Help for BICUBIC

BICUBIC is a Vicar program that enlarges images by integer zoom factors using
convolutional reconstruction techniques.  A piecewise cubic convolutional 
filter is used.

USAGE:
	BICUBIC input output [size] ZOOM=zoom [parms]

OPERATION:

BICUBIC determines the output image size from the specified zoom factor(s) and
the input image size or sub-area (i.e. NLO = NL*ZOOM(1) and NS = NSO*ZOOM(2)).
The output image is then written line by line, pixel by pixel.  The value of an
output pixel is determined by convolving the input image with a piecewise cubic
interpolation function.  For details on convolutional interpolation, the user
is referred to [HOU78], [KEY81], [MIT88] and [PAR83].

In order to perform the interpolation near the image (or sub-area) edges, the
pixels need to be extrapolated in all directions.  This is done using a simple
linear extrapolation of the two edge pixels in any column or row.
The piecewise cubic interpolation filters are described in [MIT88].  The 
piecewise cubic function is:

	           / (12 - 9*B - 6*C)*|x|**3 +		for |x| < 1
		  /	(-18 + 12*B + 6*C)*|x|**2  		 /	(6 - 2*B)
	 	/
	f(x) = <     (-B - 6*C)*|x|**3 + 		for 1 <= |x| < 2
	 	\	(6*B + 30*C)*|x|**2  	 	 \	(-12*C - 48*C)*|x| + (8*B + 24*C)
		  \
		   \ 0					otherwise

where B and C are parameters that control the shape of the cubic curves
and thus the appearance of the output image (see CONTROLLING THE CUBIC 
FUNCTION below).
CONTROLLING THE CUBIC FUNCTION:

The piecewise cubic interpolation function can be controlled by two parameters
(B and C) as shown in the above formulas.  The effects of changing these
parameters is generalized in the graph of B and C on the following page.  This
graph is the result of a subjective test in which nine expert observers were
shown images reconstructed with random values of B and C [MIT88].  The
observers were asked to classify the appearence of the images into one of
these categories: blurring, ringing, anisotropy, and satisfactory.  The graph
shows the results of the test over the range of 0.0 to 1.0.  Of course B and C
may be outside this range, the effects will be more noticeable.
		1.0 +-------------+--------------- 		    |             |               |
		    |             |               |
	     B	0.8 +           _~ \      II      |	Regions:
	      	    |     I    /    \             |	-------
	     P	    |         <      ~-_         _ 	     a	0.6 +          >        \     _-~ |	I   - Blurring
	     r	    |      __-~         |~-_-~    |	II  - Anisotropy
	     a	    +__--~~            /          |	III - Ringing
	     m	0.4 + \               |           |	IV  - Anisotropy
	     e	    |  \       V     /            |	V   - Satisfactory
	     t	    |   \           /      III    |
	     e	0.2 +    ~\         |             |
	     r	    |  IV  \        \             |
		    |       ~~~\     |            |
		0.0 +-----+-----+----++-----+----- 		   0.0   0.2   0.4   0.6   0.8   1.0
			      C Parameter 

The default values for (B, C) are (1/3, 1/3) which is recommended by [MIT88].
Other interesting values are:

	(1, 0)	 - Equivalent to the Cubic B-Spline,
	(0, 0.5) - Equivalent to the Catmull-Rom Spline,
	(0, C)	 - The family of Cardinal Cubic Splines,
	(B, 0)   - Duff's tensioned B-Splines [DUF86].

EXAMPLES:

	BICUBIC TEST.IMG TEST2.IMG ZOOM=3

	BICUBIC TEST.IMG TEST3.IMG ZOOM=(5,4)

	BICUBIC TEST.IMG TEST4.IMG ZOOM=7 B=1 C=0

REFERENCES:

[DUF86]	Duff, Tom, "Splines in Animation and Modeling", State of the Art in
	Image Synthesis, SIGGRAPH 86 Course Notes.

[HOU78]	Hou, Hsieh S. and Andrews, Harry C., "Cubic Splines for Image
	Interpolation and Digital Filtering", IEEE Trans. Acoustics, Speech,
	and Signal Processing, Vol. ASSP-26, No. 6, December 1978, pp.
	508-517.

[KEY81]	Keys, Robert G., "Cubic Convolution Interpolation for Digital Image
	Processing", IEEE Trans. Acoustics, Speech, and Signal Processing,
	Vol. ASSP-29, No. 6, December 1981, pp. 1153-1160.

[MIT88]	Mitchell, Don P. and Netravali, Arun N., "Reconstruction Filters in
	Computer Graphics", Computer Graphics, Vol. 22, No. 4, August 1988,
	pp. 221-228.

[PAR83]	Park, Stephen K. and Schowengerdt, Robert A., "Image Reconstruction by
	Parametric Cubic Convolution", Computer Vision, Graphics and Image
	Processing", Vol. 23, No. 3, September 1983, pp. 258-272.
HISTORY:

Written by:		R. Mortensen
Cognizant Programmer:	R. Mortensen

Revisions:
	Initial Version		7-Oct-88