Inequality Constrained Spline InterpolationOverview of Talk 1 Some Variational Spline Problems 2...

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Inequality Constrained Spline Interpolation

Scott Kersey

Workshop on Spline Approximation and Applicationson Carl de Boor’s 80th Birthday

Institute for Mathematical SciencesNational University of Singapore

December Dec 4–6, 2017

December 5, 2017

(IMS Spline Workshop) Spline Interpolation December 5, 2017 1 / 58

Abstract

The variational problem of best inequality constrained spline interpolation offers ageneralization of spline interpolation that smooths rough data while keeping precisecontrol on tolerances (error) at data points. While first studied in the 1960s, thisproblem has received just modest attention compared to it’s contemporaries – bestspline interpolation, the smoothing spline, least squares splines, splines in tension andquasi-interpolation – resulting in just a handful of papers and specializedimplementations. While no commercial implementations seem readily available (suchas IMSL), the problem is justifiably nonlinear and easily handled by methods ofoptimization. However, we think that by exploiting specific structure of splines onemay produce a simpler and/or more efficient implementation. It is the aim of this workto attempt this.In this talk we describe an implementation based on the active set method inoptimization combined with solutions to the problem of best spline interpolation. Wealso show how inequality constrained splines can be used to produce good or optimalknots, and we describe applications to parametric curves and surfaces in CAGD.

Distinguished Lecturer Series

What is a “Spline”?

——————————–

Overview of Talk

1 Some Variational Spline Problems

2 Quadratic Programming

3 Minimal Properties and Optimality for Inequality Splines

4 Ranking Knots – choosing good ones

5 Applications to Parametric Curves

Overview of Talk

Part I: Some Classical (Variational) Spline Problems

Best linearized spline interpolation(Schoenberg,Holloday,de Boor)

Smoothing splines (Schoenberg,Reinch, Wahba)

Least squares splines (de Boor andRice)

Splines in tension (Schweikart,Nielson)

Quasi-interpolation (de Boor andFix)

Best inequality constrained splineinterpolation

1Titanium Heat data from: Carl de Boor and John Rice, Least Squares Cubic Spline Approximation I –Fixed Knots

Remark

All these problems were developed by the 1960’s or early 70’s.

The first 5 problems have gotten much more attention (more literature) than theinequality problem.

I can’t think of any standard software package that solves the inequality problem.

Maybe the reason is that the inequality problem is difficult due to non-lineararities.

Or, perhaps the thought is optimization software will solve it.

The goal of this study is to develop an efficient program to solve the inequalityproblem by combining spline and optimization methods.

In doing so, we also want to generate an “optimal” set of knots, and generalize toparametric curves and surfaces.

Remark

Best Inequality Constrained Spline Interpolation

Linearized bending energy:

E(f ) :=

|Dmf (t)|2 dt.

Data:(ai , bi ) : i = 0, . . . , n.

Goal:minimize{E(f ) : ai ≤ f (ti ) ≤ bi}

Note that we can express the data as tolerances yi ± εi . Hence, the terminology“near-interpolation”.

Carl suggested to call these “gates”. Thus, we can say “best interpolation to gates”.

Perhaps we can say “set-valued” interpolation, borrowing terminology from NiraDyn.

E(f ) :=

|Dmf (t)|2 dt.

Data:(ai , bi ) : i = 0, . . . , n.

E(f ) :=

|Dmf (t)|2 dt.

Data:(ai , bi ) : i = 0, . . . , n.

E(f ) :=

|Dmf (t)|2 dt.

Data:(ai , bi ) : i = 0, . . . , n.

E(f ) :=

|Dmf (t)|2 dt.

Data:(ai , bi ) : i = 0, . . . , n.

E(f ) :=

|Dmf (t)|2 dt.

Data:(ai , bi ) : i = 0, . . . , n.

Literature on Inequality Constrained Splines

Inequality Splines (Atteia ’67)

Splines on Convex Sets (Atteia ’67)

Splines on Convex Sets (Laurent ’69)

Spline via Optimal Control (Mangasarian and Schumaker ’69)

Construction of Smoothing Splines by Quadratic Programming (Budkina 2000)

Near Interpolation for Parametric Curves (Kersey 2000, ’03a, ’03b)

Multivariate Smoothing Splines with Obstacles (Leetma and Oja 2009)

Immediate Benefits of the Inequality-Constrained Spline

Benefits:

The inequality spline can smooth rough data (like a smoothing spline and leastsquares spline).

