COBE Quadrilateralized Spherical Cube
All COBE map data are presented in a quadrilateralized spherical projection, an approximately equal-area projection (to within a few percent) in which the celestial sphere is projected onto an inscribed cube. Unlike the Aitoff, Mollweide, and Global-sinusoidal projections, the "quad-sphere" projection does not have polar singularities. The following PDF/Postscript/GIF figures depict various aspects of the CSC on an unfolded cube:
Tables that relate pixel numbers to ecliptic, Galactic, and equatorial coordinates according to the two numbering schemes are given in the FITS binary table files DIRBE_SKYMAP_INFO.FITS and DMR_FIRAS_SKYMAP_INFO.FITS. Note that there is only one file for DMR and FIRAS because the pixel size is the same in both cases. These files can be accessed from the Ancilary data product section of the DIRBE Product Table, DMR Product Table or FIRAS Product Table for DMR_FIRAS ancil files.
Note that the skymap_info FITS files do not contain scientific data. The two files differ only in terms of pixel resolution. The DIRBE instrument has a 42' x 42' square instantaneous field of view; a 21' pixel size (quad-sphere resolution index 9) was adopted for presentation of the DIRBE map data. The FIRAS and DMR instruments each have 7 degree beams and the corresponding map data are presented using 2.8 degree pixels (quad-sphere resolution index 6).
A detailed description of the CSC projection is given in the FITS headers. The header from DIRBE_SKYMAP_INFO.FITS is given below for convenient reference:
SIMPLE = T / File conforms to FITS standard BITPIX = 32 / Quad-sphere (QS) pixel nos. are 32-bit integers NAXIS = 3 / Data given as a cube of stacked faces NAXIS1 = 256 / x axis of face = number of QS pixels on side NAXIS2 = 256 / y axis of face = number of QS pixels on side NAXIS3 = 6 / Number of faces [faces numbered from 0 to 5] EXTEND = T / Binary table extension follows primary array. COMMENT / The binary table gives the coordinates COMMENT / for the center of the pixels. ORIGIN = 'CDAC ' / NASA Cosmology Data Analysis Center DATE = '14/10/94' / FITS file creation date (dd/mm/yy) TELESCOP= 'COBE ' / COsmic Background Explorer satellite INSTRUME= 'DIRBE ' / COBE instruments for which the Quad-sphere COMMENT / pixel array diagram in the data array is COMMENT / appropriate. OBJECT = 'ALL-SKY ' / Part of sky given in the array and extensions COMMENT COBE project keywords (non- FITS reserved): PRODUCT = 'PIXINFO9' / Pixel information file for given resolution PROJECTN= 'QUADSPHR' / Projection used is a Quad-sphere projection PIXORDER= 'QUADTREE' / Pixel ordering used PIX0LOCN= 'BRC ' / Location of initial pixel on face COMMENT / BRC = "Bottom Right Corner" COORDSYS= 'E ' / Pixelization coordinate system COMMENT Coordinate system is one of E = Ecliptic, G = Galactic, COMMENT C = Celestial, i.e. Equatorial, or S = Special PIXSIZE = 0.3239008 / Size in deg. of each side of the avg. QS pixel PIXINDEX= 9 / Quad-sphere pixel resolution index COMMENT / PIXINDEX = 6 for DMR/FIRAS and 9 for DIRBE COMMENT COMMENT This file is intended to familiarize users of the COBE data sets with COMMENT the COBE sky cube representation and to facilitate usage of the data COMMENT sets by providing sky coordinate information. The COBE sky cube is COMMENT based upon the quadrilateralized spherical cube (quad-sphere) COMMENT projection and the quad-tree pixel ordering scheme. COMMENT COMMENT NOTE: THIS FILE DOES NOT CONTAIN SCIENCE DATA FROM A COBE INSTRUMENT. COMMENT FURTHERMORE, THE COORDINATES GIVEN IN THE BINARY TABLE EXTENSION COMMENT CORRESPOND TO