Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1137 Distance Functions and Their Use in Adaptive Mathematical Morphology VLADIMIR ĆURIĆ ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 UPPSALA ISBN 978-91-554-8923-6 2014 urn:nbn:se:uu:diva-221568 Dissertation presented at Uppsala University to be publicly examined in 2347, Lägerhyddsvägen 2, Hus 2, Uppsala, Friday, 23 May 2014 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Hugues Talbot (University Paris-Est - ESIEE). Abstract Ćurić, V. 2014. Distance Functions and Their Use in Adaptive Mathematical Morphology. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1137. 88 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-8923-6. One of the main problems in image analysis is a comparison of different shapes in images. It is often desirable to determine the extent to which one shape differs from another. This is usually a difficult task because shapes vary in size, length, contrast, texture, orientation, etc. Shapes can be described using sets of points, crisp of fuzzy. Hence, distance functions between sets have been used for comparing different shapes. Mathematical morphology is a non-linear theory related to the shape or morphology of features in the image, and morphological operators are defined by the interaction between an image and a small set called a structuring element. Although morphological operators have been extensively used to differentiate shapes by their size, it is not an easy task to differentiate shapes with respect to other features such as contrast or orientation. One approach for differentiation on these type of features is to use data-dependent structuring elements. In this thesis, we investigate the usefulness of various distance functions for: (i) shape registration and recognition; and (ii) construction of adaptive structuring elements and functions. We examine existing distance functions between sets, and propose a new one, called the Complement weighted sum of minimal distances, where the contribution of each point to the distance function is determined by the position of the point within the set. The usefulness of the new distance function is shown for different image registration and shape recognition problems. Furthermore, we extend the new distance function to fuzzy sets and show its applicability to classification of fuzzy objects. We propose two different types of adaptive structuring elements from the salience map of the edge strength: (i) the shape of a structuring element is predefined, and its size is determined from the salience map; (ii) the shape and size of a structuring element are dependent on the salience map. Using this salience map, we also define adaptive structuring functions. We also present the applicability of adaptive mathematical morphology to image regularization. The connection between adaptive mathematical morphology and Lasry-Lions regularization of non- smooth functions provides an elegant tool for image regularization. Keywords: Image analysis, Distance functions, Mathematical morphology, Adaptive mathematical morphology, Image regularization Vladimir Ćurić, Department of Information Technology, Division of Visual Information and Interaction, Box 337, Uppsala University, SE-751 05 Uppsala, Sweden. © Vladimir Ćurić 2014 ISSN 1651-6214 ISBN 978-91-554-8923-6 urn:nbn:se:uu:diva-221568 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-221568) To my mother Stana and my father Zarkoˇ Mami i tati List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Lindblad, J., Curi´c,´ V., Sladoje, N. (2009) On set distances and their application to image registration, In Proceedings of the 6th International Symposium on Image and Signal Processing and Analysis (ISPA 2009), IEEE, pp. 449–454. II Curi´c,´ V., Lindblad, J., Sladoje, N., Sarve, H., Borgefors, G. (2014) A new set distance and its application to shape registration, Pattern Analysis and Applications, Volume 17, Issue 1, pp. 141–152. III Curi´c,´ V., Lindblad, J., Sladoje, N. (2011) Distance Measures between Digital Fuzzy Objects and Their Applicability in Image Processing, In Proceedings of the 14th International Workshop on Combinatorial Image Analysis (IWCIA 2011), LNCS–6636, pp. 385–397. IV Curi´c,´ V., Luengo Hendriks, C.L., Borgefors, G. (2012) Salience Adaptive Structuring Elements, IEEE Journal of Selected Topics in Signal Processing, Special Issue on Filtering and Segmentation in Mathematical Morphology, Volume 6, Issue 7, pp. 809–819. V Curi´c,´ V., Luengo Hendriks, C.L., (2012) Adaptive Structuring Elements Based on Salience Information, In Proceedings of the International Conference on Computer Vision and Graphics (ICCVG 