D-t13 351 THE C2 CONTINUITY OF PIECEWISE CUBIC ERMITE

POLYNOMIALS WITH UNEQUAL INTERYALS(U) ARMY ARMAMENTUNCLSSIFE~tRESEARCH DEVELOPMENT AND ENGINEERING CENTER NAT..L 7 CN SHN JU 97 RCCOTR-8919F/U 12/1 L

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TECHNICAL REPORT ARCCB-TR-87019 flTU'UI lb

THE C2 CONTINUITY OF PIECEWISE CUBIC HERMITE

POLYNOMIALS WITH UNEQUAL INTERVALS

I-

C. N. SHEN00

US ARYAMMETREEAC9DVEOMN

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THE C' CONTINUITY OF PIECEWISE CUBIC HERMITE FinalPOLYNOMIALS WITH UNEQUAL INTERVALS

S. PERFORMING ONG. REPORT NUMBER

7. AUTNO"(a4) S. CONTRACT OR GRANT NUMDEWlt.)

C. N. Shen

S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMERSUS Amy ARDEC AMCMS No. 6111.01.91A0.0

Banat Weapons Laboratory, SMCAR-CCB-TL PRON No. 1A6AZ601NHLCWatervliet, NY 12189-4050II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

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It. SUPPLEMENTARY NOTES

Presented at the Fourth Army Conference on Applied Mathematics and Computing,Cornell University, Ithaca, New York, 27-30 May 1986.Published in Proceedings of the Conference.

IS. KEY WORDS (Contnueon roerm aide It necoosma Md identit y block nwmhbr)

Hermite PolynomialsSpline Functions

Data SmoothingLoser Vision System

N ANSTRACT (COMM w1 01 N 08" mdert by lock M )'

)Cubic hermite polynomials are usually C' continuous. With the introduction ofsmoothing within the intervals, the second derivatives can be made continuous.This may be applied to the autonomous vehicle problem with unequal laserscanning.

In using a loser range finder to measure the range, the direction of these pZlaser rays can be subjected to angular errors. These errors in the direction /"

DOPOM43 raTI O7v5asom.rE UNCLASSIFIEDSECUIRTY CLASSIFICATION OF THIS PAGE (1ib Doe. Bnlevad)

.... .. . , X e'-4 )

MCUNY CLASIICAION OF THIS PAGKMUh Defta hAmft

20. ABSTRACT (CONT'D)

of the elevation angle, affect the determination of in-path slopes fornavigation of autonomous vehicles. A nonuniform grid may be employed to computeby the spline function method with cubic hermits polynomials. For the purposeof smoothing, it is essential t9 obtain continuous second derivatives at thegrid point from both sides.

UNCLASSIFIED116CUMITY CLASSIFICATION OF THIS PAGKCWhef Does Entere) 91

TABLE OF CONTENTS

?Age

I NTRODUCTI ON 1

RECURSIVE FILTERING AND SMOO0THING PROCEDURE 1

EXAMPLE FOR C2 CONTINUITY 4

CONCLUSION 6

REFERENCES 7

APPENDIX 8

Accession For

IVTIS GRA&iDTIC TAB

Ullannounced C3justification

Distributionl/--

Avaiilbility CodesAvwill and/or

Dist Special DI

copy)INSCTED

6

INTRODUCTION

The smoothing of gradients can be obtained by using an optimization method

for approximation involving spline functions. A nonuniform grid may be employed

to compute by the spline function method with cubic hermite polynomials.

Continuous second derivatives at the grid point from both sides are essential

for the purpose of smoothing. This method can be applied to solve the following

problems: whether the platform can climb on the estimated in-path slope or

whether it will tip over the estimated cross-path slope.

RECURSIVE FILTERING AND SMOOTHING PROCEDURE

A spline function s(t) is a solution to the optimization problem

N -I N AiJ* = Min. E [h(pi)-mi]TRi [h(Ai)-mi] + p Z f [h]2df} (1)

h c C i=1 i=2 Ai-I

where for clarity and simplicity in discussion, we only consider the cubic

spline case. A higher order polynomial spline can also be treated in a similar

manner with more complicated computations.

A cubic spline, s, is a piecewise polynomial of class C2 which has many

good properties, such as the minimum norm property and local base property (refs

1,2). From the approximation theory, we know that for each set A = {a, ..... aN,

a'1, a'N), there exists a unique cubic spline s(t;A) such that

S(Al;A) = ai , i = 1,2,...,N (2)

s( i;A) = a'i, i = 1,N (3)

where s is the first derivative of the function s. The above equations can be

lAhlberg, J. H., Nilson, E. N., and Walsh, J. L., The Theory of Splines andTheir Applications, Academic Press, Inc., 1967.

