SIAM Conference on Geometric DeSIAM Conference on Geometric Desing & Computingsing & Computing

Approximation of spatial Approximation of spatial data with shape data with shape

constraints constraints

Maria Lucia SampoliMaria Lucia SampoliUniversity of Siena, ItalyUniversity of Siena, Italy

22

OverviewOverview

Methods for data approximation with Methods for data approximation with shape control shape control

Univariate case: variable degree Univariate case: variable degree polynomial spline spacespolynomial spline spaces

Spatial data: shape information by Spatial data: shape information by zerozero moment analysismoment analysis

Approximation by tensor product surfacesApproximation by tensor product surfaces Approximation by triangular surfacesApproximation by triangular surfaces Examples and conclusionExamples and conclusion

33

Related Publications Related Publications • 19971997 P. CostantiniP. Costantini, Variable degree polynomial splines, in , Variable degree polynomial splines, in Curves and Curves and

Surfaces with Applications in CAGDSurfaces with Applications in CAGD, 85-94., 85-94.• 20002000 P. CostantiniP. Costantini, Curve and surface construction using variable degree , Curve and surface construction using variable degree

polynomial splines, polynomial splines, CAGD CAGD 17, 419-446.17, 419-446.• 20002000 P. Costantini and C. ManniP. Costantini and C. Manni, Interpolating polynomial macro-, Interpolating polynomial macro-

elements with tension properties, in elements with tension properties, in Curves and Surface fitting: Saint-Curves and Surface fitting: Saint-Malo 1999,Malo 1999, 143-152. 143-152.

• 20012001 P. Costantini and F. PelosiP. Costantini and F. Pelosi, Shape preserving approximation by , Shape preserving approximation by space curves, space curves, Num. Alg. Num. Alg. 27, 237-264.27, 237-264.

• 20032003 P. CostantiniP. Costantini, Properties and applications of new polynomial , Properties and applications of new polynomial spaces, Tech.Rep. Univ. Siena, to appear in spaces, Tech.Rep. Univ. Siena, to appear in Inter. Jour. Wavel. Mult. Infor Inter. Jour. Wavel. Mult. Infor Proc.Proc.

• 20042004 P. Costantini and F. PelosiP. Costantini and F. Pelosi, Shape preserving approximation of , Shape preserving approximation of spatial data, spatial data, Adv. Comp. Math. Adv. Comp. Math. 20, 25-51.20, 25-51.

• 20052005 P. Costantini and F. PelosiP. Costantini and F. Pelosi, Shape preserving data, Shape preserving data approximation approximation using new spline spaces, in using new spline spaces, in Mathematical Methods for Curves and Mathematical Methods for Curves and Surfaces: Tromsø 2004Surfaces: Tromsø 2004,, 81-92. 81-92.

• 20052005 I. Cravero and C. ManniI. Cravero and C. Manni, Detecting shape of spatial data via zero , Detecting shape of spatial data via zero moments, in moments, in Mathematical Methods for Curves and Surfaces: Tromsø Mathematical Methods for Curves and Surfaces: Tromsø 20042004, 93-102., 93-102.

• 20052005 I. Cravero and F. PelosiI. Cravero and F. Pelosi, Zero moment analysis for bivariate data, , Zero moment analysis for bivariate data, preprint.preprint.

• 20052005 P.Costantini and F. Pelosi, Tensor Product surface approximation, P.Costantini and F. Pelosi, Tensor Product surface approximation, preprint.preprint.

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Tension methodsTension methods

Introduced for a control on the shape Introduced for a control on the shape

3 33 1(1 ) , ,span t t P P

3 0 1 1( , ), , ( , )span t t TP P

are shapeare shape parameters. For limit values of parameters. For limit values of i

0 1( , ), ( , ) 0t t

3 1 TP P

0 1,

55

Variable degree polynomial splinesVariable degree polynomial splines

1 1,1(1 ,) ,i iii kk k k

span t t 3VP P

Costantini 1997Costantini 1997 Costantini 2000Costantini 2000

Given the knot vector { },Given the knot vector { }, consider the polynomialconsider the polynomial space inspace in

For limit values of shape parameters:For limit values of shape parameters:the functions vanish in any the functions vanish in any