2012), LNCS–7594, pp. 321–328. VI Curi´c,´ V., Luengo Hendriks, C.L., (2013) Salience-Based Parabolic Structuring Functions, In Proceedings of the 11th International Symposium on Mathematical Morphology (ISMM 2013), LNCS–7883, pp. 181–192. VII Curi´c,´ V., Angulo, J., Morphological Image Regularization Using Adaptive Structuring Functions, Manuscript for journal publication. VIII Curi´c,´ V., Landstr¨om, A., Thurley, M., Luengo Hendriks, C.L. Adaptive Mathematical Morphology – a Survey of the Field, Accepted to Pattern Recognition Letters Reprints were made with permission from the publishers. The author has contributed considerably to method development, im- plementations and writing of Papers II-VIII. The author also contributed to Paper I, but to a lesser extent. The work presented in Papers I-III were developed under close discussions with Joakim Lindblad and Nataˇsa Sladoje, who also contributed in writing. The code for multi-modal im- age registration in Paper II was mostly developed by Hamid Sarve, and Gunilla Borgefors contributed in writing. The work in Papers IV-VI was developed in close discussions with Cris L. Luengo Hendriks. The au- thor developed and implemented the methods, and wrote the papers, but with comments and advices from the coauthors. The method in Paper VII was developed in close discussions with Jes´us Angulo. The author implemented the method and wrote the paper. Paper VIII was written in a close collaboration with Anders Landstr¨om. The implemen- tations and writing was split between the two. The other coauthors contributed in discussions and comments. Related work In addition to the papers included in this thesis, the author has also written or contributed to the following publications. 1. Lindblad, J., Sladoje, N., Curi´c,´ V., Sarve, H., Johansson, C.B., Borgefors, G. (2009) Improved quantification of bone remodelling by utilizing fuzzy based segmentation, In Proceedings of the 16th Scandinavian Conference on Image Analysis (SCIA 2009), LNCS– 5575, pp. 750–759. 2. Curi´c,´ V., Heili¨o, M., Kreji´c, N., Nedeljkov, M., (2010) Mathemati- cal Model for Efficient Water Flow Management, Nonlinear Analysis – Real World Applications, Volume 11, Issue 3, pp. 1600–1612. 3. Curi´c,´ V., Lindblad, J., Sladoje, N. (2010) The Sum of minimal distances as a useful distance measure for image registration, In Proceedings of the Swedish Symposium on Image Analysis (SSBA 2010), pp. 55–58. 4. Allalou, A., Curi´c,´ V., Pardo Martin, C., Yanik, M.F., W¨ahlby, C. (2011) Approaches for increasing throughput and information con- tent of image-based zebrafish screens, In Proceedings of the Swedish Symposium on Image Analysis (SSBA 2011), pp. 5–8. 5. Curi´c,´ V., Luengo Hendriks, C.L., Borgefors G. (2012) Adaptive structuring elements based on salience distance transform, In Pro- ceedings of the Swedish Symposium on Image Analysis (SSBA 2012), pp. 127–130. 6. Gonz´alez-Castro, V., Debayle, J., Curi´c,´ V. (2014) Pixel Classifica- tion using General Adaptive Neighbourhood-based Features, To ap- pear in Proceedings of the 22th International Conference on Pat- tern Recognition (ICPR 2014), IEEE. Contents 1 Introduction ................................................... ......................................... 11 1.1 Motivation ................................................... ................................ 11 1.2 Thesisoutline ................................................... .......................... 12 2 Distance Functions ................................................... .............................. 13 2.1 Brief Introduction to Distance Functions .............................. 13 2.2 Distance Functions Between Sets ........................................... 15 2.3 Distance Functions Between Fuzzy Sets ............................... 19 2.4 Distance Transforms ................................................... ............... 22 2.4.1 Distance Transform ................................................... 22 2.4.2 Gray-Weighted Distance Transform ....................... 24 2.4.3 Salience Distance Transform ................................... 25 2.5 Applications to Image Registration ........................................ 27 3 Mathematical Morphology ................................................... ................ 33 3.1 Basic Morphological Operators ............................................... 34 3.2 Adjunction Property ................................................... ............... 36 3.3 Opening and Closing ................................................... .............. 37 3.4 Other morphological operators ............................................... 39 3.5 On the Selection