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Edge detection with trigonometric polynomial shearlets

In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.


Introduction
In many applications in signal or image processing, great importance is attached to precise information about the location and order of singularities of signals. In one dimension, this corresponds to functions which are smooth apart from pointwise singularities. Many authors discussed this problem when the Fourier coefficients of a periodic function are given; see, e.g., [1,6,7,27,31,34].
Because of their localization properties in the time and frequency domain, wavelet expansions provide a powerful tool for detecting and analyzing point discontinuities in one or more dimensions [18,26]. The reason is that only very few wavelet coefficients of translates near the location of the singularity are large in magnitude, while all other wavelet coefficients corresponding to translates which are further away from the point discontinuity decay rapidly. A framework for univariate periodic wavelets was investigated by several authors [19,[28][29][30], and some of these constructions were successfully used for the detection of pointwise singularities of periodic functions [27].
In two dimensions, the situation is more complex since not only point singularities can occur but also discontinuities along curves. To deal with these types of singularities, along with many other constructions, the theory of the continuous shearlet transform was developed [5,11,22] and defined as the mapping: f → SH ψ f (a, s, p) = f, ψ a,s,p with scale parameter a > 0, orientation parameter s ∈ R and translation parameter p ∈ R 2 . The shearlets ψ a,s,p are well-localized functions in the time and frequency domain and provide directional sensitivity controlled by the parameter s. It turned out that continuous shearlets provide a suitable tool to precisely describe different types of discontinuities along curves with asymptotic estimates. In particular, let T ⊂ R 2 be a set with a smooth boundary ∂T . If either p / ∈ ∂T or if s = s 0 does not correspond to the normal direction of ∂T at p, then lim a→0 + a −N SH ψ χ T (a, s 0 , p) = 0 forallN > 0.
Otherwise, if p ∈ ∂T and s = s 0 corresponds to the normal direction of ∂T at p, then lim a→0 + a −3/4 SH ψ χ T (a, s 0 , p) = C > 0.
The results were shown for continuous shearlets, which are compactly supported in the time [23] or frequency domain [9,13,14,21]. Based on these theoretical results, practical applications for the detection of edges in images were developed [35]. Therefore, discrete frames of shearlets were constructed by sampling the parameters of the continuous shearlet systems in a suitable way [20]. Based on the result for curvelets [4], it was possible to show that discrete shearlet systems are essentially optimal for the sparse approximation of so-called cartoon-like functions [12]. This result implies the upper estimate: for some constant C > 0 independent of the scale parameter j . In [17], the authors showed the existence of a lower estimate χ T , ψ k ≥ C 2 −3j/2 if the localization and orientation of the discrete shearlet are sufficiently close to the boundary curve and its normal direction. These two estimates are the discrete analogs of Eqs. 1 and 2 implying that discrete shearlets are able to detect step discontinuities along boundary curves of characteristic functions. The framework of multivariate periodic wavelets was developed for example in [8,25]. In [3,24], the corresponding wavelet functions were trigonometric polynomials of Dirichlet and de la Vallée Poussin-type, which can be well localized in the time and frequency domain. The construction allows for fast decomposition algorithms [2] with many different dilation matrices on each scale, including shearing. This gives rise to directional decompositions of the frequency domain similar to the tilling of the frequency plane in the case of discrete shearlet systems [4,17].
In this paper, we use the latter construction to prove two main theorems which provide upper and lower bounds similar to [17], but this time for a discrete system of periodic de la Vallée Poussin-type wavelets that are trigonometric polynomials. The upper estimate in Theorem 1 refines the estimate Eq. 3 by including the localization and orientation dependency of the shearlet coefficients in the decay estimate. Theorem 2 is the analog of the main result in [17] and implies that the constructed trigonometric polynomial shearlets in this paper are able to detect step discontinuities along boundary curves of periodic functions.
The paper is organized as follows. We start with the construction of a special case of directional de la Vallée Poussin wavelets in Section 2 which we will call trigonometric polynomial shearlets and state the two main theorems of this paper in Section 3. Section 4 provides a numerical example to illustrate the main results. After some preliminaries, Section 5 is devoted to formulate and to prove all auxiliary lemmata which are needed for the proof of the main results. In Section 6, the proofs for the upper and lower bounds of the corresponding inner products are given. Finally, we consider the extension of the construction to higher dimensions and discuss possible results in the case of corner points and smooth functions.

