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Distance Functions and Their Use in Adaptive Mathematical Morphology
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Visual Information and Interaction. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computerized Image Analysis and Human-Computer Interaction.
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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.

##### Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2014. , 88 p.
##### Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1137
##### Keyword [en]
Image analysis, Distance functions, Mathematical morphology, Adaptive mathematical morphology, Image regularization
##### National Category
Computer Vision and Robotics (Autonomous Systems)
##### Research subject
Computerized Image Processing
##### Identifiers
ISBN: 978-91-554-8923-6 (print)OAI: oai:DiVA.org:uu-221568DiVA: diva2:709476
##### Public defence
2014-05-23, Room 2347, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 13:15 (English)
##### Supervisors
Available from: 2014-04-28 Created: 2014-04-01 Last updated: 2014-07-21
##### List of papers
1. On set distances and their application to image registration
Open this publication in new window or tab >>On set distances and their application to image registration
2009 (English)In: Proc. 6th International Symposium on Image and Signal Processing and Analysis, Piscataway, NJ: IEEE , 2009, 449-454 p.Conference paper, Published paper (Refereed)
##### Abstract [en]

In this paper we study set distances that are used in image processing. We propose a generalization of Sum of minimal distances and show that its special cases include a metric by Symmetric difference. The Hausdorff metric and the Chamfer matching distances are also closely related with the presented framework. In addition, we define the Complement set distance of a given distance. We evaluate the observed distance with respect to applicability to image object registration. We perform comparative evaluations with respect to noise sensitivity, as well as with respect to rigid body transformations. We conclude that the family of Generalized sum of minimal distances has many desirable properties for this application.

##### Place, publisher, year, edition, pages
Piscataway, NJ: IEEE, 2009
##### National Category
Computer Vision and Robotics (Autonomous Systems)
##### Research subject
Computerized Image Analysis
##### Identifiers
urn:nbn:se:uu:diva-110684 (URN)978-953-184-135-1 (ISBN)
##### Conference
6th International Symposium on Image and Signal Processing and Analysis, Salzburg, Austria, 16-18 September, 2009
Available from: 2009-11-26 Created: 2009-11-23 Last updated: 2014-04-29Bibliographically approved
2. A new set distance and its application to shape registration
Open this publication in new window or tab >>A new set distance and its application to shape registration
2014 (English)In: Pattern Analysis and Applications, ISSN 1433-7541, E-ISSN 1433-755X, Vol. 17, no 1, 141-152 p.Article in journal (Refereed) Published
##### National Category
Discrete Mathematics
##### Identifiers
urn:nbn:se:uu:diva-220413 (URN)10.1007/s10044-012-0290-x (DOI)000330839400011 ()
Available from: 2012-08-23 Created: 2014-03-13 Last updated: 2017-12-05Bibliographically approved
3. Distance measures between digital fuzzy objects and their applicability in image processing
Open this publication in new window or tab >>Distance measures between digital fuzzy objects and their applicability in image processing
2011 (English)In: Combinatorial Image Analysis / [ed] Jake Aggarwal, Reneta Barneva, Valentin Brimkov, Kostadin Koroutchev, Elka Koroutcheva, Springer , 2011, 385-397 p.Conference paper, Published paper (Refereed)
Springer, 2011
##### National Category
Computer Vision and Robotics (Autonomous Systems)
##### Research subject
Computerized Image Analysis
##### Identifiers
urn:nbn:se:uu:diva-157186 (URN)
##### Conference
Internatiional Workshop on Combinatorial Image Analysis, IWCIA 2011
##### Note
We present two different extensions of the Sum of minimal distances and the Complement weighted sum of minimal distances to distances between fuzzy sets. We evaluate to what extent the proposed distances show monotonic behavior with respect to increasing translation and rotation of digital objects, in noise free, as well as in noisy conditions. Tests show that one of the extension approaches leads to distances exhibiting very good performance. Furthermore, we evaluate distance based classification of crisp and fuzzy representations of objects at a range of resolutions. We conclude that the proposed distances are able to utilize the additional information available in a fuzzy representation, thereby leading to improved performance of related image processing tasks. Available from: 2011-08-18 Created: 2011-08-18 Last updated: 2014-04-29
Open this publication in new window or tab >>Salience adaptive structuring elements
2012 (English)In: IEEE Journal on Selected Topics in Signal Processing, ISSN 1932-4553, E-ISSN 1941-0484, Vol. 6, no 7, 809-819 p.Article in journal (Refereed) Published
##### Abstract [en]

Spatially adaptive structuring elements adjust their shape to the local structures in the image, and are often defined by a ball in a geodesic distance or gray-weighted distance metric space. This paper introduces salience adaptive structuring elements as spatially variant structuring elements that modify not only their shape, but also their size according to the salience of the edges in the image. Morphological operators with salience adaptive structuring elements shift edges with high salience to a less extent than those with low salience. Salience adaptive structuring elements are less flexible than morphological amoebas and their shape is less affected by noise in the image. Consequently, morphological operators using salience adaptive structuring elements have better properties.

