sheenltc.cpp

// materials/sheenltc.cpp*
#include "materials/sheenltc.h"
#include "paramset.h"
#include "reflection.h"
#include "sampling.h"

namespace pbrt {

/*
    Reference implementation for

        "Practical Multiple-Scattering Sheen Using Linearly Transformed Cosines"
        by Tizian Zeltner, Brent Burley, and Matt Jen-Yuan Chiang 2022.

    See
        "Real-Time Polygonal-Light Shading with Linearly Transformed Cosines"
        by Heitz et al. 2016

    for more details about linearly transformed cosine distributions (LTCs).

    The LTCs here were fitted to a volumetric sheen layer of fiber-like
    particles with a SGGX microflake phase function. See

        "The SGGX Microflake Distribution"
        by Heitz et al. 2015.

    The reference volume is non-absorptive (with single-scattering albedo = 1)
    and has unit density and thickness. These choices are somewhat arbitrary but
    reproduced the overall sheen look from explicit fibers that we were going
    after.

    LTC coefficients are stored in a table parameterized by
        - Elevation angle, parameterized as `cos(theta)`
        - Sheen roughness `alpha`, which is related to the SGGX fiber cross
          section `sigma` as `alpha = sqrt(sigma)`

    Two versions of the LTC fit are available:

    - type == "volume"
        Direct fit to the volumetric sheen model.

    - type == "approx"
        Fit of an analytic exponential approximation that closely resembles
        the original model, albeit with slight differences in BRDF shape and
        reflectance.
*/

class SheenLTC : public BxDF {
  public:
    SheenLTC(int version, const Spectrum &Csheen, Float alpha)
        : BxDF(BxDFType(BSDF_REFLECTION | BSDF_GLOSSY)),
          version(version),
          Csheen(Csheen),
          alpha(alpha) {}
    Spectrum f(const Vector3f &wo, const Vector3f &wi) const;
    Spectrum Sample_f(const Vector3f &wo, Vector3f *wi, const Point2f &u,
                      Float *pdf, BxDFType *sampledType) const;
    Float Pdf(const Vector3f &wo, const Vector3f &wi) const;
    std::string ToString() const;

  private:
    int version;
    Spectrum Csheen;
    Float alpha;

    Float evalLTC(const Vector3f &wi, const Vector3f &ltcCoeffs) const;
    Vector3f sampleLTC(const Vector3f &ltcCoeffs, const Point2f &u) const;
    Vector3f fetchCoeffs(const Vector3f &wo) const;

    static constexpr int ltcRes = 32;
    static const Vector3f _ltcParamTableVolume[32][32];
    static const Vector3f _ltcParamTableApprox[32][32];
};

/* Two helper functions for evaluating / sampling from LTCs that are defined
   based on a standard coordinate system aligned with `phi = 0`. */

Float phi(const Vector3f &v) {
    Float p = std::atan2(v.y, v.x);
    if (p < 0) {
        p += 2*Pi;
    }
    return p;
}

Vector3f rotateVector(const Vector3f &v, const Vector3f &axis, float angle) {
    Float s = std::sin(angle),
          c = std::cos(angle);
    return v*c + axis*Dot(v, axis)*(1.f - c) + s*Cross(axis, v);
}

// Evaluate the BRDF.
Spectrum SheenLTC::f(const Vector3f &wo, const Vector3f &wi) const {
    Float cosThetaO = CosTheta(wo),
          cosThetaI = CosTheta(wi);
    if (cosThetaO < 0 || cosThetaI < 0) return 0.f;

    // Rotate coordinate frame to align with incident direction wo.
    Float phiStd = phi(wo);
    Vector3f wiStd = rotateVector(wi, Vector3f(0, 0, 1), -phiStd);

    // Evaluate LTC distribution in aligned coordinates.
    Vector3f ltcCoeffs = fetchCoeffs(wo);
    Spectrum value = evalLTC(wiStd, ltcCoeffs);

    /* Also consider the overall reflectance `R` and the artist-specified sheen
       scale `Csheen`. */
    Float R = ltcCoeffs[2];
    value *= Csheen * R;

    /* Compared to other rendering systems (e.g. Mitsuba), pbrt-v3 accounts
       for the cosine foreshortening factor in the integrator instead of the
       BSDF. As it is included in the LTC, we need to cancel it out here. */
    return cosThetaI > 0.f ? value / cosThetaI : Spectrum(0.f);
}

/* Sample proportionally to the BRDF. The LTC representation allows
   perfect importance sampling. */
Spectrum SheenLTC::Sample_f(const Vector3f &wo, Vector3f *wi,
                            const Point2f &u, Float *pdf,
                            BxDFType *sampledType) const {
    Float cosThetaO = CosTheta(wo);
    if (cosThetaO < 0) return 0.f;

    // Sample from the LTC distribution in aligned coordinates.
    Vector3f wiStd = sampleLTC(fetchCoeffs(wo), u);

    // Rotate coordinate frame based on incident direction wo.
    Float phiStd = phi(wo);
    *wi = rotateVector(wiStd, Vector3f(0, 0, 1), +phiStd);

    if (!SameHemisphere(wo, *wi)) return Spectrum(0.f);

    // Also return the associated sampling density.
    *pdf = Pdf(wo, *wi);
    return f(wo, *wi);
}

// Sampling density associated with `Sample_f` above. Used for MIS.
Float SheenLTC::Pdf(const Vector3f &wo, const Vector3f &wi) const {
    Float cosThetaO = CosTheta(wo),
          cosThetaI = CosTheta(wi);
    if (cosThetaO < 0 || cosThetaI < 0) return 0.f;

    // Rotate coordinate frame to align with incident direction wo.
    Float phiStd = phi(wo);
    Vector3f wiStd = rotateVector(wi, Vector3f(0, 0, 1), -phiStd);

    // Evaluate LTC distribution in aligned coordinates.
    return evalLTC(wiStd, fetchCoeffs(wo));
}

// String description
std::string SheenLTC::ToString() const {
    return StringPrintf("[ SheenLTC Csheen: %s alpha: %f]", Csheen.ToString().c_str(), alpha);
}

// Evaluate the LTC distribution in its default coordinate system.
Float SheenLTC::evalLTC(const Vector3f &wi, const Vector3f &ltcCoeffs) const {
    /*
        The (inverse) transform matrix `M^{-1}` is given by:

                     [[aInv 0    bInv]
            M^{-1} =  [0    aInv 0   ]
                      [0    0    1   ]]

        with `aInv = ltcCoeffs[0]`, `bInv = ltcCoeffs[1]` fetched from the
        table. The transformed direction `wiOriginal` is therefore:

                                       [[aInv * wi.x + bInv * wi.z]
            wiOriginal = M^{-1} * wi =  [aInv * wi.y              ]
                                        [wi.z                     ]]

        which is subsequently normalized. The determinant of the matrix is

            |M^{-1}| = aInv * aInv

        which is used to compute the Jacobian determinant of the complete
        mapping including the normalization.

        See the original paper [Heitz et al. 2016] for details about the LTC
        itself.
    */

    Float aInv = ltcCoeffs[0],
          bInv = ltcCoeffs[1];
    Vector3f wiOriginal = Vector3f(aInv * wi.x + bInv * wi.z,
                                   aInv * wi.y,
                                   wi.z);
    Float length = wiOriginal.Length();
    wiOriginal /= length;

    Float det = aInv*aInv;
    Float jacobian = det / (length*length*length);

    return CosineHemispherePdf(CosTheta(wiOriginal)) * jacobian;
}

// Sample from the LTC distribution in its default coordinate system.
Vector3f SheenLTC::sampleLTC(const Vector3f &ltcCoeffs, const Point2f &u) const {
    /*  The (inverse) transform matrix `M^{-1}` is given by:

                     [[aInv 0    bInv]
            M^{-1} =  [0    aInv 0   ]
                      [0    0    1   ]]

        with `aInv = ltcCoeffs[0]`, `bInv = ltcCoeffs[1]` fetched from the
        table. The non-inverted matrix `M` is therefore:

                [[1/aInv 0      -bInv/aInv]
            M =  [0      1/aInv  0        ]
                 [0      0       1        ]]

        and the transformed direction wi is:

                                  [[wiOriginal.x/aInv - wiOriginal.z*bInv/aInv]
            wi = M * wiOriginal =  [wiOriginal.y/aInv                         ]
                                   [wiOriginal.z                              ]]

        which is subsequently normalized.