The error is precisely controlled at the data sites (unlike smoothing and least squaressplines). This may be useful when designing parts with given tolerances, or whendata is inaccurate.

The spline is determined by the “active” knots. Other knots fall away naturally,leaving a sparser set of optimal knots.

Disadvantage:

How to choose the ai , bi (or tolerances εi ).

Benefits:

Disadvantage:

Benefits:

Disadvantage:

Benefits:

Disadvantage:

Benefits:

Disadvantage:

Good Set of Knots

Original data sites: 49. Interior knots: 13.

Optimal Set of Knots

Application: Data with Error

Interpolation Near-Interpolation

Part II: Quadratic Programming

Spline:f = Nα.

E(f ) :=

|Dk f (t)|2 dt = αTHα.

for H symmetric positive semidefinite.

ai ≤ f (ti ) ≤ bi =⇒ Aα ≤ B.

Quadratic Programming Problem:

minimize{αTHα : Aα ≤ B}

Spline:f = Nα.

E(f ) :=

ai ≤ f (ti ) ≤ bi =⇒ Aα ≤ B.

Spline:f = Nα.

E(f ) :=

ai ≤ f (ti ) ≤ bi =⇒ Aα ≤ B.

Spline:f = Nα.

E(f ) :=

ai ≤ f (ti ) ≤ bi =⇒ Aα ≤ B.

Spline:f = Nα.

E(f ) :=

ai ≤ f (ti ) ≤ bi =⇒ Aα ≤ B.

First Order Optimality Conditions of Karush-Kuhn Tucker (KKT):

2Hα + ATµ = 0

Aα = B

µ2(B − Aα) = 0

µ ≥ 0

Solving Quadratic Programming Problems by the Active Set Method

Start with an initial feasible point α0.For j = 0, 1, ...

1 Get the active set (set where inequality constraints are equality).

2 Solve a constrained least squares program on the active set.

3 Compute Lagrange multipliers µ.

4 Add or remove just one active constraint.

5 Set up new active set and go back to step 2.

6 Terminate when µ ≥ 0 for all inequality constraints.

Remark

The active set converges in finite steps (under certain conditions).

The constrained least squares problem is “best spline interpolation”.

What is the meaning of the Lagrange mutipliers µi?

Remark

Part III: Minimimal Properties and Optimality

Best Interpolation:

E(f ) :=

|Dk f (t)|2 dt.

Data: (x0, y0), . . . , (xn, yn).

Problem:minimize{E(f ) : f (ti ) = yi , f ∈W k,2(a, b)}.

Solution: f is a natural spline of order 2k and smoothness k.

Best Interpolation:

E(f ) :=

|Dk f (t)|2 dt.

Data: (x0, y0), . . . , (xn, yn).

Best Interpolation:

E(f ) :=

|Dk f (t)|2 dt.

Data: (x0, y0), . . . , (xn, yn).

Best Interpolation:

E(f ) :=

|Dk f (t)|2 dt.

Data: (x0, y0), . . . , (xn, yn).

Literature on Best Interpolation:

Demonstrates for k = 2, Best Linearized Cubic Spline Interpolation (Schoenberg1946)

First Minimum Property for k = 2 (Holladay 1957)

Minimum Properties and Best Approximation for general k (de Boor 1963)

Abstract Splines and Projections (de Boor and Lynch 1965)

Spline Interpolation using Lagrange Multipler theory

Best interpolationminimize{E(f ) : f (ti ) = yi}.

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

µi (f (ti )− yi )

= 〈Tf ,Tf 〉L2+ 〈µ,Λf − y〉`2

Theorem

The best spline satisfies:2T ∗Tf + Λ∗µ = 0.

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

= 〈Tf ,Tf 〉L2+ 〈µ,Λf − y〉`2

Theorem

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

= 〈Tf ,Tf 〉L2+ 〈µ,Λf − y〉`2

Theorem

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

= 〈Tf ,Tf 〉L2+ 〈µ,Λf − y〉`2

Theorem

Proof of Theorem

Proof.

