Dual-Paraboloid Shadow Maps

Last time I introduced using dual-paraboloid environment mapping for reflections. Well now we're going to apply the same process to shadows. So if you haven't looked at my previous post, read it over before going on.

Creating the depth/shadow maps is exactly the same as when we created the reflection maps with one exception. Instead of outputting color in the pixel shader, we output the depth of the 3d pixel, like so:

return depth.x / depth.y;

Where depth.x is the depth of the pixel and depth.y is the w component. And here is the resulting depth/shadow map for the front hemisphere.

Now, to map the shadows the process is also very similar to how we generated the reflections. We follow a similar process in the pixel shader:

• Generate the texture coordinates for the front and rear paraboloids
• Generate the depth of the pixel
• Test to see if the pixel is in shadow

We generate the texture coordinates exactly as when we generated the reflection texture coordinates. To generate the depth of the pixel we take the length of the vector from the vertex to the origin of the paraboloid (0, 0, 0) and divide by the light attenuation. Also to check which hemisphere we are in, we calculate an alpha that is the Z value of the transformed vertex and offset by .5f;

float L = length(pos);
float3 P0 = pos / L;

float alpha = .5f + pos.z / LightAttenuation;
//generate texture coords for the front hemisphere
P0.z = P0.z + 1;
P0.x = P0.x / P0.z;
P0.y = P0.y / P0.z;
P0.z = L / LightAttenuation;

P0.x = .5f * P0.x + .5f;
P0.y = -.5f * P0.y + .5f;

float3 P1 = pos / L;
//generate texture coords for the rear hemisphere
P1.z = 1 - P1.z;
P1.x = P1.x / P1.z;
P1.y = P1.y / P1.z;
P1.z = L / LightAttenuation;

P1.x = .5f * P1.x + .5f;
P1.y = -.5f * P1.y + .5f;

Now that we have generated our texture coordinates we need to test the depth of the pixel against the depth in the shadow map. To do this we index either the front or rear shadow map with the texture coordinates we generated to get the depth and compare this to our depth. If the depth is less than our depth, then the pixel is in shadow.

float depth;
float mydepth;
if(alpha >= 0.5f)
{
    depth = tex2D(ShadowFrontS, P0.xy).x;
    mydepth = P0.z;
}
else
{
    depth = tex2D(ShadowBackS, P1.xy).x;
    mydepth = P1.z;
}

//lighten the shadow just a bit so it isn't completely black
if((depth + SHADOW_EPSILON) < mydepth)
    texColor.xyz *= 0.3f;

return texColor;

And that's it. Now we have dual-paraboloid shadow mapping. If you have a pixel shader 3.0 graphics card, then the shadow also has a percentage closer filter applied to it. You also may notice seams in the shadows. This is because the splitting plane of the paraboloids is the x-axis (since the paraboloids look down the +/- z-axis). This is one of the problems of using paraboloid mapping for shadows. One has to be careful where they place the split plane to avoid this situation. Pixels that are in the center of either hemisphere suffer little distortion. But this is just a tutorial so I didn't worry too much about it.

Also you're graphics card must be able to support R32F or R16F surface formats to run the demo out of the box (sorry Charles ;) ). Otherwise, you must use the ARGB32 format and pack the depth values in all 4 channels. Here is some code to pack/unpack to/from an ARGB32 surface format. You pass the depth value to the pack method when you render to the shadow maps, and you pass the float4 color to the unpack method when you fetch from the shadow maps. I decided not to implement this so the code wouldn't become complicated by something that doesn't add to the tutorial.

//pack the depth in a 32-bit rgba color
float4 mapDepthToARGB32(const float value)
{
    const float4 bitSh = float4(256.0 * 256.0 * 256.0, 256.0 * 256.0, 256.0, 1.0);
    const float4 mask = float4(0.0, 1.0 / 256.0, 1.0 / 256.0, 1.0 / 256.0);
    float4 res = frac(value * bitSh);
    res -= res.xxyz * mask;
    return res;
}

//unpack the depth from a 32-bit rgba color
float getDepthFromARGB32(const float4 value)
{
    const float4 bitSh = float4(1.0 / (256.0 * 256.0 * 256.0), 1.0 / (256.0 * 256.0), 1.0 / 256.0, 1.0);
    return(dot(value, bitSh));
}

Next time I'll introduce using variance shadow mapping with our dual-paraboloid shadow mapping to give nice soft shadows that we can still use with pixel shader 2.0 cards.