Trigonometric polynomial shearlets
If a nonnegative and even function g : R → R with supp g = − 2 3 , 2 3 satisfies the property: we call it window function and write g ∈ W. If additionally g is q-times continuously differentiable, we use the notation g ∈ W q . We remark that a consequence of the properties of a window function is g(x) = 1 for x ∈ − 1 3 , 1 3 and g is monotonically increasing for x ∈ − 2 3 , − 1 3 and monotonically decreasing for x ∈ 1 3 , 2 3 . Furthermore, we introduce functions g : R → R given by g(x) := g x 2 − g(x). As an example of a window function, we consider: where b > 0 and define s(x) = r 2 3 + x r 2 3 − x . Then for we have g b ∈ W ∞ and this function is visualized in Fig. 1 for b = 0.025. We denote two-dimensional vectors by x = (x 1 , x 2 ) T with the inner product x T y := x 1 y 1 + x 2 y 2 and the usual Euclidean norm |x| 2 := √ x T x. Let C(A) denote the space of all continuous functions on a set A ⊆ R 2 equipped with the norm f A,∞ := f C(A) := sup x∈A |f (x)|. For x ∈ R 2 and r = (r 1 , r 2 ) T ∈ N 2 0 and a sufficiently smooth function f , we use the notation: and the space of all q-times continuously differentiable compactly supported functions will be denoted by with the norm For i ∈ {h, v}, we consider bivariate horizontal (vertical) window functions Ψ (i) : R 2 → R given by We remark that for g ∈ W q we have Ψ (i) ∈ C q 0 (R 2 ) and in this case use the notation From the support properties of the function g ∈ W, it follows that: For even j ∈ N 0 and ∈ Z with | | ≤ 2 j/2 , we define the matrices: and the corresponding discrete angles Note that these matrices occur in the construction of discrete shearlet systems, for example in [12,17]. Based on this, we introduce the notation: and, since det In polar coordinates, we define the sets and based on ideas from [12, Proposition 2.1] we show the following lemma, which is visualized on the right side of Fig. 1.
Proof We show only the case i = h since the other one is similar. In Eq. 6, we defined with the support property and, assuming that ξ 1 ∈ supp g(2 −j ·), we have In the following, we introduce polar coordinates with the notation ξ := ρ Θ(θ ), where Θ(θ ) := (cos θ, sin θ) T . Recalling the discrete angles θ (h) = arctan 2 −j/2 we have +2 .
Since ρ 2 = ξ 2 1 1 + tan 2 θ and | | ≤ 2 j/2 we can show where the last inequality holds for j ≥ 10. As a lower bound for the radius ρ, we obtain: The pattern of a regular matrix M ∈ Z 2×2 is defined by P(M) . As a consequence of [24,Lemma 2.4], the patterns of the matrices in Eq. 5 are independent of the parameter and have the tensor product structure: For i ∈ {h, v} and Ψ (i) ∈ W q 2 the translates of the de la Vallée Poussin wavelet functions (see [3]) on the pattern points y ∈ P(N (i) ) are trigonometric polynomials given by ψ In the following, we call the functions ψ (i) y trigonometric polynomial shearlets.

Main results
Let and let γ : [0, 2π) → (−π, π) 2 be a closed curve with which is a parametrization of the boundary of a set T ⊂ (−π, π) 2 . The space C u (κ) is defined as the collection of all functions of the form where Following the ideas from [4,12], let Q j , j ∈ N 0 , be the set of dyadic squares Q ⊆ [−π, π) 2 of the form (Fig. 2) with n 1 , n 2 = 0, . . . , 2 j/2 − 1. Let Q ∈ Q 1 j ⊆ Q j if ∂T ∩ Q = ∅ and for the nonintersecting squares we define Q 0 j := Q j \ Q 1 j . We remark that Q 0 j ≤ C 2 j and Q 1 j ≤ C 2 2 j/2 (see [4,12]). For Lebesgue measurable sets A ⊆ R 2 and functions f : and let L p (A) denote the collection of functions satisfying f A,p < ∞. In particular, two-dimensional 2π -periodic functions f : T 2 → R are defined on the torus Recall that the usual inner product of the Hilbert space L 2 (T 2 ) is given by: , the 2π -periodization of f .
The main results of this paper are stated in the following two theorems.
j let x 0 := x 0 (Q) ∈ ∂T ∩ Q and γ := γ (x 0 ) such that (cos γ, sin γ ) T is the normal direction of the boundary curve ∂T in x 0 . Then, we have: If y ∈ P N (i) is sufficiently far away from the boundary curve, Theorem 1 For the special case f 0 = 0 and f 1 = 1 in Eq. 8 we define T = χ T and denote by T 2π the 2π -periodization of T .
2 for sufficiently large q ∈ N and y ∈ P N (i) for large j . If there exists x 0 ∈ ∂T with the normal direction (cos γ, sin γ ) T and curvature A 0 in that point,

