Tamtam Proceedings - lamsin

Smoothing and compressing surfaces 357Figure 2.2.2. Decomposition of an Hermite interpolantIn this example, we describe the decomposition of the tensor product Hermite splineinterpolant f 2,2 that interpolates the values and the derivatives of the function f(x, y) =cos( x2 −y2), defined on Ω = [0, 4] × [−2, 2], at vertices (x i , y j ) = (i, j) of Ω ∩ Z 2 .According to Section 5, we have f 2,2 = f 0,0 + g0,0 1 + g0,0 2 + g0,0 3 + g1,1 1 + g1,1 2 + g1,1. 3 InFigure 3, we give the graphs of f, f 2,2 , f 0,0 and some detail functions.2.3. Compression of Hermite dataWe consider the function of the above example, i.e.,( x 2 )− yf(x, y) = cos2defined on Ω = [0, 4] × [−2, 2]. (x i , y j ) = (i, j) ∈ Ω ∩ Z 2 .For this function, the Hermite interpolant f 2,2 is completely determined by 368 coefficients.In the following table, we show the numbers of coefficients removed after thresholdingwith different values of ɛ = (ɛ 1 , ɛ 2 ).ɛ 2 \ɛ 1 0.01 0.02 0.03 0.08 0.15 0.2 0.5 0.70.01 49 52 56 72 93 95 104 1080.04 59 62 66 82 103 105 114 1180.1 61 64 68 84 105 107 116 1200.3 72 75 79 95 116 118 127 131The following graphs of f c 2,2 and er = f 2,2 − f c 2,2 are related to the thresholding valueɛ = (0.01, 0.01). f c 2,2 denoted the compressed Hermite interpolant obtained using ourmethod.TAMTAM –Tunis– 2005