        See the original paper [Heitz et al. 2016] for details about the LTC
        itself.
    */

    Vector3f wiOriginal = CosineSampleHemisphere(u);

   Float aInv = ltcCoeffs[0],
         bInv = ltcCoeffs[1];
    Vector3f wi = Vector3f(wiOriginal.x / aInv - wiOriginal.z*bInv / aInv,
                           wiOriginal.y / aInv,
                           wiOriginal.z);
    return Normalize(wi);
}

/* Fetch the LTC coefficients by bilinearly interpolating entries in a 32x32
   lookup table. */
Vector3f SheenLTC::fetchCoeffs(const Vector3f &wo) const {
    // Compute table indices and interpolation factors.
    Float row = std::max(0.f, std::min(alpha,        OneMinusEpsilon)) * (ltcRes - 1),
          col = std::max(0.f, std::min(CosTheta(wo), OneMinusEpsilon)) * (ltcRes - 1),
          r   = std::floor(row),
          c   = std::floor(col),
          rf  = row - r,
          cf  = col - c;
    int ri = int(r),
        ci = int(c);

    // Bilinear interpolation
    if (version == 0) {
        return (_ltcParamTableVolume[ri][ci]     * (1.f - cf) + _ltcParamTableVolume[ri][ci + 1]     * cf) * (1.f - rf) +
               (_ltcParamTableVolume[ri + 1][ci] * (1.f - cf) + _ltcParamTableVolume[ri + 1][ci + 1] * cf) * rf;
    } else {
        return (_ltcParamTableApprox[ri][ci]     * (1.f - cf) + _ltcParamTableApprox[ri][ci + 1]     * cf) * (1.f - rf) +
               (_ltcParamTableApprox[ri + 1][ci] * (1.f - cf) + _ltcParamTableApprox[ri + 1][ci + 1] * cf) * rf;
    }
}


void SheenLTCMaterial::ComputeScatteringFunctions(SurfaceInteraction *si,
                                                  MemoryArena &arena,
                                                  TransportMode mode,
                                                  bool allowMultipleLobes) const {
    // Perform bump mapping with _bumpMap_, if present
    if (bumpMap) Bump(bumpMap, si);

    // Evaluate textures for _SheenLTCMaterial_ material and allocate BRDF
    si->bsdf = ARENA_ALLOC(arena, BSDF)(*si);
    Spectrum c = Csheen->Evaluate(*si);
    Float a = Clamp(alpha->Evaluate(*si), 0, 1);
    if (!c.IsBlack()) {
        int version = -1;
        if (type == "volume") {
            version = 0;
        } else if (type == "approx") {
            version = 1;
        } else {
            Error("SheenLTCMaterial: type \"%s\" unknown", type.c_str());
        }

        si->bsdf->Add(ARENA_ALLOC(arena, SheenLTC)(version, c, a));
    }
}

SheenLTCMaterial *CreateSheenLTCMaterial(const TextureParams &mp) {
    std::string type = mp.FindString("type");
    std::shared_ptr<Texture<Spectrum>> Csheen = mp.GetSpectrumTexture("Csheen", Spectrum(1.f));
    std::shared_ptr<Texture<Float>> alpha     = mp.GetFloatTexture("alpha", 0.5f);
    std::shared_ptr<Texture<Float>> bumpMap   = mp.GetFloatTextureOrNull("bumpmap");
    return new SheenLTCMaterial(type, Csheen, alpha, bumpMap);
}


