%!PS

% USING CUBIC SPLINE BASIS FUNCTIONS
% ==================================
% by  Don Lancaster

%  Copyright c 2001 by Don Lancaster & Synergetics, Box 809, Thatcher, AZ, 85552
%  (928) 428-4073 Email: don@tinaja.com Website: http://www.tinaja.com
%  Consulting services available http://www.tinaja.com/info01.html 

%  All commercial rights and all electronic media rights ~fully~ reserved.
%  Linking usually welcome. Reposting expressly forbidden. Version 1.5 

%  Use of Bernstein Polynominals and similar basis functions can greatly ease
%  the understanding and calculation of cubic splines and Bezier curves. This
%  tutorial gives some fundamentals and guidelines.

%  IMPORTANT NOTE: Don Lancaster's file gonzo.ps is required for this program.
%  After obvious location mods, uncomment ONE of the following two lines:

   (C:\\windows\\desktop\\gonzo\\gonzo.ps) run  % use internal gonzo
%  (A:\\gonzo.ps) run  % use external gonzo

%  NOTE THAT ALL PS FILENAME STRINGS !!!DEMAND!!! DOUBLE REVERSE SLASHES.

%  GONZO20A Guru Gonzo PostScript power tools (Interim release)
%  Includes gonzo justification and layout utilities.

%  Copyright c 1990, 1996, 2001 by Don Lancaster and Synergetics, Box 809,
%  Thatcher Arizona, 5552 (928) 428-4073  don@tinaja.com  support
%  via http://www.tinaja.com  All commercial rights and all electronic
%  media rights **FULLY** reserved. Reposting is expressly forbidden.

%%%%%%%%%%% links menu directory %%%%%%%%%%%%

% This code locks pdfmark commands out of a PostScript printer.

/pdfmark where {pop}{userdict /pdfmark /cleartomark load put} ifelse

% /surl "start url" marks the beginning of a text sequence to be urled. 
% It also paints the text blue...

/surl {mark /blue cvx 0.33 /setgray cvx     % change text to blue
       /currentpoint cvx                    % remember box start
       /urly /exch cvx /store cvx
       /urlx /exch cvx /store cvx
         ] cvx                              % complete deferred command

       printlist exch 3 index exch put      % stuff into gonzo printlist
       exch 1 add exch                      % increment gonzo list count
} def

% /eurl "end url" unmarks the end of a text sequence and sets up
% the pdfmark needed to define the Acrobat web link. 

/eurl {mark                               % start deferred proc      
       exch                               % position url string
       % /black cvx 0  /setgray cvx         % turn blue marker off 
       % maintextcolor /setrgbcolor  cvx    % reset to main text color???

       /aqua cvx /black cvx
       /makeurl cvx                       % defer call of url builder
        ] cvx                             % complete deferred proc

        printlist exch 3 index exch       % stuff into gonzo printlist
        put exch 1 add exch               % increment gonzo list count      
        } def


% /makeurl generates the pdfmark, receiving a {(urlstring) makeurl}.
% Note that it is not called until formatted printlist time...

/urlover 0.2 def                    % fraction of hot area over bounds

/makeurl { /cururlname exch store          % save the url string
            mark                           % start pdfmark

currentfont /ScaleMatrix get 3 get /fsize exch store  % guess height

  /Rect  [ urlx fsize urlover mul sub      % set box left x
           urly fsize urlover mul sub      % set box left y
           currentpoint
           exch fsize urlover mul add exch
           fsize add
           ]
           /Border [ 0 0 0]   % [0 0 0 ] = none; [0 0 2] = debug 
           /Color [ .7 0 0 ]
           /Action <</Subtype /URI /URI cururlname>>
           /Subtype /Link
           /ANN                            % annotation type
           pdfmark                        % call pdf operators    
} def

% /makeurlx generates the box specific pdfmark, receiving 
% a {(urlstring) makeurl}.
% Note that it is not called until formatted printlist time...