1,

iikk

iu

11

[ , ], ii i

i i

t uu u t

u u

1(1 ,) i ik kt t 1[ , ] [ , ],i ia b u u

3 1 VP P

1

1

,3, 0 |[ , ] 3: [ , ] : i i

i i

k kr rk n u us C u u s

VS VP

3,k 01VS S

Equivalent to Equivalent to 3P : “cubic like” variable degree polynomials

66

Approximation with Variable Degree splinesApproximation with Variable Degree splines Given select a Given select a

sequence of sequence of significant knotssignificant knots Construct the piecewise linear of best Construct the piecewise linear of best

approximation approximation λλ on on Use Use λλ to define the to define the shapeshape of the data of the data Find the degrees such that Find the degrees such that

the spline curve of best the spline curve of best approximation in has the same approximation in has the same shape of shape of λλ

The structure of variable degree splines allows a The structure of variable degree splines allows a geometric geometric controlcontrol of the shape constraints of the shape constraints

DRAWBACKDRAWBACK: : Good for Good for interpolationinterpolation problems but not problems but not always good for always good for approximationapproximation

0 0, ,..., ,N Nt tP P

0 0,..., ,...,n Nu u t t

0,..., nu u

0,..., nk k

3,kVS

77

The shape of many real objects from which the data are The shape of many real objects from which the data are taken cannot be represented by linear curves or bilinear taken cannot be represented by linear curves or bilinear surfacessurfaces

!!

IDEA: IDEA: change the limit spacechange the limit space1998 Kaklis & Sapidis1998 Kaklis & Sapidis

1 1,2(1 ,) ,i iii kk k k

span t t 4VP P

admits a totally positive, normalized, “Bernstein-admits a totally positive, normalized, “Bernstein-Bézier” basis Bézier” basis equivalence to equivalence to

1,i ik k 4VP

4P

1,1, max ,i ik k

i ik k 4 k kVP P

ForFor limit values of shape parameterslimit values of shape parameters 1,i ik k 1,

4 2i ik k VP P

• Costantini 2003Costantini 2003

1

1

,2 24, 0 |[ , ] 4: [ , ] : i i

i i

k kk n u us C u u s

VS VP

02 S

k

88

Approximation with new variable Approximation with new variable degree splinesdegree splines

Given select a Given select a sequence of sequence of significant knotssignificant knots

Different approach according to Different approach according to the new limit spacethe new limit space

Construct Construct σσ in of best in of best approximation to data on approximation to data on

Use Use σσ to define the to define the shapeshape of of the data the data

Find the degrees such Find the degrees such that the spline curve of best that the spline curve of best approximation in has the approximation in has the same shape of same shape of σσ

The structure of variable degree splines allows a The structure of variable degree splines allows a geometric geometric controlcontrol of the shape constraints of the shape constraints

0 0, ,..., ,N Nt tP P

0 0,..., ,...,n Nu u t t

0,..., nu u

0,..., nk k

24,kVS

• Costantini & Pelosi 2005Costantini & Pelosi 2005

02S

99

admits a positive, normalized, “Bernstein-Bézier” admits a positive, normalized, “Bernstein-Bézier” basis basis equivalence to equivalence to

ExtensionExtension 1 1 11,

21(1 ) (, , , ,(11 ) )i i i ii ik k kk k k

span t t t t t t 6VP P

1,i ik k

1,6 2

i ik k VP P

6P

1,1, max ,i ik k

i ik k 6 k kVP P

1,i ik k 6VP

For limit values of shape parametersFor limit values of shape parameters

More flexibility due toMore flexibility due to more degrees of freedommore degrees of freedom

1010

Let be a parametric curve. Let be a parametric curve. For consider For consider , a ball of radius , a ball of radius εε with center .with center .The The zero momentzero moment of of ss in in tt is given by the is given by the barycenter barycenter

Selection of significant knotsSelection of significant knots New approach based on New approach based on Zero MomentZero Moment

analysisanalysis automatic procedureautomatic procedure

stable with respect to noisy datastable with respect to noisy data

easy to implement even for bivariate dataeasy to implement even for bivariate data

: ds RT

• Clarenz,Rumpf &Telea 2004Clarenz,Rumpf &Telea 2004• Cravero & Manni 2005Cravero & Manni 2005

t T ( ( ))B t s( )ts

0( ( )), of ( ) ( ( ))M t B t s s sT

1111

The sign of vector gives information about concavity The sign of vector gives information about concavity and convexity and convexity

The vector The vector 0( ) ( ( )) ( )t M t t n s sis called is called εε-normal, it is independent of translations -normal, it is independent of translations and is such that and is such that

22( ) ( ) ( )

6t t o

n

can be used to select corners or sharp can be used to select corners or sharp changes in data points (considering the changes in data points (considering the zero-moment analysis to the piecewise zero-moment analysis to the piecewise line connecting the dataline connecting the data

