In computational geometry, the minimum bounding rectangle (MBR), also known as bounding box (BBOX) or envelope, is an expression of the maximum extents of a two-dimensional object (e.g. point, line, polygon) or set of objects within its x-y coordinate system; in other words min(x), max(x), min(y), max(y). The MBR is a 2-dimensional case of the minimum bounding box.
MBRs are frequently used as an indication of the general position of a geographic feature or dataset, for either display, first-approximation spatial query, or spatial indexing purposes.
The degree to which an "overlapping rectangles" query based on MBRs will be satisfactory (in other words, produce a low number of "false positive" hits) will depend on the extent to which individual spatial objects occupy (fill) their associated MBR. If the MBR is full or nearly so (for example, a mapsheet aligned with axes of latitude and longitude will normally entirely fill its associated MBR in the same coordinate space), then the "overlapping rectangles" test will be entirely reliable for that and similar spatial objects. On the other hand, if the MBR describes a dataset consisting of a diagonal line, or a small number of disjunct points (patchy data), then most of the MBR will be empty and an "overlapping rectangles" test will produce a high number of false positives. One system that attempts to deal with this problem, particularly for patchy data, is c-squares.
MBRs are also an essential prerequisite for the R-tree method of spatial indexing.
Owing to their simplicity of expression and ease of use for searching, MBRs (frequently as "bounding box" or "bounding coordinates") are also commonly included in relevant standards for geospatial metadata, i.e. metadata that describes spatial (geographic) objects; examples include DCMI Box as an extension to the Dublin Core metadata scheme, "Bounding Coordinates" in the (U.S.) FGDC metadata standard, and "Geographic Bounding Box" in the (2003–current) ISO 19115 Metadata Standard for geographic information (ISO/TC 211). It is also (as "boundingBox") an element in Geography Markup Language (GML), that is utilised by a range of Web Service specifications from the Open Geospatial Consortium (OGC). In the ISO 19107 Spatial Schema (ISO/TC 211), MBR appears as the datatype GM_Envelope that is returned by the envelope() operation on the root class GM_Object.
The Geography Markup Language (GML) is the XML grammar defined by the Open Geospatial Consortium (OGC) to express geographical features. GML serves as a modeling language for geographic systems as well as an open interchange format for geographic transactions on the Internet. Key to GML's utility is its ability to integrate all forms of geographic information, including not only conventional "vector" or discrete objects, but coverages and sensor data.
A coverage is the digital representation of some spatio-temporal phenomenon. ISO 19123 provides the definition:
A GIS file format is a standard for encoding geographical information into a computer file, as a specialized type of file format for use in geographic information systems (GIS) and other geospatial applications. Since the 1970s, dozens of formats have been created based on various data models for various purposes. They have been created by government mapping agencies, GIS software vendors, standards bodies such as the Open Geospatial Consortium, informal user communities, and even individual developers.
In computer graphics and computational geometry, a bounding volume for a set of objects is a closed region that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operations, such as by using simple regions, having simpler ways to test for overlap.
R-trees are tree data structures used for spatial access methods, i.e., for indexing multi-dimensional information such as geographical coordinates, rectangles or polygons. The R-tree was proposed by Antonin Guttman in 1984 and has found significant use in both theoretical and applied contexts. A common real-world usage for an R-tree might be to store spatial objects such as restaurant locations or the polygons that typical maps are made of: streets, buildings, outlines of lakes, coastlines, etc. and then find answers quickly to queries such as "Find all museums within 2 km of my current location", "retrieve all road segments within 2 km of my location" or "find the nearest gas station". The R-tree can also accelerate nearest neighbor search for various distance metrics, including great-circle distance.
An R+ tree is a method for looking up data using a location, often coordinates, and often for locations on the surface of the Earth. Searching on one number is a solved problem; searching on two or more, and asking for locations that are nearby in both x and y directions, requires craftier algorithms.
In computer science tree data structures, an X-tree is an index tree structure based on the R-tree used for storing data in many dimensions. It appeared in 1996, and differs from R-trees (1984), R+-trees (1987) and R*-trees (1990) because it emphasizes prevention of overlap in the bounding boxes, which increasingly becomes a problem in high dimensions. In cases where nodes cannot be split without preventing overlap, the node split will be deferred, resulting in super-nodes. In extreme cases, the tree will linearize, which defends against worst-case behaviors observed in some other data structures.
