New Hermite cubic interpolator for two-dimensional curve generation

Transcription

1 New Hermite cubic interolator for two-dimensional curve generation R. Szeliski, B.Eng., M.Sc., and M.R. Ito, Ph.D. Indexing term: Comuter grahics Abstract: The roblem of generating a smooth two-dimensional curve through a set of samle oints is examined. Such curve generation techniques can be used in curve coding for transmission (e.g. 'Telewri~ing') and i.n curve design. The aer examines some arametric interolators which will generate such cur~es, with emh~sls on techniques based on the Hermite cubic (or cubic subsline) interolator. A new method IS resen.ted which gives the interolator a erformance similar to that obtained with finite imulse res?onse (FIR) fil.tenng. Comarisons are made with existing techniques using both visual results and comutational conll~xlty measures. The new method is found to have better locality, smoothing characteristics and a lower comutational cost than alternative aroaches. 1 Introduction The roblem of generating a smooth curve through an ordered set of two-dimensional oints is one that often arises in comuter grahics. The curves generated can reresent handwriting, sketching, contour lines or other grahical objects. Recent aers have resented such interolating methods based on either low-ass filtering [1J or cubic interolation [2, 3J. This aer resents a new method based on Hermite cubic (first-derivative continuous) interolation that has the desirable features of both cubic interolators and of low-ass filters while being comutationally simler than either of these aroaches. The method is designed for non-uniformly saced oints ('non-uniform grids'); a simlified version, which can be used with uniform grids, is also resented. Several alternative interolating techniques are reviewed and comared with the new method. The requirements for a suitable curve generator include simlicity of secification, smoothness, ease of calculation and locality. Simlicity of secification is achieved by requiring that the curve must interolate (ass through) the given data oints. This requirement allows the method to be used to reconstruct time or sace sub-samled curves such as handwriting. The smoothness criterion requires that the generated curves be visually leasing when dislayed on the outut device. The locality requirement means that only a few neighbouring oints are needed to calculate a curve segment. Ease of calculation requires that the algorithm be simle and fast enough to be imlemented on a microrocessor. The combination of these latter two requirements makes the method useful for the realtime coding of line grahics (such as sketches), since the reconstructed curve can be raidly generated with a delay of only a few oints. The locality criterion also ensures that the overall curve is not sensitive to the lacement of anyone samle oint, and also facilitates local curve editing. Previous work in curve generation has been in the field of curve design, with commonly used methods such as Bezier curves [4J requiring interactive adjustment of the curve. Dube [5J resented such a curve design method based on the same Hermite cubic basis which is used here. However, his interolator uses the full cubic sline solution as a starting oint, and is thus not sufficiently local. Paer 4860E (Cl), received 12th March 1985 Mr Szeliski is with the Deartment of Comuter Science, Carnegie-Mellon University, Schenley Park, Pittsburgh, Pennsylvania 15213, USA. Dr Ito is with the Deartment of Electrical Engineering, University of British Colombia, Vancouver, B.C., Canada V6T 1W5 Shlien and Allard [1J have resented an interolating method based on low-ass filtering which gives results visually similar to the ones resented here. Their method, however, requires large window sizes to eliminate certain artifacts. Renner and Pocho [2J have roosed a local cubic interolator suitable for grahics which have a mixture of straight line and curved segments. Harada and Nakamae [3J have recently resented another four-oint cubic interolator. Both methods are similar to the new method resented here but do not generate curves that are as smooth. These and other interolating methods are discussed later in the aer. The rimary alication for the new interolating method resented here is for the reconstruction of subsamled curves such as digitised handwriting. The method can thus be directly alied to the coding of sketches for instantaneous transmission ('Telewriting'), by samling the writing at a fixed frequency and using the cubic interolator to regenerate the curve. The interolator is sufficiently flexible so that a controlled lacement of the samle oints would also allow it to be used for the coding of other line-drawing grahics such as contour lines or tyeface fonts. The simlicity of the method and its strict locality also make it a suitable candidate for a new geometric rimitive for Videotex (the PLP 'Sline' rimitive which is yet to be standardised [7J). The emergence of secialised chis such as the Geometry Engine [8J which incororate the generation of cubics should make this cubic interolator attractive in general comuter grahics alications. The remainder of the aer is structured as follows. Section 2 resents the arametric formulation for the curve based on the Hermite cubic functions. Section 3 reviews a variety of methods for determining the curve derivatives ('sloes') at each samle oint which are necessary to secify comletely the interolator. The new interolator resented in this aer, which is based on a iecewise cubic aroximation to the low-ass filter, is introduced. Section 4 comares the new method with true low-ass filtering and also gives a simlified version of the algorithm which can be used for uniform grids. Section 5 discusses some methods for imlementing the interolator using integer arithmetic on very simle systems (such as a microrocessor). Section 6 discusses the locality against smoothing trade off inherent in the new interolator and resents some of the visual results obtained. The various interolating methods examined in the aer are comared. The final Section summarises the advantages of the new interolating method and some of its ossible alications.

