Definition of Keywords in the Image Map Projection Group for ISIS

Certain keywords will always appear in the IMAGE_MAP_PROJECTION group in ISIS cube file labels if the cube is in a map projection. They are:

MAP_PROJECTION_TYPE - The full map projection name These projections are described by John Snyder in the U.S.G.S. Professional paper 1395 titled "Map projections - A Working Manual".

A_AXIS_RADIUS - The major equatorial radius in kilometers

B_AXIS_RADIUS - The minor equatorial radius in kilometers (the map projection software does not currently use this value)

C_AXIS_RADIUS - The polar radius in kilometers

POSITIVE_LONGITUDE_DIRECTION - The direction of positive longitude. The value of this keyword is either EAST or WEST.

MINIMUM_LATITUDE - The lowest latitude boundary. Data is guaranteed to exist within the latitude and longitude boundaries on the label. There may be positions within the cube that have latitude or longitude values that are outside the ranges listed on the labels because the projection is not rectangular. For example, a north polar projection which includes 360 degrees of longitude and includes the pole is only bounded by the minimum latitude.

MAXIMUM_LATITUDE - The highest latitude boundary

EASTERNMOST_LONGITUDE - The right longitude boundary (at least for rectangular projections)

WESTERNMOST_LONGITUDE - The left longitude boundary (at least for rectangular projections)

MAP_SCALE - The map resolution of the projection in kilometers/pixel at the center latitude and center longitude or at one or two standard parallels depending on the projection

MAP_RESOLUTION - The map resolution of the projection in pixels/degree at the center latitude and center longitude or at one or two standard parallels depending on the projection

REFERENCE_LATITUDE - The latitude of the pole of a rotated spherical coordinate system

REFERENCE_LONGITUDE - The longitude of the pole of a rotated spherical coordinate system

MAP_PROJECTION_ROTATION - The rotation of the map projection x-y coordinate system. This defines which way is "up". Usually north is at the top of a cube. A positive value indicates that the cube has been rotated in a clockwise direction.

LINE_PROJECTION_OFFSET - the line translation from the origin of a projection to the center of line 1 of the cube.

SAMPLE_PROJECTION_OFFSET - the sample translation from the origin of a projection to the center of sample 1 of the cube.

Keywords that are specific to certain map projections will appear in the IMAGE_MAP_PROJECTION group when those map projections are used. The following table lists all possible projections, the projection abbreviation, and the (abbreviated) specific parameters needed for each projection.

Keyword  Projection                      Parameters required
ALBE     Albers Conical Equal-area       CLON,CLAT,PAR1,PAR2
AZEQ     Azimuthal Equidistant           CLON,CLAT
CYLI     Cylindrical Equal-area          CLAT*
ECON     Equidistant Conic               CLON,CLAT,PAR1,PAR2
GNOM     Gnomonic                        CLON,CLAT
LAMA     Lambert Azimuthal Equal-area    CLON,CLAT
         (+90 to -90 from CLON,CLAT)
LAMB     Lambert Conformal               CLON,CLAT,PAR1,PAR2
LAMG     Lambert Azimuthal Equal-area    CLON,CLAT
         (+170 to -170 from CLON,CLAT)
MERC     Mercator                        CLON,CLAT*
MILL     Miller Cylindrical              CLON
MSC      Modified Stereographic Conformal
OMER     Oblique Mercator or Hotine      CLON,CLAT,SCFA,INCL
                                         LAT1,LON1,LAT2,LON2
ORTH     Orthographic                    CLON,CLAT
POIN     Point Perspective               CLON,CLAT,DIST
POLA     Polar Stereographic             CLON,CLAT*
POLY     Polyconic                       CLON,CLAT
ROBI     Robinson                        CLON
SIMP     Simple Cylindrical              CLON,CLAT*
         or Equirectangular
SINU     Sinusoidal                      CLON
SOM      Space Oblique Mercator          TIM1,TIM2,INCL,ASCN
STER     Stereographic                   CLON,CLAT
TRAN     Transverse Mercator             CLON,CLAT,SCFA
UTM      Universal Transverse Mercator   CLON,CLAT,SCFA
VANG     Van der Grinten I               CLON

*CLAT defines the latitude of true scale for the following projections: CYLI, MERC, POLA and SIMP.

