## Friday, February 24, 2012

### Parallax/POM mapping and no tangent space.

I thought I would do yet another follow up post regarding thoughts on Parallax/POM mapping without the use of conventional tangent space. A lot of the terms involved are terms we already have when doing bump mapping using either derivative or height maps.

// terms shared between bump and pom
float2 TexDx = ddx(In.texST);
float2 TexDy = ddy(In.texST);
float3 vSigmaX = ddx(surf_pos);
float3 vSigmaY = ddy(surf_pos);
float3 vN = surf_norm;     // normalized
float3 vR1 = cross(vSigmaY, vN);
float3 vR2 = cross(vN, vSigmaX);
float fDet = dot(vSigmaX, vR1);

// specific to Parallax/POM
float3 vV = vView;   // normalized view vector in same space as surf_pos and vN
float2 vProjVscr = (1/fDet) * float2( dot(vR1, vV), dot(vR2, vV) );
float2 vProjVtex = TexDx*vProjVscr.x + TexDy*vProjVscr.y;

The resulting 2D vector vProjVtex is the offset vector in normalized texture space which corresponds to moving along the surface of the object by the plane projected view vector which is exactly what we want for POM. The remaining work is done the usual way.

The magnitude of vProjVtex (in normalized texture space) will correspond to the magnitude of the projected view vector at the surface. To obtain the third component of the transformed view vector the applied bump_scale must be taken into account. This is done using the following line of code:

float vProjVtexZ = dot(vN, vV) / bump_scale;

If we consider T the texture coordinate and the surface gradient of T a 2x3 matrix then an alternative way to think of how we obtain vProjVtex is through the use of surface gradients. One per component of the texture coordinate since each of these represent a scalar field.

float2 vProjVtex = mul(SurfGrad(T), vView)

The first row of SurfGrad(T) is equal to (1/fDet)*(TexDx.x*vR1 + TexDy.x*vR2) and similar for the second row but using the .y components of TexDx and TexDy. In practice it doesn't really simplify the code much unless we need to transform multiple vectors but it's a fun fact :)

Note that one of the observations made in the paper is that the surface gradient of a given scalar field can be obtained using any parametrization of the surface and the field (eq. 3). For a scalar field defined on a volume we can (though not required) use eq. 2 instead to obtain the surface gradient.

## Tuesday, January 3, 2012

### How to do more generic mixing of derivative maps?

I have added a shader to the demo which shows how to do mixing of derivative maps in a more typical scenario in which auto bump scale is used and all derivative maps are using same base unwrap but using different scales and offsets on the texture coordinate. The auto bump scale is what allows us to achieve scale invariance like we are used to with normal mapping.

The location of the demo is still the same: source and binary. Also notice that dHