<chapter>
  <title>Weaving for inkjet printers</title>
<!-- by Charles Briscoe-Smith and Robert Krawitz. -->
<!-- converted to Texinfo and DocBook by Roger Leigh -->
  <sect1>
    <title>Introduction</title>
    <para>
      The Epson Stylus Color/Photo printers don't have memory to print
      using all of the nozzles in the print head.  For example, the
      Stylus Photo 700/EX has 32 nozzles.  At 720 dpi, with an 8" wide
      image, a single line requires ((8 &times; 720 &times; 6) / 8)
      bytes, or 4320 bytes (because the Stylus Photo printers have 6
      ink colors).  To use 32 nozzles per color would require 138240
      bytes.  It's actually worse than that, though, because the
      nozzles are spaced 8 rows apart.  Therefore, in order to store
      enough data to permit sending the page as a simple raster, the
      printer would require enough memory to store 256 rows, or
      1105920 bytes.  Considering that the Photo EX can print 11"
      wide, we're looking at more like 1.5 MB.  In fact, these
      printers are capable of 1440 dpi horizontal resolution.  This
      would require 3 MB.  The printers actually have 64K-256K.
    </para>
    <para>
      With the newer (740/750 and later) printers it's even worse,
      since these printers support multiple dot sizes; of course, the
      even newer 2880&times;720 printers don't help either.
    </para>
    <para>
      Older Epson printers had a mode called
      MicroWeave<superscript>&trade;</superscript>.  In this mode, the
      host fed the printer individual rows of dots, and the printer
      bundled them up and sent them to the print head in the correct
      order to achieve high quality.  This MicroWeave mode still works
      in new printers, but in some cases the implementation is very
      minimal: the printer uses exactly one nozzle of each color (the
      first one).  This makes printing extremely slow (more than 30
      minutes for one 8.5&times;11" page), although the quality is
      extremely high with no visible banding whatsoever.  It's not
      good for the print head, though, since no ink is flowing through
      the other nozzles.  This leads to drying of ink and possible
      permanent damage to the print head.
    </para>
    <para>
      By the way, although the Epson manual says that microweave mode
      should be used at 720 dpi, 360 dpi continues to work in much the
      same way.  At 360 dpi, data is fed to the printer one row at a
      time on all Epson printers.  The pattern that the printer uses
      to print is very prone to banding.  However, 360 dpi is
      inherently a low quality mode; if you're using it, presumably
      you don't much care about quality.  It is possible to do
      microweave at 360 DPI, with significantly improved quality.
    </para>
    <para>
      Except for the Stylus Pro printers (5000, 5500, 7000, 7500,
      9000, 9500, and when it's released the 10000), which can do
      microweave at any resolution, printers from roughly the Stylus
      Color 600 and later do not have the capability to do MicroWeave
      correctly in many cases (some printers can do MicroWeave
      correctly at 720 DPI).  Instead, the host must arrange the
      output in the order that it will be sent to the print head.
      This is a very complex process; the jets in the print head are
      spaced more than one row (1/720") apart, so we can't simply send
      consecutive rows of dots to the printer.  Instead, we have to
      pass e. g. the first, ninth, 17th, 25th... rows in order for
      them to print in the correct position on the paper.  This
      interleaving process is called "soft" weaving.
    </para>
    <para>
      This decision was probably made to save money on memory in the
      printer.  It certainly makes the driver code far more
      complicated than it would be if the printer could arrange the
      output.  Is that a bad thing?  Usually this takes far less CPU
      time than the dithering process, and it does allow us more
      control over the printing process, e.g. to reduce banding.
      Conceivably, we could even use this ability to map out bad jets.
    </para>
    <para>
      Interestingly, apparently the Windows (and presumably Macintosh)
      drivers for most or all Epson printers still list a
      &ldquo;microweave&rdquo; mode.  Experiments have demonstrated
      that this does not in fact use the &ldquo;microweave&rdquo; mode
      of the printer.  Possibly it does nothing, or it uses a
      different weave pattern from what the
      &ldquo;non-microweave&rdquo; mode does.  This is unnecessarily
      confusing, at least for people who write drivers who try to
      explain them to people who don't.
    </para>
    <para>
      What makes this interesting is that there are many different
      ways of of accomplishing this goal.  The naive way would be to
      divide the image up into groups of 256 rows (for a printer with
      32 jets and a separation of 8 rows), and print all the mod8=0
      rows in the first pass, mod8=1 rows in the second, and so forth.
      The problem with this approach is that the individual ink jets
      are not perfectly uniform; some emit slightly bigger or smaller
      drops than others.  Since each group of 8 adjacent rows is
      printed with the same nozzle, that means that there will be
      distinct streaks of lighter and darker bands within the image (8
      rows is 1/90", which is visible; 1/720" is not).  Possibly worse
      is that these patterns will repeat every 256 rows.  This creates
      banding patterns that are about 1/3" wide.
    </para>
    <para>
       So we have to do something to break up this patterning.
    </para>
    <para>
      Epson do not publish the weaving algorithms that they use in
      their bundled drivers.  Indeed, their developer web site
      (http://www.ercipd.com/isv/edr_docs.htm) does not even describe
      how to do this weaving at all; it says that the only way to
      achieve 720 dpi is to use MicroWeave.  It does note (correctly)
      that 1440 dpi horizontal can only be achieved by the driver
      (i. e. in software).  The manual actually makes it fairly clear
      how to do this (it requires two passes with horizontal head
      movement between passes), and it is presumably possible to do
      this with MicroWeave.
    </para>
    <para>
      The information about how to do this is apparently available
      under non-disclosure agreement (NDA).  It's actually easy enough
      to reverse engineer what's inside a print file with a simple
      Perl script, which is supplied with the Gimp-Print distribution
      as tests/parse-escp2.  In any event, we weren't particularly
      interested in the weaving patterns Epson used.  There are many
      factors that go into choosing a good weaving pattern; we're
      learning them as we go along.  Issues such as drying time
      (giving the ink a few seconds more or less to dry can have
      highly visible effects) affect the quality of the output.
    </para>
    <para>
      The Uniprint GhostScript driver has been able to do weaving for
      a long time.  It uses patterns that must be specified for each
      choice of resolution and printer.  We preferred an algorithmic
      approach that computes a weave pattern for any given choice of
      inputs.  This obviously requires extensive testing; we developed
      a test suite specifically for this purpose.
    </para>
  </sect1>
  <sect1>
    <title>Weaving algorithms</title>
    <para>
      I considered a few algorithms to perform the weave.  The first
      one I devised let me use only (jets &minus;
      distance_between_jets &plus; 1) nozzles, or 25.  This is OK in
      principle, but it's slower than using all nozzles.  By playing
      around with it some more, I came up with an algorithm that lets
      me use all of the nozzles, except near the top and bottom of the
      page.
    </para>
    <para>
      This still produces some banding, though.  Even better quality
      can be achieved by using multiple nozzles on the same line.  How
      do we do this?  In 1440&times;720 mode, we're printing two
      output lines at the same vertical position.  However, if we want
      four passes, we have to effectively print each line twice.
      Actually doing this would increase the density, so what we do is
      print half the dots on each pass.  This produces near-perfect
      output, and it's far faster than using (pseudo)
      &ldquo;MicroWeave&rdquo;.
    </para>
    <para>
      Yet another complication is how to get near the top and bottom
      of the page.  This algorithm lets us print to within one head
      width of the top of the page, and a bit more than one head width
      from the bottom.  That leaves a lot of blank space.  Doing the
      weave properly outside of this region is increasingly difficult
      as we get closer to the edge of the paper; in the interior
      region, any nozzle can print any line, but near the top and
      bottom edges, only some nozzles can print.  We originally
      handled this by using the naive way mentioned above near the
      borders, and switching over to the high quality method in the
      interior.  Unfortunately, this meant that the quality is quite
      visibly degraded near the top and bottom of the page.  We have
      since devised better algorithms that allow printing to the
      extreme top and bottom of the region that can physically be
      printed, with only minimal loss of quality.
    </para>
    <para>
      Epson does not advertise that the printers can print at the very
      top of the page, although in practice most of them can.  The
      quality is degraded to some degree, and we have observed that in
      some cases not all of the dots get printed.  Epson may have
      decided that the degradation in quality is sufficient that
      printing in that region should not be allowed.  That is a valid
      decision, although we have taken another approach.
    </para>
    <sect2>
      <title>Simple weaving algorithms</title>
      <para>
        The initial problem is to calculate the starting position of
        each pass; the row number of the printer's top jet when
        printing that pass.  Since we assume the paper cannot be
        reverse-fed, the print head must, for each pass, start
        either further down the page than the previous pass or at
        the same position.  Each pass's start point is therefore at
        a non-negative offset from the previous pass's start point.
      </para>
      <para>
        Once we have a formula for the starting row of each pass, we
        then turn that &ldquo;inside out&rdquo; to get a formula for
        the pass number containing each row.
      </para>
      <para>
        First, let's define how our printer works.  We measure
        vertical position on the paper in &ldquo;rows&rdquo;; the
        resolution with which the printer can position the paper
        vertically.  The print head contains J ink jets, which are
        spaced S rows apart.
      </para>
      <para>
        Consider a very simple case: we want to print a page as
        quickly as possible, and we mostly don't care how sparse the
        printing is, so long as it's fairly even.
      </para>
      <para>
        It's pretty obvious how to do this.  We make one pass with
        the print head, printing J lines of data, each line S rows
        after the previous one.  We then advance the paper by S
        &times; J rows and print the next row.  For example, if J =
        7 and S = 4, this method can be illustrated like this:
      </para>
      <informalexample>
<screen>pass number
| row number------->
| |         111111111122222222223333333333444444444455555555556666666666
| 0123456789012345678901234567890123456789012345678901234567890123456789
0 *---*---*---*---*---*---*
1                             *---*---*---*---*---*---*
2 \-----------------------/                               *---*---*---*---*---*-
        7 jets              \---/
                            4 rows offset from one jet to the next
\---------------------------/
   7*4=28 rows offset from one pass to the next</screen>
      </informalexample>
      <para>
        In these examples, the vertical axis can be thought of as the
        time axis, with the pass number shown at the left margin,
        while the row number runs horizontally.  A
        <computeroutput>*</computeroutput> shows each row printed by a
        pass, and a row of <computeroutput>-</computeroutput> is used
        to link together the rows printed by one pass of the print
        head.  The first pass is numbered
        <computeroutput>0</computeroutput> and starts at row 0.  Each
        subsequent pass p starts at row p &times; S &times; J.  Each
        pass prints J lines, each line being S rows after the previous
        one.  (For ease of viewing this file on a standard terminal,
        I'm clipping the examples at column 80.)
      </para>
      <para>
        This method covers the whole page with lines printed evenly
        S rows apart.  However, we want to fill in all the other
        rows with printing to get a full-density page (we're
        ignoring oversampling at this stage).  Where we have
        previously printed a single pass, we'll now print a
        &ldquo;pass block&rdquo;: we print extra passes to fill in
        the empty rows.  A naive implementation might look like
        this:
      </para>
      <informalexample>
<screen>0 *---*---*---*---*---*---*
1  *---*---*---*---*---*---*
2   *---*---*---*---*---*---*
3    *---*---*---*---*---*---*
4                             *---*---*---*---*---*---*
5                              *---*---*---*---*---*---*
6                               *---*---*---*---*---*---*
7                                *---*---*---*---*---*---*
8                                                         *---*---*---*---*---*-
9                                                          *---*---*---*---*---*
10                                                          *---*---*---*---*---
11                                                           *---*---*---*---*--</screen>
      </informalexample>
      <para>
        (Now you can see why this process is called
        &ldquo;weaving&rdquo;!)
      </para>
    </sect2>
    <sect2>
      <title>Perfect weaving</title>
      <para>
        This simple weave pattern prints every row, but will give
        conspicuous banding patterns for the reasons discussed
        above.
      </para>
      <para>
        Let's start improving this for our simple case.  We can
        reduce banding by making sure that any given jet never
        prints a row too close to another row printed by the same
        jet.  This means we want to space the rows printed by a
        given jet evenly down the page.  In turn, this implies we
        want to advance the paper by as nearly an equal amount after
        each pass as possible.
      </para>
      <para>
        Each pass block prints S &times; J lines in S passes.  The
        first line printed in each pass block is S &times; J rows
        lower on the page than the first line printed in the
        previous pass block.  Therefore, if we advance the paper by
        J rows between each pass, we can print the right number of
        passes in each block and advance the paper perfectly evenly.
      </para>
      <para>
        Here's what this &ldquo;perfect&rdquo; weave looks like:
      </para>
      <informalexample>
<screen>                    start of full weave
                    |
0 *---*---*---*---*---*---*
1        *---*---*---*---*---*---*
2               *---*---*---*---*---*---*
3                      *---*---*---*---*---*---*
4                             *---*---*---*---*---*---*
5                                    *---*---*---*---*---*---*
6                                           *---*---*---*---*---*---*
7                                                  *---*---*---*---*---*---*
8                                                         *---*---*---*---*---*-
9                                                                *---*---*---*--
10                                                                      *---*---
11                                                                             *</screen>
      </informalexample>
      <para>
        You'll notice that, for the first few rows, this weave is
        too sparse.  It is not until the row marked &ldquo;start of
        full weave&rdquo; that every subsequent row is printed.  We
        can calculate this start position as follows:
      </para>
      <informalexample>
<screen>start = (S &minus; 1) &times; (J &minus; 1)</screen>
      </informalexample>
      <para>
        For the moment, we will ignore this problem with the weave.
        We'll consider later how to fill in the missing rows.
      </para>
      <para>
        Let's look at a few more examples of perfect weaves:
      </para>
      <informalexample>
        <para>
          S = 2, J = 7, start = (2&minus;1) &times; (7&minus;1) = 6:
        </para>
<screen>        starting row of full weave
        |
0 *-*-*-*-*-*-*
1        *-*-*-*-*-*-*
2               *-*-*-*-*-*-*
3                      *-*-*-*-*-*-*
4                             *-*-*-*-*-*-*
5                                    *-*-*-*-*-*-*
6                                           *-*-*-*-*-*-*
7                                                  *-*-*-*-*-*-*</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 7, J = 2, start = 6:
        </para>
<screen>        start
        |
0 *------*
1   *------*
2     *------*
3       *------*
4         *------*
5           *------*
6             *------*
7               *------*
8                 *------*
9                   *------*</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 4, J = 13, start = 36:
        </para>
<screen>                                      start
                                      |
0 *---*---*---*---*---*---*---*---*---*---*---*---*
1              *---*---*---*---*---*---*---*---*---*---*---*---*
2                           *---*---*---*---*---*---*---*---*---*---*---*---*
3                                        *---*---*---*---*---*---*---*---*---*--
4                                                     *---*---*---*---*---*---*-
5                                                                  *---*---*---*</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 13, J = 4, start = 36:
        </para>
<screen>                                      start
                                      |
0 *------------*------------*------------*
1     *------------*------------*------------*
2         *------------*------------*------------*
3             *------------*------------*------------*
4                 *------------*------------*------------*
5                     *------------*------------*------------*
6                         *------------*------------*------------*
7                             *------------*------------*------------*
8                                 *------------*------------*------------*
9                                     *------------*------------*------------*
10                                        *------------*------------*-----------
11                                            *------------*------------*-------
12                                                *------------*------------*---
13                                                    *------------*------------
14                                                        *------------*--------
15                                                            *------------*----
16                                                                *------------*
17                                                                    *---------
18                                                                        *-----
19                                                                            *-</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 8, J = 5, start = 28:
        </para>
<screen>                              start
                              |
0 *-------*-------*-------*-------*
1      *-------*-------*-------*-------*
2           *-------*-------*-------*-------*
3                *-------*-------*-------*-------*
4                     *-------*-------*-------*-------*
5                          *-------*-------*-------*-------*
6                               *-------*-------*-------*-------*
7                                    *-------*-------*-------*-------*
8                                         *-------*-------*-------*-------*
9                                              *-------*-------*-------*-------*
10                                                  *-------*-------*-------*---
11                                                       *-------*-------*------
12                                                            *-------*-------*-
13                                                                 *-------*----
14                                                                      *-------
15                                                                           *--</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 9, J = 5, start = 32:
        </para>
<screen>                                  start
                                  |
0 *--------*--------*--------*--------*
1      *--------*--------*--------*--------*
2           *--------*--------*--------*--------*
3                *--------*--------*--------*--------*
4                     *--------*--------*--------*--------*
5                          *--------*--------*--------*--------*
6                               *--------*--------*--------*--------*
7                                    *--------*--------*--------*--------*
8                                         *--------*--------*--------*--------*
9                                              *--------*--------*--------*-----
10                                                  *--------*--------*--------*
11                                                       *--------*--------*----
12                                                            *--------*--------
13                                                                 *--------*---
14                                                                      *-------
15                                                                           *--</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 6, J = 7, start = 30:
        </para>
<screen>                                start
                                |
0 *-----*-----*-----*-----*-----*-----*
1        *-----*-----*-----*-----*-----*-----*
2               *-----*-----*-----*-----*-----*-----*
3                      *-----*-----*-----*-----*-----*-----*
4                             *-----*-----*-----*-----*-----*-----*
5                                    *-----*-----*-----*-----*-----*-----*
6                                           *-----*-----*-----*-----*-----*-----
7                                                  *-----*-----*-----*-----*----
8                                                         *-----*-----*-----*---
9                                                                *-----*-----*--
10                                                                      *-----*-
11                                                                             *</screen>
      </informalexample>
    </sect2>
    <sect2>
      <title>Weaving collisions</title>
      <para>
        A perfect weave is not possible in all cases.  Let's look at
        another example:
      </para>
      <informalexample>
        <para>
          S = 6, J = 4:
        </para>
<screen>0 *-----*-----*-----*
1     *-----*-----*-----*
2         *-----*-----*-----*
3             *-----*-----*-----*
4             ^   *-^---*-----*-----*
5             |   ^ | *-^---*-----*-----*
              OUCH!   ^ |   ^
                      |     |</screen>
      </informalexample>
      <para>
        Here we have a collision.  Some lines printed in later
        passes overprint lines printed by earlier passes.  We can
        see why by considering which row number is printed by a
        given jet number j (numbered from 0) of a given pass, p:
      </para>
      <informalexample>
<screen>row(p, j) = (p &times; J) + (j &times; S)</screen>
      </informalexample>
      <para>
        Because J = 4 and S = 6 have a common factor of 2, jet 2 of
        pass 0 prints the same row as jet 0 of pass 3:
      </para>
      <informalexample>
<screen>row(0, 2) = (0 &times; 4) + (2 &times; 6) = 12
row(3, 0) = (3 &times; 4) + (0 &times; 6) = 12</screen>
      </informalexample>
      <para>
        In fact, with this particular weave pattern, jets 0 and 1 of
        pass p + 3 always overprint jets 2 and 3 of pass p.  We'll
        represent overprinting rows by a
        <computeroutput>^</computeroutput> in our diagrams, and
        correct rows by <computeroutput>*</computeroutput>:
      </para>
      <informalexample>
        <para>
          S = 6, J = 4:
        </para>
<screen>0 *-----*-----*-----*
1     *-----*-----*-----*
2         *-----*-----*-----*
3             ^-----^-----*-----*
4                 ^-----^-----*-----*
5                     ^-----^-----*-----*</screen>
      </informalexample>
    </sect2>
    <sect2>
      <title>What makes a &ldquo;perfect&rdquo; weave?</title>
      <para>
        So what causes the perfect weave cases to be perfect, and
        the other cases not to be?  In all the perfect cases above,
        S and J are relatively prime (i.e. their greatest common
        divisor (GCD) is 1).  As we mentioned above, S = 6 and J = 4
        have a common factor, which causes the overprinting.  Where
        S and J have a GCD of 1, they have no common factor other
        than 1 and, as a result, no overprinting occurs.  If S and J
        are not relatively prime, their common factor will cause
        overprinting.
      </para>
      <para>
        We can work out the greatest common divisor of a pair of
        natural numbers using Euler's algorithm:
      </para>
      <orderedlist>
        <listitem>
          <para>
            Start with the two numbers: (e.g.) 9, 24
          </para>
        </listitem>
        <listitem>
          <para>
            Swap them if necessary so that the larger one comes
            first: 24, 9
          </para>
        </listitem>
        <listitem>
          <para>
            Subtract the second number from the first: 15, 9
          </para>
        </listitem>
        <listitem>
          <para>
            Repeat until the first number becomes smaller: 6, 9
          </para>
        </listitem>
        <listitem>
          <para>
            Swap the numbers again, so the larger one comes first:
            9, 6
          </para>
        </listitem>
        <listitem>
          <para>
            Subtract again: 3, 6
          </para>
        </listitem>
        <listitem>
          <para>
            Swap: 6, 3
          </para>
        </listitem>
        <listitem>
          <para>
            Subtract: 3, 3
          </para>
        </listitem>
        <listitem>
          <para>
            And again: 0, 3
          </para>
        </listitem>
        <listitem>
          <para>
            When one of the numbers becomes 0, the other number is
            the GCD of the two numbers you started with.
          </para>
        </listitem>
      </orderedlist>
      <para>
        These repeated subtractions can be done with C's <literal>%</literal>
        operator, so we can write this in C as follows:
      </para>
      <informalexample>
<programlisting>unsigned int
gcd(unsigned int x, unsigned int y)
{
  if (y == 0)
      return x;
  while (x != 0) {
      if (y > x)
          swap (&amp;x, &amp;y);
      x %= y;
  }
  return y;
}</programlisting>
      </informalexample>
      <para>
        <literal>gcd(S,J)</literal> will feature quite prominently in our
        weaving algorithm.
      </para>
      <para>
        If 0 &le; j &lt; J, there should only be a single pair (p, j)
        for any given row number.  If S and J are not relatively
        prime, this assumption breaks down.  (For conciseness, let G
        = GCD(S,J).)
      </para>
      <informalexample>
        <para>
          S = 8, J = 6, G = 2:
        </para>
<screen>0 *-------*-------*-------*-------*-------*
1       *-------*-------*-------*-------*-------*
2             *-------*-------*-------*-------*-------*
3                   *-------*-------*-------*-------*-------*
4                         ^-------^-------^-------*-------*-------*
5                               ^-------^-------^-------*-------*-------*</screen>
      </informalexample>
      <para>
        In this case, jets 0, 1 and 2 of pass p + 4 collide with
        jets 3, 4 and 5 of pass p.
      </para>
      <para>
        How can we calculate these numbers?  Suppose we were to
        print using fewer jets, say J / G jets.  The greatest common
        divisor of J / G and S is 1, enabling a perfect weave.  But
        to get a perfect weave, we also have to advance the paper by
        a factor of G less:
      </para>
      <informalexample>
<screen>0 *-------*-------*       -       -       -
1    *-------*-------*       -       -       -
2       *-------*-------*       -       -       -
3          *-------*-------*       -       -       -
4             *-------*-------*       -       -       -
5                *-------*-------*       -       -       -</screen>
      </informalexample>
      <para>
        If we left the paper advance alone, we'd get a sparse weave;
        only one row can be printed every G rows:
      </para>
      <informalexample>
<screen>0 *-------*-------*       -       -       -
1       *-------*-------*       -       -       -
2             *-------*-------*       -       -       -
3                   *-------*-------*       -       -       -
4                         *-------*-------*       -       -       -
5                               *-------*-------*       -       -       -
             ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
            These rows need filling in.</screen>
      </informalexample>
      <para>
        The rows that would have been printed by the jets we've now
        omitted (shown as <computeroutput>-</computeroutput>) are
        printed by other jets on later passes.
      </para>
      <para>
        Let's analyse this.  Consider how a pass p could collide
        with pass 0.  Pass p starts at offset p &times; J.  Pass 0
        prints at rows which are multiples of S.  If p &times; J is
        exactly divisible by S, a collision has occurred, unless (p
        &times;J) &ge; J &times; S (which will happen when we
        finish a pass block).
      </para>
      <para>
        So, we want to find p and q such that p &times; J = q
        &times; S and p is minimised.  Then p is the number of rows
        before a collision, and q is the number of jets in pass 0
        which are not involved in the collision.  To do this, we
        find the lowest common multiple of J and S, which is L = (J
        &times; S) / G.  L / J is the number of rows before a
        collision, and L / S is the number of jets in the first pass
        not involved in the collision.
      </para>
      <para>
        Thus, we see that the first J / G rows printed by a given
        pass are not overprinted by any later pass.  However, the
        rest of the rows printed by pass p are overprinted by the
        first J &minus; (J / G) jets of pass p + (S / G).  We will
        use C to refer to S / G, the number of rows after which a
        collision occurs.
