395 lines
14 KiB
HLSL
395 lines
14 KiB
HLSL
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#ifndef LIGHTWEIGHT_MOMENT_MATH_INCLUDED
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#define LIGHTWEIGHT_MOMENT_MATH_INCLUDED
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/*! Code taken from the blog "Moments in Graphics" by Christoph Peters.
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http://momentsingraphics.de/?p=105
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This function computes the three real roots of a cubic polynomial
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Coefficient[0]+Coefficient[1]*x+Coefficient[2]*x^2+Coefficient[3]*x^3.*/
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float3 SolveCubic(float4 Coefficient) {
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// Normalize the polynomial
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Coefficient.xyz /= Coefficient.w;
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// Divide middle coefficients by three
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Coefficient.yz /= 3.0f;
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// Compute the Hessian and the discrimant
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float3 Delta = float3(
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mad(-Coefficient.z, Coefficient.z, Coefficient.y),
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mad(-Coefficient.y, Coefficient.z, Coefficient.x),
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dot(float2(Coefficient.z, -Coefficient.y), Coefficient.xy)
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);
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float Discriminant = dot(float2(4.0f * Delta.x, -Delta.y), Delta.zy);
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// Compute coefficients of the depressed cubic
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// (third is zero, fourth is one)
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float2 Depressed = float2(
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mad(-2.0f * Coefficient.z, Delta.x, Delta.y),
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Delta.x
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);
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// Take the cubic root of a normalized complex number
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float Theta = atan2(sqrt(Discriminant), -Depressed.x) / 3.0f;
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float2 CubicRoot;
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sincos(Theta, CubicRoot.y, CubicRoot.x);
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// Compute the three roots, scale appropriately and
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// revert the depression transform
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float3 Root = float3(
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CubicRoot.x,
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dot(float2(-0.5f, -0.5f * sqrt(3.0f)), CubicRoot),
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dot(float2(-0.5f, 0.5f * sqrt(3.0f)), CubicRoot)
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);
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Root = mad(2.0f * sqrt(-Depressed.y), Root, -Coefficient.z);
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return Root;
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}
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/*! Given coefficients of a quadratic polynomial A*x^2+B*x+C, this function
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outputs its two real roots.*/
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float2 solveQuadratic(float3 coeffs)
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{
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coeffs[1] *= 0.5;
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float x1, x2, tmp;
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tmp = (coeffs[1] * coeffs[1] - coeffs[0] * coeffs[2]);
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if (coeffs[1] >= 0) {
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tmp = sqrt(tmp);
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x1 = (-coeffs[2]) / (coeffs[1] + tmp);
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x2 = (-coeffs[1] - tmp) / coeffs[0];
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}
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else {
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tmp = sqrt(tmp);
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x1 = (-coeffs[1] + tmp) / coeffs[0];
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x2 = coeffs[2] / (-coeffs[1] + tmp);
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}
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return float2(x1, x2);
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}
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/*! Given coefficients of a cubic polynomial
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coeffs[0]+coeffs[1]*x+coeffs[2]*x^2+coeffs[3]*x^3 with three real roots,
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this function returns the root of least magnitude.*/
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float solveCubicBlinnSmallest(float4 coeffs)
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{
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coeffs.xyz /= coeffs.w;
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coeffs.yz /= 3.0;
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float3 delta = float3(mad(-coeffs.z, coeffs.z, coeffs.y), mad(-coeffs.z, coeffs.y, coeffs.x), coeffs.z * coeffs.x - coeffs.y * coeffs.y);
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float discriminant = 4.0 * delta.x * delta.z - delta.y * delta.y;
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float2 depressed = float2(delta.z, -coeffs.x * delta.y + 2.0 * coeffs.y * delta.z);
