1 / 53

Splines IV – B- spline Curves

Splines IV – B- spline Curves. based on: Michael Gleicher : Curves , chapter 15 in Fundamentals of Computer Graphics, 3 rd ed. (Shirley & Marschner ) Slides by Marc van Kreveld. Interpolation vs. approximation.

axl
Download Presentation

Splines IV – B- spline Curves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Splines IV – B-spline Curves based on: Michael Gleicher: Curves, chapter 15 in Fundamentals of Computer Graphics, 3rd ed. (Shirley & Marschner) Slides by Marc van Kreveld

  2. Interpolation vs. approximation • Interpolation means passing through given points, approximation means getting “close” to given points • Bézier curves and B-spline curves p2 p2 p1 p1 p0 p3 p0 p3 p1and p2are interpolated p1and p2are approximated

  3. B-spline curves • Approximates all control points • Polynomials of any degree d • Cd-1 continuous everywhere

  4. B-spline curves • A linear combination ofthe control points • The bi(t) are the basis functionsand show how to blend the points • The basis functions are splines themselves  B-splines the contribution of point pi to the curve depending on the parameter value t bi(t) t

  5. B-splines and B-spline curves pi-1 pi pi+1 bi(t) bi-1(t) bi+1(t) 0.55 0.3 0.15 t t’

  6. B-splines and B-spline curves pi-1 pi bi-1(t’)=0.15 bi(t’)=0.55 pi+1 bi(t) bi-1(t) bi+1(t) 0.55 bi+1(t’)=0.3 0.3 0.15 t t’

  7. B-splines and B-spline curves pi-1 pi pi+1 bi(t) bi-1(t) bi+1(t) 0.55 the blended point determined by pi-1, pi , pi+1 and their basis functions 0.3 0.15 t t’

  8. B-spline curves • A B-spline curve of n points and parameter value k • is Ck-2 continuous • is made of polynomials of degree k – 1 • has local control: any location on the curve is determined by only k control points • is bounded by the convex hull of the control points • has the variation diminishing property (any line intersects the B-spline curve at most as often as that line intersects the control polygon)

  9. Types of B-splines • Uniform linear B-splines • Uniform quadratic B-splines • Uniform cubic B-splines • Non-uniform B-splines • NURBS (non-uniform rational B-splines)  To define the B-splines is to define the B-spline curve

  10. Uniform linear B-splines • Basis functions are piecewise linear, for example: i ti+1 i+1 ti+2 otherwise 1 b1,2(t) 1 2 3 4 5 knots

  11. Uniform linear B-splines 1 b1,2(t) 1 2 3 4 5 1 b2,2(t) 1 2 3 4 5 1 b3,2(t) 1 2 3 4 5

  12. Uniform linear B-splines • Basis functions are piecewise linear, for example: • The B-spline curve can be used for parameter values t [2, n+1] (the t where the bi,2(t) sum up to 1), given points p1, p2, …, pn • The B-spline curve is just the polygonal line through the control points; only two B-splines are non-zero for any t i ti+1 i+1 ti+2 otherwise

  13. Uniform linear B-splines • Each B-spline has 3 knots and they are uniformly spaced  They are shifted copies of each other • In total there are n+2 knots (at 1, 2, …, n+2) 1 b1,2(t) 1 2 3 4 5 1 b2,2(t) 1 2 3 4 5 1 b3,2(t) 1 2 3 4 5

  14. Uniform quadratic B-splines • Uniform quadratic B-splines bi,3(t) have knots at i, i+1, i+2, and i+3 • They are shifted copies of each other 1 b1,3(t) 1 2 3 4 5 t

  15. Uniform quadratic B-splines if t [i, i+1] u = t – i if t [i+1, i+2] u = t – (i+1) if t [i+2,i+3] u = t – (i+2) otherwise u is set to be in [0,1] for the t-range 1 b1,3(t) 0.5 1 2 3 4 5 t

  16. Quadratic B-spline curve knot p6 p5 p3 p2 p5p6p7 p2p3p4 p1p2p3 p4p5p6 p3p4p5 p1 p7 p4 control points defining this piece of curve between the knots

  17. Uniform quadratic B-splines • At the 4 knots, the left and right derivatives are equal C1 continuous B-splines  C1 continuous B-spline curve 1 b2,3(t) 0.5 1 2 3 4 5 t – 0.5 b’2,3(t) (derivative) – 1

  18. Uniform quadratic B-splines • At the 4 knots, the left and right derivatives are equal C1 continuous B-splines  C1 continuous B-spline curve • Starting at t = 3, the B-splines sum up to exactly 1 b2,3(t) b3,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 t

  19. Uniform quadratic B-splines • Suppose ncontrol points • Then nB-splines and n+3 knots • The B-spline curve has n – 2 quadratic pieces, starting at the 3rd knot and ending at the n+1st knot

  20. Uniform quadratic B-splines • What is the starting point and what is the ending point of the uniform quadratic B-spline curve using control points p1, p2, …, pn ?