PIXEL CENTERS AND MAY NOT BE APPLICABLE TO SOME OF THE COMMENT COBE DATA PRODUCTS (SEE THE INSTRUMENT EXPLANATORY SUPPLEMENTS FOR COMMENT SPECIFIC ADVICE). WHEN AVAILABLE, USE THE COORDINATES GIVEN IN THE COMMENT DATA FILES RATHER THAN THE ONES GIVEN HERE. COMMENT COMMENT The primary data array for this file gives the quad-sphere (QS) COMMENT pixel number as a function of location on the COBE quad-sphere COMMENT cube faces. The QS pixels are ordered according to the quad-tree COMMENT nearest neighbor pixel numbering scheme, which is hierarchical. COMMENT Because of this, the QS pixel numbers are not sequential along a COMMENT row or column. COMMENT COMMENT Data in the primary FITS array are ordered such that the first COMMENT row of pixel numbers is at the bottom and the last row of pixel COMMENT numbers on a face is at the top, with the first QS pixel per COMMENT face being in the lower right corner as shown in the diagram below. COMMENT (See the section "Quad-tree pixelization scheme" below for more COMMENT information on how the pixel numbers fall on the sky cube.) COMMENT COMMENT row ------------------- COMMENT n | | n = NAXIS1 = NAXIS2 COMMENT ^ | | COMMENT | | | COMMENT 2 | | COMMENT 1 | | COMMENT ------------------- COMMENT n <--- 2 1 COMMENT column COMMENT COMMENT This FITS file also contains one binary table extension after COMMENT the primary data array. There are no other extensions. The COMMENT binary table gives the coordinates at the QS pixel centers. COMMENT Coordinate information is given in the binary table for various COMMENT reasons. The primary reason is that the quad-tree ordering of COMMENT the quad-sphere pixels is not linear. The CD matrix convention COMMENT might handle this, but it is not yet a part of standard FITS. COMMENT Since COBE data will be analyzed on a pixel-by-pixel basis as COMMENT well as by analyzing the full sky, the binary table format seems COMMENT to be the most conceptually clear. Furthermore, it is useful to COMMENT give the coordinates of the pixel centers in equatorial and COMMENT Galactic coordinates as well as the COBE standard ecliptic 2000, COMMENT and this is readily done via the binary table format. COMMENT COMMENT COBE Data Sets: COMMENT COMMENT The FITS files containing COBE data consist either of a COMMENT primary FITS header and primary data array, OR, a primary FITS COMMENT header, a binary table extension header, and a binary table COMMENT extension. The COBE data files, unlike this information file, COMMENT do not have both a primary data array and a binary table extension. COMMENT For those COBE data files that contain binary table extensions, COMMENT there is only one binary table per file, and there are no other COMMENT extensions in these files. In binary tables, the data consists of COMMENT various parameters in fields (table columns) sequenced by quad-tree COMMENT pixel number. COMMENT COMMENT Quadrilateralized Sky Cube Projection: COMMENT COMMENT COBE datasets are lists of records sequenced by pixel numbers COMMENT defined in a quad-tree manner (see "Quad-tree Pixelization Scheme" COMMENT below) on the faces of a cube. The cube faces are tangent plane COMMENT projections of the celestial sphere. This is the quadrilateralized COMMENT spherical (sky) cube (or quad-sphere) projection which gives COMMENT equally spaced pixels of approximately equal area on the cube, COMMENT which represent approximately equal areas on the sky. COMMENT COMMENT The COBE quadrilateralized spherical cube (CSC) projection COMMENT conserves area to approximately 1% across the whole cube with COMMENT a maximum discrepancy