cristina neacsu, karen daniels university of massachusetts lowell

Cristina Neacsu, Karen Daniels University of Massachusetts Lowell

B-Spline Curves & SurfacesB-Spline Curves & Surfaces

• Used in geometric modeling, graphics

Gerald Farin. Curves and Surfaces for Computer Aided Geometric Design, Academic Press, 1988

www.cs.columbia.edu/ ~laza/heart3D/ www.aerohydro.com/products/rg/ papers/nasa_rg1/nasaf1.htmTheo Pavlidis. Algorithms for Graphics and Image Processing, ComputerScience Press, 1982

B-Spline CurvesB-Spline Curves• B-Splines are defined by control polygons• Periodic Cubic B-Spline Curves

– Use a special set of cubic blending functions that have only local influence and each curve segment depends only on the 4 neighboring control points.

• Current work assumption:– non-self-intersecting control polygons (at most one inflection point per curve segment).

Sample Application AreasSample Application Areas

CADCAD

Sensors

Lethal ActionLocate, Identify,Track, Observe

Sensor CoverageSensor CoverageTargetingTargeting

Spline Covering ProblemSpline Covering Problem

mj

j jBA

1

)( mj ,,,, 1

• Covering Items:

• set of spline bounded regions B = {B1, …, Bj, …, Bm}

• Target Item: spline bounded region A

• Output: a solution such that

Sample A and B Translated Bs cover A

A B1 B2

NP-Hard

ApproachApproach

Construct

Inner polygonal approximation to B

Outer polygonal approximation to A

Combinatorial Covering Algorithm

feasible infeasible

Original instance feasible

Construct Cover

?

Construct

Outer polygonal approximation to B

Inner polygonal approximation to A

Combinatorial Covering Algorithm

feasible infeasible

? Original instance infeasible

Previous Work - CoveringPrevious Work - Covering• Solving an apparel trim placement problem using a maximum cover problem approach (IIE Transactions 1999 Grinde, Daniels)

• Translational polygon covering using intersection graphs (CCCG’01 Daniels, Inkulu)

• A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering (CCCG’03 Daniels, Mathur, Grinde)

Rigid, 2D, Exact, Polygonal & Point, Translation

13 {3} 1 {1}12 {3}

6 {1}

5 {3}

4 {2}

3 {2}

2 {1}

11 {2}

10 {2}

9 {3}

8 {1} 7 {3}

P

Previous Work - SplinesPrevious Work - Splines• Convex hull property & variation diminishing property[Gerald Farin. Curves andSurfaces for CAD. AcademicPress 1988]• Trisection method[Theo Pavlidis. Algorithms for Graphics and Image Processing. Computer Science Press 1982]• Monotone partitioning of spline-bounded shapes [Daniels 92]

• Estimate on the distance between a B-spline curve and its control polygon [D. Lutterkort and J. Peters. Linear Envelopes for Uniform B-Spline Curves. 2000]

L

M

A

B

C

D

P1

P2

P3P4

l1

l2

Curve Polygonal ApproximationCurve Polygonal Approximation

• Monotone partitioning of spline-bounded shapes [Daniels 92]

• Piecewise-linear approximations using midpoints, inflection points, starting points for each curve segment, tangents at these points, and intersection between these tangents.

Polygonal ApproximationPolygonal ApproximationNumber of VerticesNumber of Vertices

• The total number of vertices of the polygonal approximation is:

where n is the number of control points of the control polygon, nN is the number of curve segments with an N-shaped control polyline, nV is the number of curve segments with a V-shaped control polyline, and nS is the number of curve segments with an S-shaped control polyline.

SnVnNnnSnVnNn

n 24

7

232

22

Polygonal ApproximationsPolygonal Approximations

• K. Daniels, R. D. Bergeron, G. Grinstein. Line-monotonic Partitioning of Planar Cubic B-Splines. Computer and Graphics 16, 1 (1992), 55:68.