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float theta = abs(atan2(coeffs.x * sqrt(discriminant), -depressed.y)) / 3.0;
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float2 sin_cos;
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sincos(theta, sin_cos.x, sin_cos.y);
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float tmp = 2.0 * sqrt(-depressed.x);
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float2 x = float2(tmp * sin_cos.y, tmp * (-0.5 * sin_cos.y - 0.5 * sqrt(3.0) * sin_cos.x));
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float2 s = (x.x + x.y < 2.0 * coeffs.y) ? float2(-coeffs.x, x.x + coeffs.y) : float2(-coeffs.x, x.y + coeffs.y);
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return s.x / s.y;
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}
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/*! Given coefficients of a quartic polynomial
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coeffs[0]+coeffs[1]*x+coeffs[2]*x^2+coeffs[3]*x^3+coeffs[4]*x^4 with four
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real roots, this function returns all roots.*/
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float4 solveQuarticNeumark(float coeffs[5])
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{
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// Normalization
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float B = coeffs[3] / coeffs[4];
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float C = coeffs[2] / coeffs[4];
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float D = coeffs[1] / coeffs[4];
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float E = coeffs[0] / coeffs[4];
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// Compute coefficients of the cubic resolvent
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float P = -2.0 * C;
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float Q = C * C + B * D - 4.0 * E;
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float R = D * D + B * B * E - B * C * D;
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// Obtain the smallest cubic root
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float y = solveCubicBlinnSmallest(float4(R, Q, P, 1.0));
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float BB = B * B;
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float fy = 4.0 * y;
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float BB_fy = BB - fy;
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float Z = C - y;
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float ZZ = Z * Z;
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float fE = 4.0 * E;
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float ZZ_fE = ZZ - fE;
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float G, g, H, h;
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// Compute the coefficients of the quadratics adaptively using the two
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// proposed factorizations by Neumark. Choose the appropriate
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// factorizations using the heuristic proposed by Herbison-Evans.
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if (y < 0 || (ZZ + fE) * BB_fy > ZZ_fE * (BB + fy)) {
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float tmp = sqrt(BB_fy);
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G = (B + tmp) * 0.5;
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g = (B - tmp) * 0.5;
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tmp = (B * Z - 2.0 * D) / (2.0 * tmp);
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H = mad(Z, 0.5, tmp);
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h = mad(Z, 0.5, -tmp);
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}
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else {
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float tmp = sqrt(ZZ_fE);
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H = (Z + tmp) * 0.5;
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h = (Z - tmp) * 0.5;
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tmp = (B * Z - 2.0 * D) / (2.0 * tmp);
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G = mad(B, 0.5, tmp);
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g = mad(B, 0.5, -tmp);
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}
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// Solve the quadratics
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return float4(solveQuadratic(float3(1.0, G, H)), solveQuadratic(float3(1.0, g, h)));
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}
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/*! This function reconstructs the transmittance at the given depth from four
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normalized power moments and the given zeroth moment.*/
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float ComputeTransmittance(float b_0, float2 b_even, float2 b_odd, float depth, float bias, float overestimation, float4 bias_vector)
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{
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float4 b = float4(b_odd.x, b_even.x, b_odd.y, b_even.y);
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// Bias input data to avoid artifacts
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b = lerp(b, bias_vector, bias);
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float3 z;
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z[0] = depth;
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// Compute a Cholesky factorization of the Hankel matrix B storing only non-
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// trivial entries or related products
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float L21D11 = mad(-b[0], b[1], b[2]);