// Lookup tables for LTC coefficients

const Vector3f SheenLTC::_ltcParamTableVolume[32][32] = {
    {
        Vector3f(0.01415, 0.00060, 0.00001), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000)
    },
    {
        Vector3f(0.01941, -0.00232, 0.05839), Vector3f(0.01741, -0.00581, 0.00071), Vector3f(0.04610, -0.00769, 0.00007),
        Vector3f(0.10367, -0.00740, 0.00002), Vector3f(0.06244, -0.02445, 0.00000), Vector3f(0.23927, -0.00242, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000)
    },
    {
        Vector3f(0.01927, -0.01424, 0.38834), Vector3f(0.01895, -0.00218, 0.09768), Vector3f(0.03002, -0.00194, 0.01072),
        Vector3f(0.03912, -0.00384, 0.00150), Vector3f(0.04938, -0.00668, 0.00039), Vector3f(0.05239, -0.01107, 0.00012),
        Vector3f(0.06018, -0.00746, 0.00006), Vector3f(0.06520, -0.01591, 0.00003), Vector3f(0.08253, -0.01052, 0.00002),
        Vector3f(0.21093, -0.01495, 0.00002), Vector3f(0.12785, -0.01530, 0.00001), Vector3f(0.19030, -0.01428, 0.00001),
        Vector3f(0.15254, -0.01276, 0.00000), Vector3f(0.16585, -0.02071, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000)
    },
    {
        Vector3f(0.03084, -0.04909, 0.55348), Vector3f(0.03764, -0.00710, 0.29827), Vector3f(0.03952, -0.00236, 0.11755),
        Vector3f(0.04092, -0.00201, 0.03677), Vector3f(0.04433, -0.00298, 0.00983), Vector3f(0.05014, -0.00546, 0.00288),
        Vector3f(0.05570, -0.00834, 0.00101), Vector3f(0.06215, -0.01121, 0.00046), Vector3f(0.06660, -0.01294, 0.00023),
        Vector3f(0.07902, -0.01692, 0.00014), Vector3f(0.10099, -0.01639, 0.00010), Vector3f(0.10794, -0.01738, 0.00006),
        Vector3f(0.10632, -0.02032, 0.00004), Vector3f(0.12623, -0.01947, 0.00003), Vector3f(0.13931, -0.02354, 0.00002),
        Vector3f(0.15353, -0.02910, 0.00002), Vector3f(0.16109, -0.02565, 0.00001), Vector3f(0.14583, -0.02903, 0.00001),
        Vector3f(0.27891, -0.03066, 0.00001), Vector3f(0.22622, -0.03044, 0.00001), Vector3f(0.18932, -0.04045, 0.00001),
        Vector3f(0.20219, -0.03226, 0.00000), Vector3f(0.30269, -0.03443, 0.00000), Vector3f(0.38379, -0.03023, 0.00000),
        Vector3f(0.39038, -0.03610, 0.00000), Vector3f(0.46310, -0.02022, 0.00000), Vector3f(0.44663, -0.02590, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000),
        Vector3f(0.00000, 0.00000, 0.00000), Vector3f(0.00000, 0.00000, 0.00000)
    },
    {
        Vector3f(0.04118, -0.10668, 0.63273), Vector3f(0.05152, -0.02772, 0.42999), Vector3f(0.05724, -0.00717, 0.24931),
        Vector3f(0.05863, -0.00421, 0.13040), Vector3f(0.05952, -0.00351, 0.05910), Vector3f(0.06149, -0.00399, 0.02430),
        Vector3f(0.06448, -0.00522, 0.00981), Vector3f(0.07004, -0.00676, 0.00433), Vector3f(0.07774, -0.00866, 0.00200),
        Vector3f(0.08632, -0.01099, 0.00105), Vector3f(0.09629, -0.01332, 0.00062), Vector3f(0.10592, -0.01590, 0.00039),
        Vector3f(0.10718, -0.01724, 0.00024), Vector3f(0.12207, -0.02041, 0.00018), Vector3f(0.13413, -0.02237, 0.00013),
        Vector3f(0.13702, -0.02503, 0.00009), Vector3f(0.15294, -0.02664, 0.00008), Vector3f(0.15121, -0.02803, 0.00006),
        Vector3f(0.17652, -0.03188, 0.00005), Vector3f(0.19532, -0.03147, 0.00004), Vector3f(0.20831, -0.03346, 0.00003),
        Vector3f(0.19762, -0.03476, 0.00002), Vector3f(0.24202, -0.03464, 0.00002), Vector3f(0.32995, -0.03125, 0.00002),
        Vector3f(0.30857, -0.03303, 0.00002), Vector3f(0.39596, -0.03009, 0.00002), Vector3f(0.38346, -0.03198, 0.00001),
        Vector3f(0.42503, -0.02518, 0.00001), Vector3f(0.41592, -0.03195, 0.00001), Vector3f(0.42512, -0.01668, 0.00001),
        Vector3f(0.36714, -0.02978, 0.00001), Vector3f(0.46502, -0.00394, 0.00001)
    },
    {
        Vector3f(0.05088, -0.16006, 0.67021), Vector3f(0.06485, -0.05552, 0.50797), Vector3f(0.07697, -0.01274, 0.34475),
        Vector3f(0.08334, -0.00932, 0.22264), Vector3f(0.08654, -0.00582, 0.13300), Vector3f(0.08950, -0.00507, 0.07375),
        Vector3f(0.09263, -0.00531, 0.03867), Vector3f(0.09711, -0.00616, 0.02006), Vector3f(0.10113, -0.00792, 0.01032),
        Vector3f(0.10913, -0.00949, 0.00570), Vector3f(0.11586, -0.01167, 0.00324), Vector3f(0.12425, -0.01421, 0.00197),
        Vector3f(0.12986, -0.01635, 0.00122), Vector3f(0.13530, -0.01858, 0.00080), Vector3f(0.15037, -0.02237, 0.00059),
        Vector3f(0.15165, -0.02450, 0.00040), Vector3f(0.15858, -0.02714, 0.00030), Vector3f(0.16660, -0.03145, 0.00022),
        Vector3f(0.18051, -0.03321, 0.00018), Vector3f(0.18022, -0.03295, 0.00013), Vector3f(0.19896, -0.03492, 0.00012),
        Vector3f(0.21095, -0.03365, 0.00009), Vector3f(0.21862, -0.03733, 0.00008), Vector3f(0.23861, -0.03930, 0.00007),
        Vector3f(0.25384, -0.03879, 0.00006), Vector3f(0.27394, -0.03580, 0.00005), Vector3f(0.28563, -0.04089, 0.00004),
        Vector3f(0.29160, -0.03604, 0.00003), Vector3f(0.29300, -0.03863, 0.00003), Vector3f(0.33458, -0.03575, 0.00002),
        Vector3f(0.36514, -0.02621, 0.00002), Vector3f(0.38746, 0.00124, 0.00002)
    },
    {
        Vector3f(0.06054, -0.18133, 0.68765), Vector3f(0.07710, -0.08871, 0.55387), Vector3f(0.09569, -0.02800, 0.41042),
        Vector3f(0.10917, -0.01456, 0.29613), Vector3f(0.11813, -0.01052, 0.20481), Vector3f(0.12469, -0.00904, 0.13463),
        Vector3f(0.12958, -0.00864, 0.08432), Vector3f(0.13406, -0.00898, 0.05135), Vector3f(0.13801, -0.01019, 0.03073),
        Vector3f(0.14324, -0.01110, 0.01861), Vector3f(0.14801, -0.01354, 0.01132), Vector3f(0.15359, -0.01614, 0.00706),
        Vector3f(0.15945, -0.01922, 0.00454), Vector3f(0.16688, -0.02116, 0.00304), Vector3f(0.17552, -0.02363, 0.00212),
        Vector3f(0.17956, -0.02606, 0.00145), Vector3f(0.18275, -0.02887, 0.00102), Vector3f(0.19293, -0.03270, 0.00078),
        Vector3f(0.20081, -0.03492, 0.00059), Vector3f(0.20817, -0.03607, 0.00046), Vector3f(0.21658, -0.03714, 0.00036),
        Vector3f(0.22866, -0.03827, 0.00030), Vector3f(0.23912, -0.03832, 0.00025), Vector3f(0.24736, -0.03918, 0.00020),
        Vector3f(0.26573, -0.04071, 0.00017), Vector3f(0.26821, -0.04382, 0.00013), Vector3f(0.28767, -0.03860, 0.00012),
        Vector3f(0.30592, -0.03690, 0.00010), Vector3f(0.31228, -0.04032, 0.00008), Vector3f(0.35297, -0.03136, 0.00008),
        Vector3f(0.35570, -0.02365, 0.00006), Vector3f(0.37077, -0.00281, 0.00006)
    },
    {
        Vector3f(0.07075, -0.18042, 0.69478), Vector3f(0.08974, -0.10806, 0.58263), Vector3f(0.11306, -0.04959, 0.45689),
        Vector3f(0.13241, -0.02418, 0.35318), Vector3f(0.14704, -0.01632, 0.26617), Vector3f(0.15763, -0.01333, 0.19342),
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        Vector3f(0.86039, -0.01119, 0.37756), Vector3f(0.86098, -0.01040, 0.36805), Vector3f(0.86192, -0.00953, 0.35846),
        Vector3f(0.86278, -0.00857, 0.34911), Vector3f(0.86425, -0.00751, 0.33962), Vector3f(0.86491, -0.00614, 0.33144),
        Vector3f(0.86581, -0.00433, 0.32300), Vector3f(0.86677, 0.00002, 0.31508)
    },
    {
        Vector3f(0.72363, 0.00000, 0.74973), Vector3f(0.74617, -0.00002, 0.73527), Vector3f(0.76908, 0.00000, 0.71668),
        Vector3f(0.78618, -0.00002, 0.69953), Vector3f(0.79981, -0.00003, 0.68291), Vector3f(0.81110, -0.00001, 0.66623),
        Vector3f(0.82060, -0.00001, 0.64940), Vector3f(0.82853, -0.00004, 0.63212), Vector3f(0.83504, -0.00002, 0.61545),
        Vector3f(0.84058, -0.00003, 0.59849), Vector3f(0.84521, -0.00005, 0.58226), Vector3f(0.84932, -0.00002, 0.56605),
        Vector3f(0.85275, -0.00000, 0.55034), Vector3f(0.85605, -0.00001, 0.53461), Vector3f(0.85877, -0.00001, 0.51974),
        Vector3f(0.86129, -0.00001, 0.50561), Vector3f(0.86320, -0.00005, 0.49191), Vector3f(0.86521, -0.00004, 0.47875),
        Vector3f(0.86654, -0.00002, 0.46632), Vector3f(0.86846, -0.00002, 0.45401), Vector3f(0.86997, -0.00001, 0.44209),
        Vector3f(0.87153, 0.00003, 0.43044), Vector3f(0.87288, 0.00002, 0.42005), Vector3f(0.87352, 0.00003, 0.40990),
        Vector3f(0.87465, 0.00004, 0.39997), Vector3f(0.87549, 0.00003, 0.39069), Vector3f(0.87626, 0.00006, 0.38171),
        Vector3f(0.87676, 0.00004, 0.37342), Vector3f(0.87714, 0.00006, 0.36523), Vector3f(0.87859, 0.00006, 0.35675),
        Vector3f(0.87952, 0.00009, 0.34897), Vector3f(0.87958, 0.00003, 0.34187)
    }
};