2 2 Hermite cubic interolation The interolator used to construct the curve generally has a arametric reresentation, i.e. the curve is a twodimensional function of an underlying arameters s, x == j'(s) (1) which asses through the collection of samle oints {Xi}' This notation allows for easy extension to three dimensions [4J. The arametric notation also gives an interolator that is isotroic, i.e. invariant with resect to orientation. The arameter s is usually related to the arc length along the curve. However, using the chord length s, + 1 == s, -t d, where di == Id, I == I Xi Xii (3) with s varying linearly between control oints, will give equally good results [9J. A further simlification, s, == i, can be made if the oint sacing is fairly uniform (see Section 4). The interolation requirement recludes the use of some functions such as quadratic B-slines [1 OJ which are otherwise suitable curve generators. One simle class of functions that both interolates and can be made sufficiently smooth are the cubic slines and sub-slines. These functions are iecewise cubic, i.e. on any interval (Si' s,+ 1) the function is a cubic For sufficient smoothness, we require that the curve be first derivative (C 1) continuous. Such curves are variously known as sub-slines or Hermite cubics [4J. They are comletely defined by the control oints {Xi} and the curve derivatives ('sloes') at these oints {x;} (Fig. 1). Each (2) (4) Hermite cubic interolant function /(s) is thus comosed of iecewise cubic segments h(s) with first-derivative continuity across interval boundaries o Fig. 2 Hermite cubic hasis (or 'blendinq'[functions 3 Sloe determination To generate the cubic interolator, the sloes {i';} must be determined. One common way to secify the sloes is to require second derivative (e 2 ) continuity. The resulting system of tri-diagonal equations can then be solved using various iterative or direct methods [11, 12J, with the resulting interolator being known as the full cubic sline. However, this calculation requires all of the oints on the curve to be known, and it is thus not sufficiently local to be used for real-time reconstruction. The sloes can also be adjusted interactively. Methods such as Bezier curves [4J have been designed to do this naturally by secifying additional control oints. This is not a satisfactory aroach if the curve generation is to roceed automatically from a set of samle oints without oerator intervention. What is required instead is a method to estimate the individual sloes using only a few of the neighbouring oints. Several such methods are discussed in this section, leading to a new method based on a iecewise-cubic aroximation to finite imulse resonse (FIR) filtering. The simlest method for determining the sloe locally is to use a arabola through a samle oint and its left and right neighbours to determine the sloe at the oint (modifications can be made for the end oints of the curve). The arabolic equation g(s) = ~ (s - S;)2 + x(s - s;) + Xi Fig. 1 Cuhic segment defined hy its end oints and end-oint derivatives cubic segment can also be reresented as a linear combination of four basis or 'blending' [5J functions weighted by the end oints and the end-oint derivatives with g(si-1) == X i- 1 and g(si+ 1) == X i + 1 (7) can be solved to yield. di - 1m i. i+ 1 + di m i - 1, i x == ' '-- (8) d i 1+di The value m i j is the divided difference of the curve (Fig. 3), h(s) == X /P o(t) + x/p dt) - Xi + 1 cp d1 - t) + Xi + 1 S - cpo(l - t), t == -d.- E [0, IJ (5) l These basis functions, the Hermite Cubic basis functions, are cpo(t) == 2t 3-3t cp 1(t) == t( 1 -- t) 2 and are shown in Fig. 2. Since the samle oints {x;} are given, only the sloes {x;} need to be determined in order to secify comletely the interolator. The sloe values Xi are the true derivative values of the interolator /(s), but can only be an estimate of the derivatives of the original curve. The overall Si (6) x: -I Fig. 3 Divided differences of a curve used to calculate the sloes and is defined as the sloe of a line connecting two samle oints: ~ mij (for short) (9)

