Lecture 15. Interpolation II. 2 Piecewise polynomial interpolation Hermite splines

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1 Lecture 5 Interpolation II Introduction In te previous lecture we focused primarily on polynomial interpolation of a set of n points. A difficulty we observed is tat wen n is large, our polynomial as to be of ig order, namely n. Unfortunately, ig-order polynomials tend to suffer from wiggle, and tis limits teir practical usefulness for interpolation. In tis lecture we will explore ow we can use polynomials of moderate order to acieve smoot interpolations wile avoiding te problems associate wit ig-order polynomials. Piecewise polynomial interpolation Hermite splines Wat we've previously called linear interpolation is more precisely piecewise-linear interpolation. We don't interpolate te entire set of points wit a single line. Instead, we use different line segments over different intervals between sample points Fig.. Te complete interpolation is built by tying togeter tese lines. In fact te sample points were lines join or tie togeter, x i, y i, i=,,, x n, are appropriately called knots. Fig. : Piecewise interpolation: linear left and cubic rigt. Te sample points at wic pieces join or tie togeter are called knots. Te entire interpolation function is described by y= f x =a i x+bi if xi x< xi + and te a i,b i values are determined by te conditions y i=a i xi +bi y i+=ai x i++bi We can express a linear function in different forms, one of wic migt be more convenient for determining coefficients. We could write te it segment in te form of a st order Taylor series expanded about te point x i, EE Numerical Computing

2 Lecture 5: Interpolation II / y= y i+ y i x x i Tis trivially satisfies y= y i wen x= xi. For x= xi + te requirement y i+= y i+ y i x i+ x i provides te value te y i value y i= y i+ y i x i+ x i Alternately, te Lagrange interpolating polynomial can be written by inspection as y= y i x x i+ x x i + yi + x i x i+ x i+ x i We now extend tis idea of piecewise interpolation to polynomials. Wile a piecewise linear interpolation is continuous, te derivative is clearly not continuous at te sample points. Suppose now tat for eac x i we know bot te function value y i and te function slope y i. Let's build a piecewise polynomial interpolation tat as te specified function and slope values at te knots. Tese polynomial pieces are known as splines. Tis term comes from te practice of bending strips of wood or plastic to form smoot curves, a tecnique often used in sip building and precomputer-era drafting. For eac segment we ave four equations to satisfy, te two endpoint function values and te two endpoint slope values. Our interpolation function must terefore ave four unknown coefficients. Since a rd order polynomial as four coefficients we write Fig. f x=a i x +bi x +ci x+d i if x i x<x i+ In eac interval we ave four unknowns a i,b i, c i, d i satisfying four equations y i=a i xi +bi x i +ci x i+d i y i = ai xi + bi x i +c i y i+=ai x i++bi x i++c i x i++d i + y i+= a i x i++ bi x i++c i We could solve tese four equations in four unknowns directly. Instead we'll apply some bookkeeping to obtain a bit cleaner approac. To simplify analysis over an arbitrary interval x i x<x i+ it's a good idea to form te normalized variable u= x x i x x i = xi + x i i As x varies over x i x x i+, u varies over 0 u. Note tat du = dx i so EE Numerical Computing

3 Lecture 5: Interpolation II / y= dy dy du dy = = dx du dx i du or dy =i y du We now write y=a+b u+c u +d u dy =b+ c u+ d u du Our system of equations is now evaluating te above equations at u=0, y i=a i yi =b y i+=a+b+c+d i yi +=b+ c+ d Te simplification from is significant. Tese can easily be solved to give a= yi b= i y i c= y i i y i + y i+ i y i+ d = y i+i y i yi + +i y i+ and te interpolating cubic is y= y i+i y i u+ yi i y i + y i+ i y i+ u + y i+i y i y i++i y i+u It can be convenient to separate out te various y terms to obtain y= y i u + u +i yi u u +u+ y i+ u u +i yi + u +u A bit of factoring puts tis in a more compact form y= u [ y i + u+i yi u]+u [ y i+ u i yi + u] Te result is te so-called Hermite spline interpolation algoritm. Hermite spline interpolation Given x, y, y, x, y, y,, x n, y n, y n, wit increasing x values For a value x Find x i suc tat x i< x<x i+ Set i =x i+ xi and u= x x i/ i Calculate y= u [ y i + u+i yi u]+u [ y i+ u i yi + u] EE Numerical Computing