      </para>
      <para>
        Another example:
      </para>
      <informalexample>
        <para>
          S = 6, J = 9, G = 3, C = S / G = 2:
        </para>
<screen>0 *-----*-----*-----*-----*-----*-----*-----*-----*
1          *-----*-----*-----*-----*-----*-----*-----*-----*
2                   ^-----^-----^-----^-----^-----^-----*-----*-----*
3                            ^-----^-----^-----^-----^-----^-----*-----*-----*
4                                     ^-----^-----^-----^-----^-----^-----*-----
5                                              ^-----^-----^-----^-----^-----^--
       ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^
            These rows need filling in.</screen>
      </informalexample>
      <para>
        In this case, the first J &minus; (J / G) = 9 &minus; (9 /
        3) = 6 jets of pass p + (6 / 3) = p + 2 collide with the
        last 6 jets of pass p.  Only one row in every G = 2 rows is
        printed by this weave.
      </para>
      <informalexample>
        <para>
          S = 9, J = 6, G = 3, C = 3:
        </para>
<screen>
0 *--------*--------*--------*--------*--------*
1       *--------*--------*--------*--------*--------*
2             *--------*--------*--------*--------*--------*
3                   ^--------^--------^--------^--------*--------*
4                         ^--------^--------^--------^--------*--------*
5                               ^--------^--------^--------^--------*--------*</screen>
        <para>
          Here, the first J - (J / G) = 6 - (6 / 3) = 4 jets of pass
          p + (9 / 3) = p + 3 collide with the last 4 jets of pass
          p.
        </para>
        <para>
          Note that, in these overprinting cases, only rows
          divisible by G are ever printed.  The other rows, those
          not divisible by G, are not touched by this weave.
        </para>
        <para>
          We can modify our weave pattern to avoid overprinting any
          rows and simultaneously fill in the missing rows.  Instead
          of using J alone to determine the start of each pass from
          the previous pass, we adjust the starting position of some
          passes.  As mentioned before, we will divide the page into
          pass blocks, with S passes in each block.  This ensures
          that the first jet of the first pass in a block prints the
          row which the Jth jet of the first pass of the previous
          block would have printed, if the print head had one extra
          jet.
        </para>
        <para>
          Looking back at an example of a perfect weave, we can
          divide it into pass blocks:
        </para>
        <informalexample>
          <para>
            S = 7, J = 2, G = 1:
          </para>
<screen>              imaginary extra jet
                |
0 *------*      *      &lt;--start of pass block 0
1   *------*    |
2     *------*  |
3       *------*|
4         *-----|*
5           *---|--*
6             *-|----*
                |
7               *------*  &lt;--start of pass block 1
8                 *------*
9                   *------*</screen>
        </informalexample>
        <para>
          We can now calculate the start of a given pass by
          reference to its pass block.  The first pass of pass block
          b always starts at row (b &times; S &times; J).  The start
          row of each of the other passes in the block are
          calculated using offsets from this row.
        </para>
        <para>
          For the example above, there are 7 passes in each pass
          block, and their offsets are 0, 2, 4, 6, 8, 10 and 12.
          The next pass block is offset S &times; J = 14 rows from
          the start of the current pass block.
        </para>
        <para>
          The simplest way to modify the &ldquo;perfect&rdquo; weave
          pattern to give a correct weave in cases where G &ne; 1
          is to simply change any offsets which would result in a
          collision, until the collision disappears.  Every printed
          row in the weave, as we have shown it up to now, is
          separated from each of its neighbouring printed rows by G
          blank rows.  We will add an extra offset to each colliding
          pass in such a way that we push the pass onto these
          otherwise blank rows.
        </para>
        <para>
          We have seen that, unless G = 1, the plain weave pattern
          results in each pass colliding with the pass S / G passes
          before.  We will now subdivide our pass block into
          subblocks, each consisting of B = S / G passes.  There are
          therefore G subblocks in a pass block.
        </para>
        <para>
          For each subblock, the passes in that subblock have a
          constant offset added to them.  The offset is different
          for each subblock in a block.  There are many ways we can
          choose the offsets, but the simplest is to make the offset
          equal to the subblock number (starting from 0).
        </para>
        <para>
          Thus, the passes in the first subblock in each pass block
          remain at the offsets we've already calculated from J.
          The passes in the second subblock each have 1 added to
          their offset, the passes in the third subblock have 2
          added, and so on.  Thus, the offset of pass p (numbered
          relative to the start of its pass block) is p &times; J +
          floor(p / B).
        </para>
        <para>
          This gives us a weave pattern looking like this:
        </para>
        <informalexample>
          <para>
            S = 6, J = 9, G = 3, B = 2:
          </para>
<screen>0 *-----*-----*-----*-----*-----*-----*-----*-----*
1 ^        *-----*-----*-----*-----*-----*-----*-----*-----*
2 |              +-> *-----*-----*-----*-----*-----*-----*-----*-----*
3 |              |            *-----*-----*-----*-----*-----*-----*-----*-----*
4 |              |                  +-> *-----*-----*-----*-----*-----*-----*---
5 |              |                  |            *-----*-----*-----*-----*-----*
6 |              |                  |               +-> *-----*-----*-----*-----
7 |              |                  |               |            *-----*-----*--
  |              |                  |             start of pass block 1
  |              |                  |             (offset returns to 0)
  |              |                  start of subblock 2 (offset 2 rows)
  |              start of subblock 1 (following passes offset by 1 row)
start of passblock 0, subblock 0 (pass start calculated as p*J)</screen>
        </informalexample>
        <informalexample>
          <para>
            S = 9,  J = 6,  G = 3,  B = 3:
          </para>
<screen>0 *--------*--------*--------*--------*--------*
1       *--------*--------*--------*--------*--------*
2             *--------*--------*--------*--------*--------*
3                    *--------*--------*--------*--------*--------*
4                          *--------*--------*--------*--------*--------*
5                                *--------*--------*--------*--------*--------*
6                                       *--------*--------*--------*--------*---
7                                             *--------*--------*--------*------
8                                                   *--------*--------*--------*
9                                                       *--------*--------*-----
10                                                  \---/     *--------*--------
11                                               small offset       *--------*--
12                                                                         *----</screen>
        </informalexample>
        <para>
          This method of choosing offsets for subblocks can result
          in an occasional small offset (as shown above) between one
          pass and the next, particularly when G is large compared
          to J.  For example:
        </para>
        <informalexample>
          <para>
            S = 8, J = 4, G = 4, B = 2:
          </para>
<screen>0 *-------*-------*-------*
1     *-------*-------*-------*
2          *-------*-------*-------*
3              *-------*-------*-------*
4                   *-------*-------*-------*
5                       *-------*-------*-------*
6                            *-------*-------*-------*
7                                *-------*-------*-------*
8                                 *-------*-------*-------*
9                                \/   *-------*-------*-------*
                            very small offset!</screen>
        </informalexample>
        <para>
          We can plot the offset against the subblock number as
          follows:
        </para>
        <informalexample>
<screen>subblock number
| offset
| |
| 0123
0 *
1  *
2   *
3    *
0 *
1  *
2   *
3    *</screen>
        </informalexample>
        <para>
          The discontinuity in this plot results in the small offset
          between passes.
