FITS File Calculator Expressions

  General Syntax

    The expression can  be an arbitrarily  complex  series of operations
    performed on constants, keyword values,  and column data taken  from
    the specified FITS TABLE extension.  For  the case of row filtering,
    the expression must evaluate to a single boolean  value for each row
    of the table.

    Keyword and   column data  are referenced by   name.  Any  string of
    characters not surrounded by    quotes (ie, a constant  string)   or
    followed by   an open parentheses (ie,   a  function name)   will be
    initially interpretted   as a column  name and  its contents for the
    current row inserted into the expression.  If no such column exists,
    a keyword of that  name will be searched for  and its value used, if
    found.  To force the  name to be  interpretted as a keyword (in case
    there is both a column and keyword with the  same name), precede the
    keyword name with a single pound sign, '#', as in '#NAXIS2'.  Due to
    the generalities of FITS column and  keyword names, if the column or
    keyword name  contains a space or a  character which might appear as
    an arithmetic  term then inclose  the  name in '$'  characters as in
    $MAX PHA$ or #$MAX-PHA$.  Names are case insensitive.

    To access a table entry in a row other  than the current one, follow
    the  column's name  with  a row  offset  within  curly  braces.  For
    example, 'PHA\{-3\}' will evaluate to the value  of column PHA, 3 rows
    above  the  row currently  being processed.   One  cannot specify an
    absolute row number, only a relative offset.  Rows that fall outside
    the table will be treated as undefined, or NULLs.
    
    Boolean   operators can be  used in  the expression  in either their
    Fortran or C forms.  The following boolean operators are available:

    "equal"         .eq. .EQ. ==  "not equal"          .ne.  .NE.  !=
    "less than"     .lt. .LT. <   "less than/equal"    .le.  .LE.  <= =<
    "greater than"  .gt. .GT. >   "greater than/equal" .ge.  .GE.  >= =>
    "or"            .or. .OR. ||  "and"                .and. .AND. &&
    "negation"     .not. .NOT. !  "approx. equal(1e-7)"  ~

    The expression may  also include arithmetic operators and functions.
    Trigonometric  functions use  radians,  not degrees.  The  following
    arithmetic  operators and  functions  can be  used in the expression
    (function names are case insensitive). A null value will be returned
    in case of illegal operations such as divide by zero, sqrt(negative)
    log(negative), log10(negative), arccos(.gt. 1), arcsin(.gt. 1).

    "addition"           +          "subtraction"          -
    "multiplication"     *          "division"             /
    "negation"           -          "exponentiation"       **  ^
    "modulus"            i % j      "absolute value"       abs(x)
    "sine"               sin(x)     "arc sine"             arcsin(x)
    "cosine"             cos(x)     "arc cosine"           arccos(x)
    "tangent"            tan(x)     "arc tangent"          arctan(x)
    "hyperbolic sine"    sinh(x)    "arc tangent (2)"      arctan2(y,x)
    "hyperbolic cosine"  cosh(x)    "hyperbolic tangent"   tanh(x)
    "exponential"        exp(x)     "square root"          sqrt(x)
    "natural log"        log(x)     "common log"           log10(x)
    "round to nearest int" round(x) "random # [0.0,1.0)"   random()
    "round up to int"    ceil(x)    "round down to int"    floor(x)
    "minimum"            min(x,y)   "maximum"              max(x,y)
    "cumulative sum"    accum(x)   "sequential difference" seqdiff(x)
    "angular separation"  angsep(ra1,dec1,ra2,de2) (all in degrees)
    "if-then-else"      b?x:y


    An alternate syntax for the min and max functions  has only a single
    argument which  should be  a  vector value (see  below).  The result
    will be the minimum/maximum element contained within the vector.

    The accum(x) function forms the cumulative sum of x, element by element.
    Vector columns are supported simply by performing the summation process
    through all the values.  Null values are treated as 0.  The seqdiff(x)
    function forms the sequential difference of x, element by element.
    The first value of seqdiff is the first value of x.  A single null
    value in x causes a pair of nulls in the output.  The seqdiff and
    accum functions are functional inverses, i.e., seqdiff(accum(x)) == x
    as long as no null values are present.

    The angsep function computes the angular separation in degrees
    between 2 celestial positions, where the first 2 parameters
    give the RA-like and Dec-like coordinates (in decimal degrees)
    of the first position, and the 3rd and 4th parameters give the
    coordinates of the second position.

    In the  if-then-else expression, "b?x:y",  b is an  explicit boolean
    value or  expression.  There is  no automatic type  conversion from
    numeric to boolean values, so  one needs to use "iVal!=0" instead of
    merely "iVal"  as the boolean  argument. x and  y can be  any scalar
    data type (including string).