2Schumaker, L. L., Spline Functions: Basic Theory, John Wiley & Sons, 1981.

1I)

thought of as boundary conditions for the piecewise cubic spline interpolation

given a set of data (Pi,ai), for i a 1,2,...,N. Thus, solving the problem in

Eq. (1) is equivalent to determining a set of constraints A for the optimization

problem:

N N AiJ* = Min { E [S(i;A)-mi]TRi [s(ti;A)-mi] + p E i [s(C;A)]2df} (4)

A i=1 i-2 Pi-1

Instead of taking a direct approach to find an optimal set of constraints for

the problem above, it is proposed to further transform this problem into a form

which is convenient to be solved. From the theory of numerical analysis (ref

3), it is well known that a piecewise cubic Hermite polynomial p(t) is in the

family of C'. For each set B = AuAc, where Ac is a complement of A, i.e., Ac =

(a'i, i = 2,3,...,N-11, then B = {ai,a' i , i=1,2,... ,Nj, there exists a unique

piecewise cubic Hermite polynomial p(C;A) such that

p(Ai;B) = ai , i = 1,2,...,N (5)

P(Ai;B) = a'i , i = 2,...,N (6)

where p is the first derivative of p.

It should also be noted that for each set A, there are an infinite number

of piecewise Hermite polynomials p(t;A) such that

p(Ai;A) = ai , i = 1,2,...,N (7)

p(Ai;A) : a'i , i = 1,N (8)

Let a set of p(C;A) which satisfies the constraints in the equations above be P,

i.e.,P = {p(C;A):(5),(6) satisfiedl (9)

3Burden, R. L. et al., Numerical Analysis, Prindle, Weber, & Schmidt, 1978.

2

Referring to the paper by de Boor (ref 4), it is noted that there exists a

unique cubic spline s(C;A) in the set P. Also from the minimum norm property of

a cubic spline, we have the following relation:

N Pi N Pir f [s(C;A)]2 4 E f [p(C;A)]t (9)i.2 Ai-1 i=2 Pi-I

That isNr i [s(C;A)]2 = inf Jp(p) (10)i-2 Ai-1 peP

whereN i "

Jp = r f [p(C;A)]2 (11)i=2 Ai-1

Since a cubic spline s(C;A) is unique, a piecewise cubic Hermite polynomial

p(C;A) which minimizes the smoothing integral Jp in the above equation with

respect to Ac becomes a cubic spline s(t;A). To be more precise, we have the

following theorem.

THEOREM: Let P represent a set of piecewise cubic Hermite polynomials p

which satisfies the constraints below:

p(Ai;Ac) = ai , i = 1,2,...N (12)

p(Ai;Ac) = a'i , i = 1,N (13)

where p e C1 , A, and Ac are the same as mentioned before. Then there exists a

unique cubic spline s(f) such that

N Ai N Ai£ f [s(f)]2 dC = Min E f [p(C,Ac)]2dC (14)i=2 Ai-1 Ac i=2 Ai-1

where s and p are the second derivatives of functions s and p and s C C2. A

simple example with N a 3 is given next.

4de Boor, C., "Bicubid Spline Interpolation," J. Math Phys., Vol. 41, 1962,pp. 212-218.

3

EXAMPLE FOR CE CONTINUITY

For convenience and simplicity, we only consider a special case with N = 3.

The node points are given as A,, A2, and A3. The intervals are not equal, i.e.,

Let a set of piecewise cubic Hermite polynomials p be

P = p(t;Ac) P p C' [tl,t 3J, p(t2) = a, a c AC) (16)

which satisfies the constraints in the equations below:

p(ti;Ac) =ai ,for i = 1,2,3

p(ti;Ac) =a'i ,for i = 1,3 (17)

In this special case, a set Ac = a'2 = a-

We want to show here that the cubic Hermite polynomial p(t;Ac), which is

obtained by minimizing the smoothing integral, will become a cubic spline func-

tion s(t) C C2[tl,t31

t 2- .. t3J= Min (f [jp(t;A 2)]2dt + f [ p(t;Ac)]adtIAc t2

t 2 - t3=Min {f [p(t;a)]2dt + f + p(t;a)]2dtl (18)a t2 t2

From Eq. (A14) of the Appendix, the smoothing integral above can be written as

J(a) = (x2-Alx 1)TS 1 '(x2-Alxl) + (x3-A2x2 )TB 2 (x3-A2x2) (19)

where Ai, Bi , and xij are defined in the Appendix, and

x= (ai,ali)T ,with a'2 = a ,i = 1,2,3 (20)