        </para>
        <para>
          As we said at the beginning, we want the offsets from each
          pass to the next to be as similar as possible.  We can fix
          this by calculating the offset for a given subblock b as
          follows:
        </para>
        <informalexample>
<screen>offset(b) = 2*b             , if b &lt; ceiling(G/2)
          = 2*(G-b)-1       , otherwise</screen>
        </informalexample>
        <para>
          We can visualise this as follows, for G = 10:
        </para>
        <informalexample>
<screen>0123456789
0 *
1   *
2     *
3       *
4         *
5          *
6        *
7      *
8    *
9  *
0 *
1   *
2     *
3       *
4         *
5          *
6        *
7      *
8    *
9  *</screen>
        </informalexample>
        <para>
          and for G = 11:
        </para>
        <informalexample>
<screen>           1
 01234567890
 0 *
 1   *
 2     *
 3       *
 4         *
 5           *
 6          *
 7        *
 8      *
 9    *
10  *
 0 *
 1   *
 2     *
 3       *
 4         *
 5           *
 6          *
 7        *
 8      *
 9    *
10  *</screen>
        </informalexample>
        <para>
          This gives a weave looking like this:
        </para>
        <informalexample>
          <para>
            S = 12, J = 6, G = 6, B = 2:
          </para>
<screen>0 *-----------*-----------*-----------*-----------*-----------*
1       *-----------*-----------*-----------*-----------*-----------*
2               *-----------*-----------*-----------*-----------*-----------*
3                     *-----------*-----------*-----------*-----------*---------
4                             *-----------*-----------*-----------*-----------*-
5                                   *-----------*-----------*-----------*-------
6                                          *-----------*-----------*-----------*
7                                                *-----------*-----------*------
8                                                    *-----------*-----------*--
9                                                          *-----------*--------
10                                                             *-----------*----
11                                                                   *----------
12                                                                        *-----</screen>
        </informalexample>
        <para>
          This method ensures that the offset between passes is
          always in the range [J - 2, J + 2].
        </para>
        <para>
          (This might seem odd, but it occurs to me that a good
          weave pattern might also make a good score for bell
          ringers.  When church bells are rung, a list of
          &ldquo;changes&rdquo; are used.  For example, if 8 bells
          are being used, they will, at first, be rung in order:
          12345678.  If the first change is for bells 5 and 6, the
          bells will then be rung in the order 12346578.  If the
          second change is 1 and 2, the next notes are 21346578.
          After a long list of changes, the order the bells are rung
          in can become quite complex.
        </para>
        <para>
          For a group of bell-ringers to change the order of the
          notes, they must each either delay their bell's next ring,
          hasten it, or keep it the same as the time it takes to
          ring all the bells once.  The length of time between each
          ring of a given bell can only be changed a little each
          time, though; with an ink-jet weave pattern, we want the
          same to apply to the distance between passes.)
        </para>
        <para>
          Finally, knowing the number of jets J and their separation
          S, we can calculate the starting row of any given pass p
          as follows:
          <informalexample>
<screen>passesperblock = S
passblock = floor(p / passesperblock)
offsetinpassblock = p - passblock * passesperblock
subblocksperblock = gcd(S, J)
passespersubblock = S / subblocksperblock
subpassblock = floor(offsetinpassblock / passespersubblock)
if subpassblock &lt; ceiling(subblocksperblock/2)
  subblockoffset = 2*subpassblock
else
  subblockoffset = 2*(subblocksperblock-subpassblock)-1
startingrow = passblock * S * J + offsetinpassblock * J + subblockoffset</screen>
          </informalexample>
        </para>
        <para>
          We can simplify this down to the following:
        </para>
        <informalexample>
<screen>subblocksperblock = gcd(S, J)
subpassblock = floor((p % S) * subblocksperblock / S)
if subpassblock * 2 &lt; subblocksperblock
  subblockoffset = 2*subpassblock
else
  subblockoffset = 2*(subblocksperblock-subpassblock)-1
startingrow = p * J + subblockoffset</screen>
        </informalexample>
        <para>
          So the row number of jet j of pass p is
        </para>
        <informalexample>
<screen>subblocksperblock = gcd(S, J)

subblockoffset(p)
  = 2*subpassblock       , if subpassblock * 2 &lt; subblocksperblock
  = 2*(subblocksperblock-subpassblock)-1      , otherwise
    where
    subpassblock = floor((p % S) * subblocksperblock / S)

row(j, p) = p * J + subblockoffset(p) + j * S</screen>
        </informalexample>
        <para>
          Together with the inequality 0 &le; j &lt; J, we can use
          this definition in reverse to calculate the pass number
          containing a given row, r.  Working out the inverse
          definition involves a little guesswork, but one possible
          result is as follows.  Given a row, r, which is known to
          be the first row of a pass, we can calculate the pass
          number as follows:
        </para>
        <informalexample>
<screen>subblocksperblock = gcd(S, J)
subblockoffset = r % subblocksperblock
pass = (r - subblockoffset) / J</screen>
        </informalexample>
        <para>
          If G = 1, we can determine the pass number with this
          algorithm:
        </para>
        <informalexample>
<screen>offset = r % J
pass = (r - offset) / J
while (offset % S != 0)
{
pass--
offset += J
}
jet = offset / S</screen>
          </informalexample>
          <para>
            Generalising, we come up with this algorithm.  Given r, S
            and J:
          </para>
          <informalexample>
<screen>G = gcd(S, J)
passespersubblock = S/G
subblockoffset = r % G
subpassblock = subblockoffset / 2  , if subblockoffset % 2 == 0
           = G - (subblockoffset+1)/2    , otherwise
baserow = r - subblockoffset - (subpassblock * passespersubblock * J)
offset = baserow % J
pass = (baserow - offset) / J
while (offset % S != 0)
{
offset += J
pass -= 1
}
subblockretreat = floor(pass / passespersubblock) % G
pass -= subblockretreat * passespersubblock
pass += subpassblock * passespersubblock
jet = (r - subblockoffset - pass * J) / S</screen>
        </informalexample>
        <para>