from equal area of only a few percent at COMMENT the edges of the cube. Furthermore, the projection distorts COMMENT shapes minimally. Beam shapes or point response functions will COMMENT differ only slightly depending upon location on the quad-cube face. COMMENT COMMENT The six faces of the cube, each of dimension NAXIS1 x NAXIS2, COMMENT are the six planes of the primary data array. The cube faces are COMMENT numbered 0 through 5 corresponding to planes 1 through 6 (NAXIS3) COMMENT in the primary data array, respectively. COMMENT COMMENT The dimensions of each face correspond to the COBE sky cube COMMENT resolution index number given by PIXINDEX. The dimension of a COMMENT side of each face (NAXIS1, which is necessarily the same value as COMMENT NAXIS2) is 2**(PIXINDEX-1). Thus, the number of pixels in each COMMENT face is NAXIS1 x NAXIS2 or 2**[2*(PIXINDEX-1)]. COMMENT COMMENT The all-sky, skyward looking, unfolded cube is reassembled by COMMENT arranging the faces in a sideways T with the bar on the right side: COMMENT COMMENT 0 COMMENT 4 3 2 1 COMMENT 5 COMMENT COMMENT This corresponds to a 4*NAXIS1 by 3*NAXIS2 array as follows: COMMENT COMMENT ----------------- COMMENT | b | b | b | 0 | ^ COMMENT ----------------- | COMMENT | 4 | 3 | 2 | 1 | 3*NAXIS2 COMMENT ----------------- | COMMENT | b | b | b | 5 | v COMMENT ----------------- COMMENT <-- 4*NAXIS1 ---> COMMENT COMMENT where b stands for blanks. COMMENT COMMENT COBE data can be displayed on this unfolded cube by arranging COMMENT the pixels in the binary table as they are arranged in the primary COMMENT array of this file. For example, if the binary table for the COBE COMMENT data set of interest looks like this (this is not necessarily the COMMENT fields that are in the COBE data tables; the fields and data are COMMENT for illustrative purposes only): COMMENT COMMENT QS_Pixel_Num Signal Error_in_Signal Ecl_Long Ecl_Lat COMMENT 0 22 1 315.0 36.7 COMMENT 1 43 2 317.9 38.1 COMMENT 2 36 2 312.1 38.1 COMMENT 3 31 1 315.0 39.7 COMMENT . COMMENT . COMMENT . COMMENT COMMENT then the signal will be mapped where the pixel numbers are in COMMENT face 0 (first plane) of the primary array: COMMENT COMMENT QS Pixel Numbers Signal COMMENT 3 2 | ---> 31 36 | COMMENT 1 0 | ---> 43 22 | COMMENT ------ -------- COMMENT COMMENT This is the quad-tree pixelization scheme described below. COMMENT COMMENT Note that the tabulated signal values may not necessarily refer COMMENT to measurements made at the pixel centers; thus, the coordinates COMMENT given may not be those of the pixel centers! If the coordinates COMMENT given with the data set differ from those given in the binary COMMENT table below, then they are not for pixel centers, and thus the COMMENT data will be displayed off-center. COMMENT COMMENT Quad-tree Pixelization Scheme: COMMENT COMMENT Pixel numbers begin in the lower right corner of a cube face COMMENT and increase in quad-tree fashion from right to left and bottom COMMENT to top across the face. This "Z" pattern is followed throughout COMMENT a face as shown by the following pixel numbers in the lower right COMMENT corner of face 0: COMMENT y COMMENT COMMENT 63 62 59 58 47 46 43 42 | 7 COMMENT 61 60 57 56 45 44 41 40 | 6 "Z" Pattern: COMMENT 55 54 51 50 39 38 35 34 | 5 COMMENT 53 52 49 48 37 36 33 32 | 4 <--- COMMENT 31 30 27 26 15 14 11 10 | 3 ^ COMMENT 29 28 25 24 13 12 9 8 | 2 / COMMENT 23 22 19 18 7 6 3 2 | 1 / COMMENT 21 20 17 16 5 4 1 0 | 0 <--- COMMENT ----------------------- COMMENT x 7 6 5 4 3 2 1 0 COMMENT COMMENT The quad-tree pixel ordering scheme is related to rasterized COMMENT rectangular