Not long ago, I asked for ideas on how to quickly rotate a bitmapped
image by an arbitrary angle. Quite a response!
Thanks to all, including several whose messages don't appear here
because they were subsets of other messages I got.
First, for those of you who prefer the library, here's a summary of
the references I got. Following are some of the responses, several
of which are quite detailed.

REFERENCES
Fant, K.M. (1986) IEEE Computer Graphics & Applications 6,1 7180.
"A nonaliasing, realtime spatial transform technique."
Foley & Van Dam "Fundamentals of Interactive Computer Graphics" chapter 8
Jackson, T. (1987) Graphics and Image Processing Algorithms for the
miniDAP. BSc. Project Report, Dept. Computer SCience & Statistics,
Queen Mary College, London University, Mile End Rd., London.
Alan Paeth, ``A Fast Algorithm for General Raster Rotation,'' Graphics
Interface '86, pp. 7781, May 1986.
A. Tanaka, M. Kameyama, S. Kazama, and O. Watanabe, ``A Rotation Method
for Raster Image Using Skew Transformation,'' Proc. IEEE Conference on
Computer Vision and Pattern Recognition, pp. 272277, June 1986.

From: David Fox
Organization: Columbia University CS Department
The idea is that you compose the functions "shear" and "transpose".
"Shear" takes a bitmap like this
ab
 
 
 
cd
and shears it by a given angle x to give this
ab
\ \
>\ \
\ \
> cd
Transpose rotates that about the diagonal to give this
a
\
 \
 \
b c
\ 
\ 
\d
Then a shear in the other direction gives this
> /\a
/ \
/ \
>b/ \c
\ /
\ /
\ /
d\/
Finally, another transpose yields the rotated bitmap
/\b
/ \
/ \
a/ \d
\ /
\ /
\ /
c\/
I did not invent this algorithm, so please don't attribute it to me.
Let me know what kind of replies you get, I'm curious whether this
algorithm is widely known (and whose it is), or unknown, or whether
there are other algorithms to do this. Enjoy.
David Fox
fox@cs.columbia.edu

From: jeff@lorrie.atmos.washington.edu (Jeff Bowden)
See Foley & Van Dam "Fundamentals of Interactive Computer Graphics" chapter 8
(I think).
For a 2D rotate about the origin, multiply all coordinates by this matrix:
( x y ) ( )
( cos(r) sin(r) )
( )
( sin(r) cos(r) )
( )
method 1: (slow) Pass each pixel through the above matrix.
method 2: (faster) Pass the endpoints of each horizontal line in your image
through the matrix. Use bresenham's algorithm to plot the points in between
but instead of plotting all of the pixels the same color, iterate through your
source line and use the values you find there.
method 3: (fastest) Rotate the corner coordinates of your rectangle, use
bresenham to generate the coordinates of the endpoints of your rotated
(previously horizontal) lines, and apply method 2 using these endpoints.
NOTE: I have no actual experience with these methods on bitmaps. I only use
the coordinate transformation matrix on lines and triangles. There may be
problems with bresenham not generating the same number of points for the
rotated line as the horizontal source line. If there is a difference it
should be only 1 or 2 pixels, you need to study its behaviour before you will
know. I contemplated implementing method 3 at one point and actually coded
some of it before I got sidetracked with other things. No, I don't have the
code anywhere anymore.

From: oster%SOE.Berkeley.EDU@jade.berkeley.edu (David Phillip Oster)
for gray scale images, of course, this reduces to a simple interpolation
problem: each destination pixel is the weighted sum of some small
neighborhood of source pixels. You need to compute the weights, and come
up with an efficient algorithm for marching through the bitmap.