Let g be a variation.

∣∣∣α=0

L(f + αg , µ) =d

∣∣∣α=0〈T (f + αg),T (f + αg)〉L2

+ 〈µ,Λ(f + αg)− y〉`2

= 2 〈Tf ,Tg〉L2+ 〈µ,Λg〉`2

= 2 〈T ∗Tf , g〉L2+ 〈Λ∗µ, g〉L2

= 〈2T ∗Tf + Λ∗µ, g〉L2.

Since true for all variations, we have the result.

Optimality and Lagrange Multipliers for Spline Interpolation

Theorem

The Lagrange Multipliers are

µi = 2 jmpiD2k−1f .

The best spline function satisfies:

Λ∗(2 JmpD2k−1f + µ) = 0.

Inequality Spline Interpolation using Lagrange Multipler theory

Best near-interpolation

minimize{E(f ) : ai ≤ f (ti ) ≤ bi}.

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

µ+i (f (ti )− ai )−

µ−i (f (ti )− bi ).

Theorem

µi = 2 jmpiD2k−1f

with µi := µ+i − µ

−i .

Λ∗(2 JmpD2k−1f + µ) = 0.

with at most one of µ±i nonzero for each i .

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

µ+i (f (ti )− ai )−

µ−i (f (ti )− bi ).

Theorem

−i .

Λ∗(2 JmpD2k−1f + µ) = 0.

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

µ+i (f (ti )− ai )−

µ−i (f (ti )− bi ).

Theorem

−i .

Λ∗(2 JmpD2k−1f + µ) = 0.

Lagrangian:

L(f , µ) =

|Dk f (t)|2 dt +∑i

µ+i (f (ti )− ai )−

µ−i (f (ti )− bi ).

Theorem

−i .

Λ∗(2 JmpD2k−1f + µ) = 0.

Solving Best Inequality Spline Problem by csapni()

2 Solve the problem of Best Interpolation using csape().

3 Compute Lagrange multipliers µ = Jmpf 2k−1.

Solving Best Inequality Spline Problem by csapni()

2 Solve the problem of Best Interpolation using csape().

3 Compute Lagrange multipliers µ = Jmpf 2k−1.

Remarks about csapni()

Remark

The Matlab function csapne() is very fast with linear growth. It can handle 100, 000interpolation points without a problem.

The function csapni() is faster than a standard active set method, even after doingtricks with the factorizations. However, the growth rate is still greater than linear.

Since there are approximately n/2 active constraints, there are going to be thatmany iterations on the outside loop. Since csape() has linear growth, we expectcsapni() to be order n2.

This problem can be solved in O(n) time.

Our only hope to decrease the growth rate is to handle the the interpolation problemwith constant time. As we remove or add a single knot (constraint), we caninterpolate on a fixed window (say of 100 points) rather than all n points. Due tothe exponential decay in spline interpolation, we expect that we can achieve thiswithout loss of precision, giving us an order n implementation.

Remark

Part IV: Good knots – Ranking knots

Remark

Tpyically we have more knots than we want in interpolation or near-interpolation.Hence, we want to remove those “less important”. The questions is, how do wedecide this?

What we do is choose the m most “influential” knots, and remove the rest. How dowe decide who survives?

Based on experience with near-interpolation and the smoothing spline, we guess thatthe points with larger lagrange multipliers (i.e., larger jumps in jmpti D

2k−1f ) shouldbe more valuable, so we eliminate those with smaller multipliers. Indeed, zeromultipliers can be eliminated automatically without effect on the spline.

Remark (Ranking knots)

Keeping knots corresponding to large lagrange multipliers leads to a high concentrationof knots in areas where just a couple would suffice. Hence, we need to consider a betterrank function.

Theorem (Balanced Rank Function)

The rank functionri = rank(ti ) := (ti+1 − ti−1)3|λi |

balances the polynomial error across the knot partition T .

Literature on Balanced Intervals

References (Primary Spline Approximation References Used)

[dB73] Good approximation by splines with variable knots, C. de Boor, in SplineFunctions and Approximation (1973).

[PGS78] Practical Guide to Splines, C. deBoor (1978). (The program newknt()).

