 Research
 Open Access
Directional global threepart image decomposition
 D. H. Thai^{1, 2}Email author and
 C. Gottschlich^{1}
https://doi.org/10.1186/s1364001601100
© Thai and Gottschlich. 2016
 Received: 27 October 2015
 Accepted: 16 February 2016
 Published: 14 March 2016
Abstract
We consider the task of image decomposition, and we introduce a new model coined directional global threepart decomposition (DG3PD) for solving it. As key ingredients of the DG3PD model, we introduce a discrete multidirectional total variation norm and a discrete multidirectional Gnorm. Using these novel norms, the proposed discrete DG3PD model can decompose an image into two or three parts. Existing models for image decomposition by Vese and Osher (J. Sci. Comput. 19(1–3):553–572, 2003), by Aujol and Chambolle (Int. J. Comput. Vis. 63(1):85–104, 2005), by Starck et al. (IEEE Trans. Image Process. 14(10):1570–1582, 2005), and by Thai and Gottschlich are included as special cases in the new model. Decomposition of an image by DG3PD results in a cartoon image, a texture image, and a residual image. Advantages of the DG3PD model over existing ones lie in the properties enforced on the cartoon and texture images. The geometric objects in the cartoon image have a very smooth surface and sharp edges. The texture image yields oscillating patterns on a defined scale which are both smooth and sparse. Moreover, the DG3PD method achieves the goal of perfect reconstruction by summation of all components better than the other considered methods. Relevant applications of DG3PD are a novel way of image compression as well as feature extraction for applications such as latent fingerprint processing and optical character recognition.
Keywords
 Image decomposition
 Variational calculus
 Cartoon image
 Texture image
 Image compression
 Feature extraction
 Latent fingerprint image processing
 Optical character recognition
 Fingerprint recognition
1 Introduction
Based on these observations, we define the following goals: Goal 1: the cartoon component u contains only geometrical objects with a very smooth surface, sharp boundaries, and no texture. Goal 2: the texture component v contains only geometrical objects with oscillating patterns and v shall be both smooth and sparse. Goal 2: threepart decomposition and reconstruction f=u+v+ε.
How does achieving these goals serve the tasks of feature extraction, denoising, and compression?
Extremely efficient representations of the cartoon image u and texture image v exist. These two component images are highly compressible as discussed with full details in Section 7.3. Depending on the application, u or v or both can be considered as feature images. For the application to latent fingerprints, we are especially interested in the texture image v as a feature for fingerprint segmentation and all subsequent processing steps. Example results for the very challenging task of latent fingerprint segmentation are given in Section 7.1. In optical character recognition (OCR), preprocessing includes the removal of complex background and the isolation of characters. After threepart decomposition and depending on the scale of the characters, the cartoon image u contains the information of interest for OCR (see Fig. 1 j), and the background is separated into v and ε simultaneously in the minimization procedure. As a consequence of the requirements imposed on u and v, noise and small scale objects are driven into the residual image ε during the decomposition of f. Therefore, the image u+v can be regarded as a denoised version of f and the degree of denoising can be steered by the choice of parameters.
The paper is organized as follows. In Section 2, we begin by describing notation and preliminaries. After having established these prerequisites, we define the DG3PD model in Section 3 and in Section 4; we explain its relation to existing models in the literature for twopart and threepart decomposition. In Section 5, we describe an iterative, numerical algorithm which solves the DG3PD model for practical applications to discrete, twodimensional images. In Section 6, we perform a detailed comparison of the DG3PD method to stateoftheart decomposition approaches. Applications of DG3PD, especially feature extraction and image compression, are the topics in Section 7. Discussion and conclusions are given in Section 8. An overview of the algorithm and an additional comparison is given in the Appendix.