From: mcvax!cwi.nl!edwin@uunet.UU.NET (Edwin Blake)
Organization: CWI, Amsterdam
In reply to your question I have made the following summary (its still
too long!). If you would like more implementation details please
contact me.
What I did not mention is that there is a threepass method (suited to
SIMD machines) which Jackson [see below] develops but this is probably
irrelevant for you as well.
=============
8th November, 1988 E.H.Blake
Twodimensional Image Transformations.
======================================
The general linear transformations of images (translations, rotations
shear etc) can be achieved by a vector addition, and multiplication by a
2x2 matrix.
Clearly we wish to separate out translations of images since they are by
far the easiest and cheapest transformation.
We are now left with the transformations represented by the same two
dimensional matrix applied to the coordinates of each point in the
image. It might be convenient to separate this into components which
can be dealt with in different ways: i.e. change in the size in the x
and y directions, shear in those directions and pure rotation.
Since we are going to deal with images on a raster questions about re
sampling and aliasing will arise. We shall present two fundamentally
different approaches. The one approach deals with fractional pixels and
attempts to minimise sampling errors by distributing parts of a source
pixel amongst the destination pixels. The other approach deals only
with whole pixels and reduces temporal aliasing (i.e. flicker) at the
expense of spatial resolution. When shearing images the whole pixel
method produces pixels might not be quite right for their position but
no pixels are lost or duplicated.
The whole pixel approach can also be used for overall size changes in
the images. When the size of the image changes we either cull or dupli
cate certain pixels. Particularly with size reduction this can lead to
unacceptable results because certain features in the image may disap
pear. This leads us to adopt the fractional pixel resampling approach.
There is another important difference between the two approaches how
ever: the fractional pixel approach has to examine and interpret pixel
values and must be able to combine them. The whole pixel approach for
shearing combined with a duplicate/cull method of size change is
independent of the values of the pixels. This value independence (which
is really a type independence) is important where the pixel values are
themselves isolated points in a much bigger colour space. For example,
if the pixels are pointers into a colour table then we cannot in general
expect to be able to average the two pointers and come up with a pointer
to the intermediate colour.
These simple operations repeated over a large number of pixels make them
ideal candidates for special purpose VLSI or SIMD machines. In fact the
routines described below are partly inspired by algorithms originally
intended for the miniDAP (a 32x32 SIMD machine) [Jackson, 1987] or for a
special purpose processor [Fant, 1986].
Composition of Image Transformations.
=====================================
The 2x2 transformation matrix can be written as a composition of
transformations in a number of ways. The decomposition used depends on
the order in which the transformations are applied. The basic image
transformation which leaves its source unchanged is the unit matrix.
If we first shear in the y direction, then in the x direction and
finally scale the image the decomposition is as follows:
 a b  =  m 0  *  1 x  *  1 0  (1)
 c d   0 n   0 1   y 1 
=  m + mxy mx 
 ny n 
In the decomposition y is a shear parallel to the yaxis, x is a shear
parallel to the xaxis, n represents a scaling in the y dimension and m
a scaling in the x dimension. By identifying terms we can see that:
n = d
and provided d != 0
y = c/d & m = a  bc/d (2)
and provided ad  bc is nonzero
x = db/(ad  bc)
If the determinant, ad  bc, is zero then the transformation is singu
lar. This means that the image need not be treated as a plane. If the
transformation is nonsingular but d = 0, then we can rotate the image
by ninety degrees and exchange b for d.
If on the other hand we first apply all the ydirection changes (shear
and scaling) and then the xdirection changes, the decomposition is as
follows:
 a b  =  e f  *  1 0  (3)
 c d   0 1   g h 
=  e + fg fh 
 g h 
In the decomposition g is a shear parallel to the yaxis and h the scal
ing in that direction, f is a shear parallel to the xaxis and e the
corresponding scaling. Then:
h = d
g = d (4)
and provided d != 0
f = b/d
e = bc/(a  d)
Similar considerations as in (2) regarding d apply to (4).
Fast, Pixel Preserving Image Transformation: "QUICK"
=====================================================
In order to refer to this method of image transformation easily it will
be called the "Quick" method. The quick method always moves whole pix
els without interpretation, it is thus independent of the underlying
type of the representation. There are four basic kinds of operation we
shall need:
1. Pixel shear (translation) according to position.
2. Shearing of pixels in two dimensions together, without writing out
the intermediate results.
3. Selective pixel duplication to achieve fractional size increases.
4. Selective pixel culling to achieve fractional decreases in image
size.
A common theme to all these transformations is distributing a fraction
of an integral set of operations over an interval. For example, choos
ing which pixels to duplicate to get 10% size increase, or which lines
to shift for a fractional shear. This has a very efficient solution in
Bresenham's well known algorithm, which is mainly used for line drawing
[see any textbook, e.g. Newman & Sproull].
The other problem is combining two shears in twodimensions. The sim
plest way would be to write out the intermediate image and then
transform it a second time. However, by tracing the jagged path which
an output line follows we can derive a scanline algorithm which produces
the sheared image. By duplicating or neglecting selected pixels and
then duplicating or neglecting selected lines we have an efficient and
general image transformation method. The algorithm notionally employed
the decomposition of equation 2 but without writing out any intermediate
results.
This algorithm was fully implemented and its combination of features may
very well be unique.
TwoPass, AntiAliased Image Transformation: "PERFECT"
=======================================================
This image transformation writes out its intermediate results. The
pixel values are interpreted, which means that colour errors can occur
from resampling. For grey level images however the output is much
better, because antialiased, than the previous method. For brevity it
will be referred to as the "Perfect" method.
The algorithm interpolates a selected window of input pixels to produce
the desired view of output pixels. The columns of the image are first
processed and then the rows of the resulting intermediate image.
Because of the two pass nature of the algorithm shears can be done
easily for each dimension separately. The "perfect" method is well
suited to VLSI implementations [Fant, 1986], and the reader is referred
to Fant's article for more details. The decomposition derived in equa
tions 3 and 4 apply to this algorithm. Using the decomposition into the
shear and scaling components avoids the rather laborious procedure used
by Fant which involved calculating the positions of the four corners of
an image.
Simple Averaging Image Transformation. "PRETTY"
==================================================
Because the two pass algorithm was so slow there was an attempt to
produce an intermediate algorithm. This algorithm is essentially the
same as "Quick" except that pixels are not culled when the image
decreases in size. Instead a simple averaging is performed. Successive
pixels which would be discarded are accumulated and averaged with the
normal output pixel of a location. The algorithm was only used for
images which decreased in size. This was the most common bad case for
the "Quick" method. Whether this method is worth the extra code is not
clear.

From: Alan Wm Paeth
Organization: U. of Waterloo, Ontario
I presented a paper on this in Graphics Interface '86. The code can run on a
framebuffer (we have an Adage/Ikonas) or a mainframe (Vax implementation).
Three passes over the data are made; each "shear" the data without scaling.
This allows simple resampling and all integer math (as opposed to the
previously known 2pass varients). It also is good to ninety degrees (the two
pass version breaks down rapidly after 45). In the 90 deg case, the rotate
becomes a simple "shuffle", as done on the "Blit".
I can send you the source code or the "troff" text of the paper. [Alan
said in a later note that this offer is for anyone, not just me.  MM]
/Alan "Aha! Planet!" Paeth
Computer Graphics Laboratory
University of Waterloo

Enjoy...