[Y93] An algorithm for data reduction using splines with free knots, Y. Hu, IMA J.Num. Anal. 13 (1993)

[CA93] Constructive Approximation, R. DeVore and G. Lorentz (1993)

[NA98] Nonlinear Approximation, R. DeVore, Acta Numerica (1998)

[R69] The Approximation of Functions II, J. Rice (1969)

[PP87] Rational approximation of real functions, P. Petrushev and V. Popov,Encylopedia of Math and it’s applications (28) (1987)

Definition (Fixed Knot Spline Space)

Let Sr (T ) be the space of piecewise polynomials of order r (degree r − 1) that break atthe knots

T = [0 = t0 < t1 ≤ t2 · · · ≤ tn = 1].

No smoothness is imposed on this space. This is a linear space.

Example (Piecewise Constants: The Simplist Splines)

s(t) =5∑

αi χ[ti−1,ti )

with αi = (−1)i ,

T = [0, .2, .4, .6, .8, 1]

χ[ti−1,ti ) =

{1 t ∈ [ti−1, ti )

0 otherwise.

Definition (Approximation Error)

Approximation Error to a Function f (x) (in the inf norm on [0, 1])

||s − f ||∞ := sup{|s(t)− f (t)| : t ∈ [0, 1]}

Distance to Spline Space (in the inf norm)

σT ,r (f )∞ := dist(f , Sr (T )) = inf{||s − f ||∞ : s ∈ Sr (T )}.

s is a best approximation to f from Sr (T ) if

||s − f ||∞ = σT ,r (f )∞.

Example (Approximation to f (x) =√x by Piecewise Constants)

s :=4∑

αi χ[ti−1,ti )

withα = [.25, .5, .75, 1]

andT = [0, .25, .5, .75, 1].

What is ||f − s||∞?

Is s a b.a. from S1(T )?

If not, how could s(t) be improved?

Theorem (Kahane61, Best Approximation by Piecewise Constants)

Let f ∈ C([0, 1]). Then

σn,1(f )∞ ≤M

for n = 1, 2, 3, . . . iff f ∈ BV [0, 1], with M := Var[0,1](f ), with

var[a,b] := sup{|T |∑i=1

|f (ti+1)− f (ti )| : partitions T}.

Proof of one direction of Kahane’s theorem, from D98.

Suppose that f ∈ BV [0, 1] with M := Var[0,1](f ). Since f is continuous, we can find apartition T = [0 = t0, . . . , tn < tn+1 = 1] such that Var[ti−1,ti ](f ) = M/n. Lets =

∑i αiχ[ti−1,ti ) with αi = (f (ti−1) + f (ti ))/2. Then,

||s − f ||∞ ≤M

Example (f (x) =√x)

Since f is monotonic, M = var(√·, [0, 1]) =

1. Hence,

var(√·, [0, 1])

Following the proof, we require

var(√·, [ti−1, ti ]) =

√ti −√

ti−1 =1

This gives, ti =(

)2, and so T =

[0, .04, .16, .36, .64, 1]. By choosing

αi =( i − 1

the error on each piece is 12Mn

, hence

||s − f ||∞ =M

2n= .1.

Remark (Balanced Intervals)

Kahane’s balances the error on each piece. In this case,

||f − s||∞ = maxi||f − s||∞(ti−1, ti ) ≤

Hence, the error on the entire interval is balanced by the error on the worst interval,which is about the same.

If f is continuous, these local errors are the same.. So by balancing the variation ofa function, we are balancing the polynomial error.

Theorem (Existence and Error from Fixed Knot Spline Space)

For f ∈ C k([0, 1]) with 0 ≤ k < r , there exists s ∈ Sr (T ) such that

||s − f ||∞ ≤ Cr hk ω(f (k), h).

for some constant Cr depending only on r and h := max{ti − ti−1 : i = 1, . . . , n}, with

ω(g , h) := sup{|g(x)− g(y)| : |x − y | ≤ h}.

In [PGS] deBoor shows Cr ≤⌊ r + 1

⌋when k = 0.

Corollary

For f ∈ C r ([0, 1]), there exists s ∈ Sr (T ) such that

||s − f ||∞ ≤ Cr hr ||f (r)||∞.

If T is uniform (in which case h = 1n

||s − f ||∞ ≤ Cr n−r ||f (r)||∞.

