned}$$

for all $r\ge 1$. By induction, we get

$$\begin{aligned} \frac{\textrm{d}^r}{\textrm{d}\theta ^r}(\cos \theta -\cos \phi )^r \bigg \vert _{\theta =\phi } = (-1)^r r! \, \sin ^r \phi , \quad r \ge 0. \end{aligned}$$

Thus, we conclude that

$$\begin{aligned} \frac{\textrm{d}^r}{\textrm{d}\theta ^r} \frac{(\cos \theta - \cos \phi )^r}{(1- \cos \phi )^r}\bigg \vert _{\theta = \phi } = (-1)^r r!\left( \cot \frac{\phi }{2}\right) ^{ \!\! r} \quad \text {for all} \; r \in {\mathbb {N}}_0 \end{aligned}$$

and

$$\begin{aligned} \frac{\textrm{d}^j}{\textrm{d}\theta ^j} f_{r, \phi }(\theta , \varphi )\bigg \vert _{\theta = \phi } = 0 \quad \text {for} \; j < r. \end{aligned}$$

Hence, $f_{r, \phi }$ is infinitely often differentiable except for a jump discontinuity of order r at the latitude $\theta = \phi$. As discussed in Mhaskar and Prestin (2000b), most signals of practical interest which are invariant under rotations around the $x_3$ axis can be written in the form

$$\begin{aligned} f = g + \sum _{r=0}^R \sum _{j=1}^{J_r} c_{r,j} \, f_{r, \phi _{r, j}}, \end{aligned}$$

(19)

where g is R times continuously differentiable and also axisymmetric. Furthermore, the smoothness of g causes the corresponding analysis coefficients $\langle g, {\mathcal {D}}(\alpha , \beta , \gamma )\Psi ^{\scriptscriptstyle N} \rangle$ 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).

Remark 1

For $g \in C({\mathbb {S}}^2)$ and $\Psi ^{\scriptscriptstyle N} = \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ or $\Psi ^{\scriptscriptstyle N} = \Psi _{\text {C}}^{\scriptscriptstyle N}$, it is easy to see that

$$\begin{aligned} \sup _{\alpha , \beta , \gamma } |\langle g, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi ^{\scriptscriptstyle N} \rangle |\le \Vert g \Vert _{C({\mathbb {S}}^2)} \, \Vert \Psi ^{\scriptscriptstyle N} \Vert _{L^1({\mathbb {S}}^2)}. \end{aligned}$$

In particular, since $\langle P, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi ^{\scriptscriptstyle N} \rangle = 0$ for every $P \in \Pi _{\scriptscriptstyle \lfloor t_1 N \rfloor }({\mathbb {S}}^2)$, it holds that

$$\begin{aligned} \sup _{\alpha , \beta , \gamma } |\langle g, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi ^{\scriptscriptstyle N} \rangle |\le \inf _{P \in \Pi _{\scriptscriptstyle \lfloor t_1 N \rfloor }({\mathbb {S}}^2)} \Vert g - P \Vert _{C({\mathbb {S}}^2)} \, \Vert \Psi ^{\scriptscriptstyle N} \Vert _{L^1({\mathbb {S}}^2)}. \end{aligned}$$

Now let $g \in C^r({\mathbb {S}}^2)$ and $\omega _r$ 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

$$\begin{aligned} \inf _{P \in \Pi _{\lfloor t_1 N \rfloor }} \Vert g - P \Vert _{C({\mathbb {S}}^2)}&\le c_1 N^{-r} \max _{1 \le i< j \le 3} \, \inf _{P \in \Pi _{\scriptscriptstyle \lfloor t_1 N/2 \rfloor }({\mathbb {S}}^2)} \Vert D_{i, j}^r g - P \Vert _{C({\mathbb {S}}^2)}\\&\le c_2 N^{-r} \max _{1 \le i < j \le 3} \, \omega _r \!\left( D_{i, j}^r g , \frac{2}{t_1 N} \right) _{\!\!\infty }, \end{aligned}$$

where

$$\begin{aligned} \omega _r \!\left( D_{i, j}^r g, \frac{2}{t_1 N} \right) _{\! \!\infty } \rightarrow 0 \quad \text {as} \; N \rightarrow \infty . \end{aligned}$$

From Proposition 2.4, it follows that $\Vert \Psi _{\text {C}}^{\scriptscriptstyle N} \Vert _{L^1({\mathbb {S}}^2)} \le c N^{1/4}$ and therefore

$$\begin{aligned} \lim _{N \rightarrow \infty } N^{r- 1/4} \sup _{\alpha , \beta , \gamma } |\langle g, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N} \rangle |= 0. \end{aligned}$$

In the same way,

$$\begin{aligned} \lim _{N \rightarrow \infty } N^{r} \sup _{\alpha , \beta , \gamma } |\langle g, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}\rangle |= 0 \end{aligned}$$

follows from Proposition 2.1.

In the following, we will prove upper and lower estimates for the magnitude of the inner products $\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma )\Psi ^{\scriptscriptstyle N} \rangle$. Since $f_{r, \phi }$ is invariant under rotations around the $x_3$ axis, we can neglect the Euler angle $\alpha$. Furthermore, as illustrated by Fig. , $|\phi - \beta |$ is the geodesic distance between the center of the analysis function and the singularity of $f_{r, \phi }$. Intuitively speaking, since $\Psi ^{\scriptscriptstyle N}$ is localized and has a zero mean, we expect all analysis coefficients with large absolute values to correspond to Euler angles $\beta$ that are close to $\phi$. In addition, it is obvious that the inner products depend strongly on the orientation of ${\mathcal {D}}(\alpha , \beta , \gamma )\Psi ^{\scriptscriptstyle N}$, which is determined by $\gamma$. The extreme cases $\gamma = 0$, where the orientation of the analysis function is opposite to the edge, and $\gamma = \pi /2$, where the orientations match, are visualized in Fig. 4.

