This file lists the set of papers and tech reports that were written with
the aid of the IRIT solid modeling tool.
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 Orthogonal Decomposition of Non-Uniform Bspline Spaces using Wavelets

		   Roman Kazinnik and Gershon Elber

We  take advantage  of ideas  of  an orthogonal wavelet complement  to
produce   multiresolution  orthogonal   decomposition   of  nonuniform
\Bspline{} (NUB) spaces.  The  editing of NUB  curves and surfaces can
be handled at different levels of resolutions.

Applying Multiresolution     decomposition       to,  possibly   $C^1$
discontinuous surfaces, one can preserve the general shape on one hand
and local features  on the other of   the free-form models,  including
geometric discontinuities.

The Multiresolution decomposition of the NUB tensor product surface is
computed via the symbolic computation  of inner products of \Bspline{}
basis functions.   To find a closed  form representation for the inner
product of the \Bspline{} basis functions, an equivalent interpolation
problem is solved.

As   a   one example   for   the   strength  of   the  Multiresolution
decomposition,   a  tool demonstrating   the  Multiresolution  editing
capabilities of NUB surfaces was developed and is presented as part of
this work, allowing interactive 3D editing of NUB free-form surfaces.

* To appear in Eurographics 97, Budapest, Hungary, September 1997.
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    Physically Based Adaptive Triangulation of Freeform Surfaces.

		 Patrick Chouraqui and Gershon Elber

This paper studies  an adaptive  polygonization method for  parametric
surfaces that is   physically  based.  A   set of sampled    points is
distributed by this algorithm according to the  curvature field of the
surface. A triangulation    is then  superimposed  over  this  sampled
set. The locations  of these sampled points  are obtained by employing
physically based  models of  interaction  of particles.   Two physical
models are considered  in this study:  a spring-mass model and a model
of  electrostatically   charged  particles.   The resulting  algorithm
equally  distributes the  approximation  error   of the  triangulation
throughout the surface,  once  the equilibrium  state of the  physical
model is reached.

* Presented in Computer Graphics International 1996 (CGI 96),
  Pohang, Korea, pp 144-153, June 1996.
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	Fast Iso-Surface Extraction using Marching Gradients

		  Olga Tebeleva and Gershon Elber

Given   an explicit trivariate   hyper-surface  defined  over a  three
dimensional image  data set, ${\cal D}$, and  an iso-surface  $S_1$ of
${\cal D}$ at some iso-value $v_1$, we present an algorithm to extract
a new iso-surface $S_2$ at  iso-value  $v_2$, with $v_2$  sufficiently
close to $v_1$.  Off-line continuous reconstruction of ${\cal D}$ as a
high order  Bspline trivariate is   employed, yielding  a linear  time
complexity for the  extraction  of $S_2$ in   the size of the  data of
iso-surface    $S_1$.   The end   result allows  real-time incremental
modification of the  iso-value.   Hence, users can  potentially modify
and refine  an  extracted iso-surface to  a  precise iso-value,  in an
interactive manner.

* Presented in The fourth Pacific Graphics Conference on Computer Graphics
  and Applications, Taipei, Taiwan, pp 80-90, August 1996.
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  New Approximation Methods of Planar Offset and Convolution Curves

	     In-Kwon Lee, Myung-Soo Kim, and Gershon Elber

We  present new methods to approximate  the  offset and convolution of
planar  curves.  These  methods can  be used  as fundamental tools  in
various  geometric applications such as   NC  machining and  collision
detection  of    planar  curved   objects.   Using    quadratic  curve
approximation and  tangent field matching,  the offset and convolution
curves can be approximated by polynomial or rational curves within the
tolerance of approximation error  $\epsilon  > 0$.  We suggest   three
methods  of  offset approximation,  all  of which  allow  simple error
analysis and at the  same  time provide high-precision  approximation.
Two   methods  of convolution approximation   are  also suggested that
approximate convolution curves with polynomial or rational curves.

* Presented in Theory and Practice of Geometric Modeling,
  University of Tubingen, Germany, October 1996.
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Qualitative and Quantitative Comparisons of Offset Curve Approximation Methods

		Gershon Elber, In-Kwon Lee, and Myung-Soo Kim

* To appear in CG&A
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      Inferring 3D models from freehand sketches and constraints

 Lynn Eggli, Ching-yao Hsu, Beat D. Br\"{u}derlin, and Gershon Elber

This paper describes `Quick-sketch', a 2d and 3d modeling tool for pen
based computers. Users  of this system   define a model by  simple pen
strokes, drawn directly on the screen of a pen-based PC.  Exact shapes
and geometric relationships   are interpreted  from  the sketch.   The
system can be used to also  sketch three-dimensional solid objects and
B-spline surfaces. These  objects may be refined  by defining two- and
three-dimensional  geometric    constraints.    A  novel   graph-based
constraint solver is used to establish the geometric relationships, or
to  maintain them  when  manipulating  the objects interactively.  The
approach presented here,  is a first step  towards a conceptual design
system. Quick-sketch can be used as a hand sketching front-end to more
sophisticated modeling-, rendering- or animation systems.

