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mitsuba/src/libcore/spline.cpp

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/*
This file is part of Mitsuba, a physically based rendering system.
Copyright (c) 2007-2012 by Wenzel Jakob and others.
Mitsuba is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License Version 3
as published by the Free Software Foundation.
Mitsuba is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include <mitsuba/core/spline.h>
MTS_NAMESPACE_BEGIN
Float evalCubicInterp1D(Float x, const Float *values, size_t size, Float min, Float max, bool extrapolate) {
/* Give up when given an out-of-range or NaN argument */
if (!(x >= min && x <= max) && !extrapolate)
return 0.0f;
/* Transform 'x' so that knots lie at integer positions */
Float t = ((x - min) * (size - 1)) / (max - min);
/* Find the index of the left knot in the queried subinterval, be
robust to cases where 't' lies exactly on the right endpoint */
size_t k = std::max((size_t) 0, std::min((size_t) t, size - 2));
Float f0 = values[k],
f1 = values[k+1],
d0, d1;
/* Approximate the derivatives */
if (k > 0)
d0 = 0.5f * (values[k+1] - values[k-1]);
else
d0 = values[k+1] - values[k];
if (k + 2 < size)
d1 = 0.5f * (values[k+2] - values[k]);
else
d1 = values[k+1] - values[k];
/* Compute the relative position within the interval */
t = t - (Float) k;
Float t2 = t*t, t3 = t2*t;
return
( 2*t3 - 3*t2 + 1) * f0 +
(-2*t3 + 3*t2) * f1 +
( t3 - 2*t2 + t) * d0 +
( t3 - t2) * d1;
}
Float evalCubicInterp1DN(Float x, const Float *nodes, const Float *values, size_t size, bool extrapolate) {
/* Give up when given an out-of-range or NaN argument */
if (!(x >= nodes[0] && x <= nodes[size-1]) && !extrapolate)
return 0.0f;
size_t k = (size_t) std::max((ptrdiff_t) 0, std::min((ptrdiff_t) size - 2,
std::lower_bound(nodes, nodes + size, x) - nodes - 1));
Float f0 = values[k],
f1 = values[k+1],
width = nodes[k+1] - nodes[k],
d0, d1;
/* Approximate the derivatives */
if (k > 0)
d0 = width * (f1 - values[k-1]) / (nodes[k+1] - nodes[k-1]);
else
d0 = f1 - f0;
if (k + 2 < size)
d1 = width * (values[k+2] - f0) / (nodes[k+2] - nodes[k]);
else
d1 = f1 - f0;
Float t = (x - nodes[k]) / width;
Float t2 = t*t, t3 = t2*t;
return
( 2*t3 - 3*t2 + 1) * f0 +
(-2*t3 + 3*t2) * f1 +
( t3 - 2*t2 + t) * d0 +
( t3 - t2) * d1;
}
Float integrateCubicInterp1D(size_t idx, const Float *values, size_t size, Float min, Float max) {
Float f0 = values[idx], f1 = values[idx+1], d0, d1;
/* Approximate the derivatives */
if (idx > 0)
d0 = 0.5f * (values[idx+1] - values[idx-1]);
else
d0 = values[idx+1] - values[idx];
if (idx + 2 < size)
d1 = 0.5f * (values[idx+2] - values[idx]);
else
d1 = values[idx+1] - values[idx];
return ((d0-d1) * (Float) (1.0 / 12.0) + (f0+f1) * 0.5f) * (max-min) / (size - 1);
}
Float integrateCubicInterp1DN(size_t idx, const Float *nodes, const Float *values, size_t size) {
Float f0 = values[idx],
f1 = values[idx+1],
width = nodes[idx+1] - nodes[idx],
d0, d1;
/* Approximate the derivatives */
if (idx > 0)
d0 = width * (f1 - values[idx-1]) / (nodes[idx+1] - nodes[idx-1]);
else
d0 = f1 - f0;
if (idx + 2 < size)
d1 = width * (values[idx+2] - f0) / (nodes[idx+2] - nodes[idx]);
else
d1 = f1 - f0;
return ((d0-d1) * (Float) (1.0 / 12.0) + (f0+f1) * 0.5f) * width;
}
Float sampleCubicInterp1D(size_t idx, Float *values, size_t size, Float min,
Float max, Float sample, Float *fval) {
Float f0 = values[idx], f1 = values[idx+1], d0, d1;
/* Approximate the derivatives */
if (idx > 0)
d0 = 0.5f * (values[idx+1] - values[idx-1]);
else
d0 = values[idx+1] - values[idx];
if (idx + 2 < size)
d1 = 0.5f * (values[idx+2] - values[idx]);
else
d1 = values[idx+1] - values[idx];
/* Bracketing interval and starting guess */
Float a = 0, c = 1, b;
if (f0 != f1) /* Importance sample linear interpolant */
b = (f0-math::safe_sqrt(f0*f0 + sample * (f1*f1-f0*f0))) / (f0-f1);
else
b = sample;
sample *= ((d0-d1) * (Float) (1.0 / 12.0) + (f0+f1) * 0.5f);
/* Invert CDF using Newton-Bisection */
while (true) {
if (!(b >= a && b <= c))
b = 0.5f * (a + c);
/* CDF and PDF in Horner form */
Float value = b*(f0 + b*(.5f*d0 + b*((Float) (1.0f/3.0f) * (-2*d0-d1)
+ f1 - f0 + b*(0.25f*(d0 + d1) + 0.5f * (f0 - f1))))) - sample;
Float deriv = f0 + b*(d0 + b*(-2*d0 - d1 + 3*(f1-f0) + b*(d0 + d1 + 2*(f0 - f1))));
if (std::abs(value) < 1e-6f) {
if (fval)
*fval = deriv;
return min + (idx+b) * (max-min) / (size-1);
}
if (value > 0)
c = b;
else
a = b;
b -= value / deriv;
}
}
Float sampleCubicInterp1DN(size_t idx, Float *nodes, Float *values,
size_t size, Float sample, Float *fval) {
Float f0 = values[idx],
f1 = values[idx+1],
width = nodes[idx+1] - nodes[idx],
d0, d1;
/* Approximate the derivatives */
if (idx > 0)
d0 = width * (f1 - values[idx-1]) / (nodes[idx+1] - nodes[idx-1]);
else
d0 = f1 - f0;
if (idx + 2 < size)
d1 = width * (values[idx+2] - f0) / (nodes[idx+2] - nodes[idx]);
else
d1 = f1 - f0;
/* Bracketing interval and starting guess */
Float a = 0, c = 1, b;
if (f0 != f1) /* Importance sample linear interpolant */
b = (f0-math::safe_sqrt(f0*f0 + sample * (f1*f1-f0*f0))) / (f0-f1);
else
b = sample;
sample *= ((d0-d1) * (Float) (1.0 / 12.0) + (f0+f1) * 0.5f);
/* Invert CDF using Newton-Bisection */
while (true) {
if (!(b >= a && b <= c))
b = 0.5f * (a + c);
/* CDF and PDF in Horner form */
Float value = b*(f0 + b*(.5f*d0 + b*((Float) (1.0f/3.0f) * (-2*d0-d1)
+ f1 - f0 + b*(0.25f*(d0 + d1) + 0.5f * (f0 - f1))))) - sample;
Float deriv = f0 + b*(d0 + b*(-2*d0 - d1 + 3*(f1-f0) + b*(d0 + d1 + 2*(f0 - f1))));
if (std::abs(value) < 1e-6f) {
if (fval)
*fval = deriv;
return nodes[idx] + width*b;
}
if (value > 0)
c = b;
else
a = b;
b -= value / deriv;
}
}
Float evalCubicInterp2D(const Point2 &p, const Float *values, const Size2 &size,
const Point2 &min, const Point2 &max, bool extrapolate) {
Float knotWeights[2][4];
Size2 knot;
/* Compute interpolation weights separately for each dimension */
for (int dim=0; dim<2; ++dim) {
Float *weights = knotWeights[dim];
/* Give up when given an out-of-range or NaN argument */
if (!(p[dim] >= min[dim] && p[dim] <= max[dim]) && !extrapolate)
return 0.0f;
/* Transform 'p' so that knots lie at integer positions */
Float t = ((p[dim] - min[dim]) * (size[dim] - 1))
/ (max[dim]-min[dim]);
/* Find the index of the left knot in the queried subinterval, be
robust to cases where 't' lies exactly on the right endpoint */
knot[dim] = std::min((size_t) t, size[dim] - 2);
/* Compute the relative position within the interval */