4 Lecture 5: Interpolation II 4/ x Fig. : y= e sin x. solid circles: sample points, squares: function values, dased line: linear interpolation, solid line: Hermite-spline interpolation. Derivative values were estimated numerically. Considering te entire interpolation algoritm as a function, we write y=s x. A Scilab function to perform Hermite spline interpolation is given in Appendix as interphermite, and an example of Hermite spline interpolation is sown in Fig.. If we ad samples of te form x i, y i, y i, yi we could find 5t order interpolation polynomials for eac interval and so on, in principle, for any number of known derivatives at eac sample point. If we ave function and derivative values up to d m y /dx m, te two endpoints of eac interval will provide m+ equations. A polynomial wit tis many coefficients as order n= m+. Cubic splines If we know function and derivative values at n points, we can interpolate eac interval wit Hermite splines. Often, owever, we only know te function values and not te derivative values. Tis provides only enoug information to uniquely determine a piecewise-linear interpolation. But te smootness of a piecewise-cubic interpolation is igly desirable, and we would like to find a way to keep tat property even wen we lack derivative information. We will refer to piecewise cubic interpolation witout specific derivative values as cubic splines.. Smootest Hermite spline interpolation One approac would be to treat te y i as unknowns and find te values tat optimize some desirable property of te curve. Smootness is an intuitively appealing property to ave. A smoot curve is one in wic te slope does not cange rapidly. A sudden cange in slope produces a kink in te curve, wic is about as unsmoot as you can get. Terefore te second derivative te rate of cange of te slope sould be small. Let's write te integral of te square of te second derivative of S x as a function of te unknown y i EE Numerical Computing

5 Lecture 5: Interpolation II 5/ values: xn ϕ y, y,, y = [ S x ] dx n 4 x Using te notation S i u = u [ yi + u+ i y i u ]+u [ y i+ u i yi + u] for u as given in, we write 4 as n d ϕ= S u du i dt i= i 0 For evenly space samples were i = we sow in Appendix tat minimizing ϕ leads to te equations y + y = y y, y n+ y n = y n y n 5 y i++4 y i + y i = y i+ y i for i=,,, n 6 for te y i values. Te case of nonuniform samples is similar but a bit messier because we ave to keep track of different values.. Continuity of second derivatives Anoter approac is to require tat not only te first, but also te second derivatives of te interpolation be continuous at te knots x, x,, x n. Tere are n knots, since te endpoints x, x n are not knots no oter pieces connect tere. Continuity of te second derivatives at te knots provide n equations. For uniformly spaced samples tese turn out to be 6. Tis tells us tat te smootest Hermite spline interpolation we derived previously also results in continuous second derivatives at all knots. We ten need two more equations to obtain a unique solution. So-called natural end conditions are obtained by setting S x =0 at te endpoints x, x n, in oter words, we let te interpolation go straigt at bot ends. Tis leads to equations 5. Terefore a cubic spline interpolation wit natural end conditions is precisely te optimally smoot Hermite spline interpolation we derived above. Anoter option is to specify te end-point slopes y, y n. Tis is called te fixed-slope end conditions. If we ave a good estimate of tese slopes ten tis makes sense. Oterwise te coice is arbitrary. Finally we can coose te so-called not-a-knot conditions were we require te tird derivative of te interpolation to be continuous at te first and last knots. At tese knots, terefore, te cubic functions and te first, second and tird derivatives are continuous. But cubics tat agree in tis manner are simply te same cubic; tere is no oter possibility. So wat used to be te first and last knots are no longer knots, ence te name not-a-knot. For te uniformly sampled case tese equations read EE Numerical Computing

6 Lecture 5: Interpolation II 6/ Fig. A case were natural conditions produce a more accurate interpolation tan not-aknot conditions. y +4 y + y = y y and y n +4 y n + y n= y n y n 7 Wic end conditions sould we coose? Te natural conditions are attractive because of teir maximally smoot feature. However, in many cases te not-a-knot conditions provide a more accurate interpolation. It depends on te underlying function f x see Fig. and Fig. 4. Actual functions are not necessarily as smoot as possible! Common practice is to use te nota-knot conditions. In practice te two end conditions produce very similar results except, possibly in te first and last intervals.. Optimal smootness of natural cubic spline interpolation We've spent a lot of time working wit cubic splines. We've sown tat natural cubic splines are te smootest-possible piecewise-cubic interpolation of any set of points. If smootness is so desirable, wy not try piecewise interpolation wit even iger-order polynomials? It turns out tat no oter interpolation is smooter tan a natural cubic spline. Tis rater remarkable result tells us tat, as far as smootness is concerned, cubic splines are te best we can do. Suppose tat S x is te natural cubic spline interpolation of te n samples x i, y i. Let f x be any twice-differentiable function tat also interpolates te samples tis could even be te original function from wic te samples were drawn. Ten one can sow tat xn [S x xn x ] dx [ f x ] dx x Tis is te sense in wic we can say tat no function provides a smooter interpolation of a set of data points tan does te natural cubic spline. However, as sown in Fig. and Fig. 4, smooter is not necessarily better. EE Numerical Computing

7 Lecture 5: Interpolation II 7/ Fig. 4 A case were not-a-knot conditions produce a more accurate interpolation tan natural conditions. 4 Te interp function Scilab/Matlab As for most numerical metods we study, Scilab and Matlab ave built-in functions offering state-of-te-art implementations. Te following command yp = interpx,y,xp,str; //str = 'nearest' or 'linear' or 'spline' Implements one-dimensional interpolation of eiter nearest-neigbor, linear or spline type. For spline interpolation not-a-knot end conditions are used. Here te vectors x,y are sample data, xp is te vector of x values were we want interpolations, and yp is te vector of interpolated Fig. 5: Ten randomly selected points dots. Tick green line: nint-order polynomial. Tin blue line: not-a-knot spline. Tin dased red line: natural spline. EE Numerical Computing

