Cycles: optimize ensure_valid_reflection(), reduces render time by about 1%
This is an implementation that is about 1.5-2.1 times faster. It gives a result that is on average 6° different from the old implementation. The difference is because normals (Ng, N, N') are not selected to be coplanar, but instead reflection R is lifted the least amount and the N' is computed as a bisector. Differential Revision: https://developer.blender.org/D10084
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@ -195,108 +195,31 @@ ccl_device float2 regular_polygon_sample(float corners, float rotation, float u,
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ccl_device float3 ensure_valid_reflection(float3 Ng, float3 I, float3 N)
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{
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float3 R = 2 * dot(N, I) * N - I;
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float3 R;
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float NI = dot(N, I);
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float NgR, threshold;
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/* Reflection rays may always be at least as shallow as the incoming ray. */
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float threshold = min(0.9f * dot(Ng, I), 0.01f);
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if (dot(Ng, R) >= threshold) {
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return N;
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}
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/* Check if the incident ray is coming from behind normal N. */
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if (NI > 0) {
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/* Normal reflection */
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R = (2 * NI) * N - I;
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NgR = dot(Ng, R);
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/* Form coordinate system with Ng as the Z axis and N inside the X-Z-plane.
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* The X axis is found by normalizing the component of N that's orthogonal to Ng.
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* The Y axis isn't actually needed.
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*/
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float NdotNg = dot(N, Ng);
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float3 X = normalize(N - NdotNg * Ng);
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/* Keep math expressions. */
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/* clang-format off */
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/* Calculate N.z and N.x in the local coordinate system.
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*
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* The goal of this computation is to find a N' that is rotated towards Ng just enough
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* to lift R' above the threshold (here called t), therefore dot(R', Ng) = t.
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*
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* According to the standard reflection equation,
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* this means that we want dot(2*dot(N', I)*N' - I, Ng) = t.
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*
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* Since the Z axis of our local coordinate system is Ng, dot(x, Ng) is just x.z, so we get
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* 2*dot(N', I)*N'.z - I.z = t.
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*
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* The rotation is simple to express in the coordinate system we formed -
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* since N lies in the X-Z-plane, we know that N' will also lie in the X-Z-plane,
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* so N'.y = 0 and therefore dot(N', I) = N'.x*I.x + N'.z*I.z .
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*
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* Furthermore, we want N' to be normalized, so N'.x = sqrt(1 - N'.z^2).
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*
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* With these simplifications,
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* we get the final equation 2*(sqrt(1 - N'.z^2)*I.x + N'.z*I.z)*N'.z - I.z = t.
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*
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* The only unknown here is N'.z, so we can solve for that.
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*
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* The equation has four solutions in general:
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*
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* 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))
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* We can simplify this expression a bit by grouping terms:
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*
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* a = I.x^2 + I.z^2
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* b = sqrt(I.x^2 * (a - t^2))
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* c = I.z*t + a
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* N'.z = +-sqrt(0.5*(+-b + c)/a)
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*
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* Two solutions can immediately be discarded because they're negative so N' would lie in the
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* lower hemisphere.
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*/
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/* clang-format on */
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float Ix = dot(I, X), Iz = dot(I, Ng);
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float Ix2 = sqr(Ix), Iz2 = sqr(Iz);
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float a = Ix2 + Iz2;
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float b = safe_sqrtf(Ix2 * (a - sqr(threshold)));
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float c = Iz * threshold + a;
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/* Evaluate both solutions.