Fig. 4

Directional analysis of an axisymmetric signal $f=f_{r, \phi }$ for $r=0$. The Euler angle $\beta$ controls the latitudinal position of the analysis function and, consequently, its distance $|\phi - \beta |$ to the edge. The orientation of ${\mathcal {D}}(0, \beta , \gamma )\Psi ^{\scriptscriptstyle N}$ is determined by $\gamma$ and matches the singularity curve when $\gamma = \pi /2$

Before stating our main results, we note that, by using the substitution $t = \cos \theta$ and applying (Mhaskar and Prestin 2000b, Lemma 4), the harmonic coefficients of $f_{r, \phi }$ can be written as

$$\begin{aligned} (f_{r, \phi })_{n,k}&= \int _0^{2\pi } \int _0^\phi \frac{(\cos \theta - \cos \phi )^r}{(1- \cos \phi )^r} \, Y_n^k(\theta , \varphi ) \sin \theta \, \textrm{d}\theta \, \textrm{d}\varphi \nonumber \\&= \frac{\delta _{k, 0} \, 2\pi }{(1- \cos \phi )^r} \sqrt{\frac{2n+1}{4\pi }} \int _0^\phi (\cos \theta - \cos \phi )^r P_n(\cos \theta ) \sin \theta \, \textrm{d}\theta \nonumber \\&=\frac{\delta _{k, 0} \, 2\pi }{(1- y)^r} \sqrt{\frac{2n+1}{4\pi }} \int _y^1 (t-y)^r P_n(t)\, \textrm{d}t\nonumber \\&= \frac{\delta _{k, 0} \, 2\pi }{(1- y)^r} \sqrt{\frac{2n+1}{4\pi }} \frac{(n-r-1)!r!}{2^{r+1} n!} \, P_{n-r-1}^{(r+1, r+1)} (y)\, (1-y^2)^{r+1}, \end{aligned}$$

(20)

where $y = \cos \phi$, provided that $n \ge r+1$. In particular, $(f_{r, \phi })_{n,k} =0$ for $k \ne 0$.

3.1 Singularity detection with directional wavelets

We will now discuss the analysis of higher order jump discontinuities with directional wavelets. Here, the function $\chi _{\scriptscriptstyle K}:[0, 2\pi ) \rightarrow {\mathbb {R}}$, $K \in {\mathbb {N}}$, defined by

$$\begin{aligned} \chi _{\scriptscriptstyle K}(\gamma )=\sum _{\begin{array}{c} k = 1-K\\ k \; \equiv \; K-1 \; \text {mod}\, 2 \end{array}}^{K-1} (-1)^{(k+s_K-1)/2} \, \, \overline{\zeta _{k}^{\scriptscriptstyle K}}\,\, \textrm{e}^{\textrm{i} k \gamma }, \end{aligned}$$

where $s_K = (1-(-1)^K)/2$ and $\zeta _{k}^{\scriptscriptstyle K}$ is the directionality component $\zeta _{n, k}^{\scriptscriptstyle K}$ defined in Sect. 2.3 for $n\ge K$, plays an important role since it characterizes the directional sensitivity of $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$.

Fig. 5

Directionality function $\chi _{\scriptscriptstyle K}$ for odd (left) and even (right) values of K

Theorem 3.1

Let $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ be the directional wavelet defined in (12). Furthermore, let $s_K = (1-(-1)^K)/2$, $r \in {\mathbb {N}}_0$ and $\delta >0$. Then

$$\begin{aligned} N^r&\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle = \frac{r!}{2 \pi } \left( \frac{2\pi }{t_2}\cot \frac{\phi }{2} \right) ^{\!\!r} \sqrt{\frac{2}{\pi } \frac{\sin \phi }{\sin \beta }} \, \chi _{\scriptscriptstyle K}(\gamma ) \\&\; \times \int _{\frac{2\pi t_1}{t_2}}^{2 \pi } \kappa \!\left( \frac{t_2}{2\pi } \cdot t \right) t^{-r-1} \cos \!\left( \frac{t_2N(\phi - \beta )}{2\pi } \cdot t + \frac{\phi - \beta - (r+s_K)\pi }{2} \right) \textrm{d}t\\&\; +{\mathcal {O}}(N^{-1}) \end{aligned}$$

holds uniformly for all $\beta , \phi \in [\delta , \pi - \delta ]$. In particular, if $\phi \in [\delta , \pi -\delta ]$, there exists an interval $(i_1, i_2)\subset {\mathbb {R}}$ and a constant $c_1>0$, which both depend only on $\kappa$, $s_K$, $\delta$ and r, such that

$$\begin{aligned} N^r |\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle |\ge c_1 \left( \cot \frac{\phi }{2} \right) ^{\!\!r} \,|\chi _{\scriptscriptstyle K}(\gamma ) |\quad \text {if} \; \phi - \frac{i_2}{N} \le \beta \le \phi -\frac{i_1}{N}, \end{aligned}$$

provided that N is large enough. On the other hand, there exists a constant $c_2>0$, which depends only on $\kappa , q, K, \delta$ and r, such that

$$\begin{aligned} N^r |\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle |\le \frac{c_2}{(1 + N |\phi - \beta |)^{q+1}} \end{aligned}$$

for all $\beta , \phi \in [\delta , \pi - \delta ]$.

