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Refactored Far computation of Gregory patch basis functions:

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- fixes normalization of (s,t) hidden in Bezier weight computation
    - includes computation of true derivatives for future reference
This commit is contained in:
barfowl 2015-05-15 18:52:02 -07:00
parent 2ec507e107
commit 0d7ae5c461
1 changed files with 97 additions and 41 deletions
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138
opensubdiv/far/interpolate.cpp
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@ -302,69 +302,125 @@ void GetGregoryWeights(PatchParam::BitField bits,
// P0 e0+ e1- P1 // P0 e0+ e1- P1
// //
// Indices of boundary and interior points and their corresponding Bezier points: // Indices of boundary and interior points and their corresponding Bezier points
// (this can be reduced with more direct indexing and unrolling of loops):
// //
static int const boundaryGregory[12] = { 0, 1, 7, 5, 2, 6, 16, 12, 15, 17, 11, 10 }; static int const boundaryGregory[12] = { 0, 1, 7, 5, 2, 6, 16, 12, 15, 17, 11, 10 };
static int const boundaryBezier[12] = { 0, 1, 2, 3, 4, 7, 8, 11, 12, 13, 14, 15 }; static int const boundaryBezSCol[12] = { 0, 1, 2, 3, 0, 3, 0, 3, 0, 1, 2, 3 };
static int const boundaryBezTRow[12] = { 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3 };
static int const interiorGregory[8] = { 3, 4, 8, 9, 13, 14, 18, 19 }; static int const interiorGregory[8] = { 3, 4, 8, 9, 13, 14, 18, 19 };
static int const interiorBezier[8] = { 5, 5, 6, 6, 10, 10, 9, 9 }; static int const interiorBezSCol[8] = { 1, 1, 2, 2, 2, 2, 1, 1 };
static int const interiorBezTRow[8] = { 1, 1, 1, 1, 2, 2, 2, 2 };
// Rational multipliers of the Bezier basis functions (note we need a set of weights
// for each derivative for proper differentiation -- TBD):
// //
float sComp = 1.0f - s; // Bezier basis functions are denoted with B while the rational multipliers for the
float tComp = 1.0f - t; // interior points will be denoted G -- so we have B(s), B(t) and G(s,t):
//
// Directional Bezier basis functions B at s and t:
float Bs[4], Bds[4];
float Bt[4], Bdt[4];
bits.Normalize(s,t);
Spline<BASIS_BEZIER>::GetWeights(s, Bs, deriv1 ? Bds : 0);
Spline<BASIS_BEZIER>::GetWeights(t, Bt, deriv2 ? Bdt : 0);
// Rational multipliers G at s and t:
float sC = 1.0f - s;
float tC = 1.0f - t;
// Use <= here to avoid compiler warnings -- the sums should always be non-negative: // Use <= here to avoid compiler warnings -- the sums should always be non-negative:
float df0 = s + t; if (df0 <= 0.0f) df0 = 1.0f; float df0 = s + t; df0 = (df0 <= 0.0f) ? 1.0f : (1.0f / df0);
float df1 = sComp + t; if (df1 <= 0.0f) df1 = 1.0f; float df1 = sC + t; df1 = (df1 <= 0.0f) ? 1.0f : (1.0f / df1);
float df2 = sComp + tComp; if (df2 <= 0.0f) df2 = 1.0f; float df2 = sC + tC; df2 = (df2 <= 0.0f) ? 1.0f : (1.0f / df2);
float df3 = s + tComp; if (df3 <= 0.0f) df3 = 1.0f; float df3 = s + tC; df3 = (df3 <= 0.0f) ? 1.0f : (1.0f / df3);
float interiorPointBasis[8] = { s/df0, t/df0, float G[8] = { s*df0, t*df0, t*df1, sC*df1, sC*df2, tC*df2, tC*df3, s*df3 };
t/df1, sComp/df1,
sComp/df2, tComp/df2,
tComp/df3, s/df3 };
// Weights from a bicubic Bezier patch: // Combined weights for boundary and interior points:
//
float bezierPoint[16], bezierDeriv1[16], bezierDeriv2[16];
GetBezierWeights(bits, s, t, bezierPoint, bezierDeriv1, bezierDeriv2);
// Copy basis functions (weights) for boundary points and scale the interior basis
// functions by their rational multipliers:
//
for (int i = 0; i < 12; ++i) { for (int i = 0; i < 12; ++i) {
point[boundaryGregory[i]] = bezierPoint[boundaryBezier[i]]; point[boundaryGregory[i]] = Bs[boundaryBezSCol[i]] * Bt[boundaryBezTRow[i]];
} }
for (int i = 0; i < 8; ++i) { for (int i = 0; i < 8; ++i) {
point[interiorGregory[i]] = bezierPoint[interiorBezier[i]] * interiorPointBasis[i]; point[interiorGregory[i]] = Bs[interiorBezSCol[i]] * Bt[interiorBezTRow[i]] * G[i];
} }
//
// For derivatives, the basis functions for the interior points are rational and ideally
// require appropriate differentiation, i.e. product rule for the combination of B and G
// and the quotient rule for the rational G itself. As initially proposed by Loop et al
// though, the approximation using the 16 Bezier points arising from the G(s,t) has
// proved adequate (and is what the GPU shaders use) so we continue to use that here.
