Fat Curves and Representation of Planar FiguresL.M. MestetskiiDepartment of Information Technologies, Tver State University, Tver, Russia

Computers & Graphics 24 (2000) Computer graphics in Russia

OutlineAbstractFat curvesBoundaries of fat curvesImplicit representation of fat curvesDirect rasterization of fat curvesEngraving representationApproximation of an engraving by fat Bezier curves

AbstractFat curve = curve having a widthtrace left by a moving circle of variable radiusEngravingunion of a finite number of fat curvesGoalBezier representation for fat curves2D modeling through engravingapproximation of arbitrary bitmap binary images

ProblemTransforming 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

MethodBezier performance of greasy linesDecomposition of fat curves on parts with simple envelopesScan-converting of fat curves based on Sturm polynomialsRepresentation of any binary image as fat curves on the basis of its continuous skeleton

Fat CurvesSet of circles in the Euclidean plane R2 C: [a, b] R2 [0, ) , t[a, b] Ct = {(x, y): (xu(t))2+(yv(t))2 (r(t))2, (x,y)R2}Fat curveC = t[a,b]Ctaxis: P(t)width: r(t)end circle: Ca, Cb (initial and final circles)may be considered as the trace of moving the circle Ct

Example of a Fat CurvePlanar Bezier curvea set of circles on the plane: H = {H0,H1,,Hm}circle Hi, radius Ri, Center (Ui, Vi), i = 0,,m

[Bernstein polynomials]

Example of a Fat Curveaxis: 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 Hcontrol circles: H0, H1,, H6control polygon: H

21 circles of family Ct (t = 0.05j, j = 0,,21)

Boundaries of Fat CurvesA family of circles

Under certain conditions, the family of circles, which is a family of smooth curves, has an envelope curveThe 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

Find the Envelope CurveConditionthe first condition is always satisfied the second condition can be violated (no envelopes)

Find the Envelope CurveA parametric description of two envelopes

Define

EnvelopesConsider in more detail the case when the condition is violated and envelopes do not existInterval on which is found as a result of the decomposition of a fat curve

EnvelopesConsider a fat curve for which envelopes existAn envelope of a family of circles can be exterior of interior (dont belong to the boundary of the fat curve)Criterion for distinguishing interior envelopsdirection of axis : (u, v)direction of envelope : (x, y)exterior (supporting orientation) : ux + vy > 0interior (opposing orientation) : ux +vy < 0

EnvelopesAn envelope can change its orientation from supporting to opposing and converselyx = y = 0cut a fat curve at point t[a, b] where x=y=0, we obtain fat curves with constantly oriented envelopes

EnvelopesTwo-side fat curve: both envelopes are exteriorwhen envelopes are self-intersecting or intersect each other, it must be decomposed into partsto find monotonicity intervals: u(t) = 0 or v(t) = 0One-side fat curve: one of the envelopes is interior

Rules for Decomposing Fat CurvesThree rules for decomposing fat curvesseparate fat curves for which u2+v2 >= r2separate one-side fat curves by finding singular points of envelopes, i.e., points where x1=y1=0 or x2=y2=0Separate monotone fat curves by finding points for which u=0 or v=0

Implicit Representation of Fat CurvesMembership function of the set

point belongs to the fat curve if the following condition is satisfied for a certain

Direct Rasterization of Fat CurvesThe 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 ImageObtain a continuous representation of a figure given by its discrete representation

The solution of this problem involves 3 stepsapproximate the given bitmap binary image by a polygonal figure (PF)construct a skeletal representation of the PFapproximate the skeletal representation of the PF by fat curves

Polygonal FigureEach of the PF is a polygon of the minimum perimeter that separates the black and white pixels of the bitmap imageProblemconstructing an engraving representation of the given bitmap image

construction of an engraving representation of the PFpolygonal figure of the minimum perimeter

Skeletal RepresentationConsider 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 PFcircles: inscribed empty circlesset of centers of such circles forms the skeleton of the PFskeletal representation of a bitmap image: skeleton + inscribed empty circles

Sites & BisectorPF consists of vertices and segments: sitesevery empty circle touches two or more sitesThe maximal connected set of the centers of the inscribed empty circle that touch these sites: bisector of a pair of sitesa segment of a line or a segment of a parabola

Sites & BisectorA skeleton is an almost complete engraving There possible combinations of the pairs of sitessegment-segment, point-segment, point-point Segment-segment

Sites & BisectorPoint-segmentfind z, follows fromthatsinceand, hence,

Sites & BisectorPoint-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 CurvesSkeletal 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 GThe Hausdorff metric may be conveniently measure the distance between engravings

Find an engraving G such that

BranchSkeleton structurejuncture vertices of degree 3 or higherterminal vertices of degree 1intermediate vertices of degree 2A chain of edges that have common vertices of degree 2 will be called a branchThe entire skeleton can be represented as the union of such branches

ApproximationConsider a chain of n fat curves C1,,Cn corresponding to the same branch of the skeletonfind a fat curve C in a certain class of fat curves that provides the best approximation for this sequence of circlese.g., in the class of cubic Bezier curves CB3in other word, we must solve the minimization problem

Fat Curve Fitting ProblemEmpty circles K0,Kn located at the vertices of the branchDefine

Fat Curve Fitting ProblemThe 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

Fat Curve Fitting ProblemIn 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

Result