2 Notation and preliminaries
Forward/backward difference with periodic boundary condition in convolution/matrix form and their Fourier transform
Forward difference  Backward difference  

Convolution  \(\partial ^{+}_{x} f[\!{\mathbf {k}}] = f[\!k_{1}, k_{2}+1]  f \left [k_{1}, k_{2}\right ]\)  \(\partial ^{}_{x} f[\!{\mathbf {k}}] = f[\!k_{1}, k_{2}]  f [\!k_{1}, k_{2}1]\) 
form  \(\partial ^{+}_{y} f[\!{\mathbf {k}}] = f[\!k_{1}+1, k_{2}]  f [\!k_{1}, k_{2}]\)  \(\partial ^{}_{y} f[\!{\mathbf {k}}] = f[\!k_{1}, k_{2}]  f [\!k_{1}1, k_{2}]\) 
Matrix  \(\left [ \partial ^{+}_{x} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } = {\mathbf {f}} \, {\mathbf {D}_{\mathbf {n}}^{\text {T}}}\)  \(\left [ \partial ^{}_{x} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } =  {\mathbf {f}} \, {\mathbf {D}_{\mathbf {n}}}\) 
form  \(\left [ \partial ^{+}_{y} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } = {\mathbf {D}_{\mathbf {m}}} {\mathbf {f}}\)  \(\left [ \partial ^{}_{y} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } =  {\mathbf {D}_{\mathbf {m}}^{\text {T}}} {\mathbf {f}}\) 
Fourier  \(\left (e^{j \omega _{2}}  1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\)  \( \left (e^{j\omega _{2}}  1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\) 
transform  \(\left (e^{j\omega _{1}}  1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\)  \( \left (e^{j \omega _{1}}  1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\) 
2.1 Discrete directional TV norm
2.2 Discrete directional Gnorm
3 DG3PD
where \(\mathcal {C}\) is the curvelet transform [4, 5] with the index set \(\mathcal {K}\). Please note that by setting the parameter δ=0 in (5), we obtain a twopart decomposition which can be considered as a special case of the DG3PD model. Next, we discuss how the DG3PD model relates to existing decomposition models.
4 Related work
In this section, we give an overview of prior work in twopart and threepart image decomposition in chronological order and we explain how it relates to the proposed DG3PD model.
Mumford and Shah: In 1989, Mumford and Shah [6] have proposed piecewise smooth (for image restoration) and piecewise constant (for image segmentation) models by minimizing the energy functional. This approach can be considered as a precursor for subsequent image decomposition models with texture components. However, due to the Hausdorff onedimensional measure \(\mathcal {H}^{1}\) in \(\mathbb {R}^{2}\), it poses a challenging or even NPhard problem in optimization to minimize the Mumford and Shah functional. Later, based on the Mumford and Shah model, Chan and Vese [7] have proposed an active contour for image segmentation which they solved by a level set method [8].
Rudin, Osher, and Fatemi: In 1992, Rudin et al. [1] pioneered image decomposition with a twopart model for denoising.
Meyer: The model defined by Meyer in 2001 [2] for a twopart decomposition in the continuous setting is comprised in the DG3PD model for the special case of L=S=2 and μ _{2}=δ=0 in the discrete domain.
Vese and Osher: In 2003, Vese and Osher [9] solved Meyer’s model for twopart decomposition in the continuous setting and they proposed to approximate the L _{ ∞ }norm in the Gnorm by the L _{1}norm and to apply the penalty method for reformulating the constraint. For practical application to images, they discretized their solution. This approach is extended in [10].
Aujol and Chambolle: Aujol and Chambolle in 2005 [3] adapted the work by Meyer for discrete twopart decomposition (see (5.49) in [3]), and they used the penalty method for the constraint. Their model is included in the DG3PD model with parameters L=S=2, μ _{1}=μ _{2}=0 and applying the supremum norm to the wavelet coefficients of the oscillating pattern, i.e., \({\left \\mathcal {W} \{ \cdot \}\right \}_{\ell _{\infty }}\), instead of the curvelet coefficients \({\left \ \mathcal {C} \{ \cdot \}\right \}_{\ell _{\infty }}\) as in the DG3PD model. Moreover, Aujol and Chambolle proposed a model for discrete threepart decomposition (see (6.59) in [3]) which measures texture by the Gnorm and noise by the supremum norm of wavelet coefficients and the penalty method for the constraint. Different from Vese and Osher as well as our approach explained later, they describe the Gnorm for capturing texture using the indicator function defined on a convex set and they obtain the solution by Chambolle’s projection onto this convex set. Their model is included in the DG3PD model for parameters L=S=2, μ _{2}=0 and using wavelets instead of curvelets for the residual as before.
Starck et al.: Starck et al. [11] introduced a model for twopart decomposition based on a dictionary approach. Their basic idea is to choose one appropriate dictionary for piecewise smooth objects (cartoon) and another suitable dictionary for capturing texture parts.