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* In many cases one can be immediately discarded (if N'.z would be imaginary or larger than
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* one), so check for that first. If no option is viable (might happen in extreme cases like N
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* being in the wrong hemisphere), give up and return Ng. */
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float fac = 0.5f / a;
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float N1_z2 = fac * (b + c), N2_z2 = fac * (-b + c);
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bool valid1 = (N1_z2 > 1e-5f) && (N1_z2 <= (1.0f + 1e-5f));
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bool valid2 = (N2_z2 > 1e-5f) && (N2_z2 <= (1.0f + 1e-5f));
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float2 N_new;
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if (valid1 && valid2) {
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/* If both are possible, do the expensive reflection-based check. */
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float2 N1 = make_float2(safe_sqrtf(1.0f - N1_z2), safe_sqrtf(N1_z2));
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float2 N2 = make_float2(safe_sqrtf(1.0f - N2_z2), safe_sqrtf(N2_z2));
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float R1 = 2 * (N1.x * Ix + N1.y * Iz) * N1.y - Iz;
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float R2 = 2 * (N2.x * Ix + N2.y * Iz) * N2.y - Iz;
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valid1 = (R1 >= 1e-5f);
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valid2 = (R2 >= 1e-5f);
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if (valid1 && valid2) {
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/* If both solutions are valid, return the one with the shallower reflection since it will be
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* closer to the input (if the original reflection wasn't shallow, we would not be in this
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* part of the function). */
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N_new = (R1 < R2) ? N1 : N2;
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/* Reflection rays may always be at least as shallow as the incoming ray. */
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threshold = min(0.9f * dot(Ng, I), 0.01f);
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if (NgR >= threshold) {
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return N;
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}
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else {
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/* If only one reflection is valid (= positive), pick that one. */
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N_new = (R1 > R2) ? N1 : N2;
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}
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}
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else if (valid1 || valid2) {
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/* Only one solution passes the N'.z criterium, so pick that one. */
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float Nz2 = valid1 ? N1_z2 : N2_z2;
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N_new = make_float2(safe_sqrtf(1.0f - Nz2), safe_sqrtf(Nz2));
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}
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else {
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return Ng;
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/* Bad incident */
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R = -I;
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NgR = dot(Ng, R);
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threshold = 0.01f;
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}
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return N_new.x * X + N_new.y * Ng;
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R = R + Ng * (threshold - NgR); /* Lift the reflection above the threshold. */
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return normalize(I * len(R) + R * len(I)); /* Find a bisector. */
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}
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CCL_NAMESPACE_END
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@ -84,67 +84,30 @@ closure color principled_hair(normal N,
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closure color henyey_greenstein(float g) BUILTIN;
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closure color absorption() BUILTIN;
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normal ensure_valid_reflection(normal Ng, vector I, normal N)
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normal ensure_valid_reflection(normal Ng, normal I, normal N)
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{
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/* The implementation here mirrors the one in kernel_montecarlo.h,
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* check there for an explanation of the algorithm. */
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vector R;
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float NI = dot(N, I);
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float NgR, threshold;
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float sqr(float x)
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{
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return x * x;
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}
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vector R = 2 * dot(N, I) * N - I;
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float threshold = min(0.9 * dot(Ng, I), 0.01);
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if (dot(Ng, R) >= threshold) {
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return N;
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}
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float NdotNg = dot(N, Ng);
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vector X = normalize(N - NdotNg * Ng);
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float Ix = dot(I, X), Iz = dot(I, Ng);
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float Ix2 = sqr(Ix), Iz2 = sqr(Iz);
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float a = Ix2 + Iz2;
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float b = sqrt(Ix2 * (a - sqr(threshold)));
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float c = Iz * threshold + a;
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float fac = 0.5 / a;
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float N1_z2 = fac * (b + c), N2_z2 = fac * (-b + c);
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int valid1 = (N1_z2 > 1e-5) && (N1_z2 <= (1.0 + 1e-5));
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int valid2 = (N2_z2 > 1e-5) && (N2_z2 <= (1.0 + 1e-5));
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float N_new_x, N_new_z;
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if (valid1 && valid2) {
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float N1_x = sqrt(1.0 - N1_z2), N1_z = sqrt(N1_z2);
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float N2_x = sqrt(1.0 - N2_z2), N2_z = sqrt(N2_z2);
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float R1 = 2 * (N1_x * Ix + N1_z * Iz) * N1_z - Iz;
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float R2 = 2 * (N2_x * Ix + N2_z * Iz) * N2_z - Iz;
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valid1 = (R1 >= 1e-5);
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valid2 = (R2 >= 1e-5);
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if (valid1 && valid2) {
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N_new_x = (R1 < R2) ? N1_x : N2_x;
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N_new_z = (R1 < R2) ? N1_z : N2_z;
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if (NI > 0) {
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R = (2 * NI) * N - I;
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NgR = dot(Ng, R);
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threshold = min(0.9 * dot(Ng, I), 0.01);
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if (NgR >= threshold) {
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return N;
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}
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else {
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N_new_x = (R1 > R2) ? N1_x : N2_x;
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N_new_z = (R1 > R2) ? N1_z : N2_z;
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}
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}
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else if (valid1 || valid2) {
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float Nz2 = valid1 ? N1_z2 : N2_z2;
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N_new_x = sqrt(1.0 - Nz2);
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N_new_z = sqrt(Nz2);
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}
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else {
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return Ng;
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R = -I;
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NgR = dot(Ng, R);
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threshold = 0.01;
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}
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return N_new_x * X + N_new_z * Ng;
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R = R + Ng * (threshold - NgR);
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return normalize(I * length(R) + R * length(I));
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}
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#endif /* CCL_STDOSL_H */
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