array positions as follows. The location within a COMMENT face is specified by the 2*(PIXINDEX-1) least significant bits. COMMENT (As described below, the most significant bits are used to COMMENT identify the cube face.) The even least significant bits COMMENT (0, 2, . . ., 2*(PIXINDEX-2)) specify the x, or horizontal, COMMENT raster position, and the odd bits (1, 3, . . ., (2*PIXINDEX)-3) COMMENT specify the y position. Changing the pixel resolution thus COMMENT simply requires adding or deleting pairs of least significant COMMENT bits; each pair represents a resolution change in the x and y COMMENT face coordinates by a factor of 2. COMMENT COMMENT For example (at a PIXINDEX of 4) quad-tree pixel 24 in binary is COMMENT COMMENT 0 1 1 0 0 0 COMMENT y x y x y x COMMENT COMMENT Extracting the even and odd least significant bits gives: COMMENT COMMENT y = 010 = 2 COMMENT x = 100 = 4 COMMENT COMMENT Writing the pixel numbers in binary form clarifies the mapping COMMENT hierarchy: COMMENT COMMENT Col 7 6 5 4 3 2 1 0 Row COMMENT 111111 111110 111011 111010 | 101111 101110 101011 101010 7 COMMENT 111101 111100 111001 111000 | 101101 101100 101001 101000 6 COMMENT 110111 110110 110011 110010 | 100111 100110 100011 100010 5 COMMENT 110101 110100 110001 110000 | 100101 100100 100001 100000 4 COMMENT --------------------------------+-------------------------------- COMMENT 011111 011110 011011 011010 | 001111 001110 | 001011 001010 3 COMMENT 011101 011100 011001 011000 | 001101 001100 | 001001 001000 2 COMMENT | ---------------+--------------- COMMENT 010111 010110 010011 010010 | 000111 000110 | 000011 000010 1 COMMENT 010101 010100 010001 010000 | 000101 000100 | 000001 000000 0 COMMENT COMMENT Bit numbers 543210 543210 543210 543210 543210 543210 COMMENT COMMENT The lines of dashes show how the hierarchy is built up. COMMENT COMMENT As illustrated below, a cube face is bisected (PIXINDEX-1) COMMENT times to reach the pixel resolution. The figure depicts a COMMENT face at a PIXINDEX of 4. COMMENT PIXINDEX: COMMENT -------------------------------------------- ^ ^ ^ ^ COMMENT | | | 11 | 10 | 4 | | | COMMENT | | 11 |---10----| v 3 | | COMMENT | | | 01 | 00 | | | | COMMENT | 11 |---------10---------- v 2 | COMMENT | | | | | | COMMENT | | 01 | 00 | | | COMMENT | | | | | | COMMENT |---------------------+--------------------- v 1 (Whole COMMENT | | | | Face) COMMENT | | | | COMMENT | | | | COMMENT | 01 | 00 | | COMMENT | | | | COMMENT | | | | COMMENT | | | | COMMENT -------------------------------------------- v COMMENT COMMENT The face index is specified in the remaining Most Significant COMMENT Bits, that is, in bits 2*(PIXINDEX-1) through 2*PIXINDEX. Thus COMMENT for face 0, pixel 24, the binary number would be COMMENT COMMENT 0 0 0 0 1 1 0 0 0 , COMMENT | Face | Pixel number | COMMENT COMMENT whereas for pixel number 24 on face 5, the binary number would be COMMENT COMMENT 1 0 1 0 1 1 0 0 0 . COMMENT | Face | Pixel number | COMMENT COMMENT The pixel numbers run sequentially from 0 on face 0 to (NAXIS1 COMMENT x NAXIS2 x NAXIS3) - 1 on face 5. The last pixel number in a COMMENT face is (face number+1) * 2**[2*(PIXINDEX-1)] - 1. The first COMMENT pixel number in a face is 0 for face 0, and 1 plus the last pixel COMMENT number in the previous face for the other faces. Thus, for a COMMENT PIXINDEX of 4 the pixel number illustrated above, pixel 24 on COMMENT face 5, would be pixel number 344 (binary 101011000 = decimal 344) COMMENT with respect to the whole cube. COMMENT COMMENT World Coordinate Information: COMMENT COMMENT World coordinates such as ecliptic, celestial, and