The article, Volumetric Rendering in Real-time, covered the basis of volumetric depth rendering, but at the time of the writing, no pixel shader compliant hardware was available. This supplement describes a process designed to achieve two goals, to get more precision out of an 8 bit part, and to allow the creation of concave fog volumes.
Computing the distance of fog for the convex case was relatively simple. Recall that the front side of the fog volume was subtracted away from the backside (where the depth is measured in number of units from the camera). Unfortunately, this does not work with concave fog volumes because at any given pixel, it may have two back sides and two front sides.
The solution is intuitive and has sound mathematical backing - sum all of the front sides and subtract them from the summed front sides. As shown in diagram one - this is the mathematical equivalent of breaking the volume into convex chunks and summing the up.
(B1-A1) + (B2-A2) factors to (B2+B1)-(A2+A1)
concavity is as simple as adding the multiple front sides and subtracting
them from the multiple back sides. Clearly, a meager 8 bits won't be enough
for this. Every bit added would allow another summation and subtraction,
and allow for more complex fog scenes.
There is an important assumption being made about the fog volume. Is must be a continuous, orientable hull. That is, it cannot have any holes in it. Every ray cast through the volume must enter through hull the same number of times it exits.
Although most hardware acceleration can handle 32 bits, it is really four 8-bit channels. The way most hardware works today, there is only one place where the fog depths could be summed up: The Alpha Blender.
The alpha blender is typically used to blend on alpha textures by configuring the source destination to multiply against the source alpha, and the destination to multiply against the inverse alpha. However, they can also be used to add (or subtract) the source and destination color channels. Unfortunately, there is no way to perform a carry operation here: If one channel would exceed 255 for a color value, it simply saturates to 255.
In order to perform higher bit precision additions on the Alpha Blending Unit, the incoming data has to be formatted in a way which is compatible with the way the alpha blender adds. To do this, the color channels can hold different bits of the actual result, and most importantly, be allowed some overlap in their bits.
will give us 12 bit precision in an 8 bit pipe. The Red channel will contain
the upper 8 bits, and the blue channel will contain the lower 4 -plus
3 carry spots. The upper bit should not be used for reasons which are
discussed later. So the actual value encoded is Red*16+Blue.
Now, the Alpha Blender will add multiple values in this format correctly up to 8 times before there is any possibility of a carry bit not propagating. This limits the fog hulls to ones which do not have concavity where looking on any direction a ray might pass in and out of the volume more than 8 times.
Encoding the bits in which will be added cannot be done with a pixel shader. There are two primary limitations. First, the color interpolators are 8 bit as well. Since the depth is computed on a per vertex level, this won't let higher bit values into the independent color channels. Even if the color channel had a higher precision, the pixel shader has no instruction to capture the lower bits of a higher bit value.
The alternative is to use a texture to hold the encoded depths. The advantage of this is twofold. First, texture interpolaters have much higher precision than color interpolaters, and second, no pixel shader is needed for initial step of summing the font and back sides of the fog volume.
Unfortunately, most hardware limits the dimensions of textures. 4096 is a typical limitation. This amounts to 12 bits of precision to be encoded in the texture. 12 bits, however, is vastly superior to 8 bits and can make all the difference to making fog volumes practical.
it all Up
Three important details remain: The actual summing of the fog sides, compensating for objects inside the fog, and the final subtraction.
The summing is done in three steps. First, the scene needs to be rendered to set a Z buffer. This will prevent fog pixels from being drawn which are behind some totally occluding objects. In a real application, this z could be shared from the pass which draws the geometry. The Z is then write disabled - so that fog writes will not update the z buffer.
After this, the summing is exactly as expected. The app simply draws all the forward facing polygons in one buffer, adding up their results, and then draws all the backward facing polygons in another buffer. There is one potential problem, however. In order to sum the depths of the fog volume, the alpha blend constants need to be set to one for the destination and one for the source, thereby adding the incoming pixel with the one already in the buffer.
Unfortunately, this does not take into account objects inside the fog that are acting as a surrogate fog cover. In this case - the scene itself must be added to scene since the far end of the fog would have been rejected by the Z test.
At first, this looks like an easy solution. In the previous article, the buffers were setup so that they were initialized to the scene's depth value. This way, fog depth values would replace any depth value in the scene if they were in front of it (i.e. the Z test succeeds) - but if no fog was present the scene would act as the fog cover.