/makeurlx { /cururlnamex exch store        % save the url string
            mark                           % start pdfmark

  /Rect  [ xpos                            % set box left x
           ypos 0.9 sub                    % set box left y
           xpos 2.8 add                    % box right x
           ypos 0.9 sub 2.8 add            % box right y
           ]
           /Border [ 0 0 0]   % [0 0 0 ] = none; [0 0 2] = debug 
           /Color [ .7 0 0 ]
           /Action <</Subtype /URI /URI cururlnamex>>
           /Subtype /Link
           /ANN                            % annotation type
           pdfmark                        % call pdf operators    
} def


/maintextcolor {0 0 0 } def

% text link specific data

/tinaja {(http://www.tinaja.com/magsn01.asp) eurl} def
/cubic01 {(http://www.tinaja.com/cubic01.asp) eurl} def
/hack62 {(http://www.tinaja.com/glib/hack62.pdf) eurl} def
/post01 {(http://www.tinaja.com/post01.asp) eurl} def
/increment {(http://www.tinaja.com/text/bezgen3.html) eurl} def
/interpolate {(http://members.bellatlantic.net/~vze2vrva/design.html) eurl } def
/info01 {(http://www.tinaja.com/info01.asp) eurl} def
/bezmath {(http://www.tinaja.com/text/bezmath.html) eurl } def

/table {(http://www.tinaja.com/psutils/imtable.psl) eurl } def

/maildon {(mailto:don@tinaja.com) eurl} def




%%%%%%%%%%%%%%%% New colorizer patches %%%%%%%%%%%%%%%%%%
 

 
/boxgrays 0.899 def
/grayshade boxgrays def
/mastergray boxgrays def
/staytint {0.33 setgray} def
/staytint1 {0.25 setgray} def
/showadgrays true def

/red {0 settint} def

 /Zmacro {staytint /ypos ypos 2 add def 72 300 div setlinewidth xpos 
ypos moveto
txtwide 0 rlineto stroke /ypos ypos -2 add def tintoff} def  % hair rule
 
 /amacro {(zy0)  stringmacro /ypos
  ypos ypara add def  0.33 setgray } def             % start drop cap
 /bmacro {(iFy1) stringmacro /ypos
  ypos ypara add def  black} def             % finish drop cap
 

 
% Gonzo character colorizer

/blueon {mark /blue cvx 0.33 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def

/blueoff {mark /beige cvx 0  /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def

/tinton {mark 0.33 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def

/tinton1 {mark 0.25 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def


/tinton1b {mark beige cvx     % try for aqua after url???

 0.25 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def

/tinton1a {mark /aqua cvx     % try for aqua after url???

 0.25 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def

/tinton1p {mark /burple cvx     % try for aqua after url???

 0.25 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def

/tinton1r {mark /red cvx     % try for aqua after url???

 0.25 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def





/tinton1b {mark /beige cvx     % try for beige after url???

 0.25 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def





/tintoff {mark 0 /setgray cvx] cvx
        printlist exch 3 index exch put exch 1 add exch} def