* To appear in CAD
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			 Ruled Tracing

	  Gershon Elber, Jung-Ju Choi, and Myung-Soo Kim

The traditional   ray  tracing technique     based on a   ray--surface
intersection  is  reduced to  a ruled-  or developable-surface surface
intersection problem, enabling direct freeform  surface rendering.  By
exploiting the spatial coherence    gained in  the   ruled/developable
surface  tracing approach presented  in this   work, the emulation  of
shadows, specular reflections and/or refractions in a freeform surface
environment can all be efficiently implemented.

The   approach proposed herein   provides  a  direct freeform  surface
rendering  alternative to ray tracing.  An  implementation of a direct
freeform  surface renderer that emulates shadows  as  well as specular
reflections is discussed. This renderer processes isoparametric curves
as its  basic building block, eliminating the  need for  any polygonal
approximation.

* To appear in The Visual Computer.
  Also Center for Intelligent Systems Tech. Report,
  CIS 9501, Computer Science Department, Technion.
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		   Matching of Freeform Curves

			Gershon Elber

Freeform parametric curves  are extensively employed in various fields
such  as computer graphics, computer   vision, robotics, and geometric
modeling.   While many  applications  exploit and  combine  univariate
freeform entities into more complex forms such as sculptured surfaces,
the problem of a fair or even  optimal {\em relative} parameterization
of freeforms, under some norm, has been rarely considered.

In  this work, we   present a  scheme   that closely  approximates the
optimal relative matching  between  two  or  even $n$   given freeform
curves,  under a user's prescribed norm  that is based on differential
properties of the   curves.     This matching   is computed   as     a
reparameterization   of $n-1$  of  the  curves   that  can be  applied
explicitly using   composition. The  proposed  matching  algorithm  is
completely automatic and  has been successfully employed  in different
applications with  several   demonstrated  herein:  metamorphosis   of
freeform curves   with feature preservations,  key frame interpolation
for animation, self  intersection free ruled surface construction, and
automatic matching of rail curves of blending surfaces.

* To appear in CAD.
  Also Center for Intelligent Systems Tech. Report,
  CIS 9527, Computer Science Department, Technion.
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	Planar Curve Offset Based on Circle Approximation

	  In-Kwon Lee, Myung-Soo Kim, and Gershon Elber

An algorithm is  presented to approximate  planar offset curves within
an arbitrary tolerance $\epsilon>0$.  Given  a planar parametric curve
$C(t)$ and  an offset radius $r$, the   circle of radius  $r$ is first
approximated by piecewise quadratic B\'ezier curve segments within the
tolerance  $\epsilon$.  The   exact  offset  curve   $C_r(t)$ is  then
approximated by the convolution of $C(t)$  with the quadratic B\'ezier
curve segments.   For a  polynomial curve $C(t)$  of  degree $d$,  the
offset curve   $C_r(t)$  is approximated   by planar  rational curves,
$C^a_r(t)$'s, of degree $3d-2$.  For a rational curve $C(t)$ of degree
$d$,  the offset curve  is approximated by   rational curves of degree
$5d-4$.  When they have no self-intersections, the approximated offset
curves, $C^a_r(t)$'s, are  guaranteed to be within $\epsilon$-distance
from  the exact  offset curve  $C_r(t)$.   The  effectiveness of  this
approximation technique is  demonstrated in the offset  computation of
planar   curved    objects bounded by   polynomial/rational parametric
curves.

* CAD, Vol 28, No 8, pp 617-630, August 1996.
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      Multiresolution Control for Nonuniform Bspline Curve Editing

		Gershon Elber and Craig Gotsman

The  piecewise  polynomial \bspline{}  representation   is widely used
throughout  the CAGD   community  as  the  representation  of  choice.
However, the locality of \bspline{}  curves,  while important in  many
respects, disables  global control of  the curve, preventing efficient
and  easy manipulation.  Multiresolution  representations for  uniform
\bspline{}  curves have  been   recently  proposed to alleviate   this
problem.

Herein, we extend  the use of  multiresolution representations to  non
uniform \bspline{}   (NUBS)  curves,  including  periodic  curves. Our
method  supports  local non   uniform  refinement and  (dis)continuity
preservation.  The multiresolution  decomposition of the freeform NUBS
curve is computed using least-squares approximation, based on existing
data  reduction techniques.  The   majority of contemporary   modeling
systems   that  employ the NUBS  representation   may  now employ this
multiresolution NUBS curve editing method.

The least-squares decomposition allows us  to support NUBS curves, but
it also  imposes some preprocessing penalties  in  both time and space
compared to techniques  for multiresolution uniform \bspline{} curves.
Nonetheless,  the entire process is fast  enough to enable interactive
editing of  complex NUBS curves, as  is demonstrated by an interactive
editor implemented to test our methods.