Numerical examples
In this section, we give numerical examples to underline the main results of this paper by computing the shearlet coefficients of a characteristic function of a rotated ellipse.
In order to do that, we need to compute the Fourier transform of the characteristic function of a disc, given by D(x) := 1 for |x| 2 ≤ 1, 0 else.
We transform ξ = ρ Θ(θ ) and x = r Θ(φ) into polar coordinates and use ξ T x = rρ cos (θ − φ) to obtain where J 0 is the Bessel function of the first kind and zero order. The integral identity together with the change of variable λ = rρ leads to For a, b > 0, we define D a,b (x) := D(a −1 x 1 , b −1 x 2 ) and convert the circle into a characteristic function of an ellipse with major semi-axis of length a and minor semi-axis of length b. By the scaling property of the Fourier transform, we have: If we further rotate the function D a,b by an angle γ ∈ [0, 2π) we obtain a rotated ellipse D a,b,γ (x) := D a,b R γ x and its Fourier transform is given by In order to calculate the shearlet coefficients of a rotated ellipse, we consider the 2π -periodized function D 2π a,b,γ (x). We use Eq. 15 to see that the Fourier coefficients of this function are given by In our numerical example, we calculate the inner product Eq. 11 with Mathematica 12. We fix the characteristic function of the rotated ellipse D 2π 1,3, π 6 , which is depicted on the left side in Fig. 3 (gray area). For the one-dimensional window function, we use the smooth function g 0.025 ∈ W ∞ constructed in Eq. 4. We fix the scale j = 10 and for a better visualization we consider the matrix M 10 = 2 10 I 2 , where I 2 is the two-dimensional identity matrix and compute the shearlet coefficients in Eq. 11 on the pattern P(M 10 ), which corresponds to a two-dimensional equidistant grid of 1024 × 1024 points. Thus, the images in this example are of size 1024 × 1024, where every pixel corresponds to the magnitude of the inner product with a translate of the trigonometric polynomial shearlet ψ The right image illustrates the capability of the trigonometric polynomial shearlets to detect step discontinuities along the boundary of characteristic functions. For i ∈ {h, v} and = −2 j/2 + 1, . . . , 2 j/2 − 1, we compute all the pictures of the shearlet coefficients similar to the left image in Fig. 3 and add them component-wise to get the final result. Thus, every pixel of the image is given by the sum: and one can clearly see the only significant coefficients for all the directions are exact on the boundary of D 2π 1,3, π 6 . Besides the visual representations of the detection of step discontinuities with trigonometric polynomial shearlets, we want to illustrate the upper and lower estimates given in the two main theorems. In order to do so for the upper bound, we compute the quantity: In the left graph of Fig. 5, the values U (i) are plotted for different orientation angles θ (i) . One can see that the quotient U (i) is bounded from above by a moderate constant for every j and which confirms that the estimate in Theorem 1 provides a valid upper bound. For the lower bound, we collect all pattern points y ∈ P(M j ) for which there exists x 0 ∈ ∂T with the normal direction (cos γ, sin γ ) T fulfilling |x 0 − 2π y| 2 ≤ 8,−8 . The latter two sets include pattern points which are close to boundary points x 0 ∈ ∂T with normal direction (cos γ, sin γ ) T fulfilling θ for i ∈ {h, v}. By Theorem 2, we expect the values of the shearlet coefficients corresponding to the shearlets ψ (i) 8,−8,y for y ∈ L (i) 8,−8 to be bounded from below by a constant. Therefore, we compute the values and show them in the right graph of Fig. 5 as functions of the orientation angles θ (i) .
One can clearly see that the minimal values L (i),min are bounded from below, which confirms the result of Theorem 2. In our numerical example, a parametrization of the boundary ∂D 1,3, π 6 is given by the curvature in each point is given by κ(x) = 3 (5 + 4 cos(2x)) −3/2 . On the right side of Fig. 5, the values of the curvature κ(x) are shown in the points x 0 ∈ ∂T , where the normal direction of x 0 is orthogonal to θ (i) for | | < 2 j/2 . As anticipated in Theorem 2, one can see that the magnitude of the coefficients L (i),max , L (i),min varies as the curvature of the ellipse changes. If the curvature is small, which corresponds to the "stretched" part of the boundary, the values become larger. Intuitively, this makes sense since in that case a large part of the boundary is aligned with the corresponding shearlet.