Aujol et al.: In 2006, Aujol et al. [12] proposed a twopart decomposition of an image into a structure component and a texture component using Gabor functions for the texture part.
Gilles: Gilles [13] proposed a threepart image decomposition method in 2007 which is similar to the AujolChambolle model [3], but Gnorm is used as a measurement of the residual (or noise) instead of Besov space \(\dot {B}^{\infty }_{1, \infty }\) with a local adaptability property. Their argument is that the more a function is oscillatory, the smaller is the G norm. Then, they propose a new “mergedalgorithm” with a combination of a local adaptivity behavior and Besov space.
Maragos and Evangelopoulos: In 2007, Maragos and Evangelopoulos [14] have proposed a twopart decomposition model which relies on energy responses of a bank of 2D Gabor filters for measuring the texture component. They discuss the connection between Meyer’s oscillating functions [2], Gabor filters [15], and AMFM image modeling [16, 17].
Buades et al.: In 2010, Buades et al. [18] derived a nonlinear filter pair for twopart decomposition into cartoon and texture parts. Further models for twopart decomposition are listed in Table 1 of [18].
Maurel et al. and Chikkerur et al.: In 2011, Maurel et al. [19] proposed a decomposition approach which models the texture component by local Fourier atoms. For fingerprint textures, Chikkerur et al. [20] proposed in 2007 the application of local Fourier analysis (or shorttime Fourier transform, STFT) for image enhancement. However, the usefulness of local Fourier analysis for capturing texture information depends on and is limited by the level of noise in the corresponding local window, see Figure 2c in [21] for an example in which STFT enhances some regions successfully and fails in other regions.
G3PD: A model for discrete threepart decomposition of fingerprint images has recently been proposed by Thai and Gottschlich in 2015 [25] with the aim of obtaining a texture image v which serves as a useful feature for estimating the region of interest (ROI). The G3PD model is included in the DG3PD model by choosing L=2 and replacing the directional Gnorm in the DG3PD model by the ℓ _{1}norm of curvelet coefficients (multiscale and multiorientation decomposition) to capture texture. However, a disadvantage of the ℓ _{1}norm of curvelet coefficients is a tendency to generate the halo effect on the boundary of the texture region due to the scaling factor in curvelet decomposition (see Figure 3d in [25]), whereas the directional Gnorm in the DG3PD model is capable to capture oscillating patterns (see [2]) without the halo effect.
Directional total variation and Gnorm: In Section 2.1, we introduced the discrete directional total variation norm, and in Section 2.2, we introduced the discrete directional Gnorm. Please note the aspect of summation over multiple directions in Eqs. (2) and (4).
The term “directional total variation” has previously been used by Bayram and Kamasak [26, 27] for defining and computing the TV norm in only one specific direction. They have treated the special case of images with one globally dominant direction and addressed those by twopart decomposition and for the purpose denoising. Zhang and Wang [28] proposed an extension of the work by Bayram and Kamasak for denoising images with more than one dominant direction.
5 Solution of the DG3PD model
By analogy with the work of Vese and Osher, we consider the approximation of Gnorm with ℓ _{1} norm and the anisotropic version of directional total variation norm. The minimization problem in (5) is rewritten as
and the Lagrange multipliers are updated after every step t with a rate γ. We initialize \({\mathbf {u}}^{(0)} = {\mathbf {f}} \,, {\mathbf {v}}^{(0)} = {\boldsymbol {\epsilon }}^{(0)} = \left [ {\mathbf {r}}_{l}^{(0)} \right ]_{l=0}^{L1} = \left [ {\mathbf {w}}_{s}^{(0)} \right ]_{s=0}^{S1} = \left [ {\mathbf {g}}_{s}^{(0)} \right ]_{s=0}^{S1} = \left [\boldsymbol {\lambda }_{\mathbf {1}l}^{(0)}\right ]_{l=0}^{L1} = \left [\boldsymbol {\lambda }_{\mathbf {2}a}^{(0)}\right ]_{a=0}^{S1} = \boldsymbol {\lambda }_{\mathbf {3}}^{(0)} = \boldsymbol {\lambda }_{\mathbf {4}}^{(0)} =\boldsymbol {0}\).In each iteration, we first solve the following six subproblems in the listed order and then we update the four Lagrange multipliers:
The operator Shrink(·,·) is defined in [25].
