Dithering

Syntax: | `DENSITY\DITHERING { x y } z { p1 p2 } DENSITY\DITHERING\DITHERVECTOR d { x y } z { p1 p2 } DENSITY\DITHERING\LEVELS lvl { x y } z { p1 p2 } DENSITY\DITHERING\LEVELS\DITHERVECTOR d lvl { x y } z { p1 p2 }`
Qualifiers: | `\LEGEND, \DITHERVECTOR, \CONTOURS, \LEVELS, \AREAS, \VOLUMES, \LINES, \EQUALLYSPACED, \PARTIAL, \XPROFILE, \YPROFILE, \BORDER, \AXES, \RESET`
Defaults: | `if `z` is a matrix: `x=[1;2;3;...], y=[1;2;3;...],` `p1=0, p2=1,` d=[1;1;2;1;2;2;3;2;3;3;4;3;4;4;5;5;6;6;0;0], \-PARTIAL, \-LEGEND, \-XPROFILE, \-YPROFILE, \BORDER, \AXES, \RESET, \-LEVELS, \-AREAS, \-VOLUMES, \-LINES, \-EQUALLYSPACED`

In this type of density plot, by default, the range of data values is divided into ten (10) equally spaced levels and a different dithering pattern is associated with each level. A value is interpolated at every pixel location within the bounds of the data region to determine the level for that point. The dithering pattern for that level then determines whether that pixel is to be lit up. Thus, the boundaries of the data are divided up into different dithering pattern regions.

Input variables

If `z` is a vector, the parameters `x` and `y` are expected and must be vectors. `x` and `y` are assumed to represent a scattered set of points, where `z[i]` is the altitude corresponding to the location `(x[i],y[i])`. A matrix is interpolated on these scattered points by means of a Thiessen triangulation of the plane. The three vectors must be the same length.

If `z` is a matrix, the parameters `x` and `y` default to `[1;2;3;...]`, but if entered they must be vectors. Each matrix element, `z[i,j]`, is associated with the coordinates `(x[j],y[i])`. The length of `x` must be equal to the number of columns of `z` and the length of `y` must be equal to the number of rows. The vectors `x` and `y` are used for scaling the axes.

Dithering pattern definition

The default dithering pattern vector is:

`[ 1;1; 2;1; 2;2; 3;2; 3;3; 4;3; 4;4; 5;5; 6;6; 0;0 ]`

A user defined dithering pattern can be entered by using the `\DITHERVECTOR` qualifier and entering the dithering pattern vector, `d`, as the first parameter.

The dithering pattern is determined by pairs of numbers from `d`, so the number of dithering patterns defined by `d` is ½ the length of `d`.

For pattern number `i`, every `d[2i-1]`th pixel is lit up horizontally, and every `d[2i]`th pixel is lit up vertically. For example, if `d[1]=1` and `d[2]=1`, then for level `1` every pixel is lit up, while if `d[3]=2` and `d[4]=3`, then for level `2` every second pixel is lit up horizontally and every third pixel is lit up vertically.

Changing the range of values

The optional parameters `p1` and `p2` can be used to broaden or shrink the range of data values. If `zmax` is the maximum value of the data and `zmin`} is the minimum value of the data, the full colour range will be from a minimum of `min = p1*(zmax-zmin)+zmin` to a maximum of `max = p2*(zmax-zmin)+zmin`. If `z` is a data value and if `z < p1*(zmax-zmin)+zmin`, that data value is treated as `zmin`. If `z > p1*(zmax-zmin)+zmin`, that data value is treated as `zmax`. The default values are: `p1=0` and `p2=1`.

Legend

If the `\LEGEND` qualifier is used, a legend is drawn along the right side of the axes. The legend requires the right end of the x-axis to be set to `75%` of the window, that is, `XUPPERAXIS` is set to `75%`.

When a y-profile is drawn, using the `\YPROFILE` qualifier, the right edge of the axis box must allow space for the profile as well as a possible legend. If a y-legend profile and a legend are present, then `XUPPERAXIS` is set to `65%`. If a y-legend profile is present but not a legend, then `XUPPERAXIS` is set to `85%`.

By default, `XUPPERAXIS` is reset to its former value after the `DENSITY` command. If the `/-RESET` qualifier is used, the axis location is not reset.

Contours

By default, a contour line is drawn around the boundary of each dithering pattern region. If the `\-LINES` qualifier is used, then these contour lines will not be drawn.

If the qualifier `\CONTOURS` is used, an automatically created vector named `DENSITY$CONTOURS` will be made which will contain the boundary values of each region. If there are `N` regions, the length of `DENSITY$CONTOURS` will be `N+1`.

User specified contour levels

A specific set of contour levels can be entered by using `\LEVELS` and entering a vector of contour level values, `lvl`, as the first parameter, unless the `\DITHERVECTOR` qualifier is also used, in which case the contour level vector should be the second parameter. If both are used, and the length of the dithering vector is `N`, the length of the level vector must be `N/2 - 1`. Suppose that `zmin` and `zmax` are the minimum and maximum of the data `z`. The level vector must be strictly monotonically increasing, with `lvl[1] > zmin` and `lvl[#] < zmax`.

The `\EQUALLYSPACED` qualifier only applies to the case of a dithering type density plot with legend, where the user supplies the contour levels, for example:

`DENSITY\DITHERING\LEVELS\LEGEND\EQUALLYSPACED lvl x y m`

If the `\EQUALLYSPACED` qualifier is used, the legend boxes will all be the same size, irregardless of the values specified in the levels vector, `lvl`.

Areas and volumes

If the `\AREAS` qualifier is used, an automatically created vector named `DENSITY$AREAS` will be made which will contain the percentage areas contained within each region. If the `\VOLUMES` qualifier is used, an automatically created vector named `DENSITY$VOLUMES` will be made which will contain the percentage volumes contained within each region. If there are `N` regions, the length of `DENSITY$AREAS` and `DENSITY$VOLUMES` will both be `N`. Also, the sum of the elements of each of these vectors will always be `100`.

Examples

These examples use the following vectors.

     X=[  1;  0;  1;  0;   .2; .3;   .5;  .8]
     Y=[  5;  5;  0;  0;    1;1.5;  2.5;   4]
     Z=[ 10; 10; 10; 10; -100; 10; -100; 500]
     D=[ 1;1; 2;2; 4;4; 7;7; 11;11; 0;0 ]

`DENSITY\DITHERING X Y Z`
`DENSITY\DITHERING\LEGEND X Y Z`
`DENSITY\DITHERING\DITHERVECTOR\LEGEND D X Y Z`

GRADIENTS
DIFFUSION