A,.. 1 di... = tij-tij.1 (21)

Using Eqs. (All) and (A12), the functional J(a) is written as

4

T

3 -aa2 1 dl al 12dl -6d1 a2 1 d1 a1

I 2 -1

a 0 1 a,, -6dl 4d1 a 0 1 a

T

3a3 1 d2 a 2 12d 2 -6d2 a3 1 d2 a2

-2 -1

a'3 0 1 a -6d2 4d2 a'3 0 1 a

-3 -a

J(a) = 12d I (a2-al-dla' 1)2 - 12d1 (a2-al-dla,1 )(a-a,1 )

- 1 - 3

+ 4d1 (a-a'l) 2 + 12d 2 (a3-a2-d2a)2

-12d 2 (a3-a 2 -d2a)(a' 3-a) + 4d2 (a'3-a)2 (22)

Taking the partial derivative with respect to a yields

a J_ -1 d 2 - I

.- = -12d (a2-al-dla'l) + 8d, (a-a'l)

-3 -3+ 24d 2 (a3 -a2-d2a)(-d 2 ) - 12d 2 (-d2)(a'3-a)

-12d 2 (-1)1a 3-a2-d2a) - 8d2 (a'3 -a) = 0 (23)

Solving the equation above for a, one obtains

-2 -1 -2 -2 -a*= [3d 1 (a2-al)-dl a'1 +3d2 a3-d2 3a2-d2 a'3 ]/[2(d 1 +d2 )] (24)

To show that p(t;a*)e C2[tl,t 3], we only need to show that

lim p(t;a*) = lim p(t;a*) (25)t-t2 - t-t2 +

5

-~ q *~ -. ~ ' J

That is, for a piecewise cubic Hermite polynomial p,

PI,2(t2;a*) = P2 ,3 (t2 ;a*) (26)

where P1,2 is the cubic Hermite polynomial within the interval A, and A2, and

P2,3 is the cubic Hermite polynomial within the interval A2 and A3.

Now from the definition of piecewise cubic Hermite polynomial in the

Appendix, we have

Pl,2(t2;a*) = 6d, (a1-a2) + 2di a', + 41 a* (27)

By using Eq. (24), the above equation can be expressed as

Pl,2(t2;a*) = [-6a 2 (d1 +d2 ) + 6(aldl +a3d2 ) + 2(a'l-a'3 )]/(dl+d 2 ) (28)

In a like manner, omitting the detailed derivation, we obtain easily

P2,3(t2;a*) = [-6a 2 (d1 +d2 ) + 6(aldl +a3d2 ) + 2(a'l-a'3 )]/(dl+d 2 ) (29)

Thus, Eq. (26) is always true, that is, the conclusion in the theorem is valid.

It is proved that the C2 continuity exists in the optimization procedure for

piecewise cubic Hermite polynomials with unequal intervals.

CONCLUSION

For scanning in the direction of elevation angle from the top of a mast

where a laser is located, the intervals needed in angles are small for far away

targets, while the same are large for close-by objects. The smoothing algorithm

discussed in this report indicates that piecewise cubic Hermite polynomials can

be used for unequal intervals or nonuniform grids.

6

REFERENCES

1. Ahlberg, J. H., Nilson, E. N., and Walsh, J. L., The Theory of Splines and

Their Applications, Academic Press, Inc., 1967.

2. Schumaker, L. L., Spline Functions: Basic Theory, John Wiley & Sons, 1981.

3. Burden, R. L. et al, Numerical Analysis, Prindle, Weber, & Schmidt, 1978.

4. de Boor, C., "Bicubid Spline Interpolation," J. Math Phys., Vol. 41, 1962,

pp. 212-218.

7

APPENDIX

EVALUATION OF THE SMOO0THING INTEGRAL

A piecewise cubic Hermite polynomial in the interval [fii-1,Ai] is repre-

sented in terms of the basis functions and the state vectors xi, xi-.1, where the

state vectors are defined as in Eq. (20). By changing the independent variable

below,

= t ~ - (Al)

Then the smoothing integral in the interval [A-,i becomes

Iii = f 0 [pi-l,i(t)]2dt (A2)

where Ai-.1 = tij-tij- 1 = AiAij AiJ*

With the change of the variable above, the second derivative of the Hermite

polynomial can be written as

T

*(t) iJ3

pi-i'i(t)=

*0i'0(t) Lxi..IjL..jwhere the second derivatives of the basis functions can be derived as follows.