Proof

As discussed before, the directionality component of $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ becomes independent of n for $n \ge t_1 N \ge K$. By using (11) as well as the fact that $(f_{r, \phi })_{n,k} =0$ for $k \ne 0$, we get

$$\begin{aligned} \langle f_{r, \phi }, {\mathcal {D}}&(\alpha , \beta , \gamma ) \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle \nonumber \\&= \sum _{n=0}^\infty \sum _{k=-n}^n \sum _{k'=-n}^n \sqrt{\frac{2n+1}{8 \pi ^2}}\, \kappa \!\left( \frac{n}{N} \right) \overline{\zeta _{n, k'}^{\scriptscriptstyle K}} \, (f_{r, \phi })_{n, k} \, \overline{D_{k, k'}^n(\alpha , \beta , \gamma )} \nonumber \\&= \sum _{k'=1-K}^{K-1} \overline{\zeta _{k'}^{\scriptscriptstyle K}} \, \textrm{e}^{\textrm{i} k' \gamma }\sum _{n=0}^\infty \sqrt{\frac{2n+1}{8 \pi ^2}} \, \kappa \!\left( \frac{n}{N} \right) (f_{r, \phi })_{n, 0} \, d_{0, k'}^n(\beta ). \end{aligned}$$

(21)

The symmetries (6) and (7) yield

$$\begin{aligned} d_{0, k'}^n(\beta ) = (-1)^{k'} d_{0, -k'}^n(\beta ). \end{aligned}$$

In addition, for $k'\ge 0$, we have

$$\begin{aligned} d_{0, k'}^n(\beta ) = \frac{\sqrt{(n-k')!(n+k')!}}{n!} \left( \frac{\sin \beta }{2} \right) ^{\!\! k'} P_{n-k'}^{(k', k')}(\cos \beta ), \end{aligned}$$

(22)

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

$$\begin{aligned}&P_{n-k'}^{(k', k')}(\cos \beta )\\&\quad \quad = \sqrt{\frac{2}{\pi n \, \sin \beta }} \left( \frac{2}{\sin \beta } \right) ^{\!\! k'} \cos \!\left( \left( n + \frac{1}{2} \right) \beta - \frac{k' \pi }{2} - \frac{\pi }{4} \right) + {\mathcal {O}}(n^{-3/2})\\&\quad \quad = (-1)^{(k'-s_{k'})/2}\,\sqrt{\frac{2}{\pi n \, \sin \beta }} \left( \frac{2}{\sin \beta } \right) ^{\!\! k'} \cos \! \left( \left( n + \frac{1}{2} \right) \beta - \frac{\pi }{4} -\frac{s_{k'}\pi }{2} \right) \\&\quad \quad \quad + {\mathcal {O}}(n^{-3/2}) \end{aligned}$$

holds uniformly for all $\beta \in [\delta , \pi -\delta ]$. Furthermore, it is easy to see that

$$\begin{aligned} \frac{\sqrt{(n-k')!(n+k')!}}{n!} = 1 + {\mathcal {O}}(n^{-1}) \end{aligned}$$

for any fixed $k'$. Consequently, it follows from the above considerations that

$$\begin{aligned} d_{0, k'}^n(\beta )&= (-1)^{(k'-s_{k'})/2}\,\sqrt{\frac{2}{\pi n \, \sin \beta }} \, \cos \! \left( \left( n + \frac{1}{2} \right) \beta - \frac{\pi }{4} -\frac{s_{k'}\pi }{2} \right) \nonumber \\&\quad + {\mathcal {O}}(n^{-3/2}) \end{aligned}$$

(23)

holds uniformly for all $\beta \in [\delta , \pi -\delta ]$ and is valid for all $k'$ with $|k' |\le K-1$. We note that in (21) we only need to sum over indices $k'$ wich have the same parity as $K-1$, since by definition $\zeta _{k'}^{\scriptscriptstyle K} = 0$ if $s_{k'} = s_K$. Thus, we can exchange $s_{k'}$ with $1-s_K$. By (20), we have

$$\begin{aligned} \sqrt{\frac{2n+1}{8 \pi ^2}} (f_{r, \phi })_{n, 0}&= \frac{r! \, (\sin \phi )^{2r+2}}{\sqrt{8 \pi } \, 2^r (1-\cos \phi )^r} \frac{2n+1}{2} \frac{(n-r-1)!}{n!} \nonumber \\&\; \times P_{n-r-1}^{(r+1, r+1)}(\cos \phi ), \end{aligned}$$

(24)

where we can plug in the asymptotic expressions

$$\begin{aligned} \frac{2n+1}{2} \frac{(n-r-1)!}{n!} = n^{-r} \left( 1+ {\mathcal {O}}(n^{-1})\right) , \end{aligned}$$

(25)

which, again, is easy to verify, and

$$\begin{aligned} P_{n-r-1}^{(r+1, r+1)}(\cos \phi )&= \sqrt{\frac{2}{\pi n \, \sin \phi }}\left( \frac{2}{\sin \phi } \right) ^{\!\! r+1} \cos \!\left( \left( n+\frac{1}{2}\right) \phi - \frac{r\pi }{2} - \frac{3 \pi }{4} \right) \nonumber \\&\; + {\mathcal {O}}(n^{-3/2}), \end{aligned}$$

