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Integral Geometry and its Applications in Image Recognition and Processing

ВК.3.02.03

Вибіркова дисципліна для ОП

First

2023/2024

4 Semester

5

LO02. To possess basic principles and methods of mathematical, complex and functional analysis, linear algebra and number theory, analytical geometry, theory of differential equations, in particular partial differential equations, probability theory, mathematical statistics and random processes, numerical methods. PLO 21.3. To know the fundamental topics of mathematics and computer science, in the amount necessary for mastering general professional mathematical disciplines, applied disciplines and the use of their methods in the chosen profession. PLO 24.3. Be able to independently analyze the relevant subject area, be able to develop mathematical and structural-algorithmic models.

In order to successfully master the course "Integral geometry and its application in image recognition and processing", the student must be fluent in the material of the standard courses "Mathematical analysis", "Algebra and geometry" and selected topics of the courses "Operations research" and "Probability theory". In particular, from a course "Algebra and Geometry", the student should know the general theory of linear and affine spaces, operators on them, and the basic geometric properties of Euclidean and unitary spaces. From the course "Operations Research", the student should be fluent in the concepts of convex set and convex function. From the course "Mathematical analysis" the concept of multiple integral, and from the course "Probability theory" the concept of Poisson distribution, to be able to calculate mathematical expectations and find distributions of simple random variables.

Convex stochastic and integral geometry is a classical branch of mathematics that was formed in the first half of the 20th century in the works of Minkowski, Blaschke, Santalo, etc. With the advent of modern computers, these disciplines have gained new importance in connection with the problems of image recognition and processing. This course is an introduction to stochastic and integral geometry, as well as the theory of pattern recognition, which on the one hand introduces the student to the necessary mathematical apparatus, and on the other hand demonstrates how the constructed theory can be applied in practice.

1. Descombe X. (2012). Stochastic Geometry for Image Analysis, ISTE and John Wiley & Sons. 2. Kingman J. F. (1993). Poisson processes, Oxford University Press. 3. Schneider R. (2014). Convex bodies: The Brunn-Minkowski theory, Cambridge University Press, 2nd edition. 4. Serra J. (1982). Image Analysis and Mathematical Morphology, Academic Press. 5. Schneider R., Weil W. (2008). Stochastic and integral geometry, Springer. 6. Vapnik V. (1998). Statistical Learning Theory. John Wiley & Sons. 7. Marynych О. Lecture notes, https://do.csc.knu.ua/?page_id=944 8. Molchanov I. (2017). Theory of Random Sets, Springer, 2nd edition. 9. Rockafeller R. (1972). Convex Analysis, Princeton University Press, 2nd edition. 10. Santalo L. (2010). Integral Geometry and Geometric Probability, Cambridge University Press, 2nd edition.

Lectures - 42 hours, consultations - 2 hours, tutorials – 32 hours, slef-work - 74 hours. The course includes 4 content modules, 2 module tests and 2 laboratory works. The discipline finishes with an exam in the 4th semester.

Intermediate assessment: The maximal number of available points is 60. • Test work no. 1: 15/9 points. • Test work no. 2: 15/9 points. • Laboratory work no. 1: 15/9 points. • Laboratory work no. 2: 15/9 points. Final assessment (in the form of exam): The maximal number of available points is 40. The form of exam: writing. The types of assignments are 4 writing assignments (2 theoretical and 2 practical).

Ukrainian