@Article{frederick1990:conformal,
author = {Frederick, Carl and Schwartz, Eric L.},
title = {Conformal image warping},
journal = {IEEE Computer Graphics and Applications},
year = 1990,
volume = 10,
number = 2,
pages = {54--61},
month = Mar,
datestr = 199003,
url =
{http://ieeexplore.ieee.org/xpls/abs_all.jsp?&arNumber=00050673},
INSPEC = 3638476,
IEEE = 00050673,
abstract = {This report describes numerical and computer-graphic
methods for conformal image mapping between two
simply connected regions. The immediate motivation
for this application is that the visual field is
represented in the brain by mappings which are, at
least approximately, conformal. Thus, to simulate
the imaging properties of the human visual system
(and perhaps other sensory systems), conformal image
mapping is a necessary technique. \par There are two
distinct aspects to this problem. First, one must
implement a numerical or analytic method which
allows for the computation of a given conformal
mapping, constrained by the shape of the two simply
connected regions (hereafter known simply as
regions) to be mapped, and by a single point and
orientation correspondence between them. Second, it
is necessary to apply a space-variant
texture-mapping algorithm to warp the image, once
the mapping itself has been specified. \par For
generating the conformal map, we show a method for
analytic mappings and an implementation of the Symm
algorithm for numerical conformal mapping. The first
method evaluates the inverse mapping function at
each pixel of the range, with antialiasing by
multiresolution texture prefiltering and bilinear
interpolation. The second method is based on
constructing a piecewise affine approximation of the
mapping in the form of a joint triangulation, or
triangulation map, in which only the nodes of the
triangulation are conformally mapped. The texture is
then mapped by a local affine transformation on each
pixel of the range triangulation with the same
antialiasing used in the first method. \par We
illustrate these algorithms with examples of
conformal mappings constructed analytically from
elementary mappings, such as the linear fractional
map, the complex algorithm, etc. We also show
applications of numerically generated maps between
highly irregular regions and an example of the
visual field mapping that motivates this work. \par
In addition to providing a necessary tool for
simulation of cortical architectures, these
illustrations may be of pedigogical use for students
attempting to visualize the geometric properties of
elementary conformal mappings; they may also find
applications in such areas as fluid mechanics and
electrostatics, where conformal mapping is a natural
and basic tool.},
keywords = {computerised picture processing}
}