• D. Lutterkort and J. Peters. Linear Envelopes for Uniform B-Spline Curves. Curves and Surfaces Design: Saint-Malo 1999. P. J. Laurent, et al (Eds.). Vanderbilt University Press, 2000, p. 239:246.

• D. Lutterkort and J. Peters. Tight Linear Envelopes for Splines. Numerische Mathematik 89, 4 (2001), 735:748.

Monotone envelope N (shaded) and rectangle-based envelope H (boldfaced) for non-N-shaped control polyline

Monotone envelope N (shaded) and rectangle-based envelope H (boldfaced) for N-shaped control polyline

Polygonal ApproximationPolygonal Approximation

• Curve segment with non-N-shaped control polyline

P1

P4

P2 P3

S

E M

X2

X1

P22 P2

1 Z

Theorem: For a non-N-shaped control polyline, X1 and X2 cannot both be outside C.

Corollary: For a non-N-shaped control polyline, if X1 or X2 is outside C, then H exits C through P2P2

2.

• C is the convex hull

of SP2P3E

• H = {H1, H2},

where H1 is SX1M and

H2 is MX2E

Polygonal ApproximationPolygonal Approximation

• Curve Segment with nondegenerate-N-shaped control polyline

P1

P2

P3

P4

I

S

E

l

X1

X2

Z2

Y2

P22

P21Theorem: For a nondegenerate N-shaped control

polyline, X1 and X2 cannot both be outside C.

Corollary: For a nondegenerate-N-shaped control polyline, H can exit C only through SP3 if X2 is outside C, or through EP2, if X1 is outside C.

H = N

Combinatorial Covering Procedure

Lagrangian-CoverLagrangian-CoverQj’s:Triangles: Groups:

G1

G2

Q1

Q2

G3

T1

T2

T3

T4

T5

• Grinde, Daniels. Solving an apparel trim placement problem using a maximum cover problem approach, IIE Transactions, 1999• Daniels, Mathur, Grinde. A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering, CCCG’03

Group choices:

G1 for Q1G2 for Q2

and group k has been assigned to some Qj

LagrangianLagrangianRelaxationRelaxation

Lagrangian Dual: min LR(), subject to >= 0

Lagrangian Relaxation LR()

T

i

m

j bkkjikii

kj

gat1 1 1

T

i

m

j bkkjikii

kj

gat1 1 1

Lagrange Multipliers

1

2

3

4

1

2

3

maximize

>=0 and subtracting term < 0

removing constraints

4minimize

Lower bounds come from any feasible solution to 1

kjg

kjg

kjg

• Grinde, Daniels. Solving an apparel trim placement problem using a maximum cover problem approach, IIE Transactions, 1999

• Daniels, Mathur, Grinde. A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering, CCCG’03

Lagrangian RelaxationLagrangian Relaxation

Lagrangian Relaxation LR()

T

i

m

j bkkjikii

kj

gat1 1 1

i

T

ii t

1

1 maximize

T

i

m

j bkkjiki

kj

ga1 1 1

LR() is separable

SP1 SP2

Solve: if (1-i) >=0

then set ti=1

else set ti=0Solve: Redistribute:

Solve j sub-subproblems

- compute gkj coefficients

- set to 1 gkj with largest coefficient

m

j bk

T

ikjiki

kj

ga1 1 1

For candidate values, solve SP1, SP2

kjg

kjg

Triangle SubdivisionTriangle Subdivision

Invariant: T is a triangulation of PP

uncovered triangle

T T’

I

II

T’

ResultsResults# m n l r A B k Time

1. 2 6 12 26 C C 0 7

2. 2 12 26 18 N C 0 17

3. 3 4 8 24 C C 8 21

4. 3 15 33 51 N N 33 1573

5. 3 10 22 33 N N&C 22 452

6. 4 8 16 56 C C 31 1394

4.6.

AGENDA

m = # of Bs

n = # of points in A’s control polygon

l = # of points in A’s polygonal approx.

r = total # of points in B’s polygonal app.