3 The interolator resulting from this arabolic fit, known as the Bessel interolator, is quite simle, but does not smooth as well as methods that use more of the neighbouring oints. It is, in effect, a four-oint interolator, since the oints Xi-I' Xi' X i+ 1 and X i+ 2 are needed to define the two end sloes Xi and Xi+ 1 for the segment };(s). Higher-order olynomials (such as quartics) can be fitted to determine the sloe, but the exact solution raidly becomes comlex. An assumtion of near-uniform sacing can be used to obtain some simle results which take the form of a SUlTI of divided differences weighted by fixed rational numbers (see 9.1 Aendix A). For examle, the result for the quartic is x,= ---m,.').+-nt 1 +-m, '+I-- m '+2 (10) l 6 I -.<;.. l 3 1-,1 3 1,1 6 1,1 It can be shown that the solutions to these olynomial fits are equivalent to the first ste in the iterative solution of the full cubic sline [9J. The results using these simlified olynomial fits comare favorably with the Bessel interolator (Fig. 4). Since more oints are used to determine the sloes, better smoothing is achieved. The window function w(t) is a ositive symmetric function of window size (2 + 1) which is used to limit the duration sine (t) FI R low-ass filter resonse: windowed sinejunction, = 3 of the imulse resonse and to imrove the frequency resonse. A smaller window allows for faster calculation but yields oorer smoothing. The imroer choice of window shae can lead to artifacts in the filtered outut (Fig. 7). The interolated curve is a sum of imulse a Fig. 4 Cubic interolator using arabolic and quartic fits a Parabolic b Quartic x marks the samle oints The quartic fit roduces a smoother curve since it uses more oints (6 instead of 4) Other aroaches to determining the sloe can be based uon fitting a circular arc through three neighbouring oints [6, 13J. However, the resulting curves are too rounded to be used without additional control secification (Fig. 5), and are thus not suitable for interolation. Fig. 5 rounded ~x ( '-x Cubic interolator usinq arc circular arc fit. The result is too The method resented by Rennor and Pocho [2J uses four neighbouring oints to determine the sloe. Straight line segments are reserved using an aroach based on Akima's method [14J. The method used by Harada and Nakamae is very similar to the Bessel interolant, since their derivative value is 6/7 times that obtained in eqn. 8 [3J. A comarison of the erformance of these last two aroaches with the new method is given in the Section 6. The new method for sloe determination develoed in this aer aroximates the resonse of the low-ass filter resented by Shlien and Allard [1J. Their FIR filter is an interolator whose imulse resonse is a windowed sine function (Fig. 6) Fig. 7 c d Effect ofwindow size and shae on FIRfilter a-(' Hanning window = 2, 3, 4, resectively. The larger window sizes give more smoothing d Bartlett (triangular) window = 3. Artifacts are seen due to the nonlinear recision resonse functions weighted by the samle oints /(s) = Lxih(s - i) (12) i where the arameter s is assumed to be on a uniform grid (s, = i). From this equation, the sloe of the interolator at a samle oint is. -,( 1)j- 1 (.) /. Xi - L - tv } X i+ j/j (13) j= - j*o The FIR interolator can be extended to non-uniform grids by using an asymtotic exansion for sine (t) [9J. The corresonding sloe value can be derived as Xi. = '" L ( - 1)' j -- 1 (') \V J m i, i+ j (14) j= - i r» where m i, i+j is the divided difference defined in eqn. 9. This new result roduces an interolator that works on non-uniform grids while reserving the filtering characteristics of the FIR filter. The choice of window shae and size will determine the smoothing (filtering) characteristics of this new ('sinc-tye') interolator. To ensure that the interolator has linear recision (is straight-line reserving), it is required that L [J j= - j*o (- 1j - 1w(j) = 1 (15) sin (nt) w(t) ~ 0 for I t I ~ P One such linearity reserving window that has been found h(t) = w(t) --, (11) nt lv(t) = 0 for It I > by the authors to give good results is the Hanning