Choice of parameters
where \(\phantom {\dot {i}\!}t_{{\mathbf {w}}_{a}} [\!{\mathbf {k}}]\) and t _{ v }[ k] are defined in (11) and (16), respectively. Note that the choice of \(c_{\mu _{1}}\) and \(c_{\mu _{2}}\) is adapted to specific images.
6 Comparison of DG3PD with prior art

Goal 1: u contains only geometrical objects with a very smooth surface, sharp boundaries, and no texture.

Goal 2: v contains only objects with sparse oscillating patterns and v shall be both smooth and sparse.

Goal 3: Perfect reconstruction of f, i.e., f=u+v+ε.

Twopart decomposition instead of threepart decomposition.

Quadratic penalty method (QPM) for solving the constrained minimization instead of ALM.
Goal 1: cartoon u. Regarding the cartoon u, we observe that the VO and SED models still contain texture on the scarf (see Fig. 4 b, c, respectively). For the SED, the cartoon u is blurred and there are some smallscale objects under the table (see Fig. 5 c). Comparing all five methods, VO and SED are furthest away from achieving the first goal, whereas ROF and TVG generate better cartoon images in terms of smoother surfaces without texture and sharper boundaries. Recently, Schaeffer and Osher [23] suggested a twopart decomposition approach. Note that the cartoon image by their decomposition which is shown in Figure 7a of [23] also contains texture and does not meet the first goal. The cartoon images produced by the DG3PD method come closest to the first goal, see Fig. 4 i, m and Fig. 5 e.
Goal 2: texture v. Concerning the texture v, among the stateoftheart methods, the decomposition by the SED model results in the sparsest texture (see Figs. 4 g and 5 h), while the texture images of the ROF, VO, and TVG have more coefficients different from zero. In addition, the texture component obtained by the ROF model also contains some geometry information which should have been assigned to the cartoon component, see Figs. 4 e and 5 f. The DG3PD model yields an even sparser texture than the SED model due to the \({\left \{\mathbf {v}}\right \}_{\ell _{1}}\) in the minimization (5), see the binarized versions with threshold “0” for visualization in Fig. 4 o or Fig. 2 g.
However, there is a limitation for both methods: the noise image ε contains some pieces of information due to the value of δ which defines the level of the noise. Similar to [3], we modify the classical threshold for curvelet coefficients, i.e., \(\sigma \sqrt {2 \log {\left \mathcal {K}\right }}\), with a weighting parameter η as follows \(\delta = \eta \sigma \sqrt {2 \log {\left \mathcal {K}\right }}\) and \({\left \mathcal {K}\right }\) is total number of curvelet coefficients.
Summary. We observe that the DG3PD method meets all three requirements much more closely than the other methods for images without noise, like the original Barbara image. And in particular, for images with additive noise, the DG3PD method still achieves all three goals as shown in the comparison with Aujol and Chambolle.
7 Applications
Here, we limit ourselves to consider three important applications of the DG3PD method: feature extraction, denoising, and image compression.
7.1 Feature extraction
7.2 Denoising
The DG3PD model can be used for denoising images with texture because noise and smallscale objects are moved into the residual image ε during the decomposition of f due to the supremum norm of the curvelet coefficients of ε. Therefore, the image f _{denoised}=u+v can be regarded as a denoised version of f and the degree of denoising can be steered by the choice of parameters, especially δ. For δ=0 which is equal to twopart decomposition, we obtain the original image again. As we increase δ, more noise is driven into ε and thereby removed from f _{denoised}. Denoising images with texture, in particular with texture parts on different scales, is a relevant problem which we plan to address in future works.
Jung and Kang [34] proposed a variational minimization for vectorvalued (color) image decomposition and restoration. Different from our directional total variation, their energy function involves a weighted secondorder regularization for the cartoon component also to reduce the staircase effect and to provide image restoration of higher quality. Similar to the VeseOsher model, they use the L _{2} norm for measuring the residual which is different from our proposed \({\left \{\mathcal {C}}\{ \cdot \}\right \}_{\infty }\). Moreover, their reconstructed texture is not sparse due to a lack of the assumption on its sparsity in their minimization model; see Figure 14 b in [34]. In a different view of convex minimization for image reconstruction, Tschumperle [35] proposed a curvaturepreserving method for anisotropic smoothing of multivalued images while preserving natural curvature constraints (or the edges). Then, line integral convolutions are applied for a numerical scheme to this tensordriven diffusion PDE with two main advantages: namely, it preserves the orientation of thin image structure and the cost of computation is smaller in comparison to the classical explicit scheme. This twopart decomposition scheme is applied for denoising, inpainting, and resizing of vectorvalued images.