(26)

which holds uniformly for all $\phi \in [\delta , \pi - \delta ]$. By inserting the right hand sides of (23) and (24) into (21), while also using (25) and (26), we get

$$\begin{aligned} \langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma )&\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle = \pi ^{-1} r! \left( \cot \frac{\phi }{2} \right) ^{\!\! r} \sqrt{\frac{2}{\pi } \frac{\sin \phi }{\sin \beta }} \, \chi _{\scriptscriptstyle K}(\gamma ) \\&\times \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) n^{-r-1} \cos \!\left( \left( n + \frac{1}{2} \right) \beta - \frac{\pi }{4} - \frac{(1-s_K)\pi }{2} \right) \\&\times \cos \! \left( \left( n+\frac{1}{2}\right) \phi - \frac{r\pi }{2} - \frac{3 \pi }{4} \right) \\&+ {\mathcal {O}}(N^{-r-1}) \end{aligned}$$

uniformly for all $\beta , \phi \in [\delta , \pi - \delta ]$. 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

$$\begin{aligned} \frac{1}{2}&\sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) n^{-r-1} \cos \!\left( n(\phi - \beta ) + \frac{\phi - \beta - (r+s_K)\pi }{2}\right) \\&\quad = \frac{1}{2} \left( \frac{2 \pi }{t_2 N}\right) ^{\!\! r+1} \, \sum _{n=0}^{t_2 N-1} h\!\left( \frac{2 \pi n}{t_2 N} \right) \cos \! \left( n(\phi - \beta ) + \frac{\phi - \beta - (r+s_K)\pi }{2}\right) , \end{aligned}$$

where $h(t) = \kappa ( t \cdot t_2/(2\pi ) ) \, t^{-r-1}$. Applying (Mhaskar and Prestin 2000b, Lemma 5), we obtain

$$\begin{aligned}&\frac{2 \pi }{t_2 N} \sum _{n=0}^{t_2 N-1} h\!\left( \frac{2 \pi n}{t_2 N} \right) \cos \! \left( n(\phi - \beta ) + \frac{\phi - \beta - (r+s_K)\pi }{2}\right) \\&\quad = \int _{\frac{2 \pi t_1}{t_2}}^{2\pi } h(t) \cos \! \left( \frac{t_2 N (\phi - \beta )}{2 \pi } \cdot t + \frac{\phi - \beta - (r+s_K)\pi }{2} \right) \textrm{d}t + {\mathcal {O}}(N^{-q-1}). \end{aligned}$$

The same argument followed by repeated integration by parts yields

$$\begin{aligned} \frac{2 \pi }{t_2 N}&\sum _{n=0}^{t_2 N-1} h\!\left( \frac{2 \pi n}{t_2 N} \right) \cos \! \left( n(\phi + \beta ) + \frac{\phi + \beta - (r+3-s_K)\pi }{2}\right) = {\mathcal {O}}(N^{-q-1}). \end{aligned}$$

Hence, the first statement is proven.

The lower bound of Theorem 3.1 now follows directly, since

$$\begin{aligned} z \mapsto \int _{\frac{2\pi t_1}{t_2}}^{2 \pi } \kappa \! \left( \frac{t_2}{2\pi }\cdot t \right) t^{-r-1} \cos (z t + \psi ) \, \textrm{d} t, \quad \psi \in {\mathbb {R}}, \end{aligned}$$

is an entire function and therefore we can find an interval $(i_1, i_2)\subset {\mathbb {R}}$ which is free of zeroes.

Let us now prove the last statement. We start again with formula (21). From (22) and (24), it follows that

$$\begin{aligned}&\sum _{n=0}^\infty \sqrt{\frac{2n+1}{8 \pi ^2}} \, \kappa \!\left( \frac{n}{N} \right) (f_{r, \phi })_{n, 0} \, d_{0, k'}^n(\beta ) \nonumber \\&\quad = \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) \frac{r! \, (\sin \phi )^{2r+2}}{\sqrt{8 \pi } \, 2^r (1-\cos \phi )^r} \frac{2n+1}{2} \frac{(n-r-1)!}{n!} \, P_{n-r-1}^{(r+1, r+1)}(\cos \phi ) \nonumber \\&\quad \quad \quad \quad \times \frac{\sqrt{(n-k')!(n+k')!}}{n!} \left( \frac{\sin \beta }{2} \right) ^{\!\! k'} P_{n-k'}^{(k', k')}(\cos \beta ). \end{aligned}$$

(27)

For fixed $k' \in {\mathbb {Z}}$, we have

$$\begin{aligned} \frac{\sqrt{(n-k')!(n+k')!}}{n!} = \sum _{j=0}^q d_j n^{-j} + {\mathcal {O}}(n^{-q-1}) \end{aligned}$$

for some constants $d_0, d_1,..., d_q$, which is easy to see, as well as

$$\begin{aligned} \frac{2n+1}{2} \frac{(n-r-1)!}{n!} = n^{-r} \left( \sum _{j=0}^q {\tilde{d}}_j n^{-j} + {\mathcal {O}}(n^{-q-1}) \right) \end{aligned}$$