          Let's look at some examples of imperfect but correct weave
          patterns:
        </para>
        <informalexample>
          <para>
            S = 6,  J = 4,  GCD = 2,
            passesperblock = S = 6,
            passespersubblock = S / G = 6 / 2 = 3:
          </para>
<screen>0 *-----*-----*-----*
1     *-----*-----*-----*
2         *-----*-----*-----*
3              *-----*-----*-----*
4                  *-----*-----*-----*
5                      *-----*-----*-----*
6                         *-----*-----*-----*
7                             *-----*-----*-----*
8                                 *-----*-----*-----*
9                                      *-----*-----*-----*
10                                         *-----*-----*-----*
11                                             *-----*-----*-----*
12                                                *-----*-----*-----*
13                                                    *-----*-----*-----*
14                                                        *-----*-----*-----*
15                                                             *-----*-----*----
16                                                                 *-----*-----*
17                                                                     *-----*--
18                                                                        *-----
19                                                                            *-</screen>
        </informalexample>
        <informalexample>
          <para>
            S = 8,  J = 6,  G = 2,
            passesperblock = S = 8,
            passespersubblock= S / G = 8 / 2 = 4:
          </para>
<screen>0 *-------*-------*-------*-------*-------*
1       *-------*-------*-------*-------*-------*
2             *-------*-------*-------*-------*-------*
3                   *-------*-------*-------*-------*-------*
4                          *-------*-------*-------*-------*-------*
5                                *-------*-------*-------*-------*-------*
6                                      *-------*-------*-------*-------*-------*
7                                            *-------*-------*-------*-------*--
8                                                 *-------*-------*-------*-----
9                                                       *-------*-------*-------
10                                                            *-------*-------*-
11                                                                  *-------*---
12                                                                         *----</screen>
        </informalexample>
        <informalexample>
          <para>
            S = 6,  J = 12,  G = 6,
            passesperblock = S = 6,
            passespersubblock= S / G = 6 / 6 = 1:
          </para>
<screen>0 *-----*-----*-----*-----*-----*-----*-----*-----*-----*-----*-----*
1               *-----*-----*-----*-----*-----*-----*-----*-----*-----*-----*---
2                             *-----*-----*-----*-----*-----*-----*-----*-----*-
3                                          *-----*-----*-----*-----*-----*-----*
4                                                    *-----*-----*-----*-----*--
5                                                              *-----*-----*----
6                                                                         *-----</screen>
        </informalexample>
        <para>
          We have now solved the basic weaving problem.  There are
          two further refinements we need to consider: oversampling,
          and filling in the missing rows at the start of the weave.
        </para>
      </informalexample>
    </sect2>
    <sect2>
      <title>Oversampling</title>
      <para>
        By oversampling, we mean printing on the same row more than
        once.  There are two reasons for oversampling: to increase
        the horizontal resolution of the printout and to reduce
        banding.
      </para>
      <para>
        Oversampling to increase horizontal resolution is necessary
        because, although the printer might be able to position an
        ink drop to, for example, 1/1440" horizontally, it may not
        be able to lay down two such drops 1/1440" apart.  If it can
        print two drops 1/720" apart, 2x oversampling will be
        necessary to get a 1/1440" horizontal resolution.  If it can
        only print two drops 1/360" apart, 4x oversampling will be
        necessary for a 1/1440" horizontal resolution.  The printer
        enforces this &ldquo;drop spacing&rdquo; by only accepting
        raster passes with a horizontal resolution matching the
        spacing with which it can print dots, so we must print
        passes at different horizontal positions if we are to obtain
        a higher horizontal resolution.  (Another reason it does
        this may be to reduce the amount of memory needed in the
        printer.)
      </para>
      <para>
        Oversampling can also be done to decrease the banding
        apparent in an image.  By splitting a row into two or more
        sets of dots (&ldquo;lines&rdquo;) and printing each line on
        the same row, but with a different nozzle for each line, we
        can get a smoother print.
      </para>
      <para>
        To quantify these two kinds of oversampling, we'll introduce
        two new constants: H shows how many different horizontal
        offsets we want to print at (the &ldquo;horizontal
        oversampling&rdquo;) while O shows how many times we want to
        print each row, over and above the number of times necessary
        for horizontal oversampling (the &ldquo;extra
        oversampling&rdquo;).
      </para>
      <para>
        It is necessary for all the lines printed by a given pass to
        have the same horizontal offset, but there need not be any
        relation between them in terms of extra oversampling.  For
        the moment, however, we will treat all oversampling as
        potentially requiring this alignment; all lines in one pass
        must be derived from the original row data in the same way.
        Thus, we'll assume O = 1 for now.
      </para>
      <para>
        So, how do we do this oversampling?  In fact, it can be done
        easily: advance the paper by a factor of H less between each
        pass.  We'll define a new variable, A, to show how much we
        advance the paper between passes.  Previously, we'd have
        defined A = J; we now let A = J / H.  This also affects our
        pass blocks.  Printing one pass block used to involve
        advancing the paper S &times; J rows; it now advances the
        paper (S&times;J) / H rows.  We therefore name a group of H
        pass blocks a &ldquo;band&rdquo;.  Printing one band
        involves advancing the paper S&times;J rows, as a pass
        block did before.
