Medial Axis: Exploring the Core of Computer Vision: Unveiling the Medial Axis

By Fouad Sabry

What is Medial Axis

The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape recognition. In mathematics the closure of the medial axis is known as the cut locus.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Medial Axis

Chapter 2: Curve

Chapter 3: Voronoi Diagram

Chapter 4: Incenter

Chapter 5: Linking Number

Chapter 6: Fundamental Domain

Chapter 7: Wess-Zumino-Witten Model

Chapter 8: Topological Skeleton

Chapter 9: Ridge Detection

Chapter 10: Straight Skeleton

(II) Answering the public top questions about medial axis.

(III) Real world examples for the usage of medial axis in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Medial Axis.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 12, 2024

Book preview

Chapter 1: Medial axis

The midline is the collection of locations inside an item that are closest to the boundary in more than one direction. Topological skeleton was first developed by Harry Blum in 1967 as a means of biological shape recognition. The cut locus is a mathematical object that represents the closure of the medial axis.

In 2D, the locus of the centers of circles that are tangent to curve C in two or more points is the medial axis of a subset S that is bounded by curve C in planar space, provided that all such circles are contained in S. (Therefore, S must contain the medial axis.) A simple polygon's medial axis is a tree whose branches are the vertices and whose leaves can be straight lines or parabolas.

One definition of the medial axis transform involves the radius function of the maximally inscribed discs (MAT). Reconstructing the original shape of the domain is possible thanks to the medial axis transform, which is a comprehensive shape descriptor (see also shape analysis).

The symmetry set, of which the medial axis is a part, is defined in a similar fashion, but it also contains circles that are not part of S. (Therefore, just like the Voronoi diagram of a set of points, S's symmetry set typically goes on forever.)

When 2D circles are changed to k-dimensional hyperspheres, the medial axis becomes applicable to k-dimensional hypersurfaces. Character and object identification benefit from the 2D medial axis, whereas 3D medial axis uses include physical model surface reconstruction and complex model dimensionality reduction. The supplied set is homotopy equal to the medial axis of any bounded open set in any dimension.

If S is given by a unit speed parametrisation \gamma :{\mathbf {R}}\to {\mathbf {R}}^{2} , and \underline {T}(t)={d\gamma \over dt} is the unit tangent vector at each point.

A bitangent circle will have coordinates (center, c) and (radius, r), if

(c-\gamma (s))\cdot \underline {T}(s)=(c-\gamma (t))\cdot \underline {T}(t)=0,|c-\gamma (s)|=|c-\gamma (t)|=r.\,

In most cases, a cusp can be included in a symmetry set that forms a one-dimensional curve. Each vertex of S corresponds to an endpoint of the symmetry set.

{End Chapter 1}

Chapter 2: Curve

A curve (also called a curved line) is a mathematical object with linelike properties but not necessarily straight line.

A curve can be intuitively understood as the mark made by a moving point. This is the original definition from Euclid's Elements, written almost two thousand years ago: The line that curves

In contemporary mathematical theory, a curve is defined in this way: Specifically, a curve is the projection onto a topological space of an interval by a continuous function. Parametric curves are those whose definition is given by a function known as a parametrization. To differentiate them from more restricted curves like differentiable curves, we sometimes refer to them as topological curves in this article. Except for level curves (which are unions of curves and isolated points) and algebraic curves, this description covers the vast majority of mathematically-studied curves (see below). Implicit curves are a type of level curve or algebraic curve that are typically defined by implicit equations.

However, topological curves make up a very large class, and some of them don't have the conventional appearance of curves and can't even be drawn. Space-filling curves and fractal curves are examples of this. A curve is considered to be differentiable if and only if the defining function can be differentiated. This helps guarantee the curve will always look the same.

The zero set of a polynomial in two indeterminates is an algebraic curve in the plane. The zero set of a finite set of polynomials that also meets the additional criteria of being an algebraic variety of dimension one is called an algebraic curve. The curve is considered to be defined over k if and only if the polynomial coefficients are elements of field k. Algebraic curves are the union of topological curves when k is the field of real numbers, as is the usual case. A complex algebraic curve, which is not a curve but a surface from a topological point of view and is often referred to as a Riemann surface, is obtained when one takes into account complex zeros. Algebraic curves formed over other fields have been extensively explored despite not being curves in the usual sense. Modern cryptography makes extensive use of algebraic curves over a finite field.

Curves have always been fascinating, even before they were formally studied in mathematics. Many works of art and daily things dating back to prehistoric times display their ornamental usage. It takes very little effort to draw a curve, or at least a semblance of one, with a stick and some sand.

The more contemporary term curve has been replaced historically by the phrase line. As a result, what we now simply call lines were formerly referred to as straight lines and right lines to differentiate them from curved lines. A line, for instance, is described as a breadthless length (Def. 2) in