  21. Uniform cubic B-splines • Uniform cubic B-splines bi,4(t) have knots at i, i+1, i+2, i+3 and i+4 • They are shifted copies of each other 1 b1,4(t) 2/3 0.5 1 2 3 4 5 6 t

  22. Uniform cubic B-splines if t [i, i+1] u = t – i if t [i+1, i+2] u = t – (i+1) if t [i+2,i+3] u = t – (i+2) if t [i+3,i+4] u = t – (i+3) otherwise 1 b1,4(t) 2/3 0.5 1 2 3 4 5 6 t

  23. Uniform cubic B-splines • Every location on the B-spline curve is determined by four control points and their B-splines • Starting at the 4th knot, t = 4, the B-spline curve can be used because the B-splines sum up to 1 • A cubic B-spline is C2 continuous (the 1st and 2nd derivatives from the left and right are the same at the knots)  a cubic B-spline curve is C2 continuous

  24. Uniform cubic B-splines • At what height are the second and fourth knots of a cubic B-spline? • What weight of what control points determines the first point of a B-spline curve?

  25. Closed uniform B-spline curves • Simply repeat the first k – 1 points as the last points • quadratic: p1, p2, …, pn p1, p2, …, pn, p1, p2 • cubic: p1, p2, p3, …, pn p1, p2, …, pn, p1, p2, p3

  26. Non-uniform B-splines • Uniform B-splines have knots at 1, 2, 3, …, but generally knots can have any value  non-uniform B-splines • The knot vector t = [t1, … , tn+k] is a sequence of non-decreasing values specifying where the knots for the parameter range t occur

  27. Non-uniform B-splines • We can repeat control points and combine the consecutive B-splines into one allows more local control • We can repeat knots and adapt the B-splines affected allows more local control • When we have a sequence of control points and a useful parameter range k, ..., n+1, we may want to insert an extra point (and knot) in the middle without changing the parameter range near the ends

  28. Cox-de Boor recurrence • Cox-de Boor recurrence for defining B-splines with parameter kand knot vector t = [t1, … , tn+k]: if ti t < ti+1 otherwise The recurrence shows that parameter k B-splines are a weighted interpolation of parameter k– 1 B-splines, with weights linearly dependent on t

  29. Repeating control pointsin B-spline curves • Consider quadratic B-spline curves, in the figure: • Five B-splines of control points • 5+3 = 8 knots (t = 1, 2, ..., 8), useful in interval [3, 6] p2 p3 p4 p5 p1 b2,3(t) b3,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 t

  30. Repeating control pointsin B-spline curves • Consider quadratic B-spline curves, in the figure: • Suppose p1 is repeated • 9 knots, useful interval [3, 7] p1 p2 p3 p4 p5 p1 b’1,3(t) b2,3(t) b3,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 9 t

  31. Repeating control pointsin B-spline curves • Consider quadratic B-spline curves, in the figure: • Suppose p1 is repeated • 9 knots, useful interval [3, 7] • The B-spline curve starts at p1 at knot 3 b1,3(t) + b’1,3(t) p2 p3 p4 p5 b2,3(t) b3,3(t) b4,3(t) b5,3(t) 1 p1 0.5 1 2 3 4 5 7 6 8 9 t

  32. Repeating control pointsin B-spline curves • Consider quadratic B-spline curves, in the figure: • Suppose p3 is repeated • 9 knots, useful interval [3, 7] p3 p3 p2 p4 p5 p1 b3,3(t) b’3,3(t) b2,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 9 t

  33. Repeating control pointsin B-spline curves • Consider quadratic B-spline curves, in the figure: • Suppose p3 is repeated • 9 knots, useful interval [3, 7] • The B-spline curve passes through point p3 p3 p2 p4 p5 b3,3(t) + b’3,3(t) p1 b2,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 9 t