      </para>
      <para>
        To keep our weave pattern working correctly, so that
        overprinting does not occur within a pass block, we also
        have to redefine G as GCD(S,A).  Here's an example of an
        oversampled weave pattern:
      </para>
      <informalexample>
        <para>
          S = 4,  J = 10,  H = 2,  A = J/H = 10/2 = 5,
          G= GCD(4,5) = 1,
          passesperblock = S = 4,
          passespersubblock = S/G = 4/1 = 4:
        </para>
<screen>0 *---*---*---*---*---*---*---*---*---*
1      *---*---*---*---*---*---*---*---*---*
2           *---*---*---*---*---*---*---*---*---*
3                *---*---*---*---*---*---*---*---*---*
4                     *---*---*---*---*---*---*---*---*---*
5                          *---*---*---*---*---*---*---*---*---*
6                               *---*---*---*---*---*---*---*---*---*
7                                    *---*---*---*---*---*---*---*---*---*
8                                         *---*---*---*---*---*---*---*---*---*
9                                              *---*---*---*---*---*---*---*---*
10                                                  *---*---*---*---*---*---*---
11                                                       *---*---*---*---*---*--
12                                                            *---*---*---*---*-
13                                                                 *---*---*---*
14                                                                      *---*---
15                                                                           *--</screen>
      </informalexample>
      <para>
        Now we have to determine which line is printed by each jet on
        each pass.  If we number each line generated as we split up a
        row, we can use these numbers.  We'll number the lines in our
        diagram by replacing the <computeroutput>*</computeroutput>s
        with integers in the range [0&hellip;H&minus;1].
      </para>
      <para>
        Overprinting occurs once per pass block, so we can simply
        print pass block 0 with line 0, pass block 1 with line 1,
        pass block 2 with line 2, etc, wrapping to 0 when we've run
        out of lines:
      </para>
      <informalexample>
<screen>0 0---0---0---0---0---0---0---0---0---0
1      0---0---0---0---0---0---0---0---0---0
2           0---0---0---0---0---0---0---0---0---0
3                0---0---0---0---0---0---0---0---0---0
4                     1---1---1---1---1---1---1---1---1---1
5                          1---1---1---1---1---1---1---1---1---1
6                               1---1---1---1---1---1---1---1---1---1
7                                    1---1---1---1---1---1---1---1---1---1
8                                         0---0---0---0---0---0---0---0---0---0
9                                              0---0---0---0---0---0---0---0---0
10                                                  0---0---0---0---0---0---0---
11                                                       0---0---0---0---0---0--
12                                                            1---1---1---1---1-
13                                                                 1---1---1---1
14                                                                      1---1---
15                                                                           1--</screen>
      </informalexample>
      <informalexample>
        <para>
          S = 4,  J = 12,  H = 2,  A = J/H = 12/2 = 6,
          G= GCD(4,6) = 2,
          passesperblock= S = 4,
          passespersubblock= S/G = 4/2 = 2:
        </para>
<screen>0 0---0---0---0---0---0---0---0---0---0---0---0
1       0---0---0---0---0---0---0---0---0---0---0---0
2              0---0---0---0---0---0---0---0---0---0---0---0
3                    0---0---0---0---0---0---0---0---0---0---0---0
4                         1---1---1---1---1---1---1---1---1---1---1---1
5                               1---1---1---1---1---1---1---1---1---1---1---1
6                                      1---1---1---1---1---1---1---1---1---1---1
7                                            1---1---1---1---1---1---1---1---1--
8                                                 0---0---0---0---0---0---0---0-
9                                                       0---0---0---0---0---0---
10                                                             0---0---0---0---0
11                                                                   0---0---0--
12                                                                        1---1-</screen>
      </informalexample>
      <para>
        But what do we do if J is not an exact multiple of H?  This
        is a difficult problem, which I struggled with for quite a
        few days before giving in and taking the easy (but less
        elegant) way out.  The easy solution is to round J / H down,
        then add on the accumulated error at the end of each band.
      </para>
      <informalexample>
        <para>
          S = 4,  J = 11,  H = 2,  A = floor(J/H) = floor(11/2) = 5,
          G = GCD(4,5) = 1,
          passesperblock = S = 4,
          passespersubblock = S/G = 4/1 = 4
        </para>
<screen>Band 0:
0 0---0---0---0---0---0---0---0---0---0---0
1      0---0---0---0---0---0---0---0---0---0---0
2           0---0---0---0---0---0---0---0---0---0---0
3                0---0---0---0---0---0---0---0---0---0---0
4                     1---1---1---1---1---1---1---1---1---1---1
5                          1---1---1---1---1---1---1---1---1---1---1
6                               1---1---1---1---1---1---1---1---1---1---1
7                                    1---1---1---1---1---1---1---1---1---1---

Band 1:
8 |                                           0---0---0---0---0---0---0---0---0-
9  \-----------------------------------------/     0---0---0---0---0---0---0---0
10                   S*J rows                           0---0---0---0---0---0---
11                                                           0---0---0---0---0--
12                                                                1---1---1---1-
13                                                                     1---1---1
14                                                                          1---</screen>
      </informalexample>
      <para>
        We can calculate the starting row and subpass number of a
        given pass in this scheme as follows:
      </para>
      <informalexample>
<screen>A = floor(J / H)
subblocksperblock = gcd(S, A)
subpassblock = floor((p % S) * subblocksperblock / S)
if subpassblock * 2 &lt; subblocksperblock
  subblockoffset = 2*subpassblock
else
  subblockoffset = 2*(subblocksperblock-subpassblock)-1
band = floor(P / (S * H))
passinband = P % (S * H)
startingrow = band * S * J + passinband * A + subblockoffset
subpass = passinband / S</screen>
      </informalexample>
      <para>
        So the row number of jet j of pass p is
        <informalexample>
<screen>A = floor(J / H)
subblocksperblock = gcd(S, A)

subblockoffset(p)
  = 2*subpassblock       , if subpassblock * 2 &lt; subblocksperblock
  = 2*(subblocksperblock-subpassblock)-1      , otherwise
    where
    subpassblock = floor((p % S) * subblocksperblock / S)

band(p) = floor(p / (S * H))
passinband(p) = p % (S * H)

row(j, p) = band(p) * S * J + passinband(p) * A + subblockoffset(p) + j * S
row(j, p) = p * J + subblockoffset(p) + j * S</screen>
          <para>
            To be continued&hellip;
          </para>
        </informalexample>
      </para>
    </sect2>
  </sect1>
</chapter>
