Boundary tracing, also known as contour tracing, of a binary digital region can be thought of as a segmentation technique that identifies the boundary pixels of the digital region. Boundary tracing is an important first step in the analysis of that region.Boundary is a topological notion. However, a digital image is no topological space. Therefore, it is impossible to define the notion of a boundary in a digital image mathematically exactly. Most publications about tracing the boundary of a subset S of a digital image I describe algorithms which find a set of pixels belonging to S and having in their direct neighborhood pixels belonging both to S and to its complement I - S. According to this definition the boundary of a subset S is different from the boundary of the complement I – S which is a topological paradox.To define the boundary correctly it is necessary to introduce a topological space corresponding to the given digital image. Such space can be a two-dimensional abstract cell complex. It contains cells of three dimensions: the two-dimensional cells corresponding to pixels of the digital image, the one-dimensional cells or “cracks” representing short lines lying between two adjacent pixels, and the zero-dimensional cells or “points” corresponding to the corners of pixels. The boundary of a subset S is then a sequence of cracks and points while the neighborhoods of these cracks and points intersect both the subset S and its complement I – S. The boundary defined in this way corresponds exactly to the topological definition and corresponds also to our intuitive imagination of a boundary because the boundary of S should contain neither elements of S nor those of its complement. It should contain only elements lying between S and the complement. This are exactly the cracks and points of the complex.This method of tracing boundaries is described in the book of Vladimir A. Kovalevsky and in the web site. (Wikipedia).
Proving Parallel Lines with Angle Relationships
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
What is the Corresponding Angle Converse Theorem
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
What are parallel lines and a transversal
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
How To Determine If Two Lines are Parallel to Apply Angle Theorems
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
What are the Angle Relationships for Parallel Lines and a Transversal
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
CCSS What is the definition of a Midpoint
👉 Learn how to find the midpoint between two points. The midpoint between two points is the point halfway the line joining two given points in the coordinate plane. To find the midpoint between two points we add the x-coordinates of the two given points and divide the result by 2. This giv
From playlist Points Lines and Planes
Learn to Find Corresponding Angles From a Figure
👉 Learn how to identify angles from a figure. This video explains how to solve problems using angle relationships between parallel lines and transversal. We'll determine the solution given, corresponding, alternate interior and exterior. All the angle formed by a transversal with two paral
From playlist Parallel Lines and a Transversal
Geometry - Identifying Corresponding Angles from a Figure
👉 Learn how to identify angles from a figure. This video explains how to solve problems using angle relationships between parallel lines and transversal. We'll determine the solution given, corresponding, alternate interior and exterior. All the angle formed by a transversal with two paral
From playlist Parallel Lines and a Transversal
Determining Two Angles are Consecutive Interior Angles from a Figure
👉 Learn how to identify angles from a figure. This video explains how to solve problems using angle relationships between parallel lines and transversal. We'll determine the solution given, corresponding, alternate interior and exterior. All the angle formed by a transversal with two paral
From playlist Parallel Lines and a Transversal
Probability Seminar Topic: Dimers and 3-Webs Speaker: Richard Kenyon Affiliation: Yale University Date: October 26, 2022 11:15am West Lecture Hall This is joint work with Haolin Shi (Yale). 3-webs are bipartite, trivalent, planar graphs. They were defined and studied by Kuperberg who sho
From playlist Mathematics
Bertrand Eynard: (Mixed) topological recursion and the two-matrix model - Lecture 3
Mini course of the conference "Noncommutative geometry meets topological recursion", August 2021, University of Münster. Abstract: In this series of lecture we will introduce the 2-matrix model and the issue of mixed traces, then we shall give the answers as formulas. Some formulas will be
From playlist Noncommutative geometry meets topological recursion 2021
Rebekah Palmer: Totally Geodesic Surfaces in Knot Complements with Small Crossing Number
Rebekah Palmer, Temple University Title: Totally Geodesic Surfaces in Knot Complements with Small Crossing Number Studying totally geodesic surfaces has been essential in understanding the geometry and topology of hyperbolic 3-manifolds. Recently, Bader-Fisher-Miller-Stover showed that con
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Jesse Peterson: Von Neumann algebras and lattices in higher-rank groups, Lecture 2
Mini course of the conference YMC*A, August 2021, University of Münster. Lecture 2: Some approximation properties. Abstract: We discuss some approximation properties that are common in "rank 1" groups: Weak amenability and biexactness. YMC*A is an annual conference organised for and by ma
From playlist YMC*A 2021
Learn how to use Snell's Law to analyze complex refraction scenarios. Four example problems are solved. The Snell's Law of Refraction Video Tutorial (referenced on Slide 3) can be found at: https://youtu.be/PUUQk7VPPfQ You can find more information that supports this video on our websi
From playlist Refraction and Lenses
Nonlinear descent on moduli of local systems - Junho Peter Whang
Joint IAS/Princeton University Number Theory Seminar Topic: Nonlinear descent on moduli of local systems Speaker: Junho Peter Whang Affiliation: Princeton University Date: October 31, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Ari Stern: Hybrid finite element methods preserving local symmetries and conservation laws
Abstract: Many PDEs arising in physical systems have symmetries and conservation laws that are local in space. However, classical finite element methods are described in terms of spaces of global functions, so it is difficult even to make sense of such local properties. In this talk, I wil
From playlist Numerical Analysis and Scientific Computing
Ellipses of small eccentricity are determined by their Dirichlet... - Steven Morris Zelditch
Analysis Seminar Topic: Ellipses of small eccentricity are determined by their Dirichlet (or, Neumann) spectra Speaker: Steven Morris Zelditch Affiliation: Northwestern University Date: April 28, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Physics: Optics- Thick Lenses (14 of 56) Ray Tracing Through a Thick Lens
Visit http://ilectureonline.com for more math and science lectures! In this video I will derive the refracting and the power of the refracting surface equations of a ray tracing through a thick lens on the front boundary (surface 1). Next video in this series can be seen at: https://yout
From playlist PHYSICS 55.3 THICK LENSES
Find the midpoint between two points w(–12,–7), T(–8,–4)
👉 Learn how to find the midpoint between two points. The midpoint between two points is the point halfway the line joining two given points in the coordinate plane. To find the midpoint between two points we add the x-coordinates of the two given points and divide the result by 2. This giv
From playlist Points Lines and Planes