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float D11 = mad(-b[0], b[0], b[1]);
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float InvD11 = 1.0f / D11;
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float L21 = L21D11 * InvD11;
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float SquaredDepthVariance = mad(-b[1], b[1], b[3]);
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float D22 = mad(-L21D11, L21, SquaredDepthVariance);
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// Obtain a scaled inverse image of bz=(1,z[0],z[0]*z[0])^T
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float3 c = float3(1.0f, z[0], z[0] * z[0]);
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// Forward substitution to solve L*c1=bz
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c[1] -= b.x;
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c[2] -= b.y + L21 * c[1];
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// Scaling to solve D*c2=c1
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c[1] *= InvD11;
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c[2] /= D22;
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// Backward substitution to solve L^T*c3=c2
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c[1] -= L21 * c[2];
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c[0] -= dot(c.yz, b.xy);
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// Solve the quadratic equation c[0]+c[1]*z+c[2]*z^2 to obtain solutions
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// z[1] and z[2]
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float InvC2 = 1.0f / c[2];
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float p = c[1] * InvC2;
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float q = c[0] * InvC2;
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float D = (p * p * 0.25f) - q;
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float r = sqrt(D);
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z[1] = -p * 0.5f - r;
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z[2] = -p * 0.5f + r;
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// Compute the absorbance by summing the appropriate weights
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float3 polynomial;
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float3 weight_factor = float3(overestimation, (z[1] < z[0]) ? 1.0f : 0.0f, (z[2] < z[0]) ? 1.0f : 0.0f);
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float f0 = weight_factor[0];
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float f1 = weight_factor[1];
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float f2 = weight_factor[2];
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float f01 = (f1 - f0) / (z[1] - z[0]);
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float f12 = (f2 - f1) / (z[2] - z[1]);
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float f012 = (f12 - f01) / (z[2] - z[0]);
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polynomial[0] = f012;
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polynomial[1] = polynomial[0];
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polynomial[0] = f01 - polynomial[0] * z[1];
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polynomial[2] = polynomial[1];
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polynomial[1] = polynomial[0] - polynomial[1] * z[0];
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polynomial[0] = f0 - polynomial[0] * z[0];
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float absorbance = polynomial[0] + dot(b.xy, polynomial.yz);;
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// Turn the normalized absorbance into transmittance
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return saturate(exp(-b_0 * absorbance));
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}
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/*! This function reconstructs the transmittance at the given depth from six
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normalized power moments and the given zeroth moment.*/
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float ComputeTransmittance(float b_0, float3 b_even, float3 b_odd, float depth, float bias, float overestimation, float bias_vector[6])
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{
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float b[6] = { b_odd.x, b_even.x, b_odd.y, b_even.y, b_odd.z, b_even.z };
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// Bias input data to avoid artifacts
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//[unroll]
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UNITY_UNROLL
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for (int i = 0; i != 6; ++i) {
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b[i] = lerp(b[i], bias_vector[i], bias);
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}
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float4 z;
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z[0] = depth;
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// Compute a Cholesky factorization of the Hankel matrix B storing only non-
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// trivial entries or related products
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float InvD11 = 1.0f / mad(-b[0], b[0], b[1]);
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float L21D11 = mad(-b[0], b[1], b[2]);
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float L21 = L21D11 * InvD11;
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float D22 = mad(-L21D11, L21, mad(-b[1], b[1], b[3]));
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float L31D11 = mad(-b[0], b[2], b[3]);
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float L31 = L31D11 * InvD11;
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float InvD22 = 1.0f / D22;
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float L32D22 = mad(-L21D11, L31, mad(-b[1], b[2], b[4]));
