Volume Rendering Integral with Partial Pre-Integration

This listing is a function that will, given a pair input colors (with the alpha components set to the attenuation parameter $\tau$), a ray segment length, and a $\Psi$ table (which is ubiquitous), compute the volume rendering integral with linear interpolation for the luminance and attenuation. Section 1.2 introduces the partial pre-integration technique.

The function assumes that the $\Psi$ table, which is stored in the PsiTable variable, is 1024 by 1024. The size can be changed by changing the definition of PSI_TABLE_SIZE.

Appendix 1 gives a Mathematica script that builds $\Psi$ tables.

#define PSI_TABLE_SIZE        float2(1024,1024)
float4 integrateRay(in float4 colorBack, in float4 colorFront,
                    in float distance, in sampler2D PsiTable)
{
  float2 taudbackfront;
  taudbackfront.x = distance*colorBack.a;
  taudbackfront.y = distance*colorFront.a;
  float zeta = exp(-dot(taudbackfront, float2(0.5,0.5)));

  float2 gammabackfront = taudbackfront/(1+taudbackfront);
  float Psi = tex2D(PsiTable,
                    gammabackfront + float2(0.5,0.5)/PSI_TABLE_SIZE).w;

  float4 outColor;
  outColor.rgb = colorFront.rgb*(1-Psi) + colorBack.rgb*(Psi-zeta);
  outColor.a = 1 - zeta;

  return outColor;
}