const Vector3f SheenLTC::_ltcParamTableApprox[32][32] = {
    {
        Vector3f(0.10027, -0.00000, 0.33971), Vector3f(0.10760, -0.00000, 0.35542), Vector3f(0.11991, 0.00001, 0.30888),
        Vector3f(0.13148, 0.00001, 0.23195), Vector3f(0.14227, 0.00001, 0.15949), Vector3f(0.15231, -0.00000, 0.10356),
        Vector3f(0.16168, -0.00000, 0.06466), Vector3f(0.17044, 0.00000, 0.03925), Vector3f(0.17867, 0.00001, 0.02334),
        Vector3f(0.18645, 0.00000, 0.01366), Vector3f(0.19382, -0.00000, 0.00790), Vector3f(0.20084, -0.00001, 0.00452),
        Vector3f(0.20754, 0.00001, 0.00257), Vector3f(0.21395, 0.00000, 0.00145), Vector3f(0.22011, 0.00000, 0.00081),
        Vector3f(0.22603, -0.00000, 0.00045), Vector3f(0.23174, -0.00001, 0.00025), Vector3f(0.23726, 0.00000, 0.00014),
        Vector3f(0.24259, -0.00001, 0.00008), Vector3f(0.24777, -0.00001, 0.00004), Vector3f(0.25279, -0.00001, 0.00002),
        Vector3f(0.25768, 0.00001, 0.00001), Vector3f(0.26243, 0.00001, 0.00001), Vector3f(0.26707, -0.00000, 0.00000),
        Vector3f(0.27159, -0.00000, 0.00000), Vector3f(0.27601, -0.00000, 0.00000), Vector3f(0.28033, 0.00000, 0.00000),
        Vector3f(0.28456, -0.00000, 0.00000), Vector3f(0.28870, -0.00001, 0.00000), Vector3f(0.29276, -0.00000, 0.00000),
        Vector3f(0.29676, 0.00000, 0.00000), Vector3f(0.30067, -0.00001, 0.00000)
    },
    {
        Vector3f(0.10068, -0.00013, 0.33844), Vector3f(0.10802, -0.00001, 0.35438), Vector3f(0.12031, -0.00000, 0.30859),
        Vector3f(0.13190, 0.00001, 0.23230), Vector3f(0.14269, 0.00000, 0.16016), Vector3f(0.15274, 0.00001, 0.10429),
        Vector3f(0.16211, -0.00000, 0.06529), Vector3f(0.17088, -0.00000, 0.03975), Vector3f(0.17912, -0.00000, 0.02370),
        Vector3f(0.18691, -0.00001, 0.01391), Vector3f(0.19429, -0.00000, 0.00807), Vector3f(0.20132, -0.00000, 0.00463),
        Vector3f(0.20803, 0.00001, 0.00264), Vector3f(0.21445, 0.00000, 0.00149), Vector3f(0.22061, 0.00000, 0.00084),
        Vector3f(0.22655, -0.00000, 0.00047), Vector3f(0.23226, -0.00000, 0.00026), Vector3f(0.23779, -0.00001, 0.00015),
        Vector3f(0.24314, -0.00002, 0.00008), Vector3f(0.24832, -0.00001, 0.00004), Vector3f(0.25336, -0.00002, 0.00002),
        Vector3f(0.25824, -0.00003, 0.00001), Vector3f(0.26301, -0.00005, 0.00001), Vector3f(0.26766, -0.00008, 0.00000),
        Vector3f(0.27220, -0.00012, 0.00000), Vector3f(0.27665, -0.00019, 0.00000), Vector3f(0.28101, -0.00026, 0.00000),
        Vector3f(0.28532, -0.00041, 0.00000), Vector3f(0.28960, -0.00058, 0.00000), Vector3f(0.29389, -0.00080, 0.00000),
        Vector3f(0.29830, -0.00096, 0.00000), Vector3f(0.30309, 0.00000, 0.00000)
    },
    {
        Vector3f(0.09988, -0.00743, 0.33595), Vector3f(0.10928, -0.00078, 0.35135), Vector3f(0.12169, -0.00015, 0.30790),
        Vector3f(0.13330, -0.00006, 0.23367), Vector3f(0.14412, -0.00003, 0.16253), Vector3f(0.15420, -0.00004, 0.10680),
        Vector3f(0.16360, -0.00002, 0.06750), Vector3f(0.17240, -0.00004, 0.04148), Vector3f(0.18068, -0.00003, 0.02497),
        Vector3f(0.18850, -0.00004, 0.01480), Vector3f(0.19591, -0.00005, 0.00867), Vector3f(0.20297, -0.00005, 0.00503),
        Vector3f(0.20970, -0.00008, 0.00289), Vector3f(0.21616, -0.00012, 0.00165), Vector3f(0.22235, -0.00015, 0.00094),
        Vector3f(0.22831, -0.00020, 0.00053), Vector3f(0.23407, -0.00028, 0.00030), Vector3f(0.23963, -0.00039, 0.00017),
        Vector3f(0.24503, -0.00056, 0.00009), Vector3f(0.25028, -0.00084, 0.00005), Vector3f(0.25541, -0.00124, 0.00003),
        Vector3f(0.26045, -0.00184, 0.00002), Vector3f(0.26545, -0.00274, 0.00001), Vector3f(0.27049, -0.00410, 0.00001),
        Vector3f(0.27569, -0.00612, 0.00000), Vector3f(0.28127, -0.00908, 0.00000), Vector3f(0.28762, -0.01335, 0.00000),
        Vector3f(0.29542, -0.01917, 0.00000), Vector3f(0.30593, -0.02649, 0.00000), Vector3f(0.32153, -0.03397, 0.00000),
        Vector3f(0.34697, -0.03669, 0.00000), Vector3f(0.39704, -0.00000, 0.00000)
    },
    {
        Vector3f(0.08375, -0.07516, 0.33643), Vector3f(0.10999, -0.00776, 0.34873), Vector3f(0.12392, -0.00160, 0.30771),
        Vector3f(0.13571, -0.00070, 0.23660), Vector3f(0.14661, -0.00044, 0.16701), Vector3f(0.15676, -0.00034, 0.11147),
        Vector3f(0.16621, -0.00032, 0.07157), Vector3f(0.17508, -0.00029, 0.04470), Vector3f(0.18341, -0.00033, 0.02736),
        Vector3f(0.19128, -0.00038, 0.01648), Vector3f(0.19876, -0.00045, 0.00981), Vector3f(0.20587, -0.00057, 0.00579),
        Vector3f(0.21267, -0.00074, 0.00339), Vector3f(0.21918, -0.00096, 0.00197), Vector3f(0.22544, -0.00129, 0.00114),
        Vector3f(0.23150, -0.00179, 0.00066), Vector3f(0.23736, -0.00249, 0.00038), Vector3f(0.24308, -0.00352, 0.00022),
        Vector3f(0.24871, -0.00501, 0.00012), Vector3f(0.25433, -0.00721, 0.00007), Vector3f(0.26003, -0.01042, 0.00004),
        Vector3f(0.26604, -0.01512, 0.00002), Vector3f(0.27264, -0.02192, 0.00001), Vector3f(0.28039, -0.03158, 0.00001),
        Vector3f(0.29024, -0.04491, 0.00000), Vector3f(0.30382, -0.06232, 0.00000), Vector3f(0.32395, -0.08307, 0.00000),
        Vector3f(0.35533, -0.10404, 0.00000), Vector3f(0.40491, -0.11883, 0.00000), Vector3f(0.47816, -0.11902, 0.00000),
        Vector3f(0.56774, -0.09644, 0.00000), Vector3f(0.66332, -0.00000, 0.00000)
    },
    {
