[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122587-en":3,"doc-seo-122587-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},122587,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Euclidean Equivariant Processing and Machine Learning on Position-Orientation Space","This PhD thesis explores the interplay between Euclidean equivariant processing, position-orientation space, and machine learning. The work develops theoretical tools and models that respect Euclidean symmetries while operating on the geometry of position-orientation representations. Emphasis is placed on practical impact through machine-learning components, including PDE-based neural networks and distance/invariant constructions. Experiments are presented to validate the proposed approaches and demonstrate their usefulness across concrete application scenarios.","Euclidean Equivariant Processing and Machine Learning on Position-Orientation Space  \nCitation for published version (APA):  \nBellaard, G. (2025) . Euclidean Equivariant Processing and Machine Learning on Position-Orientation Space.[Phd Thesis 1 (Research TU/e / Graduation TU/e), Mathematics and Computer Science] . Eindhoven University of Technology.  \nDocument status and date:  \nPublished: 26/11/2025  \nDocument Version:  \nPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)  \nPlease check the document version of this publication:  \n• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.  \n• The final author version and the galley proof are versions of the publication after peer review.  \n• The final published version features the final layout of the paper including the volume, issue and page numbers.  \nLink to publication  \nGeneral rights  \nCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  \n• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.  \n• You may not further distribute the material or use it for any profit-making activity or commercial gain  \n• You may freely distribute the URL identifying the publication in the public portal.  \nIf the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:  \n[www.tue.nl/taverne](www.tue.nl/taverne)  \nTake down policy  \nIf you believe that this document breaches copyright please contact us at:  \n[openaccess@tue.nl](openaccess@tue.nl)  \nproviding details and we will investigate your claim.  \nDownload date: 28. Apr. 2026  \nEuclidean Equivariant Processing and Machine Learning on Position-Orientation Space  \nProefschrift  \nter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus prof. dr. S. K. Lenaerts, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op woensdag 26 november 2025 om 13:30 uur  \ndoor  \nGijs Bellaard  \ngeboren te Geertruidenberg  \nDit proefschrift is goedgekeurd door de promotoren en desamenstelling van de promotiecommissie is als volgt:  \nvoorzitter: prof. dr. K.M. van Hee  \npromotor: [dr. ir. R. Duits](dr. ir. R. Duits)  \nco-promotor: [dr. ir. B.M.N. Smets](dr. ir. B.M.N. Smets)  \nleden: prof. dr. C. Brune (Universiteit Twente)  \nprof. dr. V. Dolean-Maini  \nprof. dr. S.F. Portegies Zwart (Universiteit Leiden)  \n[dr. ir. E.J. Bekkers](dr. ir. E.J. Bekkers) (Universiteit van Amsterdam)  \nHet onderzoek dat in dit proefschrift wordtbeschreven is uitgevoerd in overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening.  \nColophon  \nThis document was typeset using Typst 0.13.1 ([https://typst.app](https://typst.app)).  \nGijs Bellaard © 2025. All rights reserved. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, including electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder.  \nA catalogue record is available from the Eindhoven University of Technology Library.  \nISBN: 978-90-386-6537-5.  \nPrinted by ADC Nederland ([https://adc.nl](https://adc.nl)).  \n4  \n5 Contents  \n7 Introduction  \n11 Concrete Applications  \n14 Outline  \n15 Outtakes  \n17 1 Preliminaries  \n17 1.1 Group Action","cbCaimsbmumwzuJU","https://ap.wps.com/l/cbCaimsbmumwzuJU","pdf",22795197,1,216,"English","en",105,"# Contents\n## Introduction\n## Concrete Applications\n## 1 Preliminaries\n## 2 Background\n## 3 Distance Approximations on Position-Orientation Space\n## 4 Invariant Metrics on Position-Orientation Space\n## 5 Invariants between Pairs of Position-Orientations\n## 6 Geometric Adaptations of PDE-based Neural Networks\n## 7 Semifield Scale-Spaces in PDE-based Neural Networks\n## Publications\n## Bibliography\n## Repositories\n## Index\n## Figures\n## Tables\n## Acknowledgements\n## About the Author","[{\"question\":\"What is the core research theme of this thesis?\",\"answer\":\"The thesis studies how to combine Euclidean equivariant processing on position-orientation space with machine learning, with a strong emphasis on theoretical foundations and machine-learning methods.\"},{\"question\":\"Which kinds of machine learning models are highlighted?\",\"answer\":\"The thesis particularly focuses on PDE-based neural networks and geometric adaptations that incorporate invariance and scale-space representations on position-orientation space.\"},{\"question\":\"How does the thesis support its theoretical developments?\",\"answer\":\"It includes distance approximations, invariant metrics, and invariants between position-orientations, followed by experiments and conclusions for multiple chapters to validate the proposed methods.\"}]","Euclidean Equivariant Processing and Machine Learning on Position-Orientation Space | 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