2

( )t

n

Local classification for curve featuresLocal classification for curve features Segmentation of discrete curves in regions Segmentation of discrete curves in regions

with common shapewith common shape

( )tn

1212

Tensor-product surfacesTensor-product surfaces

Assigned and Assigned and Assigned andAssigned and

0,..., nu u 0 ,...,u u unk kk

0 ,...,v v vmk kk 0,..., mv v

There exists a basis with similar properties of There exists a basis with similar properties of biquartic B-spline basisbiquartic B-spline basis Classical geometric constructionClassical geometric construction If for all indicesIf for all indices

2 2

4, 4,u vk k

VS VS

2 2 0 02 24, 4,

u vk kVS VS S S

,u vi jk k

- Given - Given ,( , , )i j i jt r P

• Costantini & Pelosi 2005Costantini & Pelosi 2005

1313

The parameter values are givenThe parameter values are given The structure of data is supposed to be suitably The structure of data is supposed to be suitably approximated by a tensor product surfaceapproximated by a tensor product surfaceThe data have weak noise The data have weak noise they can be described by biquadratic patchesthey can be described by biquadratic patches

Approximation with T-P variable degree Approximation with T-P variable degree splinessplines

SchemeScheme1.1. Extract significant knot sequences from Extract significant knot sequences from

the given parameter valuesthe given parameter values2.2. Compute Compute σσ in of best approximation to data in of best approximation to data3.3. Define the Define the shapeshape of data as the shape of of data as the shape of σσ4.4. Compute of best approximation to data Compute of best approximation to data

in the least square sensein the least square sense5.5. Find degrees such that Find degrees such that s has the same shape of has the same shape of

σσ

,i jt r

0 0,..., , ,...,n mu u v v U V

0 02 2S S

2 2

4, 4, u vk k

Ss VS V

,u vk k

1414

Example 1Example 1

1515

Example 2Example 2

1616

Resulting surfaceResulting surface

1717

DrawbacksDrawbacks

1818

Equivalence toEquivalence to : : bivariate polynomial space of bivariate polynomial space of

degree 6 defined over a triangular domaindegree 6 defined over a triangular domain

Triangular patchesTriangular patches The parameter values are givenThe parameter values are given The structure of data is supposed to be suitably The structure of data is supposed to be suitably

approximated by a triangulated surfaceapproximated by a triangulated surface The data have weak noise The data have weak noise

they can be described by quadratic triangular patchesthey can be described by quadratic triangular patches

,i jt r

6Π

6,Π k V

variable degrees associated to the verticesvariable degrees associated to the vertices

6, 2" " Δ ΔKk ΠVΠ

1 2 3, ,k k kk

1919

Free parametersFree parameters

Triangular control net of degree 6Triangular control net of degree 6

2020

1 2 3k k k k

3 3 0D C B A 3Δ The points are on a The points are on a “quadratic net”“quadratic net”

2(1 ) , (1 ) , , (1 ),k k k kkspan t t t t t t 6VP P 2(1 ) , ,k k

span t t k4VP P The points are on a net which is a degree elevated net The points are on a net which is a degree elevated net

from a proper quadratic netfrom a proper quadratic net

Construction done in 3 stepsConstruction done in 3 steps -Generic patch of degree-Generic patch of degree

2121

1 1 2 22(1 ) , (1 ) , , (1 ),k k k k

span t t t t t t 6kVP P 1 2

2(1 ) , ,k kspan t t k

4VP P The points are on a net which is a degree elevated net The points are on a net which is a degree elevated net from a proper quadratic netfrom a proper quadratic net

Construction done in 3 steps: general caseConstruction done in 3 steps: general case -Generic patch of degree-Generic patch of degree -Consider the control net of -Consider the control net of

1 2 3, ,k k kk 1 2 3max , ,k k k k

2222

ExampleExample

2323

ConcludingConcluding remarksremarks

““Quartic-like” variable degree polynomial splines can Quartic-like” variable degree polynomial splines can be used efficiently to approximate curves and be used efficiently to approximate curves and tensor-product surfacestensor-product surfaces

Working with these spaces is computationally stable Working with these spaces is computationally stable and not expensive (the computational cost and not expensive (the computational cost does not does not dependdepend on the variable degrees on the variable degrees))

To define the To define the shapeshape of spatial data the of spatial data the zero momentzero moment method is considered method is considered

The extension to triangular patches is possible using The extension to triangular patches is possible using a combination of a quartic-like and a degree 6-like a combination of a quartic-like and a degree 6-like variable degree polynomial spacesvariable degree polynomial spaces

New scheme to approximate scattered data with New scheme to approximate scattered data with triangular splines of variable degreetriangular splines of variable degree

1C