(28)

for some constants ${\tilde{d}}_0, {\tilde{d}}_1,..., {\tilde{d}}_q$. Furthermore, as discussed in Mhaskar and Prestin (2000b), we can verify that there are functions $(\beta , \phi ) \mapsto w_\nu (\beta , \phi )$ and $(\beta , \phi ) \mapsto {\tilde{w}}_\nu (\beta , \phi )$, which are continuous on $[\delta , \pi -\delta ]^2$, such that

$$\begin{aligned}&P_{n-r-1}^{(r+1, r+1)}(\cos \phi ) \, P_{n-k'}^{(k', k')}(\cos \beta ) \\&\quad \quad = \sum _{\nu = 0}^q n^{-\nu -1} \Re \bigg \{{\tilde{w}}_\nu (\beta , \phi )\, \textrm{e}^{\textrm{i}n (\phi + \beta )} + w_\nu (\beta , \phi ) \, \textrm{e}^{\textrm{i}n (\phi - \beta )} \bigg \} + {\mathcal {O}}(n^{-q-2}) \end{aligned}$$

uniformly for $\beta , \phi \in [\delta , \pi - \delta ]$. By plugging the above asymptotic formulas into (27), we get

$$\begin{aligned}&\sum _{n=0}^\infty \sqrt{\frac{2n+1}{8 \pi ^2}}\, \kappa \!\left( \frac{n}{N} \right) (f_r, \phi )_{n, 0}\, d_{0, k'}^n(\beta ) \\&\quad = \sum _{\nu =0}^q \left( \Re \bigg \{ {\tilde{C}}_\nu (\beta , \phi , k') \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) n^{-\nu -r-1} \, \textrm{e}^{\textrm{i}n (\phi + \beta )} \bigg \} \right. \\&\quad \quad \quad \quad \left. +\Re \bigg \{ C_\nu (\beta , \phi , k') \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) n^{-\nu -r-1} \, \textrm{e}^{\textrm{i}n (\phi - \beta )} \bigg \} \right) + {\mathcal {O}}(N^{-q-r-1}) \end{aligned}$$

uniformly for $\beta , \phi \in [\delta , \pi - \delta ]$, where $(\beta , \phi )\mapsto {\tilde{C}}_\nu (\beta , \phi , k')$ and $(\beta , \phi )\mapsto C_\nu (\beta , \phi , k')$ are continuous functions on $[\delta , \pi - \delta ]^2$. With the same methods as in the proof of Proposition 2.4, we finally obtain

$$\begin{aligned} \bigg |\sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) n^{-\nu -r-1}\, \textrm{e}^{\textrm{i}n (\phi \pm \beta )} \bigg |\le \frac{c}{N^{r+\nu } (1 + N |\phi \pm \beta |)^{q+1}}, \end{aligned}$$

where $c>0$ depends on $\kappa , q, r$, and $\nu$. Hence, the proof is complete. $\square$

In combination with Remark 1, Theorem 3.1 states that directional wavelets are suitable for detecting higher order singularities. Furthermore, the function $\chi _{\scriptscriptstyle K}$, which is visualized in Fig. , is contained as a factor in the dominant part of $\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle$. Therefore, the directional sensitivity, with regards to detecting discontinuities, is, almost entirely, characterized by $\chi _{\scriptscriptstyle K}$ and, in particular, independent of N. Consequently, we can view $\chi _{\scriptscriptstyle K}$ 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

• 1. $\zeta _{n, k}^{\scriptscriptstyle K}$ is independent of n,

• 2. $\zeta _{n,k}^{\scriptscriptstyle K} =0$ for $|k |\ge K$,

• 3. $\zeta _{n, k}^{\scriptscriptstyle K} = \zeta _{n,- k}^{\scriptscriptstyle K}$ and, in addition, $\zeta _{n,k}^{\scriptscriptstyle K} =0$ if $K-k$ is even.

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.

3.2 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 $\beta , \gamma \in (0, \pi )$ it holds that

$$\begin{aligned} {\mathcal {D}}(0, \beta , \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N} = \frac{1}{\sqrt{2}}\sum _{n=0}^\infty \sum _{k=-n}^n \sqrt{\frac{2n+1}{8 \pi ^2}}\, \kappa \! \left( \frac{n}{N} \right) \xi _{n,k}(\beta , \gamma ) \, Y_n^k, \end{aligned}$$

where

$$\begin{aligned} \xi _{n,k}(\beta , \gamma )&= \textrm{e}^{-\textrm{i} k {{\,\textrm{arccot}\,}}(\cos \beta \, \cot \gamma )} \left( d_{k, n}^n(\arccos (- \sin \beta \, \cos \gamma )) \, \textrm{e}^{-\textrm{i} n {{\,\textrm{arccot}\,}}\left( \frac{\cot \beta }{\sin \gamma } \right) } \right. \nonumber \\&\left. \quad + \, d_{k, n}^n(\arccos ( \sin \beta \, \cos \gamma )) \, \textrm{e}^{\textrm{i} n {{\,\textrm{arccot}\,}}\left( \frac{\cot \beta }{\sin \gamma }\right) } \right) \end{aligned}$$

(29)

and, in particular,

$$\begin{aligned} \xi _{n,0}(\beta , \gamma ) = \frac{1}{2^{n-1}}\sqrt{ \left( {\begin{array}{c}2n\\ n\end{array}}\right) } \, \left( 1- \cos ^2\gamma \, \sin ^2 \beta \right) ^{\! n/2} \, \cos \!\left( n {{\,\textrm{arccot}\,}}\! \left( \frac{\cot \beta }{\sin \gamma }\right) \right) . \end{aligned}$$