//
// An implementation of the true derivatives is provided for future reference -- it is
// unclear if the approximations will hold up under surface analysis involving higher
// order differentiation.
//
if (deriv1 and deriv2) { if (deriv1 and deriv2) {
// Copy the differentiated basis functions for Bezier to the boundary points: // Remember to include derivative scaling in all assignments below:
// float dScale = (float)(1 << bits.GetDepth());
// Combined weights for boundary points -- simple (scaled) tensor products:
for (int i = 0; i < 12; ++i) { for (int i = 0; i < 12; ++i) {
deriv1[boundaryGregory[i]] = bezierDeriv1[boundaryBezier[i]]; int iDst = boundaryGregory[i];
deriv2[boundaryGregory[i]] = bezierDeriv2[boundaryBezier[i]]; int tRow = boundaryBezTRow[i];
int sCol = boundaryBezSCol[i];
deriv1[iDst] = Bds[sCol] * Bt[tRow] * dScale;
deriv2[iDst] = Bdt[tRow] * Bs[sCol] * dScale;
} }
// XXX barry: NOTE -- THIS IS NOT CORRECT (the correction is pending)... #define _USE_BEZIER_PSEUDO_DERIVATIVES
// #ifdef _USE_BEZIER_PSEUDO_DERIVATIVES
// The basis functions for the interior points are rational and require appropriate // Approximation to the true Gregory derivatives by differentiating the Bezier patch
// differentiation wrt each parametric direction. So we will need the Bezier basis // unique to the given (s,t), i.e. having F = (g^+ * f^+) + (g^- * f^-) as its four
// functions in each of s and t rather than for (s,t) combined. The correction will // interior points:
// also need to support higher order derivatives and so will be combined with that
// extension.
//
// What follows preserves what has been done with Gregory derivatives to this point.
// //
// Combined weights for interior points -- (scaled) tensor products with G+ or G-:
for (int i = 0; i < 8; ++i) { for (int i = 0; i < 8; ++i) {
deriv1[interiorGregory[i]] = bezierDeriv1[interiorBezier[i]] * interiorPointBasis[i]; int iDst = interiorGregory[i];
deriv2[interiorGregory[i]] = bezierDeriv2[interiorBezier[i]] * interiorPointBasis[i]; int tRow = interiorBezTRow[i];
int sCol = interiorBezSCol[i];
deriv1[iDst] = Bds[sCol] * Bt[tRow] * G[i] * dScale;
deriv2[iDst] = Bdt[tRow] * Bs[sCol] * G[i] * dScale;
} }
#else
// True Gregory derivatives using appropriate differentiation of composite functions:
//
// Note that for G(s,t) = N(s,t) / D(s,t), all N' and D' are trivial constants (which
// simplifies things for higher order derivatives). And while each pair of functions
// G (i.e. the G+ and G- corresponding to points f+ and f-) must sum to 1 to ensure
// Bezier equivalence (when f+ = f-), the pairs of G' must similarly sum to 0. So we
// can potentially compute only one of the pair and negate the result for the other
// (and with 4 or 8 computations involving these constants, this is all very SIMD
// friendly...) but for now we treat all 8 independently for simplicity.
//
//float N[8] = { s, t, t, sC, sC, tC, tC, s };
float D[8] = { df0, df0, df1, df1, df2, df2, df3, df3 };
static float const Nds[8] = { 1.0f, 0.0f, 0.0f, -1.0f, -1.0f, 0.0f, 0.0f, 1.0f };
static float const Ndt[8] = { 0.0f, 1.0f, 1.0f, 0.0f, 0.0f, -1.0f, -1.0f, 0.0f };
static float const Dds[8] = { 1.0f, 1.0f, -1.0f, -1.0f, -1.0f, -1.0f, 1.0f, 1.0f };
static float const Ddt[8] = { 1.0f, 1.0f, 1.0f, 1.0f, -1.0f, -1.0f, -1.0f, -1.0f };
// Combined weights for interior points -- (scaled) combinations of B, B', G and G':
for (int i = 0; i < 8; ++i) {
int iDst = interiorGregory[i];
int tRow = interiorBezTRow[i];
int sCol = interiorBezSCol[i];
// Quotient rule for G' (re-expressed in terms of G to simplify (and D = 1/D)):
float Gds = (Nds[i] - Dds[i] * G[i]) * D[i];
float Gdt = (Ndt[i] - Ddt[i] * G[i]) * D[i];
// Product rule combining B and B' with G and G' (and scaled):
deriv1[iDst] = (Bds[sCol] * G[i] + Bs[sCol] * Gds) * Bt[tRow] * dScale;
deriv2[iDst] = (Bdt[tRow] * G[i] + Bt[tRow] * Gdt) * Bs[sCol] * dScale;
}
#endif
} }
} }