8 Lecture 5: Interpolation II 8/ Fig. 6 Sample data represented as impulses or poles of eigt yi. values. Tis is sufficient for almost all one-dimensional interpolation needs. Wit regards to wiggle, te advantage of splines over ig-order polynomials for interpolation is illustrated in Fig. 5. Notice too tat te difference of te two spline end conditions is significant only in te first and last intervals. Tese data points were cosen at random so tere is no actual underlying f x. However, te spline interpolations certainly appear more realistic. 5 Lanczos convolution interpolation Te metods we ave considered so far apply to an arbitrary set of samples x i, y i. If te x values are uniformly spaced, owever, ten some ideas from signal processing can be applied. We turn to tat now. Let's suppose we ave uniformly spaced samples wit x i= x+i. We can visualize our data as sown in Fig. 6. Imagine an impulse or pole of eigt y i erected vertically wit its base on te ground at location x i. For our purposes convolution is te process of replacing eac impulse wit a common impulse response function, centered at te impulse and scaled by te eigt y i. Te triangle function Λ x is sown in Fig. 7. Fig. 7: Te triangle function EE Numerical Computing Λx

9 Lecture 5: Interpolation II 9/ Fig. 8: Convolution of impulses wit triangle function. Tick gray curve is sum of all triangle functions and interpolates te data points. If we replace eac impulse by a stretced version of te triangle function Λ x / = x / x x > 0 { scaled by y i, ten te sum of tese n f x= y i Λ i= x x i produces te interpolation sown in Fig. 8. We recognize tis as te piecewise linear interpolation of te data wit te addition of linear extrapolations at te two ends. Tis naturally leads us to wonder if using a different impulse response function migt produce a better interpolation. Tinking of x as time and y as te amplitude of an audio signal, tere is a remarkable teorem Fig. 9 Te sinc function. EE Numerical Computing

10 Lecture 5: Interpolation II 0/ due to Nyquist wic says tat provided: te signal from wic te audio samples were draw contains frequency components only witin a limited range, and te sample separation is properly cosen, ten a convolution interpolation using te sinc function pronounced sink will exactly recreate te original function f x. Matematically y= f x = y i sinc i= x x i Te sinc function is sincx = sin π x πx and is plotted in Fig. 9. Note tat sinc 0= and sinc n=0 for n a non-zero integer. Unfortunately te sinc function extends to x ±, altoug te amplitude of te bumps drop off as / x. If we are interpolating many points, we'll ave to add a contribution from eac point. A compromise proposed by Lanczos is to window te sinc function by anoter sinc function to produce te Lanczos kernel L x= { sinc x sinc x a 0 x a x >a were a is typically cosen to be a small integer most often or. Tis is plotted in Fig. 0. Te Lanczos interpolation is i+a y= j=i + a yj L x x j Fig. 0 Te Lanczos kernel for EE Numerical Computing a=,,

11 Lecture 5: Interpolation II / Fig. : Seven data points and Lanczos interpolation. were i is te index suc tat x i x<x i+. In tis expression only te a nearest sample points contribute to te interpolation at a given value of x. An example of Lanczos interpolation wit a= is sown in. 6 Appendix Scilab code 6. Hermite spline interpolation ////////////////////////////////////////////////////////////////////// // interphermite.sci // ,, for pedagogic purposes // Given n samples xi,yi,yi in te column vectors x,y,y // were yi=fxi and yi is te derivative of fx at xi, // interpolate at points xp using Hermite splines. // Note: x and xp values must be in ascending order. ////////////////////////////////////////////////////////////////////// function yp=interphermitex, y, y, xp n = lengtx; m = lengtxp; yp = zerosxp; i = ; //start linear searc at first element for j=:m wile xpj>xi+ //find j so tat xj<=ui<=xj+ i = i+; end = xi+-xi; t = xpj-xi/; ypj = t-^*yi**t++yi**t.. +t^*yi+*-*t+yi+**t-; end endfunction EE Numerical Computing

12 Lecture 5: Interpolation II / 7 Appendix Smootest Hermite spline interpolation Assume we ave n samples x i, y i, and te x values are equally spaced x i= x+i. Let n d ϕ= S t dt i i= 0 dt were [ ] [ S i t =t yi t++ y i t +t y i+ t+ y i+ t ] One can sow tat a computer algebra program elps! 0 d S t dt= y i+ y i y i++ yi y i+ y i +4 y i + y i y i++ y i+ i dt [ ] For a knot <i<n, y j appears in S i and S i. Calling d d 4 w= S t dt+ S t dt 0 dt i 0 dt i we ave [ y [ y ] y ] + y + y y w= y i yi y i+ y j y i y i + y i + y i yi + yi + y i+ i i+ + y i y i+ i i i i+ + y i+ Setting w y i = yi ++4 y i + y i + y i y i+=0 we ave y i++4 y i + y i = y i+ y i 8 For te first interval { [ ] } y y y + y y y + y + y y + y =0 y gives us y + y = y y 9 y n+ y n = y n y n 0 wile for te last interval we find EE Numerical Computing