    The  following  type  casting  operators  are  available,  where the
    inclosing parentheses are required and taken  from  the  C  language
    usage. Also, the integer to real casts values to double precision:
    
                "real to integer"    (int) x     (INT) x
                "integer to real"    (float) i   (FLOAT) i
    
    In addition, several constants are built in  for  use  in  numerical
    expressions:
    
       #pi           3.1415...         #e            2.7182...
       #deg          #pi/180           #row          current row number
       #null         undefined value   #snull        undefined string
    
    A  string constant must  be enclosed  in quotes  as in  'Crab'.  The
    "null" constants  are useful for conditionally  setting table values
    to a NULL, or undefined, value (eg., "col1==-99 ? #NULL : col1").
    
    There is also a function for testing if  two  values  are  close  to
    each  other,  i.e.,  if  they are "near" each other to within a user
    specified tolerance. The  arguments,  value_1  and  value_2  can  be
    integer  or  real  and  represent  the two values who's proximity is
    being tested to be within the specified tolerance, also  an  integer
    or real:
                    near(value_1, value_2, tolerance)
    
    When  a  NULL, or undefined, value is encountered in the FITS table,
    the expression will evaluate to NULL unless the undefined  value  is
    not   actually   required  for  evaluation,  eg.  "TRUE  .or.  NULL" 
    evaluates to TRUE. The  following  two  functions  allow  some  NULL
    detection  and  handling:

         "a null value?"              ISNULL(x)
         "define a value for null"    DEFNULL(x,y)
	 
    The former
    returns a boolean value of TRUE if the  argument  x  is  NULL.   The
    later  "defines"  a  value  to  be  substituted  for NULL values; it
    returns the value of x if x is not NULL, otherwise  it  returns  the
    value of y.

  Bit Masks

    Bit  masks can be used to select out rows from bit columns (TFORMn =
    #X) in FITS files. To represent the mask,  binary,  octal,  and  hex
    formats are allowed:
    
                 binary:   b0110xx1010000101xxxx0001
                 octal:    o720x1 -> (b111010000xxx001)
                 hex:      h0FxD  -> (b00001111xxxx1101)
    
    In  all  the  representations, an x or X is allowed in the mask as a
    wild card. Note that the x represents a  different  number  of  wild
    card  bits  in  each  representation.  All  representations are case
    insensitive.
    
    To construct the boolean expression using the mask  as  the  boolean
    equal  operator  discribed above on a bit table column. For example,
    if you had a 7 bit column named flags in a  FITS  table  and  wanted
    all  rows  having  the bit pattern 0010011, the selection expression
    would be:
    
                            flags == b0010011
    or
                            flags .eq. b10011
    
    It is also possible to test if a range of bits is  less  than,  less
    than  equal,  greater  than  and  greater than equal to a particular
    boolean value:
    
                            flags <= bxxx010xx
                            flags .gt. bxxx100xx
                            flags .le. b1xxxxxxx
    
    Notice the use of the x bit value to limit the range of  bits  being
    compared.
    
    It  is  not necessary to specify the leading (most significant) zero
    (0) bits in the mask, as shown in the second expression above.
    
    Bit wise AND, OR and NOT operations are  also  possible  on  two  or
    more  bit  fields  using  the  '&'(AND),  '|'(OR),  and the '!'(NOT)
    operators. All of these operators result in a bit  field  which  can
    then be used with the equal operator. For example:
    
                          (!flags) == b1101100
                          (flags & b1000001) == bx000001
    
    Bit  fields can be appended as well using the '+' operator.  Strings
    can be concatenated this way, too.

  Vector Columns

    Vector columns can also be used  in  building  the  expression.   No
    special  syntax  is required if one wants to operate on all elements
    of the vector.  Simply use the column name as for a  scalar  column.
    Vector  columns  can  be  freely  intermixed  with scalar columns or
    constants in virtually all expressions.  The result will be  of  the
    same dimension as the vector.  Two vectors in an expression, though,
    need to  have  the  same  number  of  elements  and  have  the  same
    dimensions.   The  only  places  a vector column cannot be used (for
    now, anyway) are the SAO  region  functions  and  the  NEAR  boolean
    function.

    Arithmetic and logical operations are all performed on an element by
    element basis.  Comparing two vector columns,  eg  "COL1  ==  COL2",
    thus  results  in  another vector of boolean values indicating which
    elements of the two vectors are equal. 
    
