Splines Vida Movahedi January 2007
Outline • Piecewise polynomials • Splines • Two approaches • B-splines • Multivariate functions • Examples
What problem splines solve? • In the simplest situation, one is given points (ti, yi) and is looking for a piecewise polynomial function f that satisfies f(ti)=yi , all i, more or less. • Exact fit Interpolation • Approximate fit least squares approximation or smoothing splines • Or we have a complex function g that we want to approximate with a piecewise polynomial function f that satisfies f(ti)=g(ti) , all i, more or less.
Why polynomials? • Easy evaluation, differentiation, and integration • If done properly, the work needed for data fitting grows only linearly with the number of data points.
Definition of Polynomial of order n • Note that a polynomial of order n has degree<n • The set of all polynomials of order n forms a linear space • <n: linear space of polynomials of order n
Why piecewise polynomials? • Jackson’s Theorem: If g has r continuous derivatives on [a..b] and n> r+1, then • The only way to make the error small is to make (b-a)/ (n-1) small partitioning [a..b] using piecewise polynomial approximation. • It is usually much more efficient to make (b-a) small than to increase n. This is the justification for piecewise polynomial approximation.
Why not higher order polynomials? • We expect the error between the function g and the polynomial approximation pn on n sites to decrease when n increases. If the sites are uniformly spaced, it can be shown that this is not true and the interpolation error increases with n for some examples. • It is shown that choosing the sites as the zeros of the Chebyshev polynomial of degree n can lead to much decreasing interpolation error with increasing n. For some examples, the error, while decreasing, decreases far too slowly.
What is a spline? • An interval [a..b] is subdivided into sufficiently small intervals [j.. j+1], with a=1<…<l+1=b, • On each such interval, a polynomial pj of relatively low degree can provide a good approximation to g. • This can even be done in such a way that the polynomial pieces blend smoothly, i.e. so that the resulting composite function s(x) that equals pj(x) for x[jj+1] , all j , has several continuous derivatives. • Any such smooth piecewise polynomial function is called a spline.
Continuous 1st derivative Continuous 2nd derivative
Two major approaches • The ppform: Use of piecewise polynomials • The B-form: Use of Basis functions
ppform • More efficient when evaluating the spline extensively • The ppform of a polynomial spline of order k provides a description in terms of its breaks1,…,l+1 and the local polynomial coefficients cji of its l pieces. • For linear approximation
Example g(x)=sin(x) Breaks (interpolation) >> x=0:0.6:pi; y=sin(x); >> figure; plot (x,y, 'o') >> sp=spapi(2,x,y); >> hold on; fnplt(sp); >> pp=sp2pp(sp); >>fnbrk(pp) The input describes a ppform breaks(1:l+1) Columns 1 through 5 0 0.6000 1.2000 1.8000 2.4000 Column 6 3.0000 coefficients(d*l,k) 0.9411 0 0.6123 0.5646 0.0697 0.9320 -0.4973 0.9738 -0.8906 0.6755 pieces number l 5 order k 2 dimension d of target 1 Interpolating spline of order 2 The ppform provides the coefficients for each of the polynomial pieces between the breaks. In the above example the coefficients for each of the lines is provided.
The B-form • Useful during construction of a spline • The B-form describes a spline as a weighted sum • of B-splines of the required order k, with their number n>k-1+l • Each Bj,k is defined on an interval [tj..tj+k] and is zero elsewhere. • tj are called knots and are provided based on the smoothness required (breaks, but sometimes repeated!) • B-splines are functions that:
Example B-splines of order 2 (uniform spacing between knots)
Example (cont.) >> fnbrk(sp) The input describes a B-form knots(1:n+k) Columns 1 through 7 0 0 0.6000 1.2000 1.8000 2.4000 3.0000 Column 8 3.0000 coefficients(d,n) 0 0.5646 0.9320 0.9738 0.6755 0.1411 number n of coefficients 6 order k 2 dimension d of target 1 Note that in ppform, we had 6 breaks, and 10 coefficients.
de Boor Control Points for B-splines Bezier Control Points Control Points • aj are called the control points • If B are B-splines aj are de Boor control points • If B are Bezier splines aj are Bezier control points
Why B-splines? • Composed of (n-k+2) curves of k-order joined Ck-2 continuously at knot values (t0,…,tn+k) smoothness and continuity • Each point is affected by k control points • Each control point affects k segments local controllability • Inside convex hull Built-in boundedness does not shoot to infinity (unlike Bezier) • Affine invariance • Any degree of detail desired Generative nature
Linear B-splines (k=2) Cubic B-splines (k=4) Uniform B-splines • If the knots are equi-spaced, for a given order k, the splines are simply shifted copies of one another.
knot multiplicity • Knots are a sequence of breaks, sometimes repeated! • The rule is Knot multiplicity + condition multiplicity = order • For example, for a B-spline of order k=3 Simple knot smoothness_cond=2 continuity of function and first derivative Double knot smoothness_cond= 1 just continuity Triple knot No condition function can even be discontinuous
Multivariate Functions • If f is a function of x, and g is a function of y, then their tensor-product p(x,y):=f(x)g(y) is a function of x and y. • For example for a bivariate spline of order h in x and k in y, we have: • Knot sequences • s= (s1,…, sm+h) • t= (t1,…,tn+k) • Coefficients (aij:i=1..m, j=1..n)
Chord-Length Method • The knots must be non-decreasing! • So, what should we do for closed curves?! • [1,4,7,10,6,3] not acceptable • t: new variable , increasing as we follow the contour • x and y will be functions of t
Representing Image Contours Contour from Paco’s grouping algorithm
Questions? • Which method is better? Interpolation or smoothing? • What order? Linear, cubic, …? • How can we calculate the error? Should we compare with the original lines? Or with image?
References • Matlab Spline Toolbox • Carl de Boor, A Practical Guide to Splines, Springer, 2001. • ‘An Interactive Introduction to Splines’, http://www.ibiblio.org/e-notes/Splines/Intro.htm • Cohen, F.S.; Wang, J.Y. , “Part I: Modeling Image Curves Using Invariant 3-D Object Curve Models- A Path to 3-D Recognition and Shape Estimation from Image Contours”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 1, Jan. 1994. • Miroslaw Kuc, COSC 4080.03 Project report: “Representing Image Contours with Splines”, Jan. 2005 • Branka Otasevic, COSC 4080.03 Project report, Winter 1999.