* The third Pacific Graphics Conference on Computer Graphics and
  Applications, Seoul, Korea, pp 267-278, August 1995.
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	Line Art Rendering via a Coverage of Isoparametric Curves

				Gershon Elber

A line-art non-photorealistic rendering  scheme of scenes  composed of
freeform   surfaces is presented.    A freeform  surface  coverage  is
constructed  using a set of isoparametric  curves.  The density of the
isoparametric  curves is set  to be a  function of the illumination of
the surface determined using a simple  shading model, or of regions of
special importance such  as  silhouettes.  The outcome  is  one way at
achieving an aesthetic and attractive  line-art rendering that employs
isoparametric  curve based drawings   that  is suitable for   printing
publication.

* IEEE Transactions on Visualization and Computer Graphics,
  Vol 1, No 3, pp 231-239, September 1995.
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		Line Illustrations $\in$ Computer Graphics

				Gershon Elber

The   revolution of the  computer graphics  field  during the last two
decades made it possible to create  high quality synthetic images that
even experts find it difficult to differentiate from real imagery.

In this  paper, we  explore  a partially  overlooked theme of computer
graphics that aims  at conveying simple  information using simple line
drawings and illustrations of polygonal as well as freeform objects.

* The Visual Computer, Vol 11, No 6, 1995.
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     Error Bounded Piecewise Linear Approximation of Freeform Surfaces.

			     Gershon Elber

We  present two models for  piecewise linear approximation of freeform
surfaces.  One model  exploits global  curvature bounds  and the other
employs an intermediate bilinear approximation. In both models, a norm
that  minimizes  the  maximal   deviation   of the  piecewise   linear
approximation from the freeform surface is used.

* CAD, Vol 28, No 1, pp 51-57, January 1996.
  Also tech report CIS #9413, Computer Science Department, Technion,
  October 1994.
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       Symbolic and Numeric Computation in Curve Interrogation.

			     Gershon Elber

The control  of shape  of curves  is of  great importance  in computer
aided geometric design.   Determination  of planar curves'  convexity,
the  detection of  inflection  points,  coincident  regions, and  self
intersection points,  the enclosed  area of  a  closed curve,  and the
locations of extreme curvature  are important features of curves  that
can affect the design, in modeling environments.

In this paper, we investigate the ability to robustly answer the above
queries and related questions using an approach which exploits both
symbolic computation and numeric analysis.

* Computer Graphics {\it forum}, Vol 14, No 1, pp 25-34, March 1995.
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	    Adaptive Isocurves Based Rendering: the Hardware Way.

				Gershon Elber

In   a recent  work~\cite{Gershon-adap-iso-rend},  an   almost optimal
algorithm to provide a coverage based on the isoparametric curves of a
surface was  presented. This approach  was combined  successfully with
curve rendering techniques and used to  directly render surfaces using
isoparametric curves instead of polygons.

In this paper,  we describes an adaptation  of the rendering algorithm
that uses adaptive isoparametric curves  as the surface coverage, to a
generic hardware. We also discuss  the feasibility of implementing the
adaptive    isocurve extraction  algorithm   itself in   hardware. The
presented results make the surface  coverage using adaptive extraction
of isoparametric    curves a tool   for   competitive freeform surface
rendering in both software and hardware.

Several  results,  including  a  videotape  recording  of a  real time
display, are demonstrated.

* Presented in Computer Graphics International 1994 (CGI 94), Melbourne,
  Australia, June 1994.
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		Metamorphosis of Freeform Curves and Surfaces.

				Gershon Elber

Metamorphosis between two freeform \bspline{} curves is considered and
several approaches to control the process are discussed. Starting with
simple convex combination, we examine two  other approaches, one based
on multiresolution  decomposition of   freeform curves and   the other
based on edge cutting of the control polygon  of curves.  The later is
improved by introducing a correspondence test for simple metamorphosis
relation.   Finally, we consider  the  possibility of  extending these
algorithms to surfaces.

* Presented in Computer Graphics International 1995 (CGI 95), Leeds, UK,
  June 1995.
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		    Sketching as a Solid Modeling Tool

	       Lynn Eggli, Beat Bruderlin, and Gershon Elber.

This paper describes 'Quick-sketch', a 2d and 3d modeling tool for pen
based  computers. Users of this  system define a  model  by simple pen
strokes  drawn  directly  on  the  screen of  a  pen-based  PC. Lines,
circles, arcs, or B-spline curves are automatically distinguished, and
interpreted  from    these strokes.   The  system   also automatically
determines relations, such as right  angles, tangencies, symmetry, and
parallelism, from the sketch  input. These relationships are then used
to  clean  up  the drawing   by making  the  approximate relationships
exact. Constraints are  established to  maintain the relationships  in
further editing.  A  constraint maintenance system,  which is based on
gestural manipulation   and   soft constraints, is   employed  in this
system. Several techniques for sketch  based definitions of 3d objects
are  provided  as well,  including  extrusion,  surface of revolution,
ruled surfaces and sweep. Features can  be sketched on the surfaces of
3d objects, using the same 2d- and  3d techniques. This way objects of
medium complexity can be sketched in seconds. The system can be viewed
as a front-end to more  sophisticated modeling, rendering or animation
environments,  serving as a hand    sketching tool in the  preliminary
design phase.

* Presented in Solid Modeling 95, Salt Lake City, Utah, May 1995. Also to
  appear in CAD.