        Vector3f(0.05655, -0.31167, 0.32420), Vector3f(0.10687, -0.03376, 0.35090), Vector3f(0.12655, -0.00804, 0.31009),
        Vector3f(0.13914, -0.00363, 0.24233), Vector3f(0.15029, -0.00227, 0.17455), Vector3f(0.16057, -0.00177, 0.11907),
        Vector3f(0.17014, -0.00156, 0.07820), Vector3f(0.17910, -0.00153, 0.04999), Vector3f(0.18752, -0.00162, 0.03132),
        Vector3f(0.19549, -0.00182, 0.01932), Vector3f(0.20304, -0.00213, 0.01178), Vector3f(0.21025, -0.00261, 0.00712),
        Vector3f(0.21716, -0.00326, 0.00427), Vector3f(0.22380, -0.00420, 0.00255), Vector3f(0.23022, -0.00554, 0.00151),
        Vector3f(0.23648, -0.00740, 0.00089), Vector3f(0.24264, -0.01005, 0.00053), Vector3f(0.24879, -0.01381, 0.00031),
        Vector3f(0.25510, -0.01911, 0.00018), Vector3f(0.26177, -0.02658, 0.00011), Vector3f(0.26918, -0.03698, 0.00006),
        Vector3f(0.27795, -0.05122, 0.00004), Vector3f(0.28910, -0.07004, 0.00002), Vector3f(0.30427, -0.09359, 0.00001),
        Vector3f(0.32606, -0.12066, 0.00001), Vector3f(0.35822, -0.14764, 0.00001), Vector3f(0.40512, -0.16863, 0.00000),
        Vector3f(0.46849, -0.17766, 0.00000), Vector3f(0.54169, -0.17178, 0.00000), Vector3f(0.61239, -0.15052, 0.00000),
        Vector3f(0.67350, -0.11117, 0.00000), Vector3f(0.73152, 0.00000, 0.00000)
    },
    {
        Vector3f(0.05336, -0.34864, 0.38172), Vector3f(0.09920, -0.08509, 0.36009), Vector3f(0.12900, -0.02477, 0.31816),
        Vector3f(0.14348, -0.01195, 0.25287), Vector3f(0.15525, -0.00763, 0.18668), Vector3f(0.16584, -0.00587, 0.13092),
        Vector3f(0.17560, -0.00512, 0.08853), Vector3f(0.18472, -0.00492, 0.05832), Vector3f(0.19329, -0.00508, 0.03768),
        Vector3f(0.20140, -0.00555, 0.02399), Vector3f(0.20910, -0.00634, 0.01510), Vector3f(0.21647, -0.00749, 0.00942),
        Vector3f(0.22355, -0.00914, 0.00584), Vector3f(0.23039, -0.01140, 0.00360), Vector3f(0.23709, -0.01450, 0.00221),
        Vector3f(0.24371, -0.01877, 0.00135), Vector3f(0.25039, -0.02458, 0.00083), Vector3f(0.25730, -0.03248, 0.00051),
        Vector3f(0.26474, -0.04313, 0.00031), Vector3f(0.27313, -0.05721, 0.00019), Vector3f(0.28319, -0.07550, 0.00012),
        Vector3f(0.29598, -0.09825, 0.00008), Vector3f(0.31310, -0.12490, 0.00005), Vector3f(0.33683, -0.15332, 0.00003),
        Vector3f(0.36995, -0.17964, 0.00002), Vector3f(0.41497, -0.19882, 0.00002), Vector3f(0.47174, -0.20686, 0.00001),
        Vector3f(0.53490, -0.20242, 0.00001), Vector3f(0.59635, -0.18603, 0.00001), Vector3f(0.65092, -0.15783, 0.00001),
        Vector3f(0.69798, -0.11426, 0.00001), Vector3f(0.74494, 0.00000, 0.00001)
    },
    {
        Vector3f(0.05749, -0.31793, 0.44455), Vector3f(0.09398, -0.14133, 0.37908), Vector3f(0.13152, -0.05344, 0.33487),
        Vector3f(0.14884, -0.02831, 0.27078), Vector3f(0.16170, -0.01861, 0.20554), Vector3f(0.17282, -0.01431, 0.14892),
        Vector3f(0.18293, -0.01233, 0.10431), Vector3f(0.19231, -0.01159, 0.07128), Vector3f(0.20110, -0.01165, 0.04782),
        Vector3f(0.20942, -0.01233, 0.03163), Vector3f(0.21733, -0.01364, 0.02070), Vector3f(0.22491, -0.01559, 0.01344),
        Vector3f(0.23224, -0.01830, 0.00867), Vector3f(0.23938, -0.02200, 0.00557), Vector3f(0.24643, -0.02692, 0.00357),
        Vector3f(0.25351, -0.03342, 0.00228), Vector3f(0.26079, -0.04192, 0.00146), Vector3f(0.26852, -0.05298, 0.00094),
        Vector3f(0.27707, -0.06708, 0.00060), Vector3f(0.28698, -0.08468, 0.00039), Vector3f(0.29907, -0.10591, 0.00026),
        Vector3f(0.31446, -0.13028, 0.00017), Vector3f(0.33466, -0.15629, 0.00012), Vector3f(0.36155, -0.18122, 0.00008),
        Vector3f(0.39700, -0.20145, 0.00006), Vector3f(0.44194, -0.21352, 0.00005), Vector3f(0.49484, -0.21535, 0.00004),
        Vector3f(0.55114, -0.20661, 0.00003), Vector3f(0.60550, -0.18774, 0.00003), Vector3f(0.65477, -0.15830, 0.00002),
        Vector3f(0.69863, -0.11427, 0.00002), Vector3f(0.74332, 0.00000, 0.00002)
    },
    {
        Vector3f(0.06502, -0.28106, 0.49493), Vector3f(0.09745, -0.17506, 0.41308), Vector3f(0.13592, -0.08778, 0.36191),
        Vector3f(0.15585, -0.05167, 0.29839), Vector3f(0.17008, -0.03538, 0.23353), Vector3f(0.18198, -0.02741, 0.17549),
        Vector3f(0.19260, -0.02338, 0.12792), Vector3f(0.20235, -0.02153, 0.09115), Vector3f(0.21146, -0.02104, 0.06384),
        Vector3f(0.22006, -0.02158, 0.04414), Vector3f(0.22825, -0.02299, 0.03022), Vector3f(0.23612, -0.02528, 0.02053),
        Vector3f(0.24374, -0.02852, 0.01387), Vector3f(0.25120, -0.03287, 0.00933), Vector3f(0.25860, -0.03853, 0.00626),
        Vector3f(0.26608, -0.04579, 0.00420), Vector3f(0.27381, -0.05493, 0.00282), Vector3f(0.28203, -0.06627, 0.00189),
        Vector3f(0.29109, -0.08016, 0.00128), Vector3f(0.30146, -0.09668, 0.00087), Vector3f(0.31381, -0.11579, 0.00060),
        Vector3f(0.32900, -0.13678, 0.00042), Vector3f(0.34816, -0.15845, 0.00030), Vector3f(0.37257, -0.17874, 0.00022),
        Vector3f(0.40358, -0.19512, 0.00016), Vector3f(0.44202, -0.20502, 0.00013), Vector3f(0.48743, -0.20660, 0.00010),
        Vector3f(0.53748, -0.19897, 0.00008), Vector3f(0.58855, -0.18183, 0.00007), Vector3f(0.63751, -0.15424, 0.00006),
        Vector3f(0.68300, -0.11197, 0.00005), Vector3f(0.72978, 0.00001, 0.00005)
    },
    {
        Vector3f(0.07528, -0.24711, 0.53455), Vector3f(0.10863, -0.18975, 0.45764), Vector3f(0.14433, -0.11815, 0.39949),