    Eight functions are available that operate on a vector and return a
    scalar result:

    "minimum"      MIN(V)          "maximum"               MAX(V)
    "average"      AVERAGE(V)      "median"                MEDIAN(V)
    "sumation"     SUM(V)          "standard deviation"    STDDEV(V)
    "# of values"  NELEM(V)        "# of non-null values"  NVALID(V)

    where V represents the name of a vector column or a manually 
    constructed vector using curly brackets as described below.  The
    first 6 of these functions ignore any null values in the vector when
    computing the result.
    
    The SUM function literally sums all  the elements in x,  returning a 
    scalar value.   If V  is  a  boolean  vector, SUM returns the number
    of TRUE elements. The NELEM function  returns the number of elements
    in vector V whereas NVALID return the number of non-null elements in
    the  vector.   (NELEM  also  operates  on  bit  and string  columns, 
    returning their column widths.)  As an example, to  test whether all 
    elements of two vectors satisfy a  given logical comparison, one can
    use the expression
    
              SUM( COL1 > COL2 ) == NELEM( COL1 )
    
    which will return TRUE if all elements  of  COL1  are  greater  than
    their corresponding elements in COL2.
    
    To  specify  a  single  element  of  a  vector, give the column name
    followed by  a  comma-separated  list  of  coordinates  enclosed  in
    square  brackets.  For example, if a vector column named PHAS exists
    in the table as a one dimensional, 256  component  list  of  numbers
    from  which  you  wanted to select the 57th component for use in the
    expression, then PHAS[57] would do the  trick.   Higher  dimensional
    arrays  of  data  may appear in a column.  But in order to interpret
    them, the TDIMn keyword must appear in the header.  Assuming that  a
    (4,4,4,4)  array  is packed into each row of a column named ARRAY4D,
    the  (1,2,3,4)  component  element  of  each  row  is  accessed   by 
    ARRAY4D[1,2,3,4].    Arrays   up   to   dimension  5  are  currently 
    supported.  Each vector index can itself be an expression,  although
    it  must  evaluate  to  an  integer  value  within the bounds of the
    vector.  Vector columns which contain spaces or arithmetic operators
    must   have   their   names  enclosed  in  "$"  characters  as  with 
    $ARRAY-4D$[1,2,3,4].
    
    A  more  C-like  syntax  for  specifying  vector  indices  is   also 
    available.   The element used in the preceding example alternatively
    could be specified with the syntax  ARRAY4D[4][3][2][1].   Note  the
    reverse  order  of  indices  (as in C), as well as the fact that the
    values are still ones-based (as  in  Fortran  --  adopted  to  avoid
    ambiguity  for  1D vectors).  With this syntax, one does not need to
    specify all of the indices.  To  extract  a  3D  slice  of  this  4D
    array, use ARRAY4D[4].
    
    Variable-length vector columns are not supported.
    
    Vectors can  be manually constructed  within the expression  using a
    comma-separated list of  elements surrounded by curly braces ('{}').
    For example, '{1,3,6,1}' is a 4-element vector containing the values
    1, 3, 6, and 1.  The  vector can contain  only boolean, integer, and
    real values (or expressions).  The elements will  be promoted to the
    highest  datatype   present.  Any   elements   which  are themselves
    vectors, will be expanded out with  each of its elements becoming an
    element in the constructed vector.

  Good Time Interval Filtering

    A common filtering method  applied to  FITS  files is a time  filter
    using a Good Time Interval (GTI)  extension.  A high-level function,
    gtifilter(a,b,c,d),  is  available   which  performs this    special
    evaluation, returning a boolean result for each time element tested.
    Its syntax is

       gtifilter( [ "filename" [, expr [, "STARTCOL", "STOPCOL" ] ] ] )

    where  each  "[]" demarks   optional parameters.   The  filename, if
    specified,  can be  blank  ("") which will    mean to use  the first
    extension  with   the name "*GTI*"  in   the current  file,  a plain
    extension  specifier (eg, "+2",  "[2]", or "[STDGTI]") which will be
    used  to  select  an extension  in  the current  file, or  a regular
    filename with or without an extension  specifier which in the latter
    case  will mean to  use the first  extension  with an extension name
    "*GTI*".  Expr can be   any arithmetic expression, including  simply
    the time  column  name.  A  vector  time expression  will  produce a
    vector boolean  result.  STARTCOL and  STOPCOL are the  names of the
    START/STOP   columns in the    GTI extension.  If   one  of them  is
    specified, they both  must be. Note that  the quotes surrounding the
    filename and START/STOP column names are required.