This cannot be done for general concavity, however. While technically correct in the convex case, in the concave case there may be pixels at which the fog volumes are rendered multiple times on the front side and multiple sides on the backside. For these pixels, if the there was part of an object in between fog layers than the front buffer would be the sum of n front sides, and the back side would be sum of n-1 back sides. But since the fog cover was replaced by the fog - there are now more entry points then exit points. The result is painfully obvious - parts of the scene suddenly loose all fog when they should have some.
The above diagram illustrates that without taking into account the object's own depth value, the depth value generated would be B1 - A1 - A2 since B2 was never drawn because it failed the Z test of the scene. This value would be negative, and no fog would get blended. In this case, C needs to be added into the equation.
The solution requires knowing which scenario's where the scene's w depth should be added and which scenarios the scene's w depth should be ignored. Fortunately, this is not difficult to find. The only situation where the scene's w depth should be added to the total fog depth are those pixels where the object is in between the front side of a fog volume and its corresponding backside.
The above question can be thought of asking the question: did the ray ever leave the fog volume? Since the fog hulls are required to be continuous, then if the answer is no then part of the scene must have blocked the ray. This test can be performed by a standard inside outside test.
To perform an inside/outside test - each time a fog pixel is rendered, the alpha value is incremented. If the alpha values of the far fog distances is subtracted to the corresponding point on the near fog distance, then values greater then 1 indicate the ray stopped inside the volume. Values of 0 indicate that the ray left the fog volume.
To set this test up, the alpha buffer of the near and far w depth buffers must be cleared to 0. Each time a fog pixel is rendered, the alpha will be incremented by the hex value 0x10. This value was used because the pixel shader must perform a 1 or 0 logical operation. A small positive value must be mapped to 1.0 in the pixel shader, a step which requires multiple shifts. Due to instruction count restraints - the intial value has to be at least 0x10 for the shifts to saturate a non-zero value to one.
The rest is straightforward - all the front sides and all the backsides are summed up in their independent buffers. The scene is also drawn in its own buffer. Then all three buffers are ran through the final pass where the scene's w depth is added on only if the differences of the alpha values is not 0.
This requires a lengthy pixel shader. A great deal of care must be taken to avoid potential precision pitfalls. The following pixel shader performs the required math, although it requires every instruction slot of the pixel shader and nearly every register. Unfortunately, with no carry bit, there is no way to achieve a full 8 bit value at the end of the computation, so it must settle for 8.
def c1, 1.0f,0.0f,0.0f,0.0f
def c4, 0.0f,0.0f,1.0f,0.0f
tex t0 //
near buffer B
tex t1 // far buffer A
tex t2 // scene buffer C
// b = low bits (a) (4 bits)
// r = high bits (b) (8 bits)
// intermediate output:
// r1.b = (a1 - a2) (can't be greater than 7 bits set )
// r1.r = (b1 - b2)
+sub_4x r1.a,t0,t1 //If this value is non zero, then
mov_4x r0.a,r1.a //the were not as many backs as
mad r1.rgb,r0.a,t2,r1 //front and must add in the scene
dp3 t0.rgba,r1,c4 // move red component into alpha
// Need to shift r1.rgb
6 bits. This could saturate
// to 255 if any other bits are set, but that is fine
// because in this case, the end result of the subtract
// would have to be saturated since we can't be
// subtracting more than 127
dp3_x4 t1.rgba,r1,c1 // move into the alpha
add_x2 r0.a,t0.a,t1.a // the subtract was in 0-127
mov_d2 r0.a,r0.a // chop off last bit else banding
+mov r0.rgb,c3 // load the fog color
This pixel shader gives an alpha value which represents the density of fog, and loads the fog color constant into the color channels. The Alpha Blending stage can now be used to blend on the fog.
Finally, there is one situation which can cause serious problems - clipping. If a part of the fog volume is clipped away by the camera because the camera is partially in the fog, then part of the scene might be in the fog. Previously, it was assumed the camera was either entirely all the way in, or all the way out of the fog. This may not always be the case.
An alternative solution is to not allow polygons to get clipped. The vertex shader can detect vertices which would get clipped away and snap them to the near clip plane. The following vertex shader clips w depths to the near clip plane, and z depths to zero.
// transform position
into projection space
max r0.z,c40.z,r0.z //clamp to 0
max r0.w,c12.x,r0.w //clamp to near clip plane
// Subtract the Near
// Scale to give us
the far clipping plane
// load depth into
texture, don't care about y
please note that full source code will be available with the release of
DX 8.1, which is imminent. The code is in the volume fog SDK sample.