t = t - (Float) knot[dim];
/* Compute node weights */
Float t2 = t*t, t3 = t2*t;
weights[0] = 0.0f;
weights[1] = 2*t3 - 3*t2 + 1;
weights[2] = -2*t3 + 3*t2;
weights[3] = 0.0f;
/* Derivative weights */
Float d0 = t3 - 2*t2 + t,
d1 = t3 - t2;
/* Turn derivative weights into node weights using
an appropriate chosen finite differences stencil */
if (knot[dim] > 0) {
weights[2] += 0.5f * d0;
weights[0] -= 0.5f * d0;
} else {
weights[2] += d0;
weights[1] -= d0;
}
if (knot[dim] + 2 < size[dim]) {
weights[3] += 0.5f * d1;
weights[1] -= 0.5f * d1;
} else {
weights[2] += d1;
weights[1] -= d1;
}
}
Float result = 0.0f;
for (int y=-1; y<=2; ++y) {
Float wy = knotWeights[1][y+1];
for (int x=-1; x<=2; ++x) {
Float wxy = knotWeights[0][x+1] * wy;
if (wxy == 0)
continue;
size_t pos = (knot[1] + y) * size[0] + knot[0] + x;
result += values[pos] * wxy;
}
}
return result;
}
Float evalCubicInterp2DN(const Point2 &p, const Float **nodes_,
const Float *values, const Size2 &size, bool extrapolate) {
Float knotWeights[2][4];
Size2 knot;
/* Compute interpolation weights separately for each dimension */
for (int dim=0; dim<2; ++dim) {
const Float *nodes = nodes_[dim];
Float *weights = knotWeights[dim];
/* Give up when given an out-of-range or NaN argument */
if (!(p[dim] >= nodes[0] && p[dim] <= nodes[size[dim]-1]) && !extrapolate)
return 0.0f;
/* Find the index of the left knot in the queried subinterval, be
robust to cases where 't' lies exactly on the right endpoint */
size_t k = (size_t) std::max((ptrdiff_t) 0, std::min((ptrdiff_t) size[dim] - 2,
std::lower_bound(nodes, nodes + size[dim], p[dim]) - nodes - 1));
knot[dim] = k;
Float width = nodes[k+1] - nodes[k];
/* Compute the relative position within the interval */
Float t = (p[dim] - nodes[k]) / width,
t2 = t*t, t3 = t2*t;
/* Compute node weights */
weights[0] = 0.0f;
weights[1] = 2*t3 - 3*t2 + 1;
weights[2] = -2*t3 + 3*t2;
weights[3] = 0.0f;
/* Derivative weights */
Float d0 = t3 - 2*t2 + t, d1 = t3 - t2;
/* Turn derivative weights into node weights using
an appropriate chosen finite differences stencil */
if (k > 0) {
Float factor = width / (nodes[k+1]-nodes[k-1]);
weights[2] += d0 * factor;
weights[0] -= d0 * factor;
} else {
weights[2] += d0;
weights[1] -= d0;
}
if (k + 2 < size[dim]) {
Float factor = width / (nodes[k+2]-nodes[k]);
weights[3] += d1 * factor;
weights[1] -= d1 * factor;
} else {
weights[2] += d1;
weights[1] -= d1;
}
}
Float result = 0.0f;
for (int y=-1; y<=2; ++y) {
Float wy = knotWeights[1][y+1];
for (int x=-1; x<=2; ++x) {
Float wxy = knotWeights[0][x+1] * wy;
if (wxy == 0)
continue;
size_t pos = (knot[1] + y) * size[0] + knot[0] + x;
result += values[pos] * wxy;
}
}
return result;
}
Float evalCubicInterp3D(const Point3 &p, const Float *values, const Size3 &size,
const Point3 &min, const Point3 &max, bool extrapolate) {
Float knotWeights[3][4];
Size3 knot;
/* Compute interpolation weights separately for each dimension */
for (int dim=0; dim<3; ++dim) {
Float *weights = knotWeights[dim];
/* Give up when given an out-of-range or NaN argument */
if (!(p[dim] >= min[dim] && p[dim] <= max[dim]) && !extrapolate)
return 0.0f;
/* Transform 'p' so that knots lie at integer positions */
Float t = ((p[dim] - min[dim]) * (size[dim] - 1))