Definition (Free Knot Spline Space)

Let Σn,r be the space of all splines of order r with n + 1 knots (at most n intervals)

Σn,r :=⋃{

Sr (T ) : T = [0 = t0 < t1 ≤ t2 · · · ≤ tn = 1]}.

Remarks:

This space is nonlinear! (I.e., The space is not closed under addition, since addingtwo splines in Σn,r with different knots typically results in a spline with 2n knots.)

We want to find the “best” knots, if possible, but usually settle for “good” knots,which is usually good enough.

The best (good) knots depend on the function.

How to choose the knots for piecewise constants?

Want bounds on approximation, and theorems of existence and uniquenss.

We can often get the same approximation error with far fewer knots.

Definition (Distance to Nonlinear Free-Knot Spline Space)

σn,r (f )p := inf{||s − f ||p : s ∈ Σn,r}

Theorem (R69,Existence of Best Approximation)

Let f ∈ Lp([0, 1]) for 0 < p <∞ or f ∈ C([0, 1]) for p =∞. There exists s ∈ Σn,r suchthat

||s − f ||p = σn,r (f )p.

Theorem (DL93,Burchard74, Smooth Approximation)

Let f ∈ Lp([0, 1]) for 0 < p ≤ ∞ and n, r ≥ 2. Then

inf{||s − f ||p : s ∈ Σn,r ∩ C r−2} = σn,r (f )p.

Hence, we can get arbitrarily close to a best piecewise polynomial approximation with asmooth spline.

Piecewise Constants: Balanced and Uniform Knots

Knots: ti = in

equally spaced.Max error occurs on the left:

||s −√·||∞ =

√t1 =

Not the optimal rate!

Knots: ti =(

More knots placed near singularity.Max error occurs on the left:

||s −√·||∞ =

√t1 =

Optimal rate.

Free-Knot Spline Approximation Estimate

Remark

The following theorem generalizes Kahane’s theorem. The statement of the theoremis taken from [D93], Chap. 12, Theorem 4.5.

Note that while in fixed knot approximation we usually see the (maximal) meshspacing h in the estimates. For free-knot splines we don’t know the mesh spacing,but we do have a term 1/n which is h for a uniform parametrization.

Theorem (Freud and Popov (1969), Subbotin and Chernykh (1970))

If r = 1, 2, . . . and f (r−1) is of bounded variation on [0, 1], then

σn,r (f )∞ ≤ Cr n−r var[0,1]f(r−1).

Remark (Conclusion of Free-Knot Spline Basics)

The key aspect from the previous discussion relevant to the remainder of this talk isthe balancing of intervals. In particular, we have showed for the simplest case ofpiecewise constant approximation that balanced intervals provide the best bound forapproximation.

Carl’s program newknt() addresses this.

Consequences:

1 Assume f (4) ∈ L1[0, 1]. Then

|λi | ≈ 2

∫ ti+1

ti−1

|f (4)(t)| dt.

2 The balancing functions have the form:

Fi ≈ ri := (ti+1 − ti−1)3|λi |/2.

By [PGS], Fi is connected to polynomial error on an interval.

3 s ′′′0 is piecewise constant with coefficients α the cumulative sum of λ. That is,

j∑i=0

Proofs.

1 For the first corollary,

|λi | = 2 |s ′′′f (t+i )− s ′′′f (t−i )|

= 2 |s ′′′0 (t−i+1)− s ′′′0 (t+i−1)|

= 2|s ′′′0 (t−i+1)− s ′′′0 (t−i−1)|

ti+1 − ti−1(ti+1 − ti−1)

∫ ti+1

ti−1

|s ′′′0 (ti )− s ′′′0 (ti−1)|ti − ti−1

∫ ti+1

ti−1

|f (4)(t)| dt.

2 The second corollary follows by

Fi ≈ (ti+1 − ti−1)3

∫ ti+1

ti−1

|f (4)(t)| dt ≈ (ti+1 − ti−1)3|λi |/2.

3 For the third corollary, we note that since s0 is a piecewise cubic spline, then s ′′′0 ispiecewise constant. Hence, the result follows since λi is the jump in the thirdderivative across the knot.

