NUBS by Brian Wyvill What’s that?

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 2 Uniform B-Splines Basis functions for B-Splines. B-splines approximate a series of m+1 Control points. (P 0, P 1.. P m ) M  3 M – 2 cubic curve segments (Q 3, Q 4.. Q m ) Parameter range for Q i is t i  t  t i+1 There is a knot between Q i and Q i+1 at t i t i =knot value. (m-1 knots) t 3 =0 to t 4 =1 to t 5 =2 etc. Q m defined by points. (P m-3, P m- 2, P m-1, P m ) over parameter range t m =m-3 to t m+1 =m-2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t9t9 t 10 Q3Q3 Q4Q4 Q5Q5 Q6Q6 Q7Q7 Q8Q8 Q9Q9 P3P3 P2P2 P1P1 P0P0 P9P9 P8P8 P7P7 P6P6 P5P5 P4P4 Y X

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 3 Uniform B-Splines Geometry Vector for B-Splines. P i-3 P i-2 P i-1 P i G Bs =, 3  i  m T= [ (t - t i ) 3 (t - t i ) 2 (t - t i ) 1 1] Q i (t) = T i * M B * G Bs, t i  t  t i+1

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 4 Non-Uniform B-Splines Effect of Multiple Knots

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 5 NUBS Piecewise continuous curve approximating control points P 0 to P m Knot Value sequence is non-decreasing sequence of knot values t 0 through t m+4 ie there are 4 more knots than control points. t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 6 NUBS Q i (t) = P i-3 * B i-3,4 (t) + P i-2 * B i-2,4 (t) + P i-1 * B i-1,4 (t) + P i * B i-4 (t) 3  i  m t i  t < t i+1 B i,1 (t) = 1, t i  t < t i+1 otherwise zero. B i,2 (t) = (t – t i ) /(t i+1 – t i )B i,1 (t) + (t i+2 – t) /(t i+2 – t i+1 )B i+1,1 (t) B i,3 (t) = B i,4 (t) = B 0,1 (t) = B 1,1 (t) =B 2,1 (t) = 0 B 3,1 (t)=1 then 0 for i>3 B 0,2 (t) = something* B 0,1 (t) + something* B 1,1 (t) = 0 B 1,2 (t) = something* B 1,1 (t) + something* B 2,1 (t) = 0 B 2,2 (t) = something* B 2,1 (t) + (t 4 – t) /(t 4 – t 3 ) * B 3,1 (t) = (1-t)*1 This is 1 when t=0 and linearly decreasing until 0 at t=1

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 7

University of Calgary GraphicsJungle Project ENEL 555 B-Splines page 8 The Haar Wavelet As