ISO/TC 211 is a standard technical committee formed within ISO, tasked with covering the areas of digital geographic information and geomatics. It is responsible for preparation of a series of International Standards and Technical Specifications numbered in the number range starting at ISO-19101. The Chair of the committee was 1994-2016: Olaf Østensen; during 2017-2018: Christina Wasström; and from 2019 Agneta Gren Engberg.
A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on the surface of Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space. A particular SRS specification comprises a choice of Earth ellipsoid, horizontal datum, map projection, origin point, and unit of measure. Thousands of coordinate systems have been specified for use around the world or in specific regions and for various purposes, necessitating transformations between different SRS.
Simple Features is a set of standards that specify a common storage and access model of geographic features made of mostly two-dimensional geometries used by geographic databases and geographic information systems. It is formalized by both the Open Geospatial Consortium (OGC) and the International Organization for Standardization (ISO).
Catalogue Service for the Web (CSW), sometimes seen as Catalogue Service - Web, is a standard for exposing a catalogue of geospatial records in XML on the Internet. The catalogue is made up of records that describe geospatial data, geospatial services, and related resources.
A spatial database is a general-purpose database that has been enhanced to include spatial data that represents objects defined in a geometric space, along with tools for querying and analyzing such data.
Well-known text (WKT) is a text markup language for representing vector geometry objects. A binary equivalent, known as well-known binary (WKB), is used to transfer and store the same information in a more compact form convenient for computer processing but that is not human-readable. The formats were originally defined by the Open Geospatial Consortium (OGC) and described in their Simple Feature Access. The current standard definition is in the ISO/IEC 13249-3:2016 standard.
Geospatial metadata is a type of metadata applicable to geographic data and information. Such objects may be stored in a geographic information system (GIS) or may simply be documents, data-sets, images or other objects, services, or related items that exist in some other native environment but whose features may be appropriate to describe in a (geographic) metadata catalog.
A Spatial Data Infrastructure (SDI), also called geospatial data infrastructure, is a data infrastructure implementing a framework of geographic data, metadata, users and tools that are interactively connected in order to use spatial data in an efficient and flexible way. Another definition is "the technology, policies, standards, human resources, and related activities necessary to acquire, process, distribute, use, maintain, and preserve spatial data".
Hilbert R-tree, an R-tree variant, is an index for multidimensional objects such as lines, regions, 3-D objects, or high-dimensional feature-based parametric objects. It can be thought of as an extension to B+-tree for multidimensional objects.
The Spatial Archive and Interchange Format was defined in the early 1990s as a self-describing, extensible format designed to support interoperability and storage of geospatial data.
The Priority R-tree is a worst-case asymptotically optimal alternative to the spatial tree R-tree. It was first proposed by Arge, De Berg, Haverkort and Yi, K. in an article from 2004. The prioritized R-tree is essentially a hybrid between a k-dimensional tree and a r-tree in that it defines a given object's N-dimensional bounding volume as a point in N-dimensions, represented by the ordered pair of the rectangles. The term prioritized arrives from the introduction of four priority-leaves that represents the most extreme values of each dimensions, included in every branch of the tree. Before answering a window-query by traversing the sub-branches, the prioritized R-tree first checks for overlap in its priority nodes. The sub-branches are traversed by checking whether the least value of the first dimension of the query is above the value of the sub-branches. This gives access to a quick indexation by the value of the first dimension of the bounding box.
Geographic data and information is defined in the ISO/TC 211 series of standards as data and information having an implicit or explicit association with a location relative to Earth. It is also called geospatial data and information, georeferenced data and information, as well as geodata and geoinformation.
The PH-tree is a tree data structure used for spatial indexing of multi-dimensional data (keys) such as geographical coordinates, points, feature vectors, rectangles or bounding boxes. The PH-tree is space partitioning index with a structure similar to that of a quadtree or octree. However, unlike quadtrees, it uses a splitting policy based on tries and similar to Crit bit trees that is based on the bit-representation of the keys. The bit-based splitting policy, when combined with the use of different internal representations for nodes, provides scalability with high-dimensional data. The bit-representation splitting policy also imposes a maximum depth, thus avoiding degenerated trees and the need for rebalancing.