%%%%%%%%%%%%%% COMPACT  VERSION OF  COLORIZER II %%%%%%%%%%%%%%%%%%%%%%

/settint {dup /currenttint exch store 5.999 mul dup floor cvi /&cbar
          exch store dup floor sub /&cwt exch store [
{/setgray [ /dup cvx 0.3 &cwt 0.59 mul add /ge cvx
 [1 /exch cvx &cwt 0.59 mul 0.30 add /sub cvx 1 &cwt sub
 0.59 mul 0.11 add /div cvx /dup cvx 1 &cwt sub /mul cvx
 &cwt /add cvx /exch cvx ] cvx [ &cwt 0.59 mul 0.3 add /div
 cvx /dup cvx &cwt /mul cvx 0] cvx /ifelse cvx /setrgbcolor
 cvx] cvx /def cvx}
{/setgray [/dup cvx 0.59 1 &cwt sub 0.3 mul add /ge cvx
 [1 &cwt sub 0.3 mul 0.59 add /sub cvx &cwt 0.3 mul 0.11 add
 /div cvx /dup cvx &cwt /mul cvx 1 &cwt sub /add cvx /exch cvx
 1 /exch cvx] cvx [1 &cwt sub 0.3 mul 0.59 add /div cvx /dup
 cvx 1 &cwt sub /mul cvx /exch cvx 0] cvx /ifelse cvx
 /setrgbcolor cvx] cvx /def cvx}
{/setgray [/dup cvx 0.59 &cwt 0.11 mul add /ge cvx [&cwt
 0.11 mul 0.59 add /sub cvx 1 &cwt sub 0.11 mul 0.30 add
 /div cvx /dup cvx 1 &cwt sub /mul cvx &cwt /add cvx 1 /exch
 cvx] cvx [0 /exch cvx &cwt 0.11 mul 0.59 add /div cvx /dup cvx
 &cwt /mul cvx] cvx /ifelse cvx /setrgbcolor cvx] cvx
 /def cvx}
{/setgray [/dup cvx 0.59 1 &cwt sub mul 0.11 add /ge cvx [1 &cwt
 sub 0.59 mul 0.11 add /sub cvx &cwt 0.59 mul 0.30 add /div cvx
 /dup cvx &cwt /mul cvx 1 &cwt sub /add cvx 1] cvx [0 /exch cvx
 1 &cwt sub 0.59 mul 0.11 add /div cvx /dup cvx 1 &cwt sub /mul
 cvx /exch cvx] cvx /ifelse cvx /setrgbcolor cvx] cvx
/def cvx}
{/setgray [/dup cvx 0.11 &cwt 0.30 mul add /ge cvx[&cwt 0.30 mul
 0.11 add /sub cvx 1 &cwt sub 0.30 mul 0.59 add /div cvx /dup
 cvx 1 &cwt sub /mul cvx &cwt /add cvx /exch cvx 1] cvx [ &cwt
 0.30 mul 0.11 add /div cvx /dup cvx &cwt /mul cvx /exch cvx 0
 /exch cvx ] cvx /ifelse cvx /setrgbcolor cvx ] cvx /def cvx}
{/setgray [ /dup cvx 0.30 1 &cwt sub 0.11 mul add /ge cvx[ 1
 /exch cvx 1 &cwt sub 0.11 mul 0.30 add /sub cvx &cwt 0.11 mul
 0.59 add /div cvx /dup cvx &cwt /mul cvx 1 &cwt sub /add cvx
 ] cvx [ 1 &cwt sub 0.11 mul 0.30 add /div cvx /dup cvx
 1 &cwt sub /mul cvx 0 /exch cvx] cvx /ifelse cvx /setrgbcolor
 cvx ] cvx /def cvx}

] &cbar get exec exec} bind def

/beige  {0.10 settint} def      % examples of convenience operators
/aqua   {0.52 settint} def
/blue    {0.67 settint} def
/lime     {0.44 settint} def
/burple {0.75 settint} def

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





/guru { gonzo begin
ps.util.1 begin printerror nuisance begin} def  % 

guru                                           % activate gonzo utilities


50 50 10 setgrid                               % create grid
% 56 61 showgrid  

/cstretch 0.015 def
/sstretch 0.015 def
/font0 /StoneSans-Bold 1.6 gonzofont
/font1 /StoneSans 0.95 gonzofont
/font3 /StoneSans-Bold 0.85 gonzofont
/font8 /StoneSans-Bold [0.65 0 0 0.65 0 0.4] gonzofont

/txtwide 36 store

/yinc 1.2 store
/kern 0.1 store

/tabs [7 -2] store

aqua 0.33 setgray
font1

black
font2



/amacro { mark /aqua cvx 0.33 /setgray cvx ] cvx
        printlist exch 3 index exch       % stuff into gonzo printlist
        put exch 1 add exch  } def

/bmacro { mark /black cvx ] cvx
        printlist exch 3 index exch       % stuff into gonzo printlist
        put exch 1 add exch  } def




% -2 50.5 (optional header here \274) cl

aqua 0.33 setgray
font0

20 47.3 0.7 add (Using Cubic Spline Basis Functions) cc
20 45.3 0.7 add (for Image Pixel Interpolation) cc
black
font2

