HEALPix
Encyclopedia
HEALPix an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-sphere
, can refer to either an algorithm
for pixelisation of the 2-sphere, an associated software package, or an associated class of map projection
s.
The HEALPix projection
is a general class of spherical projections, sharing several key properties, which map the 2-sphere
to the Euclidean plane
. Any of these can be followed by partitioning (pixelising) the resulting region of the 2-plane. In particular, when one of these projections (the H=4, K=3 HEALPix projection), is followed by a pixelisation of the 2-plane, this is generally referred as the HEALPix pixelisation, which is widely used in physical cosmology
for maps of the cosmic microwave background. This pixelisation can be thought of as mapping the sphere to twelve square facets (diamonds) on the plane followed by the binary division of these facets into pixels, though it can be derived without using the projection. The associated software package HEALPix implements the algorithm. The HEALPix projection (as a general class of spherical projections) is represented by the keyword HPX in the FITS
standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the IAU
FITS Working Group on April 26, 2006.
The spherical projection combines a cylindrical equal area projection, the Lambert cylindrical equal-area projection
, for the equatorial regions of the sphere and a pseudocylindrical equal area projection, an interrupted Collignon projection
, for the polar regions.
As the name indicates, at a given level in the hierarchy the pixels are of equal area (which is done by bisecting the square
in the case of the H=4, K=3 projection) and their centers lie on a discrete number of circles of latitude, with equal spacing on each circle. The scheme has a number of mathematical properties which make it efficient for certain computations, e.g. spherical harmonic
transform
s. In the case of the H=4, K=3 projection, the pixels are squares in the plane (which can be inversely projected back to quadrilaterals with non-geodesic sides on the 2-sphere) and every vertex joins four pixels, with the exception of eight vertices which each join only three pixels.
The pixelisation related to the H=4, K=3 projection has become widely used in cosmology for storing and manipulating maps of the cosmic microwave background.
An alternative hierarchical grid is the Hierarchical Triangular Mesh (HTM). The pixels at a given level in the hierarchy are of similar but not identical size. The scheme is good at representing complex shapes because the boundaries are all segments of circles of the sphere.
Another alternative hierarchical grid is the Quadrilateralized Spherical Cube
.
The 12 "base resolution pixels" of H=4, K=3 HEALPix projection may be thought of as the facets of a rhombic dodecahedron
.
The H=6 HEALPix has similarities to another alternative grid based on the icosahedron
.
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, can refer to either an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
for pixelisation of the 2-sphere, an associated software package, or an associated class of map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
s.
The HEALPix projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
is a general class of spherical projections, sharing several key properties, which map the 2-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
to the Euclidean plane
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. Any of these can be followed by partitioning (pixelising) the resulting region of the 2-plane. In particular, when one of these projections (the H=4, K=3 HEALPix projection), is followed by a pixelisation of the 2-plane, this is generally referred as the HEALPix pixelisation, which is widely used in physical cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
for maps of the cosmic microwave background. This pixelisation can be thought of as mapping the sphere to twelve square facets (diamonds) on the plane followed by the binary division of these facets into pixels, though it can be derived without using the projection. The associated software package HEALPix implements the algorithm. The HEALPix projection (as a general class of spherical projections) is represented by the keyword HPX in the FITS
FITS
Flexible Image Transport System is a digital file format used to store, transmit, and manipulate scientific and other images. FITS is the most commonly used digital file format in astronomy...
standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the IAU
IAU
IAU may refer to:*International Astronomical Union*International American University*International American University College of Medicine*International Association of Universities*International Association of Ultrarunners...
FITS Working Group on April 26, 2006.
The spherical projection combines a cylindrical equal area projection, the Lambert cylindrical equal-area projection
Lambert cylindrical equal-area projection
In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is acylindrical, equal area map projection...
, for the equatorial regions of the sphere and a pseudocylindrical equal area projection, an interrupted Collignon projection
Collignon projection
The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A...
, for the polar regions.
As the name indicates, at a given level in the hierarchy the pixels are of equal area (which is done by bisecting the square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
in the case of the H=4, K=3 projection) and their centers lie on a discrete number of circles of latitude, with equal spacing on each circle. The scheme has a number of mathematical properties which make it efficient for certain computations, e.g. spherical harmonic
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
transform
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...
s. In the case of the H=4, K=3 projection, the pixels are squares in the plane (which can be inversely projected back to quadrilaterals with non-geodesic sides on the 2-sphere) and every vertex joins four pixels, with the exception of eight vertices which each join only three pixels.
The pixelisation related to the H=4, K=3 projection has become widely used in cosmology for storing and manipulating maps of the cosmic microwave background.
An alternative hierarchical grid is the Hierarchical Triangular Mesh (HTM). The pixels at a given level in the hierarchy are of similar but not identical size. The scheme is good at representing complex shapes because the boundaries are all segments of circles of the sphere.
Another alternative hierarchical grid is the Quadrilateralized Spherical Cube
Quadrilateralized Spherical Cube
In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area mapping and binning scheme for data collected on a spherical surface...
.
The 12 "base resolution pixels" of H=4, K=3 HEALPix projection may be thought of as the facets of a rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...
.
The H=6 HEALPix has similarities to another alternative grid based on the icosahedron
.
External links
- Official implementation with many languages support
- Java port of original Fortran code by Nikolay Kuropatkin. Extends resolution up to 0.3 arcsec.
- Java port optimized to use RangeSet, very good for high resolutions
- Python wrapper