(30)

Proof

By using (8) and (15), we get

$$\begin{aligned} {\mathcal {D}}(0, \beta , \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N} = \frac{1}{\sqrt{2}}\sum _{n=0}^\infty \sum _{k=-n}^n \sqrt{\frac{2n+1}{8 \pi ^2}} \, \kappa \! \left( \frac{n}{N} \right) \xi _{n,k}(\beta , \gamma ) \, Y_n^k, \end{aligned}$$

in which

$$\begin{aligned} \xi _{n,k}(\beta , \gamma )= \sum _{k'=-n}^n d_{k', n}^n(\pi /2) \left( 1 + \textrm{e}^{\textrm{i} k' \pi } \right) d_{k, k'}^n(\beta ) \, \textrm{e}^{-\textrm{i} k' \gamma }. \end{aligned}$$

Now, (29) follows from the addition theorem for Wigner d-functions (Varshalovich et al 1989, p. 87). Finally, simple calculations yield (30). $\square$

Theorem 3.3

Let $\Psi _{\text {C}}^{\scriptscriptstyle N}$ be the curvelet defined in (15). Furthermore, let $r \in {\mathbb {N}}_0$ and $\delta >0$. Then

$$\begin{aligned}&N^{r-1/4}\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \pi /2)\Psi _{\text {C}}^{\scriptscriptstyle N} \rangle = \pi ^{-5/4} \left( \frac{2 \pi }{t_2} \right) ^{\!\! r-1/4} r! \left( \cot \frac{\phi }{2} \right) ^{\!\! r} \sqrt{\frac{\sin \phi }{2}} \\&\quad \quad \quad \times \int _{\frac{2 \pi t_1}{t_2}}^{2\pi } \kappa \!\left( \frac{t_2}{2\pi } \cdot t\right) t^{-r-3/4} \cos \!\left( \frac{t_2 N(\phi - \beta )}{2\pi } \cdot t + \frac{\phi - r \pi }{2} - \frac{3 \pi }{4}\right) \textrm{d}t \\&\quad \quad \quad + {\mathcal {O}}(N^{-1}) \end{aligned}$$

holds uniformly for all $\phi \in [\delta , \pi - \delta ]$ and all $\beta \in (0, \pi )$. In particular, there exists an interval $(i_1, i_2)\subset {\mathbb {R}}$ and a constant $c_1>0$, which both depend only on $\phi$, $\kappa$ and r, such that

$$\begin{aligned} N^{r- 1/4} |\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \pi /2) \Psi _{\text {C}}^{\scriptscriptstyle N} \rangle |\ge c_1 \quad \text {if} \; \phi - \frac{i_2}{N} \le \beta \le \phi -\frac{i_1}{N}, \end{aligned}$$

provided that N is large enough. Now, let us additionally assume that there is a value $z \in (t_1, t_2]$ such that $\kappa ^{(q+1)}(t) \ne 0$ for all $t \in (t_1, z)$. Then there exists a constant $c_2>0$, which depends only on $\kappa , q,\delta$ and r, such that

$$\begin{aligned} N^{r- 1/4} |\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta ,\gamma )\Psi _{\text {C}}^{\scriptscriptstyle N} \rangle |\le \frac{c_2\left( 1- \cos ^2\gamma \, \sin ^2 \beta \right) ^{\!t_1 N /2} }{\left( 1+N \Big |\phi - {{\,\textrm{arccot}\,}}\! \left( \frac{\cot \beta }{\sin \gamma }\right) \!\! \Big |\right) ^{\! q+1}} \end{aligned}$$

for all $\beta , \gamma \in (0, \pi )$ and all $\phi \in [\delta , \pi -\delta ]$.

Proof

By using Lemma 3.2 together with (20), straightforward calculations yield

$$\begin{aligned} \langle&f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \pi /2) \Psi _{\text {C}}^{\scriptscriptstyle N}\rangle =\frac{r! \, (1-y^2)^{r+1}}{\sqrt{\pi } \, 2^{r+1} (1-y)^r} \\&\quad \times \sum _{n=0}^\infty \kappa \! \left( \frac{n}{N} \right) \frac{2n+1}{2} \frac{(n-r-1)!}{n!} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) } \, P_{n-r-1}^{(r+1, r+1)}(y)\, \cos n \beta , \end{aligned}$$

where $y = \cos \phi$. As in the proof of Theorem 3.1, we use the asymptotic formulas (25) and (26) as well as

$$\begin{aligned} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) } = \frac{1}{(n \pi )^{1/4}} \left( 1 + {\mathcal {O}}(n^{-1})\right) . \end{aligned}$$