3 43 
(|3Don Lancaster
Synergetics, Box 809, Thatcher, AZ 85552
|/surl http://www.tinaja.com|/tinaja 
|/surl don@tinaja.com|/maildon 
(928) 428-4073

|h
|a|0S|1|b|jome very hairy math involving |/tinton1 |3Bernstein Polynominal Basis Functions|1|/tintoff  can greatly \
ease the understanding and use of Bezier curves and cubic splines. Basis functions also \
can very much simplify drawing the actual curves. Often, simple table lookups can \
replace complex cubic calculations. Especially when using splines to do image \
expansion or interpolation.
|h
Cubic spline fundamentals appear in my |3|/surl Cubic Spline Library|1|/cubic01  and \
include |/surl |3this|/hack62           |/surl tutorial|1|/hack62 . |/aqua In general, a cubic spline is a way to \
draw a smooth curve that starts at      |/aqua |/tinton1 |3x0,|jy0|1|/tintoff , ends at |/tinton1 |3x3,|jy3|1|/tintoff  and whose exact shape can be \
influenced or controlled by intermediate control point pairs |/tinton1 |3x1,|jy1|1|/tintoff  and |/tinton1 |3x2,|jy2|1|/tintoff . \
The intermediate points will determine the exit or entry slope and the enthuasiasm of \
initial and final travel. Four data point pairs thus define the entire curve.
|h
A cubic spline is normally defined as a pair of parametric equations in which |/tinton1 |3t|1|/tintoff usually varies from |/tinton1 |30 to 1|1|/tintoff \274
|h|C|/staytint1  
|3x = at|83|3 + bt|82|3 + ct + d
|3y = et|83|3 |j+ ft|82|3 |j+ gt + h
|h|L|/black 
|1The cubic constants |/tinton1 |3a|1|/tintoff  through |/tinton1 |3h|1|/tintoff  can be  \
related to the control points by using        |/surl |3this math|1|/hack62 . |/aqua Because \
|/aqua |/tinton1 |3t|1|/tintoff  changes faster on the "more bent" portions of the \
curve, directly relating |/aqua |/tinton1 |3y|1|/tintoff  to |/tinton1 |3x|1|/tintoff  is \ nontrivial. Especially since multiple |/tinton1 |3y|1|/tintoff  values can result inside \
a loop or a cusp.
|h
Instead, let's start with an obvious |/tinton1 |31|j=|j1|1|/tintoff  and morph it into a bizarre |/tinton1 |3t + (1|j-|jt) = 1|1|/tintoff 
|h
Now, let's cube both sides\274
|h|C|/staytint1 
|3t|83|3 + 3t|82|3(1-t) + 3t(1-t)|82|3 + (1-t)|83|3 = 1
|h|L|/black 
|1or, defining each term as a basis function\274
|h|C|/staytint1 
|3 B3(t) + B2(t) + B1(t) + B0(t) = 1
|h|L|/black 
|1Let's plot these dudes and see what they look like\274

) cl

black  beige 0.35 setgray 
21 -2.5 (|3\320|j1|j\320) cc
aqua


0 55.5 mt line1 40 r  % temp align

showpage




%% start page two

50 50 10 setgrid                               % create grid


/cstretch 0.015 def
/sstretch 0.015 def
/font0 /StoneSans-Bold 1.6 gonzofont
/font1 /StoneSans 0.95 gonzofont
/font3 /StoneSans-Bold 0.85 gonzofont
/font4 /StoneSans-Bold [1.2 0 0 1.2 0 -0.4] gonzofont
/font8 /StoneSans-Bold [0.65 0 0 0.65 0 0.4] gonzofont