Galactic COMMENT are NOT linear across the COBE sky cube. As mentioned above, COMMENT the coordinates of the center of each pixel are given in the COMMENT binary table extension rather than by a CDELT or CD matrix COMMENT convention. The following comments are given as a general COMMENT description of how certain astronomical coordinate systems COMMENT would project onto the cube. COMMENT COMMENT The COBE sky cube is defined using the ecliptic J2000 reference COMMENT frame. With reference to this coordinate system, face 0 is centered COMMENT around the North ecliptic pole (NEP), faces 1 through 4 are along COMMENT the Ecliptic (each spanning 90 degrees of ecliptic longitude), and COMMENT face 5 is centered around the South ecliptic pole (SEP). Ecliptic COMMENT longitude increases from face 1 to face 4. The vernal equinox COMMENT (0.0 ecliptic longitude and latitude) lies at the center of face 1. COMMENT (Face 1 spans ecliptic longitudes from -45 (315) to +45 degrees.) COMMENT The anti-vernal equinox point is at the center of face 3. Faces COMMENT 1-4 include the ecliptic latitudes ranging from roughly -45 to +45 COMMENT degrees. Face 0 contains all ecliptic longitudes, and the COMMENT latitudes around the North ecliptic pole, down to approximately COMMENT +45 degrees. Face 5 contains all ecliptic longitudes, and the COMMENT latitudes around the South ecliptic pole, up to approximately COMMENT -45 degrees. The latitude ranges in each face are not exact COMMENT because of the curvilinear nature of the coordinates on the COMMENT COBE sky cube. COMMENT COMMENT The North Galactic Pole is on face 3 at J2000 ecliptic COMMENT longitude 179.32095 and latitude 29.811954 degrees, and the COMMENT Galactic center is on face 4 at J2000 ecliptic longitude COMMENT 266.14097 and latitude -5.52967943 degrees. The South Galactic COMMENT Pole is on face 1, and the Galactic anti-center is on face 2. COMMENT Pixel numbers for these locations can be found in the binary COMMENT extension table of this file. COMMENT COMMENT The ecliptic coordinates of the North Celestial Pole are COMMENT (90.0,66.5607089) degrees, located on face 0. COMMENT COMMENT References: COMMENT COMMENT Chan, F.K. and O'Neill 1975. Feasibility Study of a Quadri- COMMENT lateralized Spherical Cube Earth Data Base, Computer Sciences COMMENT Corp., EPRF Tech. Report 2-75. Prepared for the Environmental COMMENT Prediction Research Facility, Monterey, Calif. COMMENT Greisen, Eric W. and Mark Calabretta 1993. Representations of COMMENT Celestial Coordinates in FITS. Draft FITS proposal, NRAO. COMMENT O'Neill, I.M. and Laubscher, R.E. 1976. Extended Studies of a COMMENT Quadrilateralized Spherical Cube Earth Data Base, Computer COMMENT Sciences Corp., EPRF Tech. Report 3-76. Prepared for the COMMENT Environmental Prediction Research Facility, Monterey, Calif. COMMENT White, R.A. and Mather, J.C. 1991. Databases from Cosmic COMMENT Background Explorer (COBE) in Databases and On-Line Data in COMMENT Astronomy, ed. M. Albrecht and D. Egret (Kluwer, Dordrecht, COMMENT the Netherlands), pp. 29-34. COMMENT White, Richard A. and Stemwedel, Sally W. 1992. "The Quadri- COMMENT lateralized Spherical Cube and Quad-Tree for All Sky Data." COMMENT In Astronomical Data Analysis Software and Systems I, Volume COMMENT 25, edited by D.M. Worrall, C. Biemesderfer, and J. Barnes, COMMENT pp. 379-381. COMMENT HISTORY Created by COBE project software facility UDF routine HISTORY UDF_CREATE_INFO_FILES using FITSIO version 3.3 routines. HISTORY Primary array created by COBE project software UIDL routine PIX2XY. COMMENT End of creation history. END