        Vector3f(0.16570, -0.07688, 0.33687), Vector3f(0.18121, -0.05511, 0.27245), Vector3f(0.19398, -0.04327, 0.21288),
        Vector3f(0.20521, -0.03673, 0.16193), Vector3f(0.21545, -0.03319, 0.12068), Vector3f(0.22496, -0.03159, 0.08855),
        Vector3f(0.23392, -0.03136, 0.06420), Vector3f(0.24244, -0.03223, 0.04612), Vector3f(0.25063, -0.03408, 0.03290),
        Vector3f(0.25857, -0.03690, 0.02335), Vector3f(0.26634, -0.04074, 0.01650), Vector3f(0.27403, -0.04570, 0.01163),
        Vector3f(0.28178, -0.05195, 0.00819), Vector3f(0.28972, -0.05961, 0.00577), Vector3f(0.29802, -0.06886, 0.00406),
        Vector3f(0.30695, -0.07982, 0.00287), Vector3f(0.31682, -0.09255, 0.00204), Vector3f(0.32809, -0.10692, 0.00146),
        Vector3f(0.34129, -0.12259, 0.00105), Vector3f(0.35714, -0.13882, 0.00077), Vector3f(0.37647, -0.15448, 0.00057),
        Vector3f(0.40023, -0.16805, 0.00043), Vector3f(0.42931, -0.17770, 0.00033), Vector3f(0.46430, -0.18158, 0.00026),
        Vector3f(0.50501, -0.17811, 0.00021), Vector3f(0.55015, -0.16592, 0.00018), Vector3f(0.59760, -0.14331, 0.00015),
        Vector3f(0.64549, -0.10572, 0.00013), Vector3f(0.69674, 0.00000, 0.00012)
    },
    {
        Vector3f(0.08968, -0.22166, 0.57013), Vector3f(0.12485, -0.19606, 0.50498), Vector3f(0.15755, -0.13952, 0.44492),
        Vector3f(0.17936, -0.09867, 0.38452), Vector3f(0.19578, -0.07393, 0.32154), Vector3f(0.20935, -0.05908, 0.26116),
        Vector3f(0.22123, -0.05009, 0.20725), Vector3f(0.23199, -0.04467, 0.16152), Vector3f(0.24197, -0.04157, 0.12414),
        Vector3f(0.25134, -0.04011, 0.09438), Vector3f(0.26024, -0.03986, 0.07115), Vector3f(0.26879, -0.04064, 0.05329),
        Vector3f(0.27706, -0.04233, 0.03970), Vector3f(0.28514, -0.04488, 0.02947), Vector3f(0.29311, -0.04830, 0.02181),
        Vector3f(0.30105, -0.05263, 0.01611), Vector3f(0.30908, -0.05790, 0.01189), Vector3f(0.31731, -0.06417, 0.00877),
        Vector3f(0.32591, -0.07150, 0.00648), Vector3f(0.33507, -0.07986, 0.00480), Vector3f(0.34502, -0.08921, 0.00356),
        Vector3f(0.35610, -0.09937, 0.00266), Vector3f(0.36867, -0.11001, 0.00200), Vector3f(0.38323, -0.12062, 0.00151),
        Vector3f(0.40031, -0.13041, 0.00116), Vector3f(0.42056, -0.13839, 0.00090), Vector3f(0.44465, -0.14322, 0.00070),
        Vector3f(0.47318, -0.14336, 0.00056), Vector3f(0.50652, -0.13700, 0.00046), Vector3f(0.54470, -0.12166, 0.00038),
        Vector3f(0.58750, -0.09227, 0.00032), Vector3f(0.63790, 0.00000, 0.00028)
    },
    {
        Vector3f(0.10928, -0.20982, 0.60377), Vector3f(0.14371, -0.19861, 0.54677), Vector3f(0.17437, -0.15359, 0.48934),
        Vector3f(0.19613, -0.11610, 0.43261), Vector3f(0.21313, -0.09068, 0.37256), Vector3f(0.22735, -0.07402, 0.31310),
        Vector3f(0.23983, -0.06310, 0.25792), Vector3f(0.25113, -0.05592, 0.20914), Vector3f(0.26159, -0.05128, 0.16750),
        Vector3f(0.27141, -0.04843, 0.13286), Vector3f(0.28075, -0.04692, 0.10458), Vector3f(0.28970, -0.04644, 0.08182),
        Vector3f(0.29835, -0.04682, 0.06371), Vector3f(0.30679, -0.04799, 0.04942), Vector3f(0.31508, -0.04984, 0.03822),
        Vector3f(0.32329, -0.05234, 0.02949), Vector3f(0.33149, -0.05550, 0.02272), Vector3f(0.33979, -0.05931, 0.01749),
        Vector3f(0.34827, -0.06373, 0.01346), Vector3f(0.35704, -0.06873, 0.01036), Vector3f(0.36626, -0.07428, 0.00799),
        Vector3f(0.37608, -0.08028, 0.00617), Vector3f(0.38672, -0.08655, 0.00478), Vector3f(0.39843, -0.09283, 0.00371),
        Vector3f(0.41148, -0.09873, 0.00290), Vector3f(0.42626, -0.10371, 0.00228), Vector3f(0.44316, -0.10703, 0.00181),
        Vector3f(0.46265, -0.10766, 0.00145), Vector3f(0.48525, -0.10415, 0.00117), Vector3f(0.51155, -0.09426, 0.00096),
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        Vector3f(0.72704, -0.06317, 0.17397), Vector3f(0.73357, -0.05880, 0.16298), Vector3f(0.73999, -0.05404, 0.15266),
        Vector3f(0.74629, -0.04872, 0.14298), Vector3f(0.75247, -0.04259, 0.13392), Vector3f(0.75857, -0.03514, 0.12544),
        Vector3f(0.76459, -0.02515, 0.11752), Vector3f(0.77054, -0.00000, 0.11012)
    },
    {
        Vector3f(0.49546, -0.13868, 0.67921), Vector3f(0.52189, -0.13841, 0.66231), Vector3f(0.54795, -0.13540, 0.63977),
        Vector3f(0.56715, -0.13163, 0.61642), Vector3f(0.58273, -0.12780, 0.59185), Vector3f(0.59613, -0.12410, 0.56640),
        Vector3f(0.60809, -0.12056, 0.54055), Vector3f(0.61905, -0.11715, 0.51472), Vector3f(0.62925, -0.11379, 0.48927),
        Vector3f(0.63886, -0.11049, 0.46444), Vector3f(0.64800, -0.10724, 0.44041), Vector3f(0.65676, -0.10400, 0.41727),
        Vector3f(0.66517, -0.10079, 0.39507), Vector3f(0.67330, -0.09757, 0.37385), Vector3f(0.68117, -0.09438, 0.35358),
        Vector3f(0.68879, -0.09119, 0.33427), Vector3f(0.69620, -0.08798, 0.31589), Vector3f(0.70340, -0.08477, 0.29841),
        Vector3f(0.71041, -0.08153, 0.28180), Vector3f(0.71724, -0.07825, 0.26604), Vector3f(0.72389, -0.07490, 0.25108),
        Vector3f(0.73037, -0.07146, 0.23691), Vector3f(0.73669, -0.06790, 0.22349), Vector3f(0.74288, -0.06418, 0.21078),
        Vector3f(0.74892, -0.06025, 0.19877), Vector3f(0.75481, -0.05603, 0.18742), Vector3f(0.76059, -0.05144, 0.17669),
        Vector3f(0.76625, -0.04631, 0.16657), Vector3f(0.77180, -0.04042, 0.15703), Vector3f(0.77725, -0.03329, 0.14803),