Variational Approach

1 Solve the inequality constrained program to find a good initial spline approximations0 to f such that ||s0 − f ||∞ ≤ ε/2 with active knots ti and Lagrange multipliers λi .

2 Set n, the number of intervals (n + 1 knots)

3 Rank the n − 1 interior knots according to:

ri = |λi |(ti+1 − ti−1)3.

4 Using the highest rank n − 1 interior knots and two end knots, find the least squaresspline fit sf .

5 If ||s0 − sf ||∞ 6< ε/2, increase n and go to step 3.

Titanium Heat Data

8 intervals (7 interior knots)Error: .015

Runge Function f (x) = 1/(1 + x2)

16 intervals (15 interior knots)Error .00071

Remark (Conclusion)

The proposed variational approach begins with a very large (dense) sampling ofdata, and hence a very large dense set of knots. Our method is to remove knots.

The first set of knots fall out easily where the constraints are inactive. This does notchange the spline fit.

We then delete more that are ranked lower according to our rank measure.

After choosing the most influential knots, we find a least square spline fit. Inpractice, it maintains the good error estimate, as well as the near-interpolant would.

We don’t feel it necessary to work harder to move the final knots to “optimal”positions, because the original set of knots was already from a dense subset. Thismay not always be acceptable, such as for the square root function where optimalknots are on the order of e−5 in magnitudes.

Part V: Applications to Parametric Curves and Surfaces

Parametric Curves

We apply to parametric curves as follows:

f (t) = (x(t), y(t)) with x(t) and y(t) spline functions.

| · | represents the Euclidean norm

Interval constraints become polygonal constraints

For the problem of near-interpolation1, circles can be approximated polygons.

We would like to provide optimal parametrizations (data sites and knots).

7S. Kersey, Near-Interpolation, Numer. Math. 94, 523–540 (2003)

Typical Inequality Constraints

Functions:

f (ti ) ≤ b

−f (ti ) ≤ −a

Parametric Curves:

Kij ={

x : x · (Nj − Pj) ≤ 0}

(Half Space)

f (ti ) ∈ Ki :=⋂j

Kij (Convex Set)

Example: Parametric Curves with Polygonal Constraints

minimizef

2|f ′′(t)|2 +

2|f ′(t)|2 dt

for quintic spline curve f under constraints f (ti ) ∈ Ki . We approximate balls usingpolygons with several sides.

Now we have a simple way to solve the problem of near-interpolation with fixed data sites.

Example: Bad Parametrizations for Spline Curve Interpolation

Best spline interpolation with free data sites (or knots) is a difficult problem. In theexample, we have badly chosen sites. We choose balls with centers at the data points.

While the interpolant is a bad fit, the near-interpolant improves things.

But can we do better?

Example: Optimizing Data Sites

Consider near-interpolant constraints

|f (ti )− zi | ≤ εi

where f is a spline curve that solves the problem of best near-interpolation. Hence, f (ti )is constrained to lie in a closed ball of radius εi . Optimal data sites satisfy the condition

f (ti )− zi ⊥ f ′(ti ).

Hence, we want a zero ofF (ti ) := (f (ti )− zi ) · f ′(ti ).

By Newton’s method, we update ti to ti + ∆ti with

∆ti := − F (ti )

F ′(ti )= − (f (ti )− zi ) · f ′(ti )

(f (ti )− zi ) · f ′′(ti ) + f ′(ti ) · f ′(ti ). (0.1)

Example: Best Interpolation with Free Data Sites

After applying the parameter update to the previous example, we get:

Example: Best Spline Interpolation with Free Data Sites

Repeating the Process: Solve the problem of near-interpolation, update knots, shrink theconstraints Ki , and repeat until close to interpolation.

Now we have a simple way to solve the problem of best spline curve interpolation withfree data sites for basic configurations. If the data sites and knots coincide, we also getthe optimal knots.

Example: Cornu Spiral

Optimal parametrization for interpolation of the Cornu spiral.

Example: Parametric Space Curves

Consider polyhedrons around each point of valence 3 (this is needed for linearindepedence of the active constraints).

We update data sites just as we did for 2-D curves.

The End.

Thank You!

Inequality Constrained Spline InterpolationOverview of Talk 1 Some Variational Spline Problems 2...

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