Auxiliary results
For two-dimensional vector norms, we use the notation: and for binary relations and exponentials of vectors we write x ≤ y if x 1 ≤ y 1 and The Fourier coefficients of a function f ∈ L 1 (T 2 ) are given by and we have the operator We recall some basic results about the Fourier transform and its connection to Fourier series via the Poisson summation formula. Let q ∈ N 0 and r ∈ N 2 It is well known that there are constants The sum in Eq. 10 converges for almost every x ∈ T 2 and f 2π ∈ L 1 (T 2 ). For the Fourier coefficients, we have: It is a consequence of Eq. 14 and [33, Corollary VII.2.6] that for a function f ∈ C q 0 (R 2 ) with q > 2, the Poisson summation formula holds true for all x ∈ R 2 .
In the following, we prepare the proof of Theorem 1 with several auxiliary lemmata. Note that in the proofs we only show the case i = h since the other case can be handled similarly.
Proof We have R γ = cos γ − sin γ sin γ cos γ and use Eq. 6 to see In this proof, we will omit the long arguments of the function of the last line and simply write g and g. For m = (m 1 , m 2 ) T with |m| 1 ≤ q, we use the chain rule to get For sufficiently smooth functions f, g : R 2 → R, we employ the multivariate Leibniz rule which together with the triangle inequality and the binomial theorem implies In the following, we use notations and ideas from [4,12] and fix a function φ ∈ C ∞ 0 ([−π, π] 2 ). Denote φ j (x) := φ 2 j/2 x and for Q ∈ Q j given by Eq. 9 we define for k 1 , k 2 = 1, . . . , 2 j/2 and assume that φ defines a smooth partition of unity The ideas of the proof of the next lemma can be found in [4,12].
and using Eq. 17 we get The function η s is s-times continuously differentiable with respect to the variable ξ 1 . For 0 ≤ t ≤ s, we can estimate: which leads to By definition of the function φ j , we have supp φ j ≤ 2 −j and with property Eq. 13 and the Plancherel theorem we get for all 0 ≤ s ≤ u. Using again Eq. 13, it follows that Fη s , which leads together with Eq. 19 to Next, we consider the function where φ r (x) := x r φ j (x) and see that φ r 2 j/2 · ∈ C ∞ 0 (R 2 ) with |supp φ r | ≤ 2 −j is fulfilled. Thus, the Fourier transform of the function f (x) φ r 2 j/2 x satisfies Eq. 20 with a constant C 6 (u, r). We use Eq. 12 to deduce

Lemma 4
For u ∈ N let f ∈ C u (R 2 ) and f j := f φ j . Moreover, for i ∈ {h, v} and q ≥ 2 let Ψ (i) ∈ W q 2 be given. Then for Q ∈ Q 0 j and any r ∈ N 2 0 with |r| 1 ≤ q we have Proof For the partial derivative of the product inside of the norm, we use the multivariate Leibniz rule Eq. 17 and obtain Lemma 2 implies that for all ξ ∈ R 2 the inequality holds, independent of the orientation parameter . Together with Lemma 3, we have Following the approach from [4, Chapter 6.1], we assume that for j ≥ j 0 the edge curve ∂T can be parametrized on the support of φ Q , Q ∈ Q 1 j , either as ( Let E j,x 0 ,γ be an arbitrary edge fragment, which tangent in the point x 0 ∈ ∂T is pointing in the direction (cos γ, sin γ ) T for γ ∈ [0, 2π). Clearly, E j,0,0 = E j is a standard edge fragment. Moreover, in [4, Corollary 6.7], it is remarked that, although an arbitrary edge fragment E j,x 0 ,γ can not be obtained via rotation and translation of a standard edge fragment, we have the connection: of their Fourier transforms. The following lemma is a consequence of [4, Corollary 6.6].