        Vector3f(0.78262, -0.02378, 0.13957), Vector3f(0.78790, -0.00000, 0.13161)
    },
    {
        Vector3f(0.53465, -0.12522, 0.68615), Vector3f(0.55925, -0.12477, 0.67008), Vector3f(0.58386, -0.12219, 0.64863),
        Vector3f(0.60211, -0.11908, 0.62649), Vector3f(0.61691, -0.11598, 0.60327), Vector3f(0.62960, -0.11298, 0.57927),
        Vector3f(0.64087, -0.11008, 0.55488), Vector3f(0.65112, -0.10725, 0.53052), Vector3f(0.66062, -0.10446, 0.50646),
        Vector3f(0.66953, -0.10170, 0.48294), Vector3f(0.67795, -0.09893, 0.46011), Vector3f(0.68598, -0.09617, 0.43805),
        Vector3f(0.69369, -0.09339, 0.41682), Vector3f(0.70109, -0.09059, 0.39644), Vector3f(0.70824, -0.08778, 0.37690),
        Vector3f(0.71515, -0.08494, 0.35821), Vector3f(0.72185, -0.08209, 0.34034), Vector3f(0.72836, -0.07918, 0.32326),
        Vector3f(0.73467, -0.07624, 0.30696), Vector3f(0.74082, -0.07324, 0.29141), Vector3f(0.74679, -0.07016, 0.27658),
        Vector3f(0.75261, -0.06698, 0.26246), Vector3f(0.75828, -0.06367, 0.24901), Vector3f(0.76381, -0.06018, 0.23621),
        Vector3f(0.76920, -0.05650, 0.22404), Vector3f(0.77446, -0.05253, 0.21247), Vector3f(0.77961, -0.04820, 0.20148),
        Vector3f(0.78464, -0.04336, 0.19105), Vector3f(0.78957, -0.03781, 0.18115), Vector3f(0.79440, -0.03111, 0.17176),
        Vector3f(0.79913, -0.02219, 0.16287), Vector3f(0.80378, -0.00000, 0.15446)
    },
    {
        Vector3f(0.57256, -0.11055, 0.69171), Vector3f(0.59531, -0.11002, 0.67633), Vector3f(0.61841, -0.10792, 0.65578),
        Vector3f(0.63563, -0.10544, 0.63470), Vector3f(0.64962, -0.10298, 0.61271), Vector3f(0.66156, -0.10060, 0.59006),
        Vector3f(0.67214, -0.09828, 0.56710), Vector3f(0.68170, -0.09602, 0.54415), Vector3f(0.69050, -0.09375, 0.52148),
        Vector3f(0.69871, -0.09149, 0.49928), Vector3f(0.70643, -0.08920, 0.47768), Vector3f(0.71376, -0.08689, 0.45676),
        Vector3f(0.72074, -0.08455, 0.43658), Vector3f(0.72745, -0.08218, 0.41714), Vector3f(0.73389, -0.07976, 0.39844),
        Vector3f(0.74011, -0.07732, 0.38048), Vector3f(0.74611, -0.07484, 0.36325), Vector3f(0.75193, -0.07230, 0.34672),
        Vector3f(0.75756, -0.06969, 0.33088), Vector3f(0.76304, -0.06702, 0.31571), Vector3f(0.76835, -0.06427, 0.30118),
        Vector3f(0.77352, -0.06140, 0.28727), Vector3f(0.77854, -0.05841, 0.27397), Vector3f(0.78343, -0.05525, 0.26124),
        Vector3f(0.78818, -0.05188, 0.24909), Vector3f(0.79283, -0.04824, 0.23747), Vector3f(0.79736, -0.04426, 0.22638),
        Vector3f(0.80178, -0.03982, 0.21580), Vector3f(0.80609, -0.03471, 0.20571), Vector3f(0.81032, -0.02854, 0.19608),
        Vector3f(0.81445, -0.02034, 0.18691), Vector3f(0.81849, 0.00001, 0.17818)
    },
    {
        Vector3f(0.60872, -0.09464, 0.69662), Vector3f(0.62966, -0.09415, 0.68177), Vector3f(0.65120, -0.09250, 0.66197),
        Vector3f(0.66737, -0.09060, 0.64182), Vector3f(0.68051, -0.08873, 0.62094), Vector3f(0.69172, -0.08690, 0.59955),
        Vector3f(0.70159, -0.08513, 0.57791), Vector3f(0.71047, -0.08335, 0.55632), Vector3f(0.71860, -0.08158, 0.53498),
        Vector3f(0.72612, -0.07978, 0.51406), Vector3f(0.73318, -0.07795, 0.49369), Vector3f(0.73983, -0.07608, 0.47393),
        Vector3f(0.74614, -0.07417, 0.45480), Vector3f(0.75217, -0.07223, 0.43634), Vector3f(0.75795, -0.07023, 0.41855),
        Vector3f(0.76350, -0.06820, 0.40140), Vector3f(0.76885, -0.06611, 0.38489), Vector3f(0.77401, -0.06395, 0.36901),
        Vector3f(0.77900, -0.06175, 0.35374), Vector3f(0.78383, -0.05946, 0.33905), Vector3f(0.78851, -0.05707, 0.32494),
        Vector3f(0.79306, -0.05459, 0.31137), Vector3f(0.79745, -0.05197, 0.29835), Vector3f(0.80173, -0.04920, 0.28584),
        Vector3f(0.80590, -0.04623, 0.27384), Vector3f(0.80993, -0.04302, 0.26232), Vector3f(0.81388, -0.03949, 0.25127),
        Vector3f(0.81771, -0.03554, 0.24068), Vector3f(0.82144, -0.03097, 0.23053), Vector3f(0.82508, -0.02547, 0.22081),
        Vector3f(0.82863, -0.01815, 0.21150), Vector3f(0.83210, 0.00000, 0.20258)
    },
    {
        Vector3f(0.64301, -0.07753, 0.70123), Vector3f(0.66223, -0.07712, 0.68683), Vector3f(0.68223, -0.07593, 0.66763),
        Vector3f(0.69736, -0.07455, 0.64827), Vector3f(0.70966, -0.07320, 0.62840), Vector3f(0.72015, -0.07187, 0.60815),
        Vector3f(0.72934, -0.07056, 0.58775), Vector3f(0.73757, -0.06924, 0.56743), Vector3f(0.74506, -0.06791, 0.54737),
        Vector3f(0.75196, -0.06655, 0.52771), Vector3f(0.75838, -0.06515, 0.50855), Vector3f(0.76441, -0.06371, 0.48992),
        Vector3f(0.77009, -0.06222, 0.47188), Vector3f(0.77550, -0.06069, 0.45443), Vector3f(0.78065, -0.05912, 0.43757),
        Vector3f(0.78559, -0.05749, 0.42129), Vector3f(0.79032, -0.05583, 0.40557), Vector3f(0.79488, -0.05409, 0.39042),
        Vector3f(0.79926, -0.05229, 0.37579), Vector3f(0.80349, -0.05042, 0.36169), Vector3f(0.80758, -0.04846, 0.34810),
        Vector3f(0.81154, -0.04640, 0.33499), Vector3f(0.81537, -0.04423, 0.32235), Vector3f(0.81906, -0.04191, 0.31018),
        Vector3f(0.82266, -0.03941, 0.29845), Vector3f(0.82614, -0.03670, 0.28716), Vector3f(0.82952, -0.03372, 0.27629),
        Vector3f(0.83279, -0.03035, 0.26582), Vector3f(0.83599, -0.02647, 0.25574), Vector3f(0.83908, -0.02178, 0.24605),