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float L32 = L32D22 * InvD22;
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float D33 = mad(-b[2], b[2], b[5]) - dot(float2(L31D11, L32D22), float2(L31, L32));
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float InvD33 = 1.0f / D33;
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// Construct the polynomial whose roots have to be points of support of the
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// canonical distribution: bz=(1,z[0],z[0]*z[0],z[0]*z[0]*z[0])^T
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float4 c;
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c[0] = 1.0f;
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c[1] = z[0];
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c[2] = c[1] * z[0];
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c[3] = c[2] * z[0];
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// Forward substitution to solve L*c1=bz
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c[1] -= b[0];
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c[2] -= mad(L21, c[1], b[1]);
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c[3] -= b[2] + dot(float2(L31, L32), c.yz);
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// Scaling to solve D*c2=c1
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c.yzw *= float3(InvD11, InvD22, InvD33);
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// Backward substitution to solve L^T*c3=c2
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c[2] -= L32 * c[3];
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c[1] -= dot(float2(L21, L31), c.zw);
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c[0] -= dot(float3(b[0], b[1], b[2]), c.yzw);
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// Solve the cubic equation
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z.yzw = SolveCubic(c);
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// Compute the absorbance by summing the appropriate weights
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float4 weigth_factor;
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weigth_factor[0] = overestimation;
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weigth_factor.yzw = (z.yzw > z.xxx) ? float3 (0.0f, 0.0f, 0.0f) : float3 (1.0f, 1.0f, 1.0f);
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// Construct an interpolation polynomial
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float f0 = weigth_factor[0];
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float f1 = weigth_factor[1];
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float f2 = weigth_factor[2];
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float f3 = weigth_factor[3];
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float f01 = (f1 - f0) / (z[1] - z[0]);
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float f12 = (f2 - f1) / (z[2] - z[1]);
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float f23 = (f3 - f2) / (z[3] - z[2]);
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float f012 = (f12 - f01) / (z[2] - z[0]);
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float f123 = (f23 - f12) / (z[3] - z[1]);
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float f0123 = (f123 - f012) / (z[3] - z[0]);
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float4 polynomial;
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// f012+f0123 *(z-z2)
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polynomial[0] = mad(-f0123, z[2], f012);
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polynomial[1] = f0123;
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// *(z-z1) +f01
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polynomial[2] = polynomial[1];
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polynomial[1] = mad(polynomial[1], -z[1], polynomial[0]);
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polynomial[0] = mad(polynomial[0], -z[1], f01);
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// *(z-z0) +f0
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polynomial[3] = polynomial[2];
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polynomial[2] = mad(polynomial[2], -z[0], polynomial[1]);
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polynomial[1] = mad(polynomial[1], -z[0], polynomial[0]);
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polynomial[0] = mad(polynomial[0], -z[0], f0);
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float absorbance = dot(polynomial, float4 (1.0, b[0], b[1], b[2]));
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// Turn the normalized absorbance into transmittance
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return saturate(exp(-b_0 * absorbance));
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}
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float ComputeTransmittance(float b_0, float4 b_even, float4 b_odd, float depth, float bias, float overestimation, float bias_vector[8])
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{
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float b[8] = { b_odd.x, b_even.x, b_odd.y, b_even.y, b_odd.z, b_even.z, b_odd.w, b_even.w };
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// Bias input data to avoid artifacts
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//[unroll]
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UNITY_UNROLL
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for (int i = 0; i != 8; ++i) {
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b[i] = lerp(b[i], bias_vector[i], bias);
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}
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float z[5];
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z[0] = depth;
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// Compute a Cholesky factorization of the Hankel matrix B storing only non-trivial entries or related products