/txtwide 36 store

/yinc 1.2 store
/kern 0.1 store

/tabs [8 -2] store

aqua 0.33 setgray
font1

black
font2
% t3 + 3t^2*(1-t) + 3t(1-t)^2 + (1-t)^3 = 1

/plotnormalbasis {gsave translate





20 10 showgrid


beige  0.5 setgray


0 10 moveto
0 0.01 1 {/tt exch store

    tt 20 mul 
    1 tt sub     dup dup mul mul 10 mul lineto

  }
for



 line3 stroke


0 0 moveto
0 0.01 1 {/tt exch store       %% (1-t)^2 *3t

    tt 20 mul 

  1 tt sub      dup mul       tt mul 3 mul 10 mul lineto
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for

 line3 stroke





0 0 moveto
0 0.01 1 {/tt exch store

    tt 20 mul 
    1 tt sub     tt dup mul     mul 3 mul 10 mul lineto
  }
for

 line3 stroke



0 0 moveto
0 0.01 1 {/tt exch store

    tt 20 mul 
    tt dup dup mul mul 10 mul lineto
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for

 line3 stroke


aqua 0.30 setgray 1 setlinecap line3

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6.667 5 ((B1)) cc
13.333 5 ((B2)) cc

17.7 9 ((B3)) cc

0 -1.5 (t|j=|j0) cc
10 -1.5 (t|j=|j0.5) cc
20 -1.5 (t|j=|j1.0) cc

-1.5 -0.3 (0.0) cc
-1.5 9.7 (1.0) cc


grestore  aqua} def


9 41 plotnormalbasis









%36 21 showgrid                                 % show grid under waveforms








3 37 (|1We can see two rather remarkable properties here:
|h|3|/staytint1 
       The basis function peaks happen at the spline's control points.
|h|/black |1
and\274
|h|3|/staytint1 
       Any (t) value is the sum of the four basis functions evaluated at \n       (t) scaled by their control point values|1.
|h|1|/black 
Or, in mathspeak\274
|h|3|/staytint1 
                x(t) = x0*B0(t) + x1*B1(t) + x2*B2(t) + x3*B3(t)
                y(t) = y0*B0(t) |j+ y1*B1(t) |j+ y2*B2(t) + y3*B3(t)
|h|1|/black 
Note that the influence points are always at |/tinton1 |3t = 1/3|1|/tintoff  and |/tinton1 |3t = 2/3|1|/tintoff , regardless of where \
they lie in |/tinton1 |3x-y|1|/tintoff  space. And that the four unscaled basis point values always add up to one for any allowed value of |/tinton1 |3t|1|/tintoff .
|h
The easiest way to actually plot a cubic spline is with the |/tinton1 |3curveto|1|/tintoff  operator in                |/surl |3PostScript|1|/post01 . A fast language independent differental method is  |/surl |3shown here|1|/increment .  

|/tinton1 |t|t|4An image interpolation application

|/tinton1 |3Image interpolation|1|/tintoff  can be needed any time you want to make a bitmap larger or smaller. Or any time you are resampling or rectifying or masking or changing geometry. The object is to make the image larger or smaller or distorted without introducing artifacts or looking bad. A very useful tutorial on cubic spline image interpolation |/surl |3appears here|1|/interpolate .
|h
The trick is to take two previous pixels and two post pixels on a scan line and \
suitably evaluate them to create one or more credible new pixels. These new intermediate \
results can then be used vertically with four scan lines to create the actual final \
interpolated pixels.
|h            
Doing the obvious of using the four pixels as control points for a cubic spline \
works. But tends to be blurry because of excessive low pass filtering. We would \
also like a "knob" that lets us adjust how much sharpening we can get.
) cl

black  beige 0.35 setgray 
21 -2.5 (|3-|j2|j-) cc
aqua


0 55.5 mt line1 40 r  % temp align

showpage




%% start page three

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/plothiresbasis {gsave translate




gsave 0 -1 translate
30 13 showgrid
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aqua 0.30 setgray 1 setlinecap line3

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gsave 0 -1 mt 13 pu 30 pr 13 pd closepath clip newpath % stop excess

0 -0.1 -1 {/aa exch store

aa  -1 mul 0.75 mul


settint 0.5 setgray




0 10 moveto
-1 0.01 2 {/tt exch store

    tt  1 add 10 mul 
   
                 %   |tC0(t) = -at|83|3 + at|82|3 

tt dup dup mul mul    aa mul    neg
tt dup mul    aa mul    add

      10 mul 2 add lineto


  }
for

 line3 stroke


0 0 moveto
-1 0.01 2 {/tt exch store      
                            %  |tC1(t) = -(a+2)t|83|3 + (2a+3)t|82|3 - at 
    tt 1 add 10 mul 

    tt dup dup mul mul 2 aa add mul neg
    tt dup mul     3 aa 2 mul add  mul add
    tt aa mul sub