7.3 Compression
Based on the DG3PD model, we propose a novel approach to image compression. The core idea is to perform image decomposition by the DG3PD method first and subsequently to compress the three component images by three different algorithms, each particularly suited for compressing the specific type of image. This scheme can be used for lossy as well as lossless compression.
7.3.1 Cartoon image compression
As stated in our definition of goals, the cartoon image consists of geometric objects with a very smooth or piecewise constant surface and sharp edges. This special kind of images is highly compressible, and a very effective approach is based on diffusion. Anisotropic diffusion [36] is useful for many purposes in image processing, e.g., fingerprint image enhancement by oriented diffusion filtering [21].
The basic idea of diffusionbased compression is store information for only a few sparse locations which encode the edges of the cartoon image. The surface areas are inpainted using a linear or nonlinear diffusion process. Please note that the cartoon image obtained by the DG3PD method is much better suited for this type of compression due to the property of sharper edges between geometric objects in comparison to the cartoon images of the other decomposition approaches. Moreover, some difficulties and drawbacks of diffusionbased compression for arbitrary images do not apply to this special case. In general, it is a challenging question how and where to select locations for diffusion seed points. In our case, this task is easily solvable because of the sharp edges between homogeneous regions in the DG3PD cartoon image. This allows for an extremely sparse selection of locations on corners and edges.
Image compression with edgeenhancing anisotropic diffusion (EED) has been studied by Galic et al. [37] and has been improved by Schmaltz et al. [38]. Compression of cartoonlike images with homogeneous diffusion has been analyzed by Mainberger et al. [39].
A viable alternative to diffusionbased compression of cartoon images is a dictionarybased approach [40] in which the dictionary is optimized for cartoon images. Another very promising possibility to compress the DG3PD cartoon component is the usage of linear splines over adaptive triangulations which has been proposed in the work of Demaret et al. [41].
7.3.2 Texture image compression
Tailormade solutions are available for texture image compression and especially for compressing oscillating patterns like fingerprints.
A third, very good compression possibility is dictionary learning [40] with optimization of the dictionary for the texture component v. For fingerprint images, this problem has recently been studied by Shao et al. [45].
7.3.3 Residual image compression
For image compression using DG3PD, we propose the following steps in this order: First, image decomposition f=u+v+ε. Second, a tailormade, lossy, high compression of the cartoon component u and the texture component v. Third, decompressing u and v in order to compute the compression residual image s=f−u _{ d }−v _{ d }, where u _{ d } is the cartoon image and v _{ d } the texture image after decompression. Fourth, compression of s.
In steps two and four, the term “compression” denotes the whole process including coefficient quantization and symbol encoding (see [46] for scalar quantization, Huffman coding, LZ77, LZW, and many other standard techniques).
Let be e _{ u }=u−u _{ d }, the difference between the cartoon component before and after compression, i.e., the compression error, and e _{ v }=v−v _{ d }, then we can rewrite r=f−u _{ d }−v _{ d }=u+v+ε−u _{ d }−v _{ d }=ε+e _{ u }+e _{ v }. Hence, the residual image s computed in step four contains the residual component ε plus the compression errors of the other two components. Now, lossless compression can be achieved by lossless compression of s. If the goal is lossy compression with a certain target quality or target compression rate, this can be achieved by adapting the lossy compression of s accordingly. See Fig. 11 for the effects of different compression rates on the decompressed cartoon u _{ d } and the decompressed texture v _{ d }.
An additional advantage of decompression beginning with u _{ d }, followed by v _{ d } and finally s _{ d } is the fast generation of a preview image which mimics the effects of interlacing. In a scenario with limited bandwidth for data transmission, e.g., sending an image to a mobile phone, the user can be shown a preview based on the compressed, transmitted, and decompressed u image. During the transmission of the compressed v and s, the user can decide whether to continue or abort the transmission.
