Minimum Bounding Box: Unveiling the Power of Spatial Optimization in Computer Vision

By Fouad Sabry

What is Minimum Bounding Box

In geometry, the minimum bounding box or smallest bounding box for a point set S in N dimensions is the box with the smallest measure within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box".

How you will benefit

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

Chapter 1: Minimum bounding box

Chapter 2: Convex hull

Chapter 3: Collision detection

Chapter 4: Computational geometry

Chapter 5: Bounding volume

Chapter 6: Bounding sphere

Chapter 7: R-tree

Chapter 8: Convex polytope

Chapter 9: Minimum bounding rectangle

Chapter 10: Convex hull algorithms

(II) Answering the public top questions about minimum bounding box.

(III) Real world examples for the usage of minimum bounding box 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 Minimum Bounding Box.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 5, 2024

Book preview

Chapter 1: Minimum bounding box

The lowest or smallest bounding or enclosing box for a point set S in N dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) that contains all the points. When different units of measurement are employed, the minimal box is typically referred to as minimum-perimeter bounding box.

The fact that the least bounding box of a point set is identical to the minimum bounding box of its convex hull can be exploited to speed up computations.

The labels box and hyperrectangle are derived from their use in the Cartesian coordinate system, where they are depicted as a rectangle (two-dimensional case), rectangular parallelepiped (three-dimensional instance), etc.

In two dimensions, it is referred to as the minimum bounding rectangle.

The axis-aligned minimum bounding box (or AABB) for a given point set is the minimal bounding box whose edges are parallel to the (Cartesian) coordinate axes. It is the Cartesian product of N intervals determined by the minimum and maximum value of the associated coordinate for each point in S.

Axis-aligned minimum bounding boxes are used to approximate an object's location and to describe its shape in a very basic manner. In computational geometry and its applications, for instance, when it is necessary to locate intersections in a group of objects, the intersections between their MBBs serve as the initial check. Due to the fact that it is typically a considerably less expensive operation than the actual intersection check (because it simply requires coordinate comparisons), it permits fast omitting checks of pairings that are far apart.

The arbitrarily oriented minimal bounding box is the minimum bounding box that is calculated with no orientation restrictions. Minimum bounding box algorithms based on the rotating calipers method can find the minimum-area or minimum-perimeter bounding box of a two-dimensional convex polygon in linear time and of a three-dimensional point set in the time required to construct its convex hull followed by a linear-time computation.

When an object has its own local coordinate system, it can be advantageous to save a bounding box relative to these axes, which does not require transformation as the object's own transformation changes.

In digital image processing, the bounding box is simply the coordinates of the rectangular border that completely encloses a digital picture when it is displayed on a page, canvas, screen, or other bidimensional background.

{End Chapter 1}

Chapter 2: Convex hull

The convex hull, convex envelope, or convex closure of a shape in geometry is the smallest convex set that contains the shape. The convex hull may be defined as the intersection of all convex sets containing a particular subset of a Euclidean space, or as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull can be seen as the form contained by an extended rubber band.

Open are the convex hulls of open sets, and compact sets have convex hulls that are compact.

Each convex compact set is the convex hull of its extremities.

The convex hull operator is a closure operator example, Every antimatroid can be represented by applying this closure operator to finite point sets.

Finding the convex hull of a finite number of points in the plane or other low-dimensional Euclidean spaces presents algorithmic challenges, and the twofold issue of overlapping half-spaces, are essential computational geometry problems.

They can be solved in time O(n\log n) for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions.

Convex hulls have also been explored for simple polygons, Brownian motion, space curves, and the epigraphs of functions, in addition to finite point sets. In mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology, convex hulls have numerous uses. Convex skull, orthogonal convex hull, convex layers, Delaunay triangulation, and Voronoi diagram are related structures.

A collection of points in a Euclidean space is convex if it contains the line segments connecting each pair of its points.

The convex hull of a given set X may be defined as

The (unique) minimal convex set containing X

The intersection of all convex sets containing X

The set of all convex combinations of points in X

The union of all simplices with vertices in X

For sets that are limited in the Euclidean plane, not in a single line, the boundary of the convex hull is the simple closed curve with minimum perimeter containing X .

One may imagine stretching a rubber band so that it surrounds the entire set S and then releasing it, allowing it to shrink; when it tightens up, it encloses the convex hull of S .

For three-dimensional objects, the initial definition of the convex hull specifies that it is the smallest possible convex bounding volume. The definition using intersections of convex sets can be extended to non-Euclidean geometry, and the definition using convex combinations can be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces; convex hulls can also be abstractly generalized to oriented matroids.

It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing X , for every X ? However, the second meaning, the intersection of all convex sets containing X , is clearly defined.

It is a subset of every other convex set Y that contains X , because Y is included among the sets being intersected.

Thus, it is exactly the unique minimal convex set containing X .

Therefore, The initial two definitions are