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float D22 = mad(-b[0], b[0], b[1]);
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float InvD22 = 1.0 / D22;
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float L32D22 = mad(-b[1], b[0], b[2]);
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float L32 = L32D22 * InvD22;
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float L42D22 = mad(-b[2], b[0], b[3]);
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float L42 = L42D22 * InvD22;
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float L52D22 = mad(-b[3], b[0], b[4]);
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float L52 = L52D22 * InvD22;
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float D33 = mad(-L32, L32D22, mad(-b[1], b[1], b[3]));
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float InvD33 = 1.0 / D33;
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float L43D33 = mad(-L42, L32D22, mad(-b[2], b[1], b[4]));
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float L43 = L43D33 * InvD33;
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float L53D33 = mad(-L52, L32D22, mad(-b[3], b[1], b[5]));
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float L53 = L53D33 * InvD33;
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float D44 = mad(-b[2], b[2], b[5]) - dot(float2(L42, L43), float2(L42D22, L43D33));
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float InvD44 = 1.0 / D44;
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float L54D44 = mad(-b[3], b[2], b[6]) - dot(float2(L52, L53), float2(L42D22, L43D33));
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float L54 = L54D44 * InvD44;
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float D55 = mad(-b[3], b[3], b[7]) - dot(float3(L52, L53, L54), float3(L52D22, L53D33, L54D44));
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float InvD55 = 1.0 / D55;
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// Construct the polynomial whose roots have to be points of support of the
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// Canonical distribution:
|
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// bz = (1,z[0],z[0]^2,z[0]^3,z[0]^4)^T
|
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float c[5];
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c[0] = 1.0;
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c[1] = z[0];
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c[2] = c[1] * z[0];
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||
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c[3] = c[2] * z[0];
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c[4] = c[3] * z[0];
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||
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// Forward substitution to solve L*c1 = bz
|
||
|
|
c[1] -= b[0];
|
||
|
|
c[2] -= mad(L32, c[1], b[1]);
|
||
|
|
c[3] -= b[2] + dot(float2(L42, L43), float2(c[1], c[2]));
|
||
|
|
c[4] -= b[3] + dot(float3(L52, L53, L54), float3(c[1], c[2], c[3]));
|
||
|
|
|
||
|
|
// Scaling to solve D*c2 = c1
|
||
|
|
//c = c .*[1, InvD22, InvD33, InvD44, InvD55];
|
||
|
|
c[1] *= InvD22;
|
||
|
|
c[2] *= InvD33;
|
||
|
|
c[3] *= InvD44;
|
||
|
|
c[4] *= InvD55;
|
||
|
|
|
||
|
|
// Backward substitution to solve L^T*c3 = c2
|
||
|
|
c[3] -= L54 * c[4];
|
||
|
|
c[2] -= dot(float2(L53, L43), float2(c[4], c[3]));
|
||
|
|
c[1] -= dot(float3(L52, L42, L32), float3(c[4], c[3], c[2]));
|
||
|
|
c[0] -= dot(float4(b[3], b[2], b[1], b[0]), float4(c[4], c[3], c[2], c[1]));
|
||
|
|
|
||
|
|
// Solve the quartic equation
|
||
|
|
float4 zz = solveQuarticNeumark(c);
|
||
|
|
z[1] = zz[0];
|
||
|
|
z[2] = zz[1];
|
||
|
|
z[3] = zz[2];
|
||
|
|
z[4] = zz[3];
|
||
|
|
|
||
|
|
// Compute the absorbance by summing the appropriate weights
|
||
|
|
float4 weigth_factor = (float4(z[1], z[2], z[3], z[4]) <= z[0].xxxx);
|
||
|
|
// Construct an interpolation polynomial
|
||
|
|
float f0 = overestimation;
|
||
|
|
float f1 = weigth_factor[0];
|
||
|
|
float f2 = weigth_factor[1];
|
||
|
|
float f3 = weigth_factor[2];
|
||
|
|
float f4 = weigth_factor[3];
|
||
|
|
float f01 = (f1 - f0) / (z[1] - z[0]);
|
||
|
|
float f12 = (f2 - f1) / (z[2] - z[1]);
|
||
|
|
float f23 = (f3 - f2) / (z[3] - z[2]);
|
||
|
|
float f34 = (f4 - f3) / (z[4] - z[3]);
|
||
|
|
float f012 = (f12 - f01) / (z[2] - z[0]);
|
||
|
|
float f123 = (f23 - f12) / (z[3] - z[1]);
|
||
|
|
float f234 = (f34 - f23) / (z[4] - z[2]);
|
||
|
|
float f0123 = (f123 - f012) / (z[3] - z[0]);
|
||
|
|
float f1234 = (f234 - f123) / (z[4] - z[1]);
|
||
|
|
float f01234 = (f1234 - f0123) / (z[4] - z[0]);
|
||
|
|
|
||
|
|
float Polynomial_0;
|
||
|
|
float4 Polynomial;
|
||
|
|
// f0123 + f01234 * (z - z3)
|
||
|
|
Polynomial_0 = mad(-f01234, z[3], f0123);
|
||
|
|
Polynomial[0] = f01234;
|
||
|
|
// * (z - z2) + f012
|
||
|
|
Polynomial[1] = Polynomial[0];
|
||
|
|
Polynomial[0] = mad(-Polynomial[0], z[2], Polynomial_0);
|
||
|
|
Polynomial_0 = mad(-Polynomial_0, z[2], f012);
|
||
|
|
// * (z - z1) + f01
|
||
|
|
Polynomial[2] = Polynomial[1];
|
||
|
|
Polynomial[1] = mad(-Polynomial[1], z[1], Polynomial[0]);
|
||
|
|
Polynomial[0] = mad(-Polynomial[0], z[1], Polynomial_0);
|
||
|
|
Polynomial_0 = mad(-Polynomial_0, z[1], f01);
|
||
|
|
// * (z - z0) + f1
|
||
|
|
Polynomial[3] = Polynomial[2];
|
||
|
|
Polynomial[2] = mad(-Polynomial[2], z[0], Polynomial[1]);
|
||
|
|
Polynomial[1] = mad(-Polynomial[1], z[0], Polynomial[0]);
|
||
|
|
Polynomial[0] = mad(-Polynomial[0], z[0], Polynomial_0);
|
||
|
|
Polynomial_0 = mad(-Polynomial_0, z[0], f0);
|
||
|
|
float absorbance = Polynomial_0 + dot(Polynomial, float4(b[0], b[1], b[2], b[3]));
|
||
|
|
// Turn the normalized absorbance into transmittance
|
||
|
|
return saturate(exp(-b_0 * absorbance));
|
||
|
|
}
|
||
|
|
// replace [unroll] with UNITY_UNROLL for metal etc compatibility
|
||
|
|
#endif
|