        Vector3f(0.84209, -0.01551, 0.23672), Vector3f(0.84502, 0.00001, 0.22776)
    },
    {
        Vector3f(0.67520, -0.05931, 0.70649), Vector3f(0.69276, -0.05903, 0.69240), Vector3f(0.71128, -0.05822, 0.67366),
        Vector3f(0.72537, -0.05730, 0.65499), Vector3f(0.73687, -0.05639, 0.63599), Vector3f(0.74664, -0.05549, 0.61679),
        Vector3f(0.75518, -0.05459, 0.59754), Vector3f(0.76280, -0.05368, 0.57840), Vector3f(0.76970, -0.05276, 0.55954),
        Vector3f(0.77600, -0.05180, 0.54106), Vector3f(0.78185, -0.05080, 0.52303), Vector3f(0.78730, -0.04977, 0.50551),
        Vector3f(0.79243, -0.04869, 0.48852), Vector3f(0.79726, -0.04757, 0.47207), Vector3f(0.80185, -0.04641, 0.45614),
        Vector3f(0.80622, -0.04521, 0.44074), Vector3f(0.81040, -0.04396, 0.42583), Vector3f(0.81441, -0.04265, 0.41143),
        Vector3f(0.81826, -0.04130, 0.39750), Vector3f(0.82195, -0.03988, 0.38404), Vector3f(0.82551, -0.03837, 0.37102),
        Vector3f(0.82892, -0.03679, 0.35844), Vector3f(0.83223, -0.03510, 0.34628), Vector3f(0.83541, -0.03331, 0.33452),
        Vector3f(0.83849, -0.03135, 0.32316), Vector3f(0.84146, -0.02922, 0.31218), Vector3f(0.84433, -0.02686, 0.30158),
        Vector3f(0.84711, -0.02421, 0.29134), Vector3f(0.84980, -0.02113, 0.28144), Vector3f(0.85241, -0.01739, 0.27189),
        Vector3f(0.85493, -0.01240, 0.26266), Vector3f(0.85737, -0.00000, 0.25376)
    },
    {
        Vector3f(0.70511, -0.04013, 0.71307), Vector3f(0.72112, -0.03997, 0.69919), Vector3f(0.73819, -0.03950, 0.68079),
        Vector3f(0.75128, -0.03897, 0.66265), Vector3f(0.76198, -0.03845, 0.64443), Vector3f(0.77108, -0.03791, 0.62615),
        Vector3f(0.77901, -0.03737, 0.60793), Vector3f(0.78606, -0.03682, 0.58989), Vector3f(0.79240, -0.03625, 0.57212),
        Vector3f(0.79819, -0.03565, 0.55474), Vector3f(0.80350, -0.03502, 0.53779), Vector3f(0.80844, -0.03437, 0.52130),
        Vector3f(0.81305, -0.03367, 0.50532), Vector3f(0.81738, -0.03295, 0.48981), Vector3f(0.82148, -0.03220, 0.47479),
        Vector3f(0.82537, -0.03142, 0.46024), Vector3f(0.82906, -0.03059, 0.44615), Vector3f(0.83259, -0.02973, 0.43250),
        Vector3f(0.83596, -0.02882, 0.41928), Vector3f(0.83918, -0.02787, 0.40648), Vector3f(0.84227, -0.02686, 0.39407),
        Vector3f(0.84523, -0.02578, 0.38205), Vector3f(0.84808, -0.02463, 0.37040), Vector3f(0.85081, -0.02339, 0.35913),
        Vector3f(0.85345, -0.02205, 0.34819), Vector3f(0.85597, -0.02057, 0.33761), Vector3f(0.85842, -0.01893, 0.32734),
        Vector3f(0.86076, -0.01707, 0.31740), Vector3f(0.86302, -0.01492, 0.30777), Vector3f(0.86519, -0.01229, 0.29845),
        Vector3f(0.86729, -0.00878, 0.28941), Vector3f(0.86932, -0.00000, 0.28066)
    },
    {
        Vector3f(0.73154, -0.02030, 0.72031), Vector3f(0.74609, -0.02024, 0.70643), Vector3f(0.76177, -0.02006, 0.68812),
        Vector3f(0.77389, -0.01984, 0.67034), Vector3f(0.78383, -0.01961, 0.65270), Vector3f(0.79228, -0.01939, 0.63519),
        Vector3f(0.79965, -0.01914, 0.61784), Vector3f(0.80618, -0.01889, 0.60072), Vector3f(0.81203, -0.01864, 0.58392),
        Vector3f(0.81735, -0.01837, 0.56751), Vector3f(0.82221, -0.01807, 0.55151), Vector3f(0.82672, -0.01776, 0.53597),
        Vector3f(0.83089, -0.01744, 0.52087), Vector3f(0.83482, -0.01710, 0.50623), Vector3f(0.83850, -0.01673, 0.49203),
        Vector3f(0.84198, -0.01635, 0.47828), Vector3f(0.84528, -0.01595, 0.46494), Vector3f(0.84842, -0.01552, 0.45201),
        Vector3f(0.85140, -0.01508, 0.43946), Vector3f(0.85424, -0.01460, 0.42730), Vector3f(0.85696, -0.01408, 0.41549),
        Vector3f(0.85955, -0.01354, 0.40404), Vector3f(0.86204, -0.01296, 0.39292), Vector3f(0.86442, -0.01233, 0.38212),
        Vector3f(0.86669, -0.01164, 0.37165), Vector3f(0.86887, -0.01087, 0.36148), Vector3f(0.87097, -0.01002, 0.35161),
        Vector3f(0.87297, -0.00905, 0.34203), Vector3f(0.87490, -0.00791, 0.33272), Vector3f(0.87673, -0.00653, 0.32369),
        Vector3f(0.87850, -0.00466, 0.31492), Vector3f(0.88020, 0.00001, 0.30640)
    },
    {
        Vector3f(0.75486, -0.00000, 0.72806), Vector3f(0.76807, 0.00000, 0.71395), Vector3f(0.78246, -0.00000, 0.69552),
        Vector3f(0.79366, -0.00000, 0.67790), Vector3f(0.80290, 0.00001, 0.66069), Vector3f(0.81077, 0.00001, 0.64378),
        Vector3f(0.81763, -0.00000, 0.62716), Vector3f(0.82368, -0.00000, 0.61086), Vector3f(0.82912, -0.00000, 0.59491),
        Vector3f(0.83404, -0.00000, 0.57936), Vector3f(0.83852, -0.00000, 0.56423), Vector3f(0.84266, 0.00000, 0.54953),
        Vector3f(0.84649, 0.00000, 0.53526), Vector3f(0.85008, -0.00000, 0.52142), Vector3f(0.85343, 0.00000, 0.50800),
        Vector3f(0.85660, -0.00000, 0.49498), Vector3f(0.85959, 0.00000, 0.48235), Vector3f(0.86241, -0.00000, 0.47011),
        Vector3f(0.86510, 0.00001, 0.45821), Vector3f(0.86766, 0.00000, 0.44666), Vector3f(0.87010, -0.00001, 0.43545),
        Vector3f(0.87242, 0.00000, 0.42456), Vector3f(0.87464, -0.00000, 0.41398), Vector3f(0.87675, 0.00000, 0.40369),
        Vector3f(0.87877, 0.00000, 0.39370), Vector3f(0.88070, -0.00000, 0.38398), Vector3f(0.88255, -0.00000, 0.37453),
        Vector3f(0.88431, 0.00000, 0.36535), Vector3f(0.88600, 0.00000, 0.35642), Vector3f(0.88761, -0.00000, 0.34773),
        Vector3f(0.88915, -0.00000, 0.33929), Vector3f(0.89063, -0.00000, 0.33107)
    }
};

}  // namespace pbrt