Using the change of variables, we rewrite the basis functions as

0i'i(t) =t2 (3Ai-1-2t)/Ai_1

3

=P'lt t 2 (t-A&i~i)IAi~1 2

Oi,o(t) = (,Ai~i-t)2(&i~j+2t)/,Ai~j3

'V~ot)=t(Ai-1-t)2Ai_1 2 (A5)

8

Aa

Then, taking the second derivative with respect to t yields

*0i,i(t) a 6h--t/i1

41i,O = 6(2t-Ai-l)/Aji 3

*iO= (6t-4Ai-l)/Aji 2 (A6)

Therefore, the integrand of the smoothing integral is expressed as

-T

where Ki...ii is defined as

go we isi t ii

ii~~~o I iI ttttIV

ititiiii it of It

*i,O(I(ij *5, ~A)ijA *, j)(i0A *is I)iO

(A8)By utilizing the above equation, the smoothing integral becomes

'i-1, 0 (A9)

Evaluating the above integral, we obtain

9

Ki-l~~t~d

0

12/Ai 1..1 -6/A.. 12 -12/hi_13 -6/Ai-.1

2

-6/Ai 1..1 4/Ai-1 6/Ai- 12 2/A1-1

-6/A 1... 1 2/Ai-it 6/Ai 1 3 6/Ai-12

-6/A-i2 2/Ai1 6hi-1 4/i-1(AlO)

Matrices B.1..1 and Ai..1 are defined as foliows:

F 1 A1...1

Ai..1 = a 1 j(All)

0 1

-3 -

1i-1 -Ai-I Ii- (A12)

notation, Eq. (A9) is rewritten as

10

T

T 1T -..1

- i- -A -~ -Tij i- - Xi ]L-

T T..=i- pij~ Ai...ji...Ai- i - (A16

TT..

=i- -A - pA..1. xi-71

-1

=x-ilx ) Bi-lxiAilx -1 (A14)

1iI i1 1i1 x -

-w 7'T

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AO-RI83 351 THE C2 CONTINUITY OF PIECEWISE CUSIC NERNITEPOLYNOMIALS WI1TH UNEQUAL INYERVALS(U) RY ARMAMENTRESEAIRCH DEVELOPMENT AND ENGINEERING CENTER NAT..

UNCLRSSIFIED C N SHEN JUL 87 ARCC-TR-871±9 F/G L2/1 NL

4..6~ %

% %

InMI

2.2.

V t- P R

dS X

VII.

k'n 11 February 1988

ERRATA SHEET(Change Notice)

Cl TO: TECHNICAL REPORT ARCCB-TR-87019

THE C2 CONTINUITY OF PIECEWISE CUBIC HERMITE

POLYNOMIALS WITH UNEQUAL INTERVALS

by

C. N. SHEN , -1

% %

Please remove pages 1 through 4 from abovepublication and insert new pages enclosed.Corrections have been made to Equations 2,9, 10, and 11 on pages 1 and 3.

,.% . .;-

US ARMY ARMAMENT RESEARCH, DEVELOPMENT,AND ENGINEERING CENTER

CLOSE COMBAT ARMAMENTS CENTER . % ,

BENET LABORATORIES

WATERVLIET, N.Y. 12189-4050 ~k

t *J.) I N

Oak_

INTRODUCTION

The smoothing of gradients can be obtained by using an optimization method

for approximation involving spline functions. A nonuniform grid may be employed

to compute by the spline function method with cubic hermite polynomials.

Continuous second derivatives at the grid point from both sides are essential t_,9

for the purpose of smoothing. This method can be applied to solve the following

problems: whether the platform can climb on the estimated in-path slope or .%

whether it will tip over the estimated cross-path slope.

RECURSIVE FILTERING AND SMOOTHING PROCEDURE

A spline function s(l) is a solution to the optimization problem

N NJ*- Mi. t (h(Ai)-mi]TRi [h(i)-miJ + p E f [hJ'df(

h c C2 i-1 i=2 Ai-1

where for clarity and simplicity in discussion, we c 1 consider the cubic

spline case. A higher order polynomial spline can also be treated in a similar

manner with more complicated computations.

A cubic spline, s, is a piecewise polynomial of class C2 which has many

good properties, such as the minimum norm property and local base property (refs

1,2). From the approximation theory, we know that for each set A = al,.. .,aN,

a'1 , a'Ni, there exists a unique cubic spline s(t;A) such that "i

s(Ai;A) = ai , i = 1,2,...,N (2)

s(Ai;A) = a'i, i = 1,N (3)

where s is the first derivative of the function s. The above equations can be

1Ahlberg, J. H., Nilson, E. N., and Walsh, J. L., The Theory of Splines and 77"!Their Applications, Academic Press, Inc., 1967. ,

2Schumaker, L. L., Spline Functions: Basic Theory, John Wiley & Sons, 1981.