4 window, w(j) == cos ' (jn/2). An examle of the resulting sloe estimator for == 3 is Xi == -4 m.:«, + 4: m.«, + 4mi. i + 1-4m i + l. i + 2 (16) which is quite similar in form and erformance to the simlified quartic fit (eqn. 10). Rewriting the above equations to take advantage of the symmetry in the window function, we obtain the final formulation for the new sine-tye interolator, which is a iecewise-cubic function (eqn. 5) with the sloes at the samle oints defined by r r. == I,,( - 1)j-l\t\) ''Ifm,._. + m,.+.} ~.l I, I} 1.1 J (17) j= I and the weighting coefficients {W j } satisfying L (- I)J -- W j == 2 and w.i ~,j ==, (18) j--e:-l Here, the derivative of the curve becomes discontinuous at the control oint if d, #- d i + I (x i -- is the derivative from the left X from the right). However, since the derivative is 1+ direction continuous, sufficiently smooth curves are still roduced if d, ~ d,+ 1[4, 9J. For this simlified interolator, the sloe becomes a simle linear combination of neighbouring oints. It now becomes ossible to create an 'imulse resonse' for the interolator by suerimosing the contributions of several neighbouring oints to one imulse value in the man?er ~f eqn. 12. Fig. 8 shows the resultant resonse, which IS Xj-2 xi+2 Fig. 8 Imulse resonse of simlified cubic interolator, calculated hy suerimosition The resonse is comarable to Fig. 6 and differs the most near the tail ends 4 Comarison with low-ass filtering The results obtained visually with this new sine-tye cubic interolator are similar to those obtained with true lowass filtering, FIR filtering uses a sum of windowed imulse resonse functions to generate the interolated curve (see eqn. 12). The windowing ensures that only a few oints in the neighbourhood of the samle oint are considered in the summation, ensuring the locality of the interolant. The greatest advantage of the cubic interolator over the FIR filter is that it has linear recision (i.e. it reserves straight lines), even for non-uniformly saced oints. This is only aroximately true of the FIR filter, even for large window sizes, and leads to artifacts for smaller windows (Fig. 7). The choice of a small window size ensures greater locality for the interolator and allows the reconstruction of the curve to roceed with less delay. In ractice, a window size of 7 ( == 3) has been found to give a good balance between sufficient smoothing and good locality (see eqn. 16). If the samle oint sacing is fairly uniform, as is usually the case for curves which were time samled at a fixed frequency, then a simlification can be made to the interolator given in eqn. 17. Using the assumtion that s, == i, a simlified "normalized' sloe estimate Xi can be calculated as with " P Xi == L ltt',j. (Xi+.i - X i--) (19) j = 1 \vj == (- 1).i- I w(j)/j (20) The required linearity condition becomes Ljwj == t (21) j= I An examle of the normalised derivative for a Hanning window ( == 3) is (22) The "normalised' derivative must still be converted to the actual derivative by Xi - == Xi. d i (23) X i + == rt i. d i + 1 similar to the true windowed sine function (Fig. 6). The sine-tye cubic interolator has similar smoothing characteristics to the FIR digital filter, and also has a bet ter linearity. As well, the calculators of the cubic interolator is faster, as is shown in the following Section. 5 Imlementation The calculation of the Hermite cubic interolator for dislaying a curve consists of two searate arts. The first is the calculation of the sloes at each control oint. This can be done on-line, since only a small number of oints have to be accumulated. If the divided differences are used to calculate the sloes (eqn. 17), then the distance values {dj must also be calculated and retained. Because fixed rational weights are used, the sloe calculation is easily imlemented using integer arithmetic. The calculations are further simlified if simle differences are used (eqn. 19). For examle, the sloe for the sirnlified interolator using a Hanning window given in eqn. 22 can be calculated using four add/subtracts and three arithmetic shifts. One secial case that arises in the sloe calculation is the treatment of end oints. Several aroaches are ossible [9J; one of the simlest is to reeat the end oints before and after the curve, which in effect sets the divided differences to zero beyond the curve. Thus, if the curve has n control oints, {Xiii == 1, n}, then we define Xi == Xl for i < 1 Xi ==.r, for i > n (24) m., == 0 for j < 1 or j > n This aroach has been found to roduce satisfactory results for the curve near its end oints. The other secial case that arises is the occurrence of reeated oints. Such double oints do not arise often in samled hand-drawn curves, excet near corners or oints of en stoage. One simle way of treating double oints is to extend the divided difference definition (eqn. 7) to be zero at such oints F[si' Sj J == mij ~ 0 when s, == Sj (25) A more useful aroach, however, is to treat the reeated oint as an end oint, and to calculate the left and right derivatives searately (ignoring contributions before or after the double oint). This roduces a discontinuity in

