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113 行
4.4 KiB
113 行
4.4 KiB
// Library header containing various useful functions for doing moment based shadow maps.
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// Supported flavors are VSM, EVSM and MSM
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// conversion helper for VSM flavors
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// Chebychev's inequality (one-tailed version)
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// P( x >= t ) <= pmax(t) := sigma^2 / (sigma^2 + (t - u)^2)
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// for us t is depth, u is E(x) i.d. the blurred depth
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float ShadowMoments_ChebyshevsInequality( float2 moments, float depth, float minVariance, float lightLeakBias )
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{
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// variance sig^2 = E(x^2) - E(x)^2
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float variance = max( moments.y - (moments.x * moments.x), minVariance );
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// probabilistic upper bound
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float mD = depth - moments.x;
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float p = variance / (variance + mD * mD);
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p = saturate( (p - lightLeakBias) / (1.0f - lightLeakBias) );
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return max( p, depth <= moments.x );
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}
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// helper for EVSM
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float2 ShadowMoments_WarpDepth( float depth, float2 exponents )
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{
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// Rescale depth into [-1;1]
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depth = 2.0 * depth - 1.0;
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float pos = exp( exponents.x * depth );
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float neg = -exp(-exponents.y * depth );
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return float2( pos, neg );
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}
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// helpers for MSM
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// Prepare the moments so there's little quantization error when storing the moments at half
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// precision. This step becomes unnecessary if the moments are stored in 32bit floats.
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float4 ShadowMoments_Encode16MSM( float depth )
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{
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float dsq = depth * depth;
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float4 moments = { depth, dsq, depth * dsq, dsq * dsq };
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float4x4 mat = { - 2.07224649 , 13.7948857237, 0.105877704 , 9.7924062118,
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32.23703778 , -59.4683975703, -1.9077466311, -33.7652110555,
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-68.571074599 , 82.0359750338, 9.3496555107, 47.9456096605,
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39.3703274134, -35.364903257 , -6.6543490743, -23.9728048165 };
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float4 optimized = mul( moments, mat );
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optimized[0] += 0.035955884801;
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return optimized;
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}
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float4 ShadowMoments_Decode16MSM( float4 moments )
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{
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moments[0] -= 0.035955884801;
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float4x4 mat = { 0.2227744146, 0.1549679261, 0.1451988946, 0.163127443,
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0.0771972861, 0.1394629426, 0.2120202157, 0.2591432266,
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0.7926986636, 0.7963415838, 0.7258694464, 0.6539092497,
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0.0319417555, -0.1722823173, -0.2758014811, -0.3376131734 };
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return mul( moments, mat );
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}
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void ShadowMoments_SolveMSM( float4 moments, float depth, float momentBias, out float3 z, out float4 b )
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{
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// Bias input data to avoid artifacts
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z[0] = depth;
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b = lerp( moments, 0.5.xxxx, momentBias );
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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 L32D22 = mad( -b[0], b[1], b[2] );
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float D22 = mad( -b[0], b[0], b[1] );
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float sqDepthVar = mad( -b[1], b[1], b[3] );
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float D33D22 = dot( float2( sqDepthVar, -L32D22 ), float2( D22, L32D22 ) );
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float InvD22 = 1.0 / D22;
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float L32 = L32D22 * InvD22;
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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( 1.0, 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 + L32 * c[1];
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// Scaling to solve D * c2 = c1
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c[1] *= InvD22;
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c[2] *= D22 / D33D22;
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// Backward substitution to solve L^T * c3 = c2
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c[1] -= L32 * 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 z[1] and z[2]
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float p = c[1] / c[2];
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float q = c[0] / c[2];
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float D = ((p*p) * 0.25) - q;
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float r = sqrt( D );
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z[1] = -(p * 0.5) - r;
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z[2] = -(p * 0.5) + r;
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}
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float ShadowMoments_SolveDelta3MSM( float3 z, float2 b, float lightLeakBias )
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{
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// Construct a solution composed of three Dirac-deltas and evaluate its CDF
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float4 switchVal = (z[2] < z[0]) ? float4( z[1], z[0], 1.0, 1.0 )
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: ((z[1] < z[0]) ? float4( z[0], z[1], 0.0, 1.0 ) : 0.0.xxxx);
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float quotient = (switchVal[0] * z[2] - b[0] * (switchVal[0] + z[2]) + b[1]) / ((z[2] - switchVal[1]) * (z[0] - z[1]));
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float attenuation = saturate( switchVal[2] + switchVal[3] * quotient );
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return saturate( ((1.0 - attenuation) - lightLeakBias) / (1.0f - lightLeakBias) );
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}
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float ShadowMoments_SolveDelta4MSM( float3 z, float4 b, float lightLeakBias)
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{
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// Use a solution made of four deltas
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float zFree = ((b[2] - b[1]) * z[0] + b[2] - b[3]) / ((b[1] - b[0]) * z[0] + b[1] - b[2]);
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float w1Factor = (z[0] > zFree) ? 1.0 : 0.0;
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float attenuation = saturate( (b[1] - b[0] + (b[2] - b[0] - (zFree + 1.0) * (b[1] - b[0])) * (zFree - w1Factor - z[0]) / (z[0] * (z[0] - zFree))) / (zFree - w1Factor) + 1.0 - b[0] );
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return saturate( ((1.0 - attenuation) - lightLeakBias) / (1.0f - lightLeakBias) );
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}
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