Wavelet Shrinkage on Paths for Denoising of Scattered Data

Heinen, Dennis; Plonka, Gerlind

doi:10.1007/s00025-012-0285-3

Cited by 6 publications

(6 citation statements)

References 17 publications

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“…Algorithm 1 shows the easy-path procedure to find a path π in a region R: starting from some point (which for example can be chosen using the lexicographical ordering), the algorithm tries always to select the closest avaiable neighbour. If there are more points equally close, it selects the one that would make for the straightest path, the rationale being that a more regular path will lead to a smoother signal (see for example [7] for a proof that, when f is sufficiently…”

Section: Easy-pathmentioning

confidence: 99%

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A region-based easy-path wavelet transform for sparse image representation

Budinich

2017

Int. J. Wavelets Multiresolut Inf. Process.

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The Easy Path Wavelet Transform is an adaptive transform for bivariate functions (in particular natural images) which has been proposed in [13]. It provides a sparse representation by finding a path in the domain of the function leveraging the local correlations of the function values. It then applies a one dimensional wavelet transform to the obtained vector, decimates the points and iterates the procedure. The main drawback of such method is the need to store, for each level of the transform, the path which vectorizes the two dimensional data. Here we propose a variation on the method which consists of firstly applying a segmentation procedure to the function domain, partitioning it into regions where the variation in the function values is low; in a second step, inside each such region, a path is found in some deterministic way, i.e. not data-dependent. This circumvents the need to store the paths at each level, while still obtaining good quality lossy compression. This method is particularly well suited to encode a Region of Interest in the image with different quality than the rest of the image.

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Section: Easy-pathmentioning

confidence: 99%

“…In [15] and [14] it was shown that, with a suitable choiche of the paths, the N -term approximation given by the EPWT is optimal for piecewise-Hölder functions. In [16] the EPWT was used as part of a hybrid method for Image Approximation while in [7] for denoising of scattered data.…”

Section: Introductionmentioning

confidence: 99%

A region-based easy-path wavelet transform for sparse image representation

Budinich

2017

Int. J. Wavelets Multiresolut Inf. Process.

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“…Hence, the EPWT is an adaptive transform in contrast to the other transforms mentioned above. Its approximation properties are well understood and its usefulness for image processing is well established, see [49,97,96].…”

Section: Shearletsmentioning

confidence: 99%

Phase Retrieval with Sparsity Constraints

Loock¹

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Tag der mündlichen Prüfung: 7. Juni 2016 welcoming atmosphere, a continuous coffee supply, and many interesting discussions. This thesis would not have been possible without the generous financial support of the collaborative research center SFB755 "Nanoscale Photonic Imaging" which also provided for many interesting workshops and fantastic travel opportunities. In this regard, I would like to express my special gratitude to Tim Salditt whose enthusiasm in bringing together different scientific disciplines is exceptional.The last years would not have been as interesting and joyful as they were without the friendship of many people. I would like to especially mention Sirach Lotz, Malte Rösner, Malte Schüler, and Bruno Schyska. They made my time in Bremen more cheerful, comforting, and inspiring than I could have ever hoped for. I am furthermore very thankful for the friendship of Jan Preibisch and Kai Feldhusen.I am eternally grateful to my parents, Susann and Otmar Loock, for their understanding, advice, and never-ending support. Their hard work and dedication provided me with incredible role models and gave me the freedom to pursue my interests.Special thanks are due to Anna Wolf whose determination was inspiring and set an example for me. It was her encouragement, support, and love which made me stick it out in the end. Thank you! Conclusion A. The Mathematical Model B. Fundamental Results from Convex Analysis Bibliography Curriculum VitaeList of Figures 5.5. Reconstructions of a real-valued object from exact measurements for different constraints with starting guess x (0) = m.

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“…In Plonka et al (2012) and Plonka et al (2013) it was shown that, with a suitable choice of the paths, the N -term approximation given by the EPWT is optimal for piecewise-Hölder functions. In Plonka et al (2011) the EPWT was used as part of a hybrid method for Image Approximation while in Heinen and Plonka (2012) for denoising of scattered data.…”

Section: Rbepwtmentioning

confidence: 99%

“…append ψ to π v = ψ − p p = ψ remove ψ from Q end while 20: return π 3 The Region Based Easy Path Wavelet Transform rationale being that a more regular path will lead to a smoother signal (see for example Heinen and Plonka (2012) for a proof that, when f is sufficiently smooth, a straighter path gives smaller wavelet coefficients). This is done by computing the scalar product of the increment vector with a preferred direction vector, which at every iteration is updated to be the last increment in the path.…”

Section: Rbepwtmentioning

confidence: 99%

Adaptive Multiscale Methods for Sparse Image Representation and Dictionary Learning

Budinich¹

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Taking the PhD position at the Institut für Numerische und Angewandte Mathematik in Göttingen was a decision that involved many unknowns: I would have to separate from my family, friends and girlfriend, I didn't know German and I hadn't yet met my supervisor. Still, I had a good feeling about it and I can now say it turned out to be the most beneficial event yet to occur in my life. I met amazing people from all over the world that helped me discover my true vocations and the way I should pursue, giving me the most precious gift of friendship when it was most needed. They have made my stay abroad not only tolerable but exciting and I now look forward to the many more adventures that await us together in the future. This thesis is, I hope, the proof that I also couldn't have hoped for a better supervisor than Gerlind Plonka, who by example taught me the value of perseverance and hard work in research. Above all, Gerlind fostered my confidence in my own abilities by always giving me great freedom and being very encouraging and appreciative of my work, even when I thought it didn't hold so much value.I want to dedicate this thesis to my whole "Vitalba" family, with whom I grew up and whom I sorely miss. In particular, I want to express my gratitude to my parents for having always been there like a trusted beacon, even (especially) when the sea was rough or the horizon covered in fog. I also want to dedicate this work to my friend Giulio Regeni, who was killed like a beast and taught me to live like a Man.

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Wavelet Shrinkage on Paths for Denoising of Scattered Data

Cited by 6 publications

References 17 publications

A region-based easy-path wavelet transform for sparse image representation

A region-based easy-path wavelet transform for sparse image representation

Phase Retrieval with Sparsity Constraints

Adaptive Multiscale Methods for Sparse Image Representation and Dictionary Learning

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