5 curvature, and can be used to lace corners, shar bends, or straight line segments, at selected locations along the curve (Fig. 9). This added flexibility enhances the interolator for use in curve design. cubic interolator [3J. Comared to the new interolator, the FIR filter exhibits some artifacts due to the limited Fig. 9 Corner, shar bends and straight lines using double oints Cubic interolator = 3 x: single oint 0: double oint The second art of the interolator calculation is the generation of each cubic segment once its end oint derivatives are known. One simle aroach is to use a look-u table for the basis functions cpo and cp 1 (eqn. 6). This is easily imlemented using integer arithmetic (four multilies er outut oint), and comares favourably with a look-u table imlementation of FI R filtering (2 multilies). If using multilications is comutationally too exensive (as for some simle microrocessors or for direct hardware imlementation), a finite algorithm can be used [9J. This aroach, also known as the digital differential analyser (DDA), can be imlemented using integer addand-shift-only arithmetic (see 9.2 Aendix B for derivation). An imlementation for a 16-oint cubic segment would require an average of 7.5 adds and 4.5 shifts er outut oint, and could also be imlemented directly in hardware. An existing hardware cubic generator, such as the one incororated in the Geometry Engine [8J, could also be used for dislaying the cubic segments. If a raster device is being used for outut, then an incremental algorithm could be designed to dislay the curve one ixel at a time. The comutational requirements for the sine-tye cubic interolator are thus quite low. Each sloe can be calculated as a weighted combination of neighbouring samle oints, and the cubic segment can be calculated using either a look-u table, a finite difference algorithm, or direct hardware imlementation. Fig. 10 c Cubic interolator with [Janning window a=2 h=3 (' = 4 d=5 The smoothing is better for larger windows 6 Results The new sine-tye cubic interolator resented here can be looked at as a family of ossible interolators, since both the window shae and window size can be varied. The window shae determines the weighting coefficients used in eqns. 17 and 18, and has not been found to be a critical factor in the erformance of the interolator. Comare, for examle, eqns. 10 and 16, which use different weighting factors, and whose resonses (shown in Figs. 4 and 11) are nearly identical. In ractice, the Hanning window, ",'(j) == cos ' (jn/2), has been found to give good results (with linear recision guaranteed). The choice of window size gives a larger variation in erformance. Better smoothing is achieved by using larger windows (Fig, 10), while a smaller window size gives better locality, and hence less delay in reconstruction. The sine-tye interolator can be comared visually (Fig. 11) to the FIR filter [IJ, the cubic interolator due to Renner and Pocho [2J, and Harada and Nakamac's Fig. 11 Comarison ofjour interolators a New cubic interolator, uniform grid ( = 3) h FIR filter ( = 3) c Renner and Pocho's interolator d Harada and Nakarnae's interolator The arrows indicate some visual artifacts window size, and both Renner and Pocho's and Harada and Nakamae's interolators are not as smooth, since they are only four-oint interolants. The alternate curve generation methods can also be comared in terms of comutational comlexity. Once the sloes have been calculated, the dislayed oints in the cubic segment (for the cubic interolators) can be calculated using four-table look-u/multilies er outut oint, as comared to 2 such oerations for the FIR filter. The three cubic-based methods can ultimately use the same cubic segment generator, so that they must be comared on the basis of arameter calculation. Comared to Renner and Pocho's tangent and arameter interval calculation [2J, the sloe determination for the new method is significantly simler. Harada and Nakamae's method, a

