## PA170 Digital Geometry

Faculty of Informatics
Autumn 2021
Extent and Intensity
2/1/0. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Pavel Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 52 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Learning outcomes
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attendance at exercises. Homework score.
Language of instruction
English
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2019
Extent and Intensity
2/1/0. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Pavel Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Timetable
Thu 12:00–14:50 A218
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 52 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Learning outcomes
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attendance at exercises. Homework score.
Language of instruction
English
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2017
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Petr Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Timetable
Wed 10:00–12:50 A318
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 23 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Learning outcomes
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2015
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Petr Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Timetable
Thu 9:00–11:50 A318
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 23 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2013
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Petr Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Timetable
Tue 9:00–11:50 C418
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 22 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2011
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Timetable
Mon 10:00–11:50 A107, Mon 12:00–13:50 A107
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 25 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and an object representation using a grid of points (e.g., in the form of digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms: digital image, pixel, voxel, image resolution, types of grids, grid scanning
• grid point and grid cell models: adjacency, incidence, connectedness, components, component labeling algorithms.
• Digitalization: digitization models, line digitization.
• Measurement in digital images: metrics, integer-valued metrics approximating Euclidean metric, distance transform, distance measurement between sets.
• Oriented adjacency graphs: border, boundary, border tracing algorithm, holes, combinatorial results for regular graphs (grids)
• Application of graph theory in image processing, graph-cut based image segmentation.
• Incidence pseudographs, open and closed regions, ordered labeling of multilevel images.
• Introduction to topology. Basic topological concepts. Definition of continuous as well as digital curve. Jordan Veblen theorem.
• Euclidean and simplex complexes (triangulation). Topological definition of surfaces and their classification. Combinatorial results. Regular tilings.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Digital straight segment recognition, digital straightness, digital convex hull and its computation.
• Image deformations: Thinning, skeletons.
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2010
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Timetable
Mon 10:00–11:50 C525, Mon 12:00–13:50 C525
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 24 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and an object representation using a grid of points (e.g., in the form of digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms: digital image, pixel, voxel, image resolution, types of grids, grid scanning
• grid point and grid cell models: adjacency, incidence, connectedness, components, component labeling algorithms.
• Digitalization: digitization models, line digitization.
• Measurement in digital images: metrics, integer-valued metrics approximating Euclidean metric, distance transform, distance measurement between sets.
• Oriented adjacency graphs: border, boundary, border tracing algorithm, holes, combinatorial results for regular graphs (grids)
• Application of graph theory in image processing, graph-cut based image segmentation.
• Incidence pseudographs, open and closed regions, ordered labeling of multilevel images.
• Introduction to topology. Basic topological concepts. Definition of continuous as well as digital curve. Jordan Veblen theorem.
• Euclidean and simplex complexes (triangulation). Topological definition of surfaces and their classification. Combinatorial results. Regular tilings.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Digital straight segment recognition, digital straightness, digital convex hull and its computation.
• Image deformations: Thinning, skeletons.
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2009
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Timetable
Fri 10:00–11:50 C525, Fri 12:00–13:50 C525
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 24 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and an object representation using a grid of points (e.g., in the form of digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms: digital image, pixel, voxel, image resolution, types of grids, grid scanning
• grid point and grid cell models: adjacency, incidence, connectedness, components, component labeling algorithms.
• Digitalization: digitization models, line digitization.
• Measurement in digital images: metrics, integer-valued metrics approximating Euclidean metric, distance transform, distance measurement between sets.
• Oriented adjacency graphs: border, boundary, border tracing algorithm, holes, combinatorial results for regular graphs (grids)
• Application of graph theory in image processing, graph-cut based image segmentation.
• Incidence pseudographs, open and closed regions, ordered labeling of multilevel images.
• Introduction to topology. Basic topological concepts. Definition of continuous as well as digital curve. Jordan Veblen theorem.
• Euclidean and simplex complexes (triangulation). Topological definition of surfaces and their classification. Combinatorial results. Regular tilings.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Digital straight segment recognition, digital straightness, digital convex hull and its computation.
• Image deformations: Thinning, skeletons.
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
The course is taught annually.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2008
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Timetable
Fri 12:00–13:50 B411, Fri 14:00–14:50 B411
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 17 fields of study the course is directly associated with, display
Course objectives
The course brings the basic overview of digital geometry. We will discuss problems that arise from object digitalization and their representation by the set of points on a grid (e.g, digital image). Especially, we will define the basic terms (adjacency, connectedness, boundary, etc.) and show how to measure geometric as well as topologic properties of digital sets (distance, length, volume, etc.).
Syllabus
• Digital image, pixel, voxel, image resolution
• Types of grids, grid point and grid cell models, adjacency vs. incidence, switch adjacency
• Connectedness and components, component labeling
• Digitalization
• Measurement in digital images: metrics, integer-valued metrics, regular metric, approximation to Euclidean metric, chamfer distance
• Distance transform
• Distance measurement between sets
• Digital sets: digital line, digital circle, etc.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Boundary and border and their computation
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Assessment methods
optional homework, written test and then oral exam
Language of instruction
Czech
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2007
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Timetable
Mon 15:00–15:50 B204, Mon 16:00–17:50 B204
Prerequisites
The knowledge of mathematics fundamentals and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 17 fields of study the course is directly associated with, display
Course objectives
The course brings the basic overview of digital geometry. We will discuss problems that arise from object digitalization and their representation by the set of points on a grid (e.g, digital image). Especially, we will define the basic terms (adjacency, connectedness, boundary, etc.) and show how to measure geometric as well as topologic properties of digital sets (distance, length, volume, etc.).
Syllabus
• Basics: digital image, pixel, voxel, image resolution
• Grids: grid point and grid cell models
• Connectedness and components, component labeling
• Digitalization models
• Measurement in digital images: metrics, integer-valued metrics, regular metric, approximation to Euclidean metric, chamfer distance
• Distance transform
• Distance measurement between sets: Hausdorf metric and its computation
• Digital sets: digital line, digital circle, etc.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Adjacency graphs and incidence pseudographs
• Boundary and border and their computation
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Assessment methods (in Czech)
Přenášky v češtině. Povinná účast na cvičeních, domácí úkoly. Písemná zkouška.
Language of instruction
Czech
Follow-Up Courses
Study Materials
The course is taught annually.
Teacher's information
http://cbia.fi.muni.cz
The course is also listed under the following terms Autumn 2006, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2006
Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Timetable
Mon 17:00–17:50 B011, Tue 17:00–18:50 B204
Prerequisites
The knowledge of mathematics fundamentals and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The course brings the basic overview of digital geometry. We will discuss problems that arise from object digitalization and their representation by the set of points on a grid (e.g, digital image). Especially, we will define the basic terms (adjacency, connectedness, boundary, etc.) and show how to measure geometric as well as topologic properties of digital sets (distance, length, volume, etc.).
Syllabus
• Basics: digital image, pixel, voxel, image resolution
• Grids: grid point and grid cell models
• Connectedness and components, component labeling
• Digitalization models
• Measurement in digital images: metrics, integer-valued metrics, regular metric, approximation to Euclidean metric, chamfer distance
• Distance transform
• Distance measurement between sets: Hausdorf metric and its computation
• Digital sets: digital line, digital circle, etc.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Adjacency graphs and incidence pseudographs
• Boundary and border and their computation
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Assessment methods (in Czech)
Přenášky v češtině. Povinná účast na cvičeních, domácí úkoly. Písemná zkouška.
Language of instruction
Czech
Follow-Up Courses
The course is taught annually.
Teacher's information
http://lom.fi.muni.cz
The course is also listed under the following terms Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2020