It follows that

$$\begin{aligned} \langle&f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \pi /2) \Psi _{\text {C}}^{\scriptscriptstyle N}\rangle = \pi ^{-5/4} \sqrt{2 \sin \phi } \,\left( \cot \frac{\phi }{2} \right) ^{\!\! r} r! \\&\quad \times \sum _{n=0}^\infty \kappa \! \left( \frac{n}{N} \right) n^{-r-3/4} \cos \!\left( \left( n + \frac{1}{2}\right) \phi - \frac{r \pi }{2} - \frac{3\pi }{4} \right) \cos n \beta \\&\quad + {\mathcal {O}}(N^{-r-3/4}) \end{aligned}$$

uniformly for all $\phi \in [\delta , \pi - \delta ]$. By the addition formula for the cosine function, we have

$$\begin{aligned}&\sum _{n=0}^\infty \kappa \! \left( \frac{n}{N} \right) n^{-r-3/4} \cos \!\left( \left( n + \frac{1}{2}\right) \phi - \frac{r \pi }{2} - \frac{3\pi }{4} \right) \cos n \beta \\&\quad = \frac{1}{2}\sum _{n=0}^\infty \kappa \! \left( \frac{n}{N} \right) n^{-r-3/4} \left( \cos \left( n(\phi - \beta ) + \psi \right) + \cos \left( n(\phi + \beta ) + \psi \right) \right) , \end{aligned}$$

where $\psi = (\phi - r \pi )/2 - 3\pi /4$. Again, we can apply (Mhaskar and Prestin 2000b, Lemma 5), which yields

$$\begin{aligned}&\sum _{n=0}^\infty \kappa \! \left( \frac{n}{N}\right) n^{-r-3/4} \cos ( n(\phi \pm \beta ) +\psi )\\&\quad = \left( \frac{2\pi }{t_2 N}\right) ^{\!\! r + 3/4} \, \sum _{n=0}^{t_2N -1} h\!\left( \frac{2\pi n}{t_2 N} \right) \cos ( n(\phi \pm \beta ) +\psi )\\&\quad = \left( \frac{2\pi }{t_2 N}\right) ^{\!\! r - 1/4} \left( \int _{\frac{2 \pi t_1}{t_2}}^{2\pi } h(t) \cos \!\left( \frac{t_2 N(\phi \pm \beta )}{2\pi } \cdot t + \psi \right) \textrm{d}t + {\mathcal {O}}(N^{-q-1}) \right) , \end{aligned}$$

in which

$$\begin{aligned} h(t) = \kappa \!\left( \frac{t_2}{2\pi } \cdot t\right) t^{-r-3/4}. \end{aligned}$$

Thus, repeated integration by parts proves the first statement. Furthermore, the lower bound follows from the same arguments as in Theorem 3.1.

Let us now prove the upper bound. By using Lemma 3.2 and (20), simple calculations yield

$$\begin{aligned} \langle&f_{r, \phi }, {\mathcal {D}}(\alpha , \beta , \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N}\rangle =\frac{r!\, (1-y^2)^{r+1}}{\sqrt{\pi } \, 2^{r+1} (1-y)^r} \\&\quad \times \sum _{n=0}^\infty \kappa \!\left( \frac{n}{N} \right) \frac{2n+1}{2} \frac{(n-r-1)!}{n!} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) } \, z^n \, P_{n-r-1}^{(r+1, r+1)}(y) \, \cos n \varphi , \end{aligned}$$

where $y = \cos \phi$ and

$$\begin{aligned} z = \sqrt{1- \cos ^2\gamma \, \sin ^2 \beta }, \quad \varphi = {{\,\textrm{arccot}\,}}\!\left( \frac{\cot \beta }{\sin \gamma }\right) . \end{aligned}$$

Again, we use (28), where we exchange q with $q+1$, as well as

$$\begin{aligned} \frac{1}{2^n} \sqrt{\left( {\begin{array}{c}2n\\ n\end{array}}\right) } = \frac{1}{(n \pi )^{1/4}} \left( \sum _{\ell =0}^{q+1} d_\ell n^{-\ell } + {\mathcal {O}}(n^{-q-2}) \right) , \end{aligned}$$

which can be derived from Stirling’s formula. As discussed in Mhaskar and Prestin (2000b), there exist functions $\phi \mapsto w_\nu (\phi )$, $\nu = 0, 1,..., q+1$, that are continuous on $[ \delta , \pi - \delta ]$, such that

$$\begin{aligned} P_{n-r-1}^{(r+1, r+1)}(y)\, \cos n \varphi&= \Re \bigg \{ \left( \textrm{e}^{\textrm{i} n (\phi - \varphi )} + \textrm{e}^{\textrm{i} n (\phi + \varphi )} \right) \sum _{\nu =0}^{q+1} w_\nu (\phi ) n^{-\nu -1/2} \bigg \}\\&\quad + {\mathcal {O}}(n^{-q-2}) \end{aligned}$$

holds uniformly for all $\phi \in [ \delta , \pi - \delta ]$. Thus, from our above considerations, we conclude that there also exist functions $\phi \mapsto C_{\nu , r}(\phi )$, $\nu = 0, 1,..., q+1$, which are, again, continuous on $[\delta , \pi - \delta ]$, such that

$$\begin{aligned} \langle f_{r, \phi }, {\mathcal {D}}(&\alpha , \beta , \gamma ) \Psi _{\text {C}}^{\scriptscriptstyle N}\rangle \\&= \Re \bigg \{ \sum _{\nu =0}^{q+1} C_{\nu , r}(\phi ) \sum _{n=0}^{\infty } \kappa \!\left( \frac{n}{N} \right) n^{-r-\nu -3/4} \, z^n \left( \textrm{e}^{\textrm{i} n (\phi - \varphi )} + \textrm{e}^{\textrm{i} n (\phi + \varphi )} \right) \bigg \} \\&\quad \;+ z^{t_1 N} \, {\mathcal {O}}(N^{-q-r-5/4}) \end{aligned}$$