6 simle arabolic fit, is very similar to the non-uniform grid cubic interolator with = 2. The sloe calculation for the new interolant is slightly simler since it uses fixed weights to combine the divided differences (comare eqns. 17 and 8). The new interolator resented in this aer can also be used for curve design. While it does not have as much flexibility as a general design curve (such as a Bezier curve) where each individual sloe can be adjusted, it offers greater simlicity in secification, since all control oints lie on the curve. Moreover, extra oints can be inserted or removed with only local effect, and double oints can be used to lace shar bends in the curve (Fig. 12). rx--...x---x----x'x ~ x x./ x-- Fig. 12 Curve design usinq the new cubic interolator. Curvature control is effected by using more oint s The interolator resented here was originally devised for the real-time coding of handwriting and sketches. The quality of the coding deends on the samling rate, and using the cubic interolator allows the use of lower rates than would be required with straight-line interolation. Fig. 13 shows the same curve samled at different fixed rates using the cubic interolator for reconstruction. This aer has examined the use of arametric cubic subslines (Hermite cubics) for two-dimensional curve generation/reconstruction. The Hermite cubic aroach encomasses a family of interolators, with the different interolants being defined by the choice of sloe (derivative) estimator. In articular, a new local sloe estimator has been resented which uses a linear combination of a few divided differences, and which has the same smoothing characteristics as the FIR filter. The six-oint interolant, which uses a combination of four divided differences (eqn. 16), has been found to give a good balance between calculation time and smoothing. A simlified algorithm which calculates the sloe as a linear combination of a few neighbouring oints can be used on uniform grids (where the oint sacing is fairly regular). Once the end-oint derivatives of a cubic curve segment are known, the oints along the segment can be calculated for dislay, using either a look-u table or a finite difference (DDA) algorithm. The new cubic interolator has several advantages over other methods. It is strictly local (as oosed to the full cubic sline), and thus requires less storage, and can be calculated on-line with only a few oints' delay. It has smoothing characteristics similar to the FIR filter (better smoothing than other local cubic methods such as Renner and Pocho's), and also has linear recision, which eliminates the artifacts associated with the FIR filtering aroach. The simlicity of calculation, both in sloe estimation and cubic segment generation, is another advantage. The interolator is also isotroic (insensitive to orientation), and can be extended to higher-dimensional curves. The curve generation method resented has several otential alications. It can be used with the coding of sketches and handwriting for transmission (Telewritingj or for storage. The coding of other curves such as contour lines or tyeface fonts is also ossible, and is enhanced by the simlicity of secification. The method can be used for general curve design, where control of the curvature is not critical. As such, it is a good candidate for selection as the PLP sline rimitive used with Videotex coding. In summary, the simlicity of the method, its strict locality and good smoothing characteristics make it a good choice for two-dimensional curve generation. 8 References Fig. 13 Cubic interolation ihanninq window, = 3) with different decimation (sub-samling) rates a Decimation J h 5 c 7 d9 The visual and comutational results resented in this section show that the new sine-tye cubic interolator has two major advantages over other methods. It is less comutationally exensive (simler), and it has desirable smoothing characteristics while maintaining linear recision. 