    10 mul 2 add lineto
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for

 line3 stroke





0 0 moveto
-1 0.01 2 {/tt exch store

    tt 1 add 10 mul

                   %  |tC2(t) =  (a+2)t|83|3 - (a+3)t|82|3 + 1 

     tt dup dup mul mul  2 aa add mul
     tt dup mul  3 aa add mul sub
     1 add

    10 mul 2 add lineto
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for

 line3 stroke



0 0 moveto
-1 0.01 2 {/tt exch store

    tt 1 add 10 mul 

                  % |tC3(t) =  at|83|3 - 2at|82|3 + at 

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for 
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} for  % cycles aa's


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aqua 0.30 setgray 1 setlinecap line3

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20 -1.5 (t|j=|j+1) cc
30 -1.5 (t|j=|j+2) cc







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/yinc 0.9 store

0 -3 (second\nprevious\npixel) cc
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20 -3 (first\nfollowing\npixel) cc
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grestore  aqua} def


6 29 plothiresbasis









%36 21 showgrid                                 % show grid under waveforms





3 50.5 (|1Instead, some new |/tinton1 |3high resolution|1|/tintoff  basis functions of\274 
|h|3|/staytint1 
     |tB0(t) =  at|83|3 - 2at|82|3 + at                                          
     |tB1(t) =  (a+2)t|83|3 - (a+3)t|82|3 + 1 
     |tB2(t) = -(a+2)t|83|3 + (2a+3)t|82|3 - at
     |tB3(t) = -at|83|3 + at|82|3 
|h|1|/black 
         \274 can be used to advantage as pixel interpolators. Here is how they plot\274



    














Note that we normally only use the region from |/tinton1 |3t|j=|j0|1|/tintoff  to |/tinton1 |3t|j=|j1|1|/tintoff .
|h
If |/tinton1 |3a|j=|j0|1|/tintoff  , we blur somewhat and use only the adjacent \
pixel info. This is pretty much the same as the older |/tinton1 |3bilineal interpolation|1|/tintoff .
|h
An |/tinton1 |3a|j=|j-0.5|1|/tintoff  selection is often \
best for data to be calculated, such as a medical or astronomical image. \
And an |/tinton1 |3a|j=|j-1.0|1|/tintoff  choice is often best for people-viewed \
results as it includes additional edge enhancement.
|h
The needed basis calculations can be done once and then used as |/surl |3fast table|/table           |/surl lookups|1|/table . \
Relating a general spline's |/tinton1 |3y|1|/tintoff  to |/tinton1 |3x|1|/tintoff  rather than \
to |/tinton1 |3t|1|/tintoff  is |/surl |3often difficult|1|/bezmath . But since the \
|/tinton1 |3x|1|/tintoff  values here will always be |/tinton1 |3-1|1|/tintoff , \
|/tinton1 |30|1|/tintoff , |/tinton1 |31|1|/tintoff , and |/tinton1 |32|1|/tintoff , our \
|3|/tinton1 x(t)|1|/tintoff  simplifies to\274
|h|3|C|/staytint1 
x(t) = -2(2a|j+|j1)t|83|3 + 3(2a|j+|j1)t|82|3 - 2at
|h|L|1|/black 
From which the inverse |/tinton1 |3t(x)|/tintoff |1 is easily found as\274 
|h|3|C|/staytint1 
t(x) = 2(2a|j+|j1)x|83|3 - 3(2a|j+|j1)x|82|3 + 2(a|j+|j1)x
|h|L|1|/black 
Underflows or overflows can sometimes happen with sudden changes near \
max or min intensity. These rare events should be suitably clipped.
|h
Additional support on these concepts can be |/surl |3found here|1|/info01 .

) cl



        %%     \274 can be used that plot something like this\274


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21 -2.5 (|3-|j3|j-) cc
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