.~ ~~~ .. .

.1

.%4 °.%,

thought of as boundary conditions for the piecewise cubic spline interpolation

given a set of data (Ai,ai), for i = 1,2,...,N. Thus, solving the problem in

Eq. (1) is equivalent to determining a set of constraints A for the optimization

problem:

N -t N iJ*= Min E [s(Ci;A)-mi]TRi [s(ti;A)-mi] + p E f [s(t;A)]2d I (4)

A i=1 i=2 Ai-1

Instead of taking a direct approach to find an optimal set of constraints for

the problem above, it is proposed to further transform this problem into a form

which is convenient to be solved. From the theory of numerical analysis (ref

3), it is well known that a piecewise cubic Hermite polynomial p(t) is in the

family of C'. For each set B = AuAc, where Ac is a complement of A, i.e., Ac =

a'i, i = 2,3,...,N-1I, then B • {ai,a'i, i=1,2,...,NI, there exists a unique

piecewise cubic Hermite polynomial p( ;A) such that

p(Ai;B) = ai , i = 1,2,...,N (5)

P(Pi;B) = a'i , i = 2,...,N (6)

where p is the first derivative of p.

It should also be noted that for each set A, there are an infinite number %

of piecewise Hermite polynomials p(4;A) such that

p(Ai;A) = a i i = 1,2,...,N (7)

p(Ai;A) - a'i , = ,N (8)

Let a set of p(t;A) which satisfies the constraints in the equations above be P,

i.e.,= p(t;A):(5),(6) satisfiedi (l)

3Burden, R. L. et al., Numerical Analysis, Prindle, Weber, & Schmidt, 1978.

•., .. ,...,

2

Ne*

Referring to the paper by de Boor (ref 4), it is noted that there exists a

unique cubic spline s(C;A) in the set P. Also from the minimum norm property of

a cubic spline, we have the following relation:

N 4iN Pi

That isN Air [ s(t;A)]2dt z inf Jp(p) (10)

where i2A-

N Ai

Since a cubic spline s(C;A) is unique, a piecewise cubic Hermite polynomial

p(C;A) which minimizes the smoothing integral Jp in the above equation with

respect to Ac becomes a cubic splin. s(t;A). To be more precise, we have the

following theorem.

THEOREM: Let P represent a set of piecewise cubic Hermite polynomials p

which satisfies the constraints below:

p(Ai;Ac) - ai . i - 1.2 .... N (12)

P(41 ;Ac) - a'l , i a 1,N (13)

where p c C', A, and Ac are the same as mentioned before. Then there exists a

unique cubic spline s(4) such that

N AiN AEI: f s(f)]'dC - Min E f (p(C,Ac)JIdj (14) .

1-2 Ai-I Ac 1.2 Ai-I

where s and p are the second derivatives of functions s and p and s c C2. A

simple example with N a 3 is given next.

4de Boor, C., 'Bicubid Spline Interpolation," J. Math Phys., Vol. 41, 1962,pp. 212-218.

3 P. #

34

J%

EXAMPLE FOR C2 CONTINUITY

For convenience and simplicity, we only consider a special case with N = 3.

The node points are given as A,, A2, and A3. The intervals are not equal, i.e.,

(2"Al) 0 (A3-A2) (15)

Let a set of piecewise cubic Hermite polynomials p be

P = [p(t;Ac) , p 6 C' [tl,t 3], p(t2 ) = a, a c Ac] (16)

which satisfies the constraints in the equations below:

p(ti;Ac) = ai , for i = 1,2,3

p(ti;Ac) =a' 1 for i - 1,3 (17)

In this special case, a set Ac = a'2 = a.

We want to show here that the cubic Hermite polynomial p(t;Ac), which is

obtained by minimizing the smoothing integral, will become a cubic spline func-

tion s(t) C C'[tl,t 3] ,'**

t2 - t3 a

J* = Min i2 [p(t;Ac))zdt)Ac ti t2+

= Min {f [p(t;a)]dt + f [p(t;a)]dt} (18)a tl t2

+

From Eq. (A14) of the Appendix, the smoothing integral above can be written as

J(a) = (x2-Alxj)T8l (x2-Alx I ) + (x3-A2x2 )TB 2 (x3-A2 x2 ) (19)-,

where Ai, Bi , and xi are defined in the Appendix, and

xi = (ai,a'i)T , with a'2 = a , i a 1,2,3 (20)

hi-I " dil - ti-ti-l (21)

Using Eqs. (All) and (A12), the functional J(a) is written as

F

4

i ,,b -: ....- -

FILI

IL IL - - -