7 Conclusions SHLIEN, S., and ALLARD, P.: "A FIR filtering aroach for the generation of smooth curves on a grahics terminal', Comut. Grahics & / maqe Processinq., 19R1, 17,. 269-~280 2 RENNER, G., and POCHOP, V.: "A new method for local smooth interolation', EUROGRAPHICS' 81 Conference, (North-Holland, New York, 1981) 3 HARADA, K., and NAKAMAE, E.: "An isotroic four-oint interolator based on cubic slines', Comut. Grahics & l maqe Process., 1982,20, BOEHM, W.: 'On cubics: a survey', ibid., 1982, 19, DUBE, R.P.: 'Preliminary secification of sline curves', / EEE Trans; 1979,C-28, KNUTH, D.E.: "METAFONT, a system for alhabet design' (Digital Press, Bedford, Mass., 1979) 7 "Presentation Level Protocol, VIDEOTEX Standard', American Telehone and Telegrah Comany, CLARK, J.H.: 'The geometry engine: A VLSI geometry system for grahics', Comut. Grahics, 1982, 16, SZELISKI, R.: 'Real time coding of hand drawn curves', M.Sc. Thesis, University of British Columbia, Vancouver, Canada LAPALME, R.S.: "An interactive data reduction technique for line drawings', M.Eng. Thesis, Royal Military College, Kingston, Canada LIOU, M-L.: "Sline fit made easy', / EEE Trans., 1976, C-25, MOSS, R., and LINGARD, A.: "Parametric sline curves in integer arithmetic designed for use in microcomuter controlled lotters', Comut. & Grahics, 1979, 4, MIDGELEY, J.E.: 'Isotroic four-oint interolation', Comut. Grahics & / mage Process: 1979, 9, AKIMA, H.: 'A new method for interolation and smooth curve fitting based on local rocedures', J. ACM, 1970,17, Aendix 9.1 Aendix A This aendix derives the sloe estimate obtained by using a quartic olynomial fit. An assumtion of quasi-uniform sacing is made in order to simlify the solution. The

7 quartic olynomial gi(s) == u(s - SJ4 + b(s - SJ3 + c(s - SJ2 + xi(s - sj + Xi is made to ass through the four neighbouring oints but with a resulting erformance which is oorer than that obtained by using a Hanning window, w(j) == cos' (jn/2m). 9.2 Aendix B g lsj) == X j, j == i - 2, i-i, i + 1, i + 2 The calculation of the cubic segment once its end-oint deriva: yes are known can be imlemented using finite dif- In matrix notation, we have or -(d i d, 1 ) 3 (d i d i _ 1 ) 2 -(d i d i - 1 ) U] [Xi_2 - Xi] -d~l d~l -d i - 1 b Xi-I-Xi d; d; d i C - X i + l - X i [ id, + d., 1)3 id, + d i + 1)2 id, + d i + 1) Xi X i Xi -(d i d i - 1 ) 2 -d i - 1 b _ m i - 1, i 'i]' 1 l][u] [m i - d i 1 'c m, '+ 1 td, + di+d 1 Xi m;:;+2 This general matrix inversion must be erformed for each oint if a non-uniform grid is assumed. However, if a uniform grid s, == i (and hence d, == 1) is assumed for the left-hand side, a simlified matrix results ference equations (also known as the digital differential analyser). To generate a cubic on the interval [0, 1J knowing the end-oint artials g(o), g(1), g'(o) and g'(1), the following set of difference equations can be used: l = ~ ~ =~ ~][:] [:;=~::] g"(t + ~t) == g"(t) + ~t. g"'(o) 1 1 lie m, Xi m;: ;+ 2 This can be solved for 1 1 x I == 1. 3 {m' I 1,I. + ttt, 1,1 '+1} - -6{m' I _ 2,I,+ ttt, 1,1 '+2J g'(t + ~t) == g'(t) + ~t. [g"(t) + g"(t + ~t)j/2 g(t + ~t) == g(t) + ~t. [g'(t) + g'(t + ~t)j/2 - ~t3. g"'(0)/12 The original second and third derivatives can be obtained from This aroach is referred to as 'quasi-uniform' grid sacing, since some of the non-uniformity in sacing is reflected in the divided differences mi,.i' Since the sloe is a g"(o) == 6g(1) - 6g(0) - 4g'(0) - 2g'(1) linear combination of these divided differences, the If the number of oints to be generated on the cubic segment is n, then ~t == l/n. Divisions by n can be relaced by arithmetic shifts if n is chosen to be a ower of two. This yields an algorithm that uses only additions, subtrac tions, and arithmetic shifts (see Reference 9 for a comlete examle including aroriate scaling). resulting interolator has linear recision, i.e. it reroduces straight lines. The aroach given here can be extended to higherorder olynomials, but is of limited usefulness. The results are in all cases similar in form to the sine-tye cubic interolator m Xi == I (- l)j~ I vvj{mi, i-j + mi, j= 1 i+ j} g"'(o) == 12{g(0) - g(l)} + 6{g'(0) + g'(l)}