CENTER_LATITUDE - (CLAT) Projections requiring a center latitude are: ALBE, AZEQ, CYLI, GNOM, LAMA, LAMB, LAMG, MERC, OMER, ORTH, POIN, POLA, POLY, SIMP, STER, TRAN and UTM. The center latitude defines the latitude of origin of the projection, except for the following projections: CYLI, MERC, POLA and SIMP where it defines the latitude of true scale.

CENTER_LONGITUDE - (CLON) Projections requiring a center longitude are: ALBE, AZEQ, ECON, GNOM, LAMA, LAMB, LAMG, MERC, MILL, OMER, ORTH, POIN, POLA, POLY, ROBI, SIMP, SINU, STER, TRAN, UTM and VANG. The center longitude defines the central meridian of the projection.

FIRST_STANDARD_PARALLEL - (PAR1) Projections requiring standard parallels are: ALBE, ECON and LAMB. The standard parallels are the parallels which are intersected by the projection. If there is only one standard parallel, then the projection is tangent to that parallel.

SECOND_STANDARD_PARALLEL - (PAR2) See FIRST_STANDARD_PARALLEL

SCALE_FACTOR - (SCFA) The projections requiring a scale factor are: OMER, TRAN and UTM. This is a multiplicative factor of the map scale.

PLANET_ROTATION - (TIM1) The SOM is the only projection requiring the time of planet rotation. TIM1 is the length of Earth's rotation with respect to the precessed ascending node of the satellite orbit in minutes. (See Snyder).

SATELLITE_ROTATION - (TIM2) The SOM is only projection requiring the time of satellite rotation. TIM2 is the time required for one revolution of the spacecraft in minutes.

ORBIT_INCLINATION - (INCL) The projections requiring the orbit inclination are: OMER and SOM. INCL is the angle of inclination between the plane of the planet's equator and the plane of the satellite orbit, measured counterclockwise from the equator to the orbital plane at the ascending node.

ASCENDING_NODE - (ASCN) The SOM is only projection requiring the longitude of the ascending node of the orbit. This is the longitude at the equator that the satellite crosses at time 0.

IMAGE_HEIGHT - (DIST) The POIN is the only projection requiring the height above the planet which is in kilometers.

LATITUDE_CONTROL_1 - (LAT1) The OMER is the only projection requiring the latitude and longitude of two control points. These control points must be on the great circle chosen for the projection. The Mercator is a special case of the Oblique Mercator where the Equator is the great circle chosen for the projection. LAT1 is the latitude of the first control point.

LONITUDE_CONTROL_1 - (LON1) The longitude of the first control point. See LATITUDE_CONTROL_1

LATITUDE_CONTROL_2 - (LAT2) The latitude of the second control point. See LATITUDE_CONTROL_1

LONGITUDE_CONTROL_2 - (LON2) The longitude of the second control point. See LATITUDE_CONTROL_1

GEOMETRIC DEFINITION OF A PIXEL IN ISIS

The purpose here is to describe the spatial or geometric definition of a pixel used in the ISIS digital products and software provided by the USGS in Flagstaff. A broad range of factors enters into this question. For example, is a pixel to be conceived of as a point or as an area. The point definition would be most convenient, for instance, when dealing with coordinate grid overlays. This results in an odd number of pixels across a map that has an even number of spatial increments. For changing scales (for instance by even powers of 2) this definition becomes a problem. In this case it makes more sense to treat a pixel as a finite area. Then an even number of pixels covers an even number of spatial increments and decreasing/increasing scales by a power of 2 becomes trivial. However, grids now fall between pixels, at least in a mathematical sense. Their treatment in the generation of hardcopy therefore becomes an issue.