Lemma 5
For j ∈ N let I j = 2 j −1 , 2 j +1 and E j be a standard edge fragment. Then for angles θ, γ ∈ [0, 2π) and r ∈ N 2 0 , we have: We can deduce the following result, whose proof uses ideas from [12, Proposition 2.1].
Then for a standard edge fragment E j , a rotation matrix R γ by the angle γ ∈ [0, 2π) and r ∈ N 2 0 we have Proof From Lemma 1, we know supp Ψ (i) ⊂ W (i) and we transform the integral into polar coordinates and use Lemma 5 to obtain: Then for a standard edge fragment E j , a rotation matrix R γ by the angle γ ∈ [0, 2π) and r ∈ N 2 0 we have Proof We repeat the steps of the proof of Lemma 4 and use Lemma 6 instead of Lemma 3 in the last step.
The Laplace operator is denoted by Δ := ∂ (2,0) + ∂ (0,2) and for q ∈ N 0 we have For the next lemma, we define the second-order differential operator L := I + 2 j Δ, which was already used in [4,12]. Using Eq. 22, we have: Proof By applying the Cauchy-Schwarz inequality twice, we obtain and get the result by inserting the corresponding upper bounds for the norm from Lemma 4 and Lemma 7.
The last part of this section consists of lemmata which are needed to proof Theorem 2. We start with some important localization properties.
Since Ψ (h) ∈ W 2q 2 , we can use the Poisson summation formula Eq. 16 to arrive at where S(n) := Let R γ be a rotation matrix by the angle γ . Then and in the integral S(0) we use this rotation matrix for a change of variable to see Since the function Ψ (h) is compactly supported, we can use q-times partial integration together with Lemma 2 and Eq. 7 to deduce Using the same idea as before, we substitute with the rotation matrices R γ n in the integrals S(n), where γ n is the direction of the vector x − 2π (y − n). Similar to Eq. 25 we use 2q-times integration by parts with respect to the first variable, Lemma 2 and Eq. 7 to obtain Observe that and with the inverse triangle inequality we can estimate |x − 2π(y − n)| 2 ≥ 2π |n| ∞ − |x − 2π y| ∞ ≥ |x − 2π y| 2 (2 |n| ∞ − 1).
Since n ∈ Z 2 ; |n| ∞ = k, k ∈ N = 8k we use Eq. 26 to conclude (27) and the infinite sum in the last line converges because q ≥ 2. We finish the proof by making use of Eqs. 25 and 27 in Eq. 24. Let y = (y 1 , y 2 ) T ∈ P N (h) . We assume that the boundary curve can be vertically parametrized by (f (x), x) T for |x − 2πy 2 | ≤ 2 −j/2 . For m := f (2πy 2 ) ∈ [−1, 1] and A := 1 2 f (2πy 2 ) let be the second-order Taylor approximation for f (x) in the point x 0 = 2πy 2 . Denote by T (h) y the modified version of T by replacing the function f (x) by the approximation T y (x) for |x − 2πy 2 | ≤ 2 −j/2 if the corresponding parametrization is a vertical curve and similarly T (v) y as the modified version of T if the parametrization is a horizontal curve. Although this notation seems to be counterintuitive, it is convenient since by Lemma 11 only the interaction of horizontal wavelets with vertical curves and vertical wavelets with horizontal curves contributes to the desired lower bound in Theorem 2. The analog of the following lemma can be found in [17] for the discrete and in [13,14] for the continuous setting.

Lemma 10
For i ∈ {h, v} and large q ∈ N let Ψ (i) ∈ W 2q 2 be given. Then for Proof We only show the proof for i = h since the other case is similar. For this proof, we define the set B j = (x 1 , x 2 ) T ∈ [−π, π) 2 : |x 2 − 2π y 2 | ≤ 2 −7j/16 and write Using the definition of T y (x) in Eq. 28, we can estimate for the area between T and T (h) y if |x − 2πy 2 | ≤ 2 −j/2 . From Lemma 9, we can obtain the uniform bound ψ (h) y (x) ≤ C(q) 2 3j/2 and we can estimate the first integral by |I 1 | ≤ C(q) 2 3j/2 |x−2πy 2 |≤2 −7j/16 |x − 2πy 2 | 3 dx ≤ C(q) 2 3j/2 2 −7j/4 = C(q) 2 −j/4 . In addition, we use again Lemma 9 but this time for the decay term in the minimum to arrive at 16−17/16) for the second integral, which shows that the lemma is proved for q ≥ 21.
From the divergence theorem, one can see that the Fourier transform of a characteristic function T = χ T is given by (29) with the outer normal vector n(x). We remind the parametrization of ∂T given by γ (x), x ∈ [0, 2π) and use polar coordinates to represent the line integral Eq. 29 as and β k (x) := n(γ k (x)) 1 + f k (x) 2 . Using this, we can conclude: where we again transformed the integral into polar coordinates and the interchange of summation and integration is valid since the function Ψ (i) has finite support.

Denote by
. . , M} and can prove the following lemma, whose idea of proof was given in [17].