The course is not taught in Autumn 2020

Extent and Intensity
2/1/0. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Pavel Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 52 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Learning outcomes
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attendance at exercises. Homework score.
Language of instruction
English
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2018

The course is not taught in Autumn 2018

Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Petr Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 23 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Learning outcomes
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2016

The course is not taught in Autumn 2016

Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Petr Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 23 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2014

The course is not taught in Autumn 2014

Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
doc. RNDr. Petr Matula, Ph.D. (assistant)
Guaranteed by
doc. RNDr. Petr Matula, Ph.D.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 22 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and object representation using a grid of points (e.g., in the form of a digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms of digital geometry
• Component labeling algrotithms
• Object digitization
• Measurements in digital spaces
• Distance maps and their computation
• Border tracing algorithms
• Topological properties of digital spaces
• Digital geometric figure recognition (line, arc, plane)
• Estimation and computation of geometric and topological properties of digital sets (volume, surface, length, curvature, etc.)
• Digital convex hull
• Thinning and skeletons
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2013, Autumn 2015, Autumn 2017, Autumn 2019, Autumn 2021.

## PA170 Digital Geometry

Faculty of Informatics
Autumn 2012

The course is not taught in Autumn 2012

Extent and Intensity
2/1. 3 credit(s) (plus extra credits for completion). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
Teacher(s)
doc. RNDr. Pavel Matula, Ph.D. (lecturer)
Guaranteed by
prof. Ing. Jiří Sochor, CSc.
Department of Visual Computing - Faculty of Informatics
Contact Person: doc. RNDr. Pavel Matula, Ph.D.
Supplier department: Department of Visual Computing - Faculty of Informatics
Timetable
Fri 10:00–11:50 C525, Fri 12:00–12:50 B311
Prerequisites
The basic knowledge of mathematics and graph theory is recommended.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 22 fields of study the course is directly associated with, display
Course objectives
At the end of the course students should be able to: understand and explain basic problems that arise after object digitization and an object representation using a grid of points (e.g., in the form of digital image); measure geometric and topological properties of digital objects (e.g., length, area, perimeter, volume, Euler characteristic, and the number of holes); compare digital metrics; efficiently implement the key algorithms of digital geometry (e.g., region labeling, border tracing, and distance map computation); identify the fundamentals of the discussed methods.
Syllabus
• Basic terms: digital image, pixel, voxel, image resolution, types of grids, grid scanning
• grid point and grid cell models: adjacency, incidence, connectedness, components, component labeling algorithms.
• Digitalization: digitization models, line digitization.
• Measurement in digital images: metrics, integer-valued metrics approximating Euclidean metric, distance transform, distance measurement between sets.
• Oriented adjacency graphs: border, boundary, border tracing algorithm, holes, combinatorial results for regular graphs (grids)
• Application of graph theory in image processing, graph-cut based image segmentation.
• Incidence pseudographs, open and closed regions, ordered labeling of multilevel images.
• Introduction to topology. Basic topological concepts. Definition of continuous as well as digital curve. Jordan Veblen theorem.
• Euclidean and simplex complexes (triangulation). Topological definition of surfaces and their classification. Combinatorial results. Regular tilings.
• Estimation and computation of geometric and topological properties of digital sets: volume, surface, area, perimeter, length, curvature, Euler characteristic, etc.
• Digital straight segment recognition, digital straightness, digital convex hull and its computation.
• Image deformations: Thinning, skeletons.
Literature
• KLETTE, Reinhard and Azriel ROSENFELD. Digital geometry: geometric methods for digital picture analysis. Amsterdam: Elsevier, 2004. 656 pp. info
Teaching methods
Lectures followed by class exercises where we will solve practical problems by taking the advantage of lecture findings. Homework.
Assessment methods
Written test, oral exam. Obligatory attandance at exercises. Homework score.
Language of instruction
Czech