8 Conclusions
The DG3PD model is a novel method for threepart image decomposition. We have shown that the DG3PD method achieves the goals defined in the introduction much better than other relevant image decomposition approaches. The DG3PD model lays the groundwork for applications such as image compression, denoising, and feature extraction for challenging tasks such as latent fingerprint processing. We follow in the footsteps of Aujol and Chambolle [3] who pioneered threepart decomposition and DG3PD generalizes their approach. We believe that threepart decomposition is the way forward to address many important problems in image processing and computer vision. Buades et al. [18] asked in 2010: “Can images be decomposed into the sum of a geometric part and a textural part?” Our answer to that question is no if an image contains other parts than cartoon and texture, i.e., noise or smallscale objects. Consider, e.g., the noisy Barbara image in Fig. 8 d. If the sum of the cartoon and texture images shall reconstruct the input image f, a twopart decomposition has to assign the noise parts either to the cartoon or to the texture component. In principle, not even the best twopart decomposition model can fully achieve both goals regarding the desired properties of the cartoon and texture component simultaneously. The solution is that noise and smallscale objects which do not belong to the cartoon or texture have to be allotted to a third component.
In our future work, we intend to optimize the DG3PD method for specific applications, especially image compression and latent fingerprint processing. Issues for improvement include the datadriven, automatic parameter selection, and the convergence rate (can the same decomposition be achieved in fewer iterations?) Furthermore, we plan to explore and evaluate specialized compression approaches for cartoon, texture, and residual images.
Additionally, a very interesting application for the residual component can be biometric liveness detection. Recently, Gragnaniello et al. [47] have concluded that highpass filtering before computing local image descriptors improves the accuracy of their proposed iris liveness detection algorithm. The residual image obtained by DG3PD contains the highfrequency components of the input image. Applications of DG3PD to iris or fingerprint liveness detection [48, 49] are therefore very promising. A survey of local image descriptors for fingerprint, iris, and face liveness detection can be found in [50].
Optimal solutions of transportation problems are the key to compute the earth mover’s distance (EMD) or Wasserstein distance [51]. Recently, Brauer and Lorenz [52] discussed an interesting connection between Meyer’s Gnorm and transportation problems. Solving transportation problems for images of dimension 512×512 pixels or larger can be a computationally extremely challenging problem (depending on the number of producers and consumers, and their distribution over the image domain) even for stateoftheart methods such as the shortlist method [51]. However, in the special case of p=1, the KantorovichRubinstein duality provides a loophole which allows it to avoid solving the associated transportation problem. Based on this property, Brauer and Lorenz [52] proposed a threepart image decomposition with transport norms. Lellmann et al. [53] proposed the use of transport norms for image denosing and twopart image decomposition. We believe that the commonalities between the VeseOsher model [9], Meyer’s Gnorm, and the recently proposed models using transport norms deserve further research.
Moreover, our experiments have shown that the curvelet transform as part of the DG3PD model is very suitable for capturing the residual component in threepart image decomposition in terms of i.i.d. (or weakly correlated) Gaussian (or nonGaussian) noise. In principle, it is also conceivable to apply the shearlet transform [54], the contourlet transform [55], or the steerable wavelet transform [56] instead of the curvelet transform, and we deem it worth investigating if especially for specific types of images one of these transforms performs considerably better than the others—as part of a DG3PD model with directional total variational norms, directional Gnorms, or transport norms.
9 Appendix
Declarations
Acknowledgements
The authors would like to thank Luminita Vese for her permission to show the images in Figs. 4(b, f), 5(b, g), 6(b, f, j, n, e, i) and 12(a, b). The authors would like to thank JeanFrancois Aujol for his permission to depict the images in Figs. 4(d, h), 5(d, i), 8(a, b, c) and 12(d, h, a, e). The authors would like to thank JeanLuc Starck for his permission to display the images in Figs. 4(c, g), 5(c, h), and 12(c, g). The authors gratefully acknowledge support by the German Research Foundation (DFG) and the Open Access Publication Funds of the University of Göttingen. D.H. Thai is supported by the National Science Foundation under Grant DMS1127914 to the Statistical and Applied Mathematical Sciences Institute. C. Gottschlich gratefully acknowledges support by the FelixBernsteinInstitute for Mathematical Statistics in the Biosciences and the Niedersachsen Vorab of the Volkswagen Foundation.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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