Fat Curves and Representation of Planar Figures

1
Fat Curves and Representation of Planar Figures

• L.M. Mestetskii
• Department of Information Technologies, Tver
  State University, Tver, Russia
• Computers Graphics 24 (2000)Computer graphics
  in Russia

2
Outline

• Abstract
• Fat curves
• Boundaries of fat curves
• Implicit representation of fat curves
• Direct rasterization of fat curves
• Engraving representation
• Approximation of an engraving by fat Bezier curves

3
Abstract

• Fat curve curve having a width
• trace left by a moving circle of variable radius
• Engraving
• union of a finite number of fat curves
• Goal
• Bezier representation for fat curves
• 2D modeling through engraving
• approximation of arbitrary bitmap binary images

4
Problem

• Transforming the engraving representation into a
  discrete one in order to render a figures on
  raster display devices
• (Inverse Problem)
• Obtaining an engraving representation of figures
  given by their discrete or boundary representation

5
Method

• Bezier performance of greasy lines
• Decomposition of fat curves on parts with simple
  envelopes
• Scan-converting of fat curves based on Sturm
  polynomials
• Representation of any binary image as fat curves
  on the basis of its continuous skeleton

6
Fat Curves

• Set of circles in the Euclidean plane R2 C a,
  b ? R2 0, 8) , t?a, bCt (x, y)
  (x-u(t))2(y-v(t))2 ? (r(t))2, (x,y)?R2
• Fat curve
• C ?t?a,bCt
• axis P(t)
• width r(t)
• end circle Ca, Cb (initial and final circles)
• may be considered as the trace of moving the
  circle Ct

7
Example of a Fat Curve

• Planar Bezier curve
• a set of circles on the plane H H0,H1,,Hm
• circle Hi, radius Ri, Center (Ui, Vi), i 0,,m

Bernstein polynomials
8
Example of a Fat Curve

• axis P(t) (u(t), v(t)), width r(t)
• axis P(t) is an ordinary Bezier curve of degree m
  with the control points formed by the centers of
  the circles from H
• control circles H0, H1,, H6
• control polygon H

21 circles of family Ct (t 0.05j, j 0,,21)
9
Boundaries of Fat Curves

• A family of circles
• Under certain conditions, the family of circles,
  which is a family of smooth curves, has an
  envelope curve
• The necessary conditions for a point (x,y)?R2 to
  the envelope of a family of curves given by the
  equation F(x, y, t) 0

10
Find the Envelope Curve
the first condition is always satisfiedthe
second condition can be violated (no envelopes)
Condition
11
Find the Envelope Curve

• A parametric description of two envelopes
• Define

12
Envelopes

• Consider in more detail the case when the
  condition is violated and
  envelopes do not exist
• Interval on which
  is found as a result of the decomposition of a
  fat curve

13
Envelopes

• Consider a fat curve for which envelopes exist
• An envelope of a family of circles can be
  exterior of interior (dont belong to the
  boundary of the fat curve)
• Criterion for distinguishing interior envelops
• direction of axis (u, v)
• direction of envelope (x, y)
• exterior (supporting orientation) ux vy
  gt 0
• interior (opposing orientation) ux vy lt 0

14
Envelopes

• An envelope can change its orientation from
  supporting to opposing and conversely
• x y 0
• cut a fat curve at point t?a, b where xy0,
  we obtain fat curves with constantly oriented
  envelopes

15
Envelopes

• Two-side fat curve both envelopes are exterior
• when envelopes are self-intersecting or intersect
  each other, it must be decomposed into parts
• to find monotonicity intervals u(t) 0 or
  v(t) 0
• One-side fat curve one of the envelopes is
  interior

16
Rules for Decomposing Fat Curves

• Three rules for decomposing fat curves
• separate fat curves for which u2v2 gt r2
• separate one-side fat curves by finding singular
  points of envelopes, i.e., points where x1y10
  or x2y20
• Separate monotone fat curves by finding points
  for which u0 or v0

17
Implicit Representation of Fat Curves

• Membership function of the set
• point belongs to the fat curve if
  the following condition is satisfied for a
  certain

18
Direct Rasterization of Fat Curves

• The discrete tracing of contour of a domain given
  by its membership function consists in an
  inspection of the points with integer coordinates
  located along this contour

19
Engraving Representation of a Binary Image

• Obtain a continuous representation of a figure
  given by its discrete representation
• The solution of this problem involves 3 steps
• approximate the given bitmap binary image by a
  polygonal figure (PF)
• construct a skeletal representation of the PF
• approximate the skeletal representation of the PF
  by fat curves

20
Polygonal Figure

• Each of the PF is a polygon of the minimum
  perimeter that separates the black and white
  pixels of the bitmap image
• Problem
• constructing an engraving representation of the
  given bitmap image
• construction of an engravingrepresentation of
  the PF

polygonal figure of the minimum perimeter
21
Skeletal Representation

• Consider the set of all circles in the plane
• all their interior point are also interior of the
  PF
• the boundary of each circle at least two boundary
  points of the PF
• circles inscribed empty circles
• set of centers of such circles forms the skeleton
  of the PF
• skeletal representation of a bitmap image
  skeleton inscribed empty circles

22
Sites Bisector

• PF consists of vertices and segments sites
• every empty circle touches two or more sites
• The maximal connected set of the centers of the
  inscribed empty circle that touch these sites
  bisector of a pair of sites
• a segment of a line or a segment of a parabola

23
Sites Bisector

• A skeleton is an almost complete engraving
• There possible combinations of the pairs of sites
• segment-segment, point-segment, point-point
• Segment-segment

24
Sites Bisector

• Point-segment

find z, follows from
that
since
and, hence,
25
Sites Bisector

• Point-point
• The engraving constructed on the basis of the
  skeletal representation of a PF will be called
  the skeletal engraving

26
Approximation of an Engraving by Fat Bezier
Curves

• Skeletal engravings provide a highly accurate
  description of bitmap binary images (too many fat
  curves)
• Considered as a problem of the approximation of a
  skeletal engraving G by another engraving G
• The Hausdorff metric may be conveniently measure
  the distance between engravings
• Find an engraving G such that

27
Branch

• Skeleton structure
• juncture vertices of degree 3 or higher
• terminal vertices of degree 1
• intermediate vertices of degree 2
• A chain of edges that have common vertices of
  degree 2 will be called a branch
• The entire skeleton can be represented as the
  union of such branches

28
Approximation

• Consider a chain of n fat curves C1,,Cn
  corresponding to the same branch of the skeleton
• find a fat curve C in a certain class of fat
  curves that provides the best approximation for
  this sequence of circles
• e.g., in the class of cubic Bezier curves C?B3
• in other word, we must solve the minimization
  problem

29
Fat Curve Fitting Problem

• Empty circles K0,Kn located at the vertices of
  the branch
• Define

30
Fat Curve Fitting Problem

• The approximation fat curve C is sought in the
  form of a Bezier curve of degree m H0,,Hm
  are the control circles of C(t)
• The problem is to find a set of control circles
  such that it minimizes the quadratic mean
  distance from the empty circles K0,,Kn

31
Fat Curve Fitting Problem

• In the optimization problem, the objective
  function
• The optimal solution if found by solving a system
  of linear equations obtained from the following
  condition
• If the fat Bezier curve with the control circles
  H0,,Hm does not provide the desired accuracy
• the chain of n fat curves C0,,Cm is partitioned
  into two shorter chains, and the approximation
  problem is solved separately for each of these
  chains

32
Result