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

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

Abstract Fat curve Fat curve = “curve having a width”  trace left by a moving circle of variable radius Engraving 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

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

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

Fat Curves Set of circles in the Euclidean plane R 2 C: [a, b] → R 2 × [0, ∞ ), t ∈ [a, b] C t = {(x, y): (x−u(t)) 2 +(y−v(t)) 2 ≦ (r(t)) 2, (x,y) ∈ R 2 } Fat curve  C = ∪ t ∈ [a,b] C t  axis: P(t)  width: r(t)  end circle: C a, C b (initial and final circles)  may be considered as the trace of moving the circle C t P(u,v) r (x,y)

Example of a Fat Curve Planar Bezier curve  a set of circles on the plane: H = {H 0,H 1,…,H m }  circle H i, radius R i, Center (U i, V i ), i = 0,…,m [Bernstein polynomials]

H0H0 H1H1 H2H2 H3H3 H4H4 H6H6 H5H5 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: H 0, H 1,…, H 6  control polygon: H 21 circles of family C t (t = 0.05j, j = 0,…,21)

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) ∈ R 2 to the envelope of a family of curves given by the equation F(x, y, t) = 0

Find the Envelope Curve (x 1,y 1 ) (x 2,y 2 ) Condition ： the first condition is always satisfied the second condition can be violated (no envelopes)

Find the Envelope Curve A parametric description of two envelopes Define (x 1,y 1 ) (x 2,y 2 )

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

Envelopes Consider a fat curve for which envelopes exist An envelope of a family of circles can be exterior of interior (don’t 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) : u’x’ + v’y’ > 0  interior (opposing orientation) : u’x’ +v’y’ < 0 interior envelope exterior envelope (x,y)r (u,v)

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 x’=y’=0, we obtain fat curves with constantly oriented envelopes

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 u’=0 v’=0

Rules for Decomposing Fat Curves Three rules for decomposing fat curves  separate fat curves for which u’ 2 +v’ 2 >= r’ 2  separate one-side fat curves by finding singular points of envelopes, i.e., points where x’ 1 =y’ 1 =0 or x’ 2 =y’ 2 =0  Separate monotone fat curves by finding points for which u’=0 or v’=0 exterior envelope (x,y)r (u,v) u’=0 v’=0

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

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

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

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 engraving representation of the PF polygonal figure of the minimum perimeter

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

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

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

Sites & Bisector Point-segment find z, follows from that sinceand, hence,

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

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

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

Approximation Consider a chain of n fat curves C 1,…,C n 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 ∈ B 3  in other word, we must solve the minimization problem

Fat Curve Fitting Problem Empty circles K 0,…K n located at the vertices of the branch Define

Fat Curve Fitting Problem The approximation fat curve C is sought in the form of a Bezier curve of degree m H 0,…,H m 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 K 0,…,K n

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 H 0,…,H m does not provide the desired accuracy  the chain of n fat curves C 0,…,C m is partitioned into two shorter chains, and the approximation problem is solved separately for each of these chains

Result