kernel_montecarlo.h

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/*
 * Parts adapted from Open Shading Language with this license:
 *
 * Copyright (c) 2009-2010 Sony Pictures Imageworks Inc., et al.
 * All Rights Reserved.
 *
 * Modifications Copyright 2011, Blender Foundation.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are
 * met:
 * * Redistributions of source code must retain the above copyright
 *   notice, this list of conditions and the following disclaimer.
 * * Redistributions in binary form must reproduce the above copyright
 *   notice, this list of conditions and the following disclaimer in the
 *   documentation and/or other materials provided with the distribution.
 * * Neither the name of Sony Pictures Imageworks nor the names of its
 *   contributors may be used to endorse or promote products derived from
 *   this software without specific prior written permission.
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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 * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */
 
#ifndef __KERNEL_MONTECARLO_CL__
#define __KERNEL_MONTECARLO_CL__
 
CCL_NAMESPACE_BEGIN
 
/* distribute uniform xy on [0,1] over unit disk [-1,1] */
ccl_device void to_unit_disk(float *x, float *y)
{
  float phi = M_2PI_F * (*x);
  float r = sqrtf(*y);
 
  *x = r * cosf(phi);
  *y = r * sinf(phi);
}
 
/* return an orthogonal tangent and bitangent given a normal and tangent that
 * may not be exactly orthogonal */
ccl_device void make_orthonormals_tangent(const float3 N, const float3 T, float3 *a, float3 *b)
{
  *b = normalize(cross(N, T));
  *a = cross(*b, N);
}
 
/* sample direction with cosine weighted distributed in hemisphere */
ccl_device_inline void sample_cos_hemisphere(
    const float3 N, float randu, float randv, float3 *omega_in, float *pdf)
{
  to_unit_disk(&randu, &randv);
  float costheta = sqrtf(max(1.0f - randu * randu - randv * randv, 0.0f));
  float3 T, B;
  make_orthonormals(N, &T, &B);
  *omega_in = randu * T + randv * B + costheta * N;
  *pdf = costheta * M_1_PI_F;
}
 
/* sample direction uniformly distributed in hemisphere */
ccl_device_inline void sample_uniform_hemisphere(
    const float3 N, float randu, float randv, float3 *omega_in, float *pdf)
{
  float z = randu;
  float r = sqrtf(max(0.0f, 1.0f - z * z));
  float phi = M_2PI_F * randv;
  float x = r * cosf(phi);
  float y = r * sinf(phi);
 
  float3 T, B;
  make_orthonormals(N, &T, &B);
  *omega_in = x * T + y * B + z * N;
  *pdf = 0.5f * M_1_PI_F;
}
 
/* sample direction uniformly distributed in cone */
ccl_device_inline void sample_uniform_cone(
    const float3 N, float angle, float randu, float randv, float3 *omega_in, float *pdf)
{
  float zMin = cosf(angle);
  float z = zMin - zMin * randu + randu;
  float r = safe_sqrtf(1.0f - sqr(z));
  float phi = M_2PI_F * randv;
  float x = r * cosf(phi);
  float y = r * sinf(phi);
 
  float3 T, B;
  make_orthonormals(N, &T, &B);
  *omega_in = x * T + y * B + z * N;
  *pdf = M_1_2PI_F / (1.0f - zMin);
}
 
ccl_device_inline float pdf_uniform_cone(const float3 N, float3 D, float angle)
{
  float zMin = cosf(angle);
  float z = dot(N, D);
  if (z > zMin) {
    return M_1_2PI_F / (1.0f - zMin);
  }
  return 0.0f;
}
 
/* sample uniform point on the surface of a sphere */
ccl_device float3 sample_uniform_sphere(float u1, float u2)
{
  float z = 1.0f - 2.0f * u1;
  float r = sqrtf(fmaxf(0.0f, 1.0f - z * z));
  float phi = M_2PI_F * u2;
  float x = r * cosf(phi);
  float y = r * sinf(phi);
 
  return make_float3(x, y, z);
}
 
ccl_device float balance_heuristic(float a, float b)
{
  return (a) / (a + b);
}
 
ccl_device float balance_heuristic_3(float a, float b, float c)
{
  return (a) / (a + b + c);
}
 
ccl_device float power_heuristic(float a, float b)
{
  return (a * a) / (a * a + b * b);
}
 
ccl_device float power_heuristic_3(float a, float b, float c)
{
  return (a * a) / (a * a + b * b + c * c);
}
 
ccl_device float max_heuristic(float a, float b)
{
  return (a > b) ? 1.0f : 0.0f;
}
 
/* distribute uniform xy on [0,1] over unit disk [-1,1], with concentric mapping
 * to better preserve stratification for some RNG sequences */
ccl_device float2 concentric_sample_disk(float u1, float u2)
{
  float phi, r;
  float a = 2.0f * u1 - 1.0f;
  float b = 2.0f * u2 - 1.0f;
 
  if (a == 0.0f && b == 0.0f) {
    return zero_float2();
  }
  else if (a * a > b * b) {
    r = a;
    phi = M_PI_4_F * (b / a);
  }
  else {
    r = b;
    phi = M_PI_2_F - M_PI_4_F * (a / b);
  }
 
  return make_float2(r * cosf(phi), r * sinf(phi));
}
 
/* sample point in unit polygon with given number of corners and rotation */
ccl_device float2 regular_polygon_sample(float corners, float rotation, float u, float v)
{
  /* sample corner number and reuse u */
  float corner = floorf(u * corners);
  u = u * corners - corner;
 