Lemma 11
For i ∈ {h, v} and q ∈ N let Ψ (i) ∈ W q 2 be given. Then for any n ∈ N there is a constant C(n) > 0 such that for every k ∈ M (i) we have Proof We show the result for i = h since the other case is similar. Suppose that Hence, the outer normal vector in x is given by n(γ k (x)) = (f k (x), −1) T leading to From the assumption on horizontal curves for i = h we have f k (x) < 1 and the support properties of the function Ψ (h) given in Lemma 1 imply for some small δ > 0. From that, we conclude: For n ∈ N, we do n-times integration by parts with respect to the variable x and obtain I k (2 j ρ, θ) ≤ 2 −jN , which leads to Lemma 1 implies supp Ψ (h) (2 j ·) ≤ C and since n ∈ N was arbitrary the lemma is proven.
The proof of the following lemma was given in [14].

Lemma 12
For i ∈ {h, v} and q ∈ N let Ψ (i) ∈ W q 2 and y = (y 1 , y 2 ) T ∈ P N (i) be given. Then for any n ∈ N there is a constant C(n) > 0 such that The following lemma is a special case of [32,Proposition 8.3], called method of stationary phase.

Lemma 13 Let φ and ϕ be smooth functions on the real line. Suppose
For x ∈ (0, ∞), we introduce the so-called Fresnel integrals show the following lemma (Fig. 6).

Lemma 14 We have
Proof It is clear that F c(0) = F s(0) = 0 and for x > 0 it is well known that F c(x) > 0 and F s(x) > 0, which implies Moreover, we have d dx Since we see that x k is a local maximum point of F + for k even and a local minimum point for k odd. To get information about global extremal points, we define k even and (cos v + sin v) < 0 for k odd. It follows that I k > 0 for k even and I k < 0 for k odd. Additionally, we have I 0 > 3.36 > 1.45 > |I 1 | and since v −1/2 is monotonically decreasing we have |I k | > |I k+1 | for k ≥ 1.
Putting these observations together, we obtain (I k + I k+1 ) > 0 for k even and (I k +I k+1 ) < 0 for k odd. From that, we deduce that x 0 = 3π 4 is the global maximum point of F + since for even k > 0 we have Similarly for k > 1 odd, we can write which shows that smallest local minimum is obtained at the point x 1 = 7π 4 and thus 1.
Similarly, one can show that x k := π 4 + kπ, k ∈ N 0 , is a local maximum point of F − for k even and a local minimum point for k odd. Since x 0 < 3π 4 < x 1 = 5π 4 and F − ( 3π 4 ) > 0.14 > 0 we have F − (x) > 0 for 0 < x < 3π 4 , which together with Eq. 32 gives the first statement of the lemma.
With similar arguments as in Eqs. 33 and 34, we have and we use the mean value theorem of integration to deduce that there exists x ∈ (0, r(λ, p, A)] such that Using Lemma 14, the proof is finished.
For the last lemma of this section, we define the following integrals:

Lemma 16
There is a constant C > 0 such that for all D ∈ − 3π 4 , 3π 4 , p ∈ − 1 4 , 1 4 and A > 0 at least one of the inequalities is true.
Proof We define a(λ, p, A) sin(Dλ) dλ and show that there exists a constant C > 0 that either one of the statements in Eq. 38 or equivalently P + (D, p, A) ≥ C is true. For the rest of the proof, the variables p ∈ − 1 4 , 1 4 and A > 0 will be arbitrary and fixed. For simplicity, we assume D ∈ [0, 3π 4 ] since the proof for negative values of D is similar. We consider different intervals for the variable D and show that at least one of the equivalent propositions, see Eq. 38 or P + (D, p, A) ≥ C , is true. From the construction of the window function, we deduce g(λ) λ −1 > 0 and from Lemma 15 we know a(λ, p, A) + b(λ, p, A) > 0 for λ ∈ 1 3 , 4 3 . Thus, for D = 0, it holds that −1 a(λ, p, A) + b(λ, p, A) dλ > 0.