It was decided that the area concept of a pixel was the better choice; we would have to live with the assymmetries introduced in things like cartographic grids. There are various solutions: (1) use two pixels for the width of a grid line, (2) stagger grid pixels back-and-forth across the mathematical position, (3) use a convention whereby grid lines are systemmatically drawn offset from their mathematical position. The next issue is the conversion between integer coordinates and real coordinates of the pixel mesh. We adopt the convention that pixels are numbered (or named if you like) beginning in the upper left corner with line 1, sample 1 (pixel 1,1); lines increase downward; samples increase to the right. (Even this is not a universal standard; some astronomical systems begin, perhaps more logically, in the lower left corner.) There are three reasonable possibilities for aligning a real, or floating point, coordinate system with the pixel mesh: the coordinate 1.0, 1.0 could be the upper left, the center, or the lower right of pixel 1,1. The convention historically used for geometric calibration files (reseau positions) and also used in the Multimission Image Processing Laboratory at the Jet Propulsion Laboratory, is that the center of the pixel is defined as its location in real coordinates. In other words, the real coordinates of the center of pixel 1,1 are 1.0, 1.0. The top left corner of the pixel is .5, .5 and the bottom right corner is 1.49999..., 1.499999. The bottom and right edge of a pixel is the mathematically open boundary. This is the standard adopted in the ISIS software and its digital products.

Cartographic conventions must also be defined. Many of our digital products are in some map projection with an associated latitude and longitude range. The projection representation of a pixel is mathematically open at the increasing (right and lower) boundaries, and mathematically closed at its left and upper boundaries. An exception occurs at the physical limits of a projection; the lower boundary of the lowest pixel is closed to include the limit of the projection (e. g. the south pole). In the case of a rectangular projection such as a Mercator or a Simple Cylindrical, the left edge of pixel 1,1 is labeled with the left end of the longitude range and the right edge of the right most pixel is labeled with the right end of the longitude range. For example, if an image is in a Simple Cylindrical projection with a longitude range of -180.0 to +180.0, a latitude range of -90.0 to +90.0 and a scale of one degree/pixel, the image will have 180 lines and 360 samples. The latitude and longitude of the top left corner of pixel 1,1 is 90.0, 180.0. (if the planet has positive longitude to the west). The latitude and longitude of the bottom right corner of pixel 180,360 is -90.0, -180.0.

Coordinates of Pixel 1,1

           longitude  180.0         179.00001
                        |               |
            latitude    |               |   line
               90.0  -- ----------------- -- .5
                        |               |
                        |               |
                        |               |
                        |               |
                        |       +       |
                        |   (1.0,1.0)   |
                        |               |
                        |               |
                        |               |
            89.00001 -- ----------------- -- 1.49999
                        |               |
                        |               |
               sample  .5            1.49999

Finally, we must select a convention for drawing grid lines for various cartographic coordinates on planetary images and maps. The convention used in ISIS is that a grid line is drawn in the pixels that contain its floating point value until the open boundary is reached and then an exception is made so that the outer range of latitude and longitude will always appear on the image. This means, in the example given above, a 10 degree grid would start on pixel 1 and be drawn on every tenth pixel (11,21,31,...) until the open boundary is reached. Then the line would be drawn on the pixel previous to the open boundary (line 180 instead of line 181, or sample 360 instead of 361).

To summarize, the ISIS conventions are:

1. Pixels are treated as areas, not as points.
2. The integer coordinates begin with 1,1 (read "line 1, sample 1") for the upper-left-most pixel; lines increase downward; samples increase to the right.
3. Integer and floating point image coordinates are the same at the center of a pixel.
4. Grids will be drawn in the pixels that contain the floating point location of the grid lines except for open boundaries, which will be drawn to the left or above the open boundary.