  /* uniform sampled triangle weights */
  u = sqrtf(u);
  v = v * u;
  u = 1.0f - u;
 
  /* point in triangle */
  float angle = M_PI_F / corners;
  float2 p = make_float2((u + v) * cosf(angle), (u - v) * sinf(angle));
 
  /* rotate */
  rotation += corner * 2.0f * angle;
 
  float cr = cosf(rotation);
  float sr = sinf(rotation);
 
  return make_float2(cr * p.x - sr * p.y, sr * p.x + cr * p.y);
}
 
ccl_device float3 ensure_valid_reflection(float3 Ng, float3 I, float3 N)
{
  float3 R = 2 * dot(N, I) * N - I;
 
  /* Reflection rays may always be at least as shallow as the incoming ray. */
  float threshold = min(0.9f * dot(Ng, I), 0.01f);
  if (dot(Ng, R) >= threshold) {
    return N;
  }
 
  /* Form coordinate system with Ng as the Z axis and N inside the X-Z-plane.
   * The X axis is found by normalizing the component of N that's orthogonal to Ng.
   * The Y axis isn't actually needed.
   */
  float NdotNg = dot(N, Ng);
  float3 X = normalize(N - NdotNg * Ng);
 
  /* Keep math expressions. */
  /* clang-format off */
  /* Calculate N.z and N.x in the local coordinate system.
   *
   * The goal of this computation is to find a N' that is rotated towards Ng just enough
   * to lift R' above the threshold (here called t), therefore dot(R', Ng) = t.
   *
   * According to the standard reflection equation,
   * this means that we want dot(2*dot(N', I)*N' - I, Ng) = t.
   *
   * Since the Z axis of our local coordinate system is Ng, dot(x, Ng) is just x.z, so we get
   * 2*dot(N', I)*N'.z - I.z = t.
   *
   * The rotation is simple to express in the coordinate system we formed -
   * since N lies in the X-Z-plane, we know that N' will also lie in the X-Z-plane,
   * so N'.y = 0 and therefore dot(N', I) = N'.x*I.x + N'.z*I.z .
   *
   * Furthermore, we want N' to be normalized, so N'.x = sqrt(1 - N'.z^2).
   *
   * With these simplifications,
   * we get the final equation 2*(sqrt(1 - N'.z^2)*I.x + N'.z*I.z)*N'.z - I.z = t.
   *
   * The only unknown here is N'.z, so we can solve for that.
   *
   * The equation has four solutions in general:
   *
   * N'.z = +-sqrt(0.5*(+-sqrt(I.x^2*(I.x^2 + I.z^2 - t^2)) + t*I.z + I.x^2 + I.z^2)/(I.x^2 + I.z^2))
   * We can simplify this expression a bit by grouping terms:
   *
   * a = I.x^2 + I.z^2
   * b = sqrt(I.x^2 * (a - t^2))
   * c = I.z*t + a
   * N'.z = +-sqrt(0.5*(+-b + c)/a)
   *
   * Two solutions can immediately be discarded because they're negative so N' would lie in the
   * lower hemisphere.
   */
  /* clang-format on */
 
  float Ix = dot(I, X), Iz = dot(I, Ng);
  float Ix2 = sqr(Ix), Iz2 = sqr(Iz);
  float a = Ix2 + Iz2;
 
  float b = safe_sqrtf(Ix2 * (a - sqr(threshold)));
  float c = Iz * threshold + a;
 
  /* Evaluate both solutions.
   * In many cases one can be immediately discarded (if N'.z would be imaginary or larger than
   * one), so check for that first. If no option is viable (might happen in extreme cases like N
   * being in the wrong hemisphere), give up and return Ng. */
  float fac = 0.5f / a;
  float N1_z2 = fac * (b + c), N2_z2 = fac * (-b + c);
  bool valid1 = (N1_z2 > 1e-5f) && (N1_z2 <= (1.0f + 1e-5f));
  bool valid2 = (N2_z2 > 1e-5f) && (N2_z2 <= (1.0f + 1e-5f));
 
  float2 N_new;
  if (valid1 && valid2) {
    /* If both are possible, do the expensive reflection-based check. */
    float2 N1 = make_float2(safe_sqrtf(1.0f - N1_z2), safe_sqrtf(N1_z2));
    float2 N2 = make_float2(safe_sqrtf(1.0f - N2_z2), safe_sqrtf(N2_z2));
 
    float R1 = 2 * (N1.x * Ix + N1.y * Iz) * N1.y - Iz;
    float R2 = 2 * (N2.x * Ix + N2.y * Iz) * N2.y - Iz;
 
    valid1 = (R1 >= 1e-5f);
    valid2 = (R2 >= 1e-5f);
    if (valid1 && valid2) {
      /* If both solutions are valid, return the one with the shallower reflection since it will be
       * closer to the input (if the original reflection wasn't shallow, we would not be in this
       * part of the function). */
      N_new = (R1 < R2) ? N1 : N2;
    }
    else {
      /* If only one reflection is valid (= positive), pick that one. */
      N_new = (R1 > R2) ? N1 : N2;
    }
  }
  else if (valid1 || valid2) {
    /* Only one solution passes the N'.z criterium, so pick that one. */
    float Nz2 = valid1 ? N1_z2 : N2_z2;
    N_new = make_float2(safe_sqrtf(1.0f - Nz2), safe_sqrtf(Nz2));
  }
  else {
    return Ng;
  }
 
  return N_new.x * X + N_new.y * Ng;
}
 
CCL_NAMESPACE_END
 
#endif /* __KERNEL_MONTECARLO_CL__ */