  34. Repeating control pointsin B-spline curves • For a B-spline curve with parameter k (degree k – 1), it will pass through the first and last control points if each occurs k – 1 times • Similarly, we can make it pass through an intermediate control point by having k – 1 copies of that control point • The level of geometric continuity, G, at an intermediate repeated control point decreases

  35. Repeating control pointsin B-spline curves • On the parameter interval [4,5], only p2 and p3 have a non-zero B-spline  the curve is a line segment! Also on parameter interval [5,6] p3 p2 p4 p5 b3,3(t) + b’3,3(t) p1 b2,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 9 t

  36. Repeating control pointsin B-spline curves p6 p5 p3 p2 p5p6p7 p4p5p6 p2p3p4 p1p2p3 p3p4p5 knot 3 p1 p7 p4 quadratic B-spline curve

  37. Repeating control pointsin B-spline curves p6 p5 p3 p2 p5p6p7 p4p5p6 p2p3p4 p1p2p3 knot 3 p1 p3p4 p4p5 p7 repeated control point p4 p4 quadratic B-spline curve with repeated control point

  38. Repeating control pointsin B-spline curves • The quadratic B-spline curve remains C1 although it is only G0 at the repeated control point p3 p2 p4 p5 b3,3(t) + b’3,3(t) p1 b2,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 9 t

  39. Repeating control pointsin B-spline curves p6 p5 p3 p2 p5p6p7 p4p5p6 p2p3p4 p1p2p3 knot 3 p1 p3p4 p4p5 p7 repeated control point p4 p4 quadratic B-spline curve with repeated control point

  40. Repeating knots in B-spline curves • Repeating a knot means that one sequence of control points no longer occurs • For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2p3p4 ) • The curve part before (based on points p1p2p3 ) directly connects to the curve part after (based on points p3p4p5) • The B-splines get different shapes to accommodate the missing part

  41. Repeating knots in B-spline curves • Repeating a knot means that one sequence of control points no longer occurs • For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2p3p4) b2,3(t) b3,3(t) b4,3(t) b5,3(t) 1 b1,3(t) 0.5 1 2 3 4 5 7 6 8 t

  42. Repeating knots in B-spline curves • Repeating a knot means that one sequence of control points no longer occurs • For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2p3p4) b3,3(t) b4,3(t) b5,3(t) b2,3(t) 1 b1,3(t) 0.5 = 1 2 3 4 5 7 6 8 t

  43. Repeating knots in B-spline curves • Repeating a knot means that one sequence of control points no longer occurs • For example, for quadratic B-spline curves, the part between knots 4 and 5 (was based on points p2p3p4) b3,3(t) b2,3(t) b4,3(t) 1 b1,3(t) b5,3(t) 0.5 1 2 3 7 6 8 5 4 t

  44. Repeating knots in B-spline curves p6 p5 p3 p2 p5p6p7 p4p5p6 p2p3p4 p1p2p3 p3p4p5 knot 3 p1 p7 p4 repeat knot 5 quadratic B-spline curve

  45. Repeating knots in B-spline curves p6 p5 p3 p2 p5p6p7 p4p5p6 p2p3p4 p1p2p3 knot 3 p1 p7 p4 knot 5 = 6 quadratic B-spline curve with repeated knot

  46. Repeating knots in B-spline curves • Repeating a control point creates line segments on either side, but repeating a knot does not have this effect on quadratic B-spline curves • Repeating a knot twice (three same knots in a sequence) creates a discontinuity in a quadratic B-spline curve • Two consecutive curve parts have no point in common • The curve part before the two disappearing parts (e.g., based on points p1p2p3 ) is disconnected from the curve part after those parts (based on points p4p5p6 )

  47. Repeating knots in B-spline curves p6 p5 p3 p2 p5p6p7 p4p5p6 p2p3p4 p1p2p3 p3p4p5 knot 3 p1 p7 p4 repeat knot 5 twice quadratic B-spline curve

  48. Repeating knots in B-spline curves p6 p5 p3 p2 p5p6p7 p2p3p4 p1p2p3 knot 3 p1 p7 p4 knot 5 = 6 = 7 quadratic B-spline curve

  49. Repeating knots in B-spline curves • In cubic B-spline curves • a double knot gives C1and G1 continuity • a triple knot gives C0and G0continuity • a quadruple knot disconnects the curve • a triple knot at the start lets p1 be the start of the curve • a triple knot at the end lets pn be the end of the curve

  50. Repeating knots in B-spline curves • For cubic B-spline curves: • a double knotgives C1continuity • a triple knotgives C0 continuity

More Related