/ (max[dim]-min[dim]);
/* Find the index of the left knot in the queried subinterval, be
robust to cases where 't' lies exactly on the right endpoint */
knot[dim] = std::min((size_t) t, size[dim] - 2);
/* Compute the relative position within the interval */
t = t - (Float) knot[dim];
/* Compute node weights */
Float t2 = t*t, t3 = t2*t;
weights[0] = 0.0f;
weights[1] = 2*t3 - 3*t2 + 1;
weights[2] = -2*t3 + 3*t2;
weights[3] = 0.0f;
/* Derivative weights */
Float d0 = t3 - 2*t2 + t,
d1 = t3 - t2;
/* Turn derivative weights into node weights using
an appropriate chosen finite differences stencil */
if (knot[dim] > 0) {
weights[2] += 0.5f * d0;
weights[0] -= 0.5f * d0;
} else {
weights[2] += d0;
weights[1] -= d0;
}
if (knot[dim] + 2 < size[dim]) {
weights[3] += 0.5f * d1;
weights[1] -= 0.5f * d1;
} else {
weights[2] += d1;
weights[1] -= d1;
}
}
Float result = 0.0f;
for (int z=-1; z<=2; ++z) {
Float wz = knotWeights[2][z+1];
for (int y=-1; y<=2; ++y) {
Float wyz = knotWeights[1][y+1] * wz;
for (int x=-1; x<=2; ++x) {
Float wxyz = knotWeights[0][x+1] * wyz;
if (wxyz == 0)
continue;
size_t pos = ((knot[2] + z) * size[1] + (knot[1] + y))
* size[0] + knot[0] + x;
result += values[pos] * wxyz;
}
}
}
return result;
}
Float evalCubicInterp3DN(const Point3 &p, const Float **nodes_,
const Float *values, const Size3 &size, bool extrapolate) {
Float knotWeights[3][4];
Size3 knot;
/* Compute interpolation weights separately for each dimension */
for (int dim=0; dim<3; ++dim) {
const Float *nodes = nodes_[dim];
Float *weights = knotWeights[dim];
/* Give up when given an out-of-range or NaN argument */
if (!(p[dim] >= nodes[0] && p[dim] <= nodes[size[dim]-1]) && !extrapolate)
return 0.0f;
/* Find the index of the left knot in the queried subinterval, be
robust to cases where 't' lies exactly on the right endpoint */
size_t k = (size_t) std::max((ptrdiff_t) 0, std::min((ptrdiff_t) size[dim] - 2,
std::lower_bound(nodes, nodes + size[dim], p[dim]) - nodes - 1));
knot[dim] = k;
Float width = nodes[k+1] - nodes[k];
/* Compute the relative position within the interval */
Float t = (p[dim] - nodes[k]) / width,
t2 = t*t, t3 = t2*t;
/* Compute node weights */
weights[0] = 0.0f;
weights[1] = 2*t3 - 3*t2 + 1;
weights[2] = -2*t3 + 3*t2;
weights[3] = 0.0f;
/* Derivative weights */
Float d0 = t3 - 2*t2 + t, d1 = t3 - t2;
/* Turn derivative weights into node weights using
an appropriate chosen finite differences stencil */
if (k > 0) {
Float factor = width / (nodes[k+1]-nodes[k-1]);
weights[2] += d0 * factor;
weights[0] -= d0 * factor;
} else {
weights[2] += d0;
weights[1] -= d0;
}
if (k + 2 < size[dim]) {
Float factor = width / (nodes[k+2]-nodes[k]);
weights[3] += d1 * factor;
weights[1] -= d1 * factor;
} else {
weights[2] += d1;
weights[1] -= d1;
}
}
Float result = 0.0f;
for (int z=-1; z<=2; ++z) {
Float wz = knotWeights[2][z+1];
for (int y=-1; y<=2; ++y) {
Float wyz = knotWeights[1][y+1] * wz;
for (int x=-1; x<=2; ++x) {
Float wxyz = knotWeights[0][x+1] * wyz;
if (wxyz == 0)
continue;
size_t pos = ((knot[2] + z) * size[1] + (knot[1] + y))
* size[0] + knot[0] + x;
result += values[pos] * wxyz;
}
}
}
return result;
}
MTS_NAMESPACE_END
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