uniformly for $\phi \in [ \delta , \pi - \delta ]$. Furthermore, for

$$\begin{aligned} h_{N, z}(t) = \kappa \!\left( \frac{t_2}{2 \pi } \cdot t \right) t^{-r-\nu - 3/4}\, z^{\frac{t_2 N}{2\pi } t}, \quad N \in {\mathbb {N}}, \end{aligned}$$

we get, by using the same arguments as in the proof of Proposition 2.4,

$$\begin{aligned}&\sum _{n=0}^{\infty } \kappa \!\left( \frac{n}{N} \right) n^{-r-\nu -3/4} \, z^n \, \textrm{e}^{\textrm{i} n (\phi \pm \varphi )}\\&\quad \quad = \left( \frac{2 \pi }{t_2 N} \right) ^{\!\! r+\nu +3/4} \, \sum _{n=0}^{t_2 N -1} h_{N, z}\!\left( \frac{2 \pi n}{t_2 N} \right) \textrm{e}^{\textrm{i}n(\phi \pm \varphi )}\\&\quad \quad = \left( \frac{2 \pi }{t_2 N} \right) ^{\!\! r+\nu -1/4} \int _{\frac{2 \pi t_1}{t_2}}^{2\pi } h_{N, z}(t)\, \textrm{e}^{\textrm{i}\frac{t_2 N (\phi \pm \varphi )}{2\pi } t} \, \textrm{d}t + z^{t_1 N}\,{\mathcal {O}}(N^{-q-r-\nu - 3/4}) \end{aligned}$$

and additionally

$$\begin{aligned} \bigg |\int _{\frac{2 \pi t_1}{t_2}}^{2\pi } h_{N, z}(t)\, \textrm{e}^{\textrm{i}\frac{t_2 N (\phi \pm \varphi )}{2\pi } t} \, \textrm{d}t \bigg |\le \frac{c \, z^{t_1 N}}{(1 + N |\phi - \varphi |)^{q+1}}, \end{aligned}$$

where $c>0$ depends on $\kappa , q, r$ and $\nu$. Hence, the upper bound is proven. $\square$

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 $\langle f_{r, \phi }, {\mathcal {D}}(\alpha , \beta ,\gamma )\Psi _{\text {C}}^{\scriptscriptstyle N} \rangle$ decay rapidly for large values of N whenever $\gamma \ne \pi /2$.

4 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 $f = {\textbf{1}}_{C({\textbf{z}}, \phi )}$ with center ${\textbf{z}} = ((5\pi -2)/10, 0)$ and opening angle $\phi = \pi /5$, as visualized on the left side of Fig. . By choosing a fixed angle $\gamma$ in (10), we determine the orientation to which our further analysis will be most sensitive. Here, we set $\gamma = \pi /4$. As discussed in Sect. 2.2, this means that the original analysis function is rotated around the $x_3$ axis by $\pi /4$ 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.

Fig. 6

Left: Indicator function $f = {\textbf{1}}_{C({\textbf{z}}, \phi )}$ 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

The wavelets $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ and curvelets $\Psi _{\text {C}}^{\scriptscriptstyle N}$ defined in (12) and (15) are uniquely determined by the localization function $\kappa$ and by the parameters K and N. Here, we choose $\kappa$ to be the kernel constructed in McEwen et al (2018) with $\text {supp}(\kappa ) = [1/2, 2]$, as shown in Fig. . The resulting pre-rotated analysis functions ${\mathcal {D}}(0, 0, \pi /4)\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ and ${\mathcal {D}}(0, 0, \pi /4) \Psi _{\text {C}}^{\scriptscriptstyle N}$ are visualized in Fig.  for the parameters $K=1, 8, 16$ and $N = 16, 32, 64, 128$. Furthermore, Fig.  shows the corresponding analysis coefficients $W_{f}^{\scriptscriptstyle K, N} :{\mathbb {S}}^2\rightarrow {\mathbb {R}}$ and $C_{f}^{\scriptscriptstyle N} :{\mathbb {S}}^2\rightarrow {\mathbb {R}}$, given by

Fig. 7

Kernel function $t \mapsto \kappa \!\left( \frac{t}{N} \right)$ for $N = 2^j$, $j=0, 1,..., 7$

$$\begin{aligned} W_{f}^{\scriptscriptstyle K, N}(\beta , \alpha ) = \langle f,{\mathcal {D}}(\alpha , \beta , \pi /4)\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N} \rangle \end{aligned}$$

and

$$\begin{aligned} C_{f}^{\scriptscriptstyle N}(\beta , \alpha ) = \langle f,{\mathcal {D}}(\alpha , \beta , \pi /4)\Psi _{\text {C}}^{\scriptscriptstyle N} \rangle . \end{aligned}$$

All values $W_{f}^{\scriptscriptstyle K, N}$ and $C_{f}^{\scriptscriptstyle N}$ 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 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 $\Psi _{\text {W}, \scriptscriptstyle 1}^{\scriptscriptstyle N}$, 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 $\chi _{\scriptscriptstyle 1}$ 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 $\Psi _{\text {W}, \scriptscriptstyle 16}^{\scriptscriptstyle N}$ is more directionally sensitive than $\Psi _{\text {W}, \scriptscriptstyle 8}^{\scriptscriptstyle N}$, 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 $\chi _{\scriptscriptstyle K}$ visualized in Fig. 5, since it appears as a factor in the dominant term of the analysis coefficients. In particular, the directional sensitivity of $\Psi _{\text {W}, \scriptscriptstyle K}^{\scriptscriptstyle N}$ 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. 8

Pre-rotated directional wavelets and curvelets for different parameters

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