C = convex

N = non-convex

k = # points in subdivision

C++, LEDA library

450 MHz SPARC Ultra

ResultsResults# m n l r A B k Time

7. 4 15 36 40 N C 36 1787

8. 4 12 28 68 N N 31 2187

9. 5 17 37 57 N N&C 37 1984

10. 5 16 36 48 N C 36 2937

11. 5 13 27 67 N N 27 1084

12. 6 15 35 72 N C 37 1846*

12.11.8.

AGENDA

m = # of Bs

n = # of points in A’s control polygon

l = # of points in A’s polygonal approx.

r = total # of points in B’s polygonal app.

C = convex

N = non-convex

k = # points in subdivision

Future WorkFuture Work

• Remove constraint of single inflection point

• Cusps, loops Construct

Inner polygonal approximation to B

Outer polygonal approximation to A

Combinatorial Covering Algorithm

feasible infeasible

Original instance feasible

Construct Cover

?

Construct

Outer polygonal approximation to B

Inner polygonal approximation to A

Combinatorial Covering Algorithm

feasible infeasible

? Original instance infeasible

Future Work Future Work – continued– continued

• Generalize further the covering problem

Rigid, 2D, Exact, Spline, Translationthis work:

flexible approximate3D rotationfuture work:

If I do not find an exact covering, can I modify Qj’s control polygons such that they can cover P?

Backup – Backup – Polygonal CoveringPolygonal Covering

Q3

Q1 Q2

Sample P and Q

P1

P2

Translated Q Covers P

• Input:

– Covering Items: Q = {Q1, Q2 , ... , Qm}

– Target Items: P = {P1, P2 , ... , Ps}

– Subgroup G of

• Output: a solution = { 1, …, j , ... , m}, , such that mj

j jQP

1

)()( nRIso

Rigid, 2D, Exact, Polygonal & Point, Translation

Gj

NP-hard

BackupBackup

Subdivision

BackupBackup

Number of points in polygonal approximation

S

E TMI

TME

M

S

E

T1

T2 Out

In

I

S

E T1

T2

Out

In

I

BackupBackup

Each segment of a cubic B-spline curve is influenced by only 4 control points, and conversely each control point influences only 4 curve segments.

pi 1 i

pi+2

pi-1

pi+1

0

p1 p2

p3

p4 p5

p0

p1

*

BackupBackupFor segment i of a periodic cubic B-spline curve we have:

24,414,34,214,1 iiiii puFpuFpuFpuFup 2:1 niwith for open curves

and where Fj,4 are called blending functions and they are:

1336

1 234,1 uuuuF

4636

1 234,2 uuuF

13336

1 234,3 uuuuF

34,4 6

1uuF

10 u

0 u

1 F2,4

F3,4

F1,4 F4,4

F

1

1 u 0 u=1

F2,4 F3,4

F1,4 F4,4

BackupBackup

Periodic Cubic B-Spline Curves

24,414,34,214,1 iiiii puFpuFpuFpuFup

for open curves 2:1 ni

Periodic cubic B-spline curves are well suited to produce closed curves. In matrix notation:

1mod2

1mod1

1mod

1mod1

4,44,34,24,1

ni

ni

ni

ni

i

p

p

p

p

uFuFuFuFup

1:1 ni for closed curves

p1 p2

p3

p4 p5

p0

1

2

5

4

3

0

BackupBackup• C0 continuity ensures that there are no gaps or breaks between a curve’s beginning and ending points.

• C1 continuity between two curve segments requires a common tangent line at their joining point; C1 continuity also requires C0 continuity.

• C2 continuity requires that the two curves possess equal curvature at their joint.

BackupBackup

Variables: Parameters:

Brought into objective function for Lagrangian Relaxation

Lagrangian Relaxation is used as a heuristic since optimal value of Lagrangian Dual is no better than Linear Programming relaxation.

exactly 1 group chosen for each Qj

value of 1 contributed to objective function for each triangle covered by a Qj, where that triangle is in a group chosen for that Qj

kjg