    In  its  simplest form, no parameters need to be provided -- default
    values will be used.  The expression "gtifilter()" is equivalent to
    
       gtifilter( "", TIME, "*START*", "*STOP*" )
    
    This will search the current file for a GTI  extension,  filter  the
    TIME  column in the current table, using START/STOP times taken from
    columns in the GTI  extension  with  names  containing  the  strings
    "START"  and "STOP".  The wildcards ('*') allow slight variations in
    naming conventions  such  as  "TSTART"  or  "STARTTIME".   The  same
    default  values  apply for unspecified parameters when the first one
    or  two  parameters  are  specified.   The  function   automatically 
    searches   for   TIMEZERO/I/F   keywords  in  the  current  and  GTI 
    extensions, applying a relative time offset, if necessary.

  Spatial Region Filtering
  
    Another common  filtering method is  a  spatial filter using  a SAO-
    style region file.  The syntax for this high-level filter is

       regfilter( "regfilename" [ , Xexpr, Yexpr [ , "wcs cols" ] ] )

    The region file name is required, but the rest is optional.  Without
    any explicit expression for the X and Y coordinates (in pixels), the
    filter will search for  and operate on columns "X"  and "Y".  If the
    region file is   in "degrees" format  instead  of "pixels" ("hhmmss"
    format is not supported, yet), the  filter will need WCS information
    to convert the region coordinates to pixels.  If supplied, the final
    parameter string contains the names of the 2 columns (space or comma
    separated) which contain   the   desired WCS information.    If  not
    supplied, the filter  will scan the X  and Y expressions for  column
    names.  If only one is found in each  expression, those columns will
    be used.  Otherwise, an error will be returned.

    The region shapes supported are (names are case insensitive):

       Point         ( X1, Y1 )               <- One pixel square region
       Line          ( X1, Y1, X2, Y2 )       <- One pixel wide region
       Polygon       ( X1, Y1, X2, Y2, ... )  <- Rest are interiors with
       Rectangle     ( X1, Y1, X2, Y2, A )       | boundaries considered
       Box           ( Xc, Yc, Wdth, Hght, A )   V within the region
       Diamond       ( Xc, Yc, Wdth, Hght, A )
       Circle        ( Xc, Yc, R )
       Annulus       ( Xc, Yc, Rin, Rout )
       Ellipse       ( Xc, Yc, Rx, Ry, A )
       Elliptannulus ( Xc, Yc, Rinx, Riny, Routx, Routy, Ain, Aout )
       Sector        ( Xc, Yc, Amin, Amax )

    where (Xc,Yc) is  the coordinate of  the shape's center; (X#,Y#) are
    the coordinates  of the shape's edges;  Rxxx are the shapes' various
    Radii or semimajor/minor  axes; and Axxx  are the angles of rotation
    (or bounding angles for Sector) in degrees.  For rotated shapes, the
    rotation angle  can  be left  off, indicating  no rotation.   Common
    alternate  names for the regions  can also be  used: rotbox <-> box;
    rotrectangle <-> rectangle;  (rot)rhombus <-> (rot)diamond;  and pie
    <-> sector.  When a  shape's name is  preceded by a minus sign, '-',
    the defined region  is instead the area  *outside* its boundary (ie,
    the region is inverted).  All the shapes within a single region file
    are AND'd together to create the region.

    There are three functions that are primarily for use with SAO region
    files, but they  can  be  used  directly.  They
    return  a  boolean true   or  false  depending   on  whether a   two
    dimensional point is in the region or not:

    "point in a circular region"
          circle(xcntr,ycntr,radius,Xcolumn,Ycolumn)
    
    "point in an elliptical region"
         ellipse(xcntr,ycntr,xhlf_wdth,yhlf_wdth,rotation,Xcolumn,Ycolumn)
    
    "point in a rectangular region"
             box(xcntr,ycntr,xfll_wdth,yfll_wdth,rotation,Xcolumn,Ycolumn)
    
    where 
       (xcntr,ycntr) are the (x,y) position of the center of the region
       (xhlf_wdth,yhlf_wdth) are the (x,y) half widths of the region
       (xfll_wdth,yfll_wdth) are the (x,y) full widths of the region
       (radius) is half the diameter of the circle
       (rotation) is the angle(degrees) that the region is rotated with
             respect to (xcntr,ycntr)
       (Xcoord,Ycoord) are the (x,y) coordinates to test, usually column
             names
       NOTE: each parameter can itself be an expression, not merely a
             column name or constant.
    

    For  complex  or commonly  used filters,  one   can also  place  the
    expression into a text file and import it  into the calculator using
    the  syntax  '@filename.txt'.  The   expression can  be  arbitrarily
    complex and extend over multiple lines of the file.

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