##### Keyword
Adaptive mathematical morphology, anisotropic filtering, morphological amoebas, salience distance transform
##### National Category
Other Mathematics
##### Identifiers
urn:nbn:se:uu:diva-181248 (URN)10.1109/JSTSP.2012.2207371 (DOI)000310138400007 ()
Available from: 2012-09-20 Created: 2012-09-20 Last updated: 2017-12-07Bibliographically approved
5. Adaptive structuring elements based on salience information
Open this publication in new window or tab >>Adaptive structuring elements based on salience information
2012 (English)In: Computer Vision and Graphics / [ed] L. Bolc, K. Wojciechowski, R. Tadeusiewicz, L.J. Chmielewski, Springer, 2012, 321-328 p.Conference paper, Published paper (Other academic)
##### Abstract [en]

Adaptive structuring elements modify their shape and size according to the image content and may outperform fixed structuring elements. Without any restrictions, they suffer from a high computational complexity, which is often higher than linear with respect to the number of pixels in the image. This paper introduces adaptive structuring elements that have predefined shape, but where the size is adjusted to the local image structures. The size of adaptive structuring elements is determined by the salience map that corresponds to the salience of the edges in the image, which can be computed in linear time. We illustrate the difference between the new adaptive structuring elements and morphological amoebas. As an example of its usefulness, we show how the new adaptive morphological operations can isolate the text in historical documents.

Springer, 2012
##### Series
Lecture Notes in Computer Science, ISSN 03029743 ; 7594
##### National Category
Other Mathematics Other Computer and Information Science
##### Identifiers
urn:nbn:se:uu:diva-181246 (URN)10.1007/978-3-642-33564-8-39 (DOI)000313005700039 ()978-3-642-33564-8 (ISBN)
##### Conference
International Conference on Computer Vision and Graphics, September 24-26, 2012, Warsaw, Poland
Available from: 2012-09-20 Created: 2012-09-20 Last updated: 2014-04-29Bibliographically approved
6. Salience-Based Parabolic Structuring Functions
Open this publication in new window or tab >>Salience-Based Parabolic Structuring Functions
2013 (English)In: Mathematical Morphology and Its Applications to Signal and Image Processing, Springer Berlin/Heidelberg, 2013, 183-194 p.Conference paper, Published paper (Refereed)
##### Abstract [en]

It has been shown that the use of the salience map based on the salience distance transform can be useful for the construction of spatially adaptive structuring elements. In this paper, we propose salience-based parabolic structuring functions that are defined for a fixed, predefined spatial support, and have low computational complexity. In addition, we discuss how to properly define adjunct morphological operators using the new spatially adaptive structuring functions. It is also possible to obtain flat adaptive structuring elements by thresholding the salience-based parabolic structuring functions.

##### Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2013
##### Series
Lecture Notes in Computer Science, ISSN 0302-9743 ; 7883
##### National Category
Other Mathematics
##### Research subject
Mathematics with specialization in Applied Mathematics; Computerized Image Analysis
##### Identifiers
urn:nbn:se:uu:diva-204715 (URN)10.1007/978-3-642-38294-9_16 (DOI)978-3-642-38293-2 (ISBN)
##### Conference
11th International Symposium on Mathematical Morphology
Available from: 2013-08-09 Created: 2013-08-09 Last updated: 2014-04-29Bibliographically approved
7. Morphological image regularization using adaptive structuring functions
Open this publication in new window or tab >>Morphological image regularization using adaptive structuring functions
##### National Category
Other Mathematics
##### Identifiers
urn:nbn:se:uu:diva-221161 (URN)
Available from: 2014-03-25 Created: 2014-03-25 Last updated: 2014-04-29
8. Adaptive Mathematical Morphology: a survey of the field
Open this publication in new window or tab >>Adaptive Mathematical Morphology: a survey of the field
2014 (English)In: Pattern Recognition Letters, ISSN 0167-8655, E-ISSN 1872-7344, Vol. 47, 18-28 p.Article in journal (Refereed) Published
##### Abstract [en]

We present an up-to-date survey on the topic of adaptive mathematical morphology. A broad review of research performed within the field is provided, as well as an in-depth summary of the theoretical advances within the field. Adaptivity can come in many different ways, based on different attributes, measures, and parameters. Similarities and differences between a few selected methods for adaptive structuring elements are considered, providing perspective on the consequences of different types of adaptivity. We also provide a brief analysis of perspectives and trends within the field, discussing possible directions for future studies.

##### National Category
Computer Vision and Robotics (Autonomous Systems)
##### Identifiers
urn:nbn:se:uu:diva-221159 (URN)10.1016/j.patrec.2014.02.022 (DOI)000339999200003 ()
Available from: 2014-03-18 Created: 2014-03-25 Last updated: 2017-12-05Bibliographically approved

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Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
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Language
• de-DE
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• en-US
• fi-FI
• nn-NO
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Output format
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