Proof of the main results
We start with the proof of Theorem 1.
Recall that we denote the set of dyadic squares Q ⊆ [−π, π) 2 of the form Eq. 9 for j ∈ N 0 by Q j and smooth functions φ Q with support on these dyadic squares with the property are defined in Eq. 18. Moreover, for u ∈ N let f ∈ C u (R 2 ) and define f Q := f φ Q for Q ∈ Q j . We can decompose For the non-intersecting squares we define Q 0 j := Q j \ Q 1 j . According to Eq. 10, we denote by f 2π Q the 2π -periodization of f Q . From the finite support of f Q , we deduce f Q ∈ L 1 (R 2 ) and from Eq. 15 we get Moreover by Eq. 12, we have Ff Q ∈ C q (R 2 ) for all q ∈ N 0 . The smoothness assumption on the window function Thus, the estimates Eq. 14 hold for this function and with Parseval's identity and the Poisson summation formula it follows: and define f ( we can use integration by parts repeatedly in both variables for every r ∈ N 2 0 with |r| 1 ≤ q to obtain With the calculation 2r and the representation Eq. 23 of the q-th order differential operator L q we have (43) A consequence from Hölder's inequality for a set A ⊂ R 2 with finite Lebesgue measure |A|, parameters 1 ≤ p ≤ s < ∞ and a function f ∈ L s (A) is the estimate From Lemma 8, we conclude L q F[f ] Ψ (i) ∈ L 2 (R 2 ) and with the estimate Eq. 44 for p = 1 and s = 2 together with the upper bound for the support size of Ψ (i) given by Eq. 7 we see that Next, Eqs. 43 and 45 and Lemma 8 for u = 2 imply We split up the infinite sum in the last line into where due to Eq. 42 the summand corresponding to n = 0 is bounded from above by C(q) 2 −jq . With the monotonicity of finite vector norms and the inverse triangle inequality, we get for n = 0, because again with Eq. 42 we have 2 y − x 1 π ∞ ≤ 1. Moreover the equation n ∈ Z 2 ; |n| ∞ = k, k ∈ N = 8k holds, leading to Using the splitting Eq. 47 and the corresponding upper bound Eq. 48 for the infinite sum in Eq. 46, we get: For Q ∈ Q 1 j , we use Eq. 21 to write where E j is a standard edge fragment. With the same arguments as in the first case and again Lemma 8, we can deduce: and we split up the infinite sum into Using the same arguments, which led to Eq. 48, we see that the infinite sum in the last equation is bounded from above by C(q) 2 −jq implying To finish the proof, we use the decomposition Eq. 41 and the fact that Q 0 j ≤ C 2 j to get We proceed with the proof of Theorem 2. For y ∈ P(N (i) ) let T From the finite support of T The smoothness assumption on the window . Similar as in the proof of Theorem 1, this product fulfills Eq. 14 and with Parseval's identity and the Poisson summation formula it follows: Using again the decomposition Eq. 41 for T we can repeat the arguments from the proof of Theorem 1 to see that Assume that we can show With the inverse triangle inequality, we can deduce and again with the inverse triangle inequality and Lemma 10 we finally get Thus, it is left to show the existence of a constant C 2 (q) > 0 such that Eq. 50 is fulfilled. From Eqs. 30 and 31, we recall the representation: and consider only the case i = h since the other case is similar. First, we use Lemma 11 and the inverse triangle inequality to see that |S(0)| is bounded from below by where the last term is negligible for large j and n ∈ N. Assume that there is k * ∈ M (v) such that 2πy 2 − π 2 −j/2 , 2πy 2 + 2 −j/2 ⊆ [a k * , b k * ]. In the following, we omit the index k * for simplicity and let ε = 2 −j/2 . Lemma 12 and the inverse triangle inequality lead to From the previous observations, we conclude that, if we want to show |S(0)| ≥ C(q), it is enough to find a constant C 2 (q) > 0 such that ∞ 0 2π 0 2πy 2 +ε We write the last integral as In the integral I 2 , we substitute θ = τ + π and use the symmetry properties of the univariate window functions g and g to see g (ρ cos(τ + π)) = g (−ρ cos τ ) = g (ρ cos τ ) and g ρ cos(τ + π)(2 j/2 tan(τ + π) − = g ρ cos τ (2 j/2 tan τ − , which lead together with Θ(τ + π) = −Θ(τ ) to I = 2 i Im(I 1 ) = 2 i Im(I 2 ) since Let us first assume A = 1 2 f (2πy 2 ) > 0. The case A < 0 is similar and will be omitted and the case A = 0 will be discussed separately in the end of the proof. With Lemma 10 we can replace the function f (x) locally for |x − 2πy 2 | < 2 −j/2 by For every where B := m+2(x 0 −2πy 2 ), C := f (2πy 2 )+m(x 0 −2πy 2 )+(x 0 −2πy 2 ) 2 and m = f (2πy 2 ) ∈ [−1, 1]. We choose x 0 ∈ ∂T such that there is ∈ −2 j/2 , . . . , 2 j/2 with 2 j/2 B + ≤ 1 4 and y 1 = 2 −j (z 1 − 1 2 ), z 1 = −2 j −1 , . . . , 2 j −1 − 1 such that 2 j (2π y 1 − C) ≤ 3π 4 .
We follow the ideas of [17,Section 3.2] and change the variable to v = x − x 0 . Thus, we can rewrite I 1 as where Λ : Hence, we can write I 1 as We apply Lemma 13 for t 0 = v θ , which leads to 2ε −2ε From Lemma 1, we have ρ ∈ 1 2 , 2 so that the notation O((2 j ρ) −1 ) can be identified with a function r(j ) such that |r(j )| ≤ C 2 2 −j as j ≥ j 0 . As explained in [17, p. 115] the constant C 2 > 0 is independent of θ, ρ, j , , y. With Eq. 53 we can write the integral in Eq. 52 as I 1 = I 11 + I 12 , where ×(ρ |A cos θ |) − 1 2 ϕ(v θ ) dθ dρ, shows that where a(λ, p, A) and b(λ, p, A) are defined in Eqs. 35  we can write the integral Eq. 57 as I 11 (D, p) = Re(I 11 (D, p)) + i Im(I 11 (D, p)) with Using the connection I 2 = −I 1 , we can start at Eq. 51, use again Eq. 13 and repeat all the previous steps for I 2  As a consequence of the relation I = 2 i Im(I 1 ) = 2 i Im(I 2 ) we see that I = 2 i Im(I 11 + I 12 ) = 2 i Im(I 21 + I 22 ). By the inverse triangle inequality, Eq. 54 and its analog for I 22 we can use Lemma 16 in order to finish the proof of the Theorem 2 for A > 0.
In the case A = 0 we see that Eq. 51 simplifies to Note that Lemma 13 can not be applied in this case. Instead we use the substitutions u = 2 j/2 v and similar to the previous case t = 2 j/2 tan θ − and λ = ρ μ together with the simplifications and notations from the last pages to arrive at the analogous integral to Eq. 57, which in this case is given by for 0 < |D| ≤ 3π 4 . For the case D = 0, we slightly modify the function g to make it odd. Then with a similar argument as before we see that I = 2Re(I 1 (0, p)) > 0.

Generalizations and possible extensions
In this paper, we showed that trigonometric polynomial shearlets based on the construction of multivariate periodic de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of characteristic functions.
For even j ∈ N 0 and = 1 2 ) T ∈ Z 2 with | 1 | ≤ 2 j/2 and | 2 | ≤ 2 j/2 the matrices analog to Eq. 5 are given by As in the two-dimensional case, this construction is similar to the classical shearlets and its higher-dimensional generalizations. The authors in [15] proved in detail that continuous shearlet systems in three dimensions are able to detect boundary curves of piecewise smooth surfaces. As remarked in [17], an analogous result holds for discrete shearlets in dimension 3. We are convinced that it should be possible to derive a similar result for trigonometric polynomial shearlets, but a detailed proof is not in the focus of this paper.
Another interesting open question is the behavior of the shearlet coefficients near corner points. If γ : [0, 2π) → ∂T is a parametrization of the boundary ∂T , we call x 0 = γ (t 0 ) ∈ ∂T a corner point, if γ (t + 0 ) = ±γ (t − 0 ). For continuous shearlets, this question was answered in [14] and in a more general setting in the context of parabolic molecules in [10]. As far as we know, there is no result for corner points in the discrete setting until now. It would be very interesting to investigate in which way the techniques of the continuous setting can be combined with the ideas of this paper to prove similar results for discrete shearlets. We will leave this question as a topic for future research.
In many applications, such as image processing, the functions to be analyzed are piecewise smooth and not characteristic functions of sets as discussed in this paper. In [10,16], it was shown that the continuous shearlet coefficients of functions of the form B(x) = f (x) χ T (x) with f ∈ C ∞ (R 2 ) exhibit the same decay rate as Eq. 1 if p / ∈ ∂T or if s = s 0 does not correspond to the normal direction of ∂T at p. If p ∈ ∂T and s = s 0 corresponds to the normal direction of ∂T at p, then 0 < lim where n denotes the number of vanishing derivatives of f at p. As in the case of corner points, there is no analogous result for discrete shearlet systems yet. To give a proof for the case of piecewise smooth functions is again beyond the scope of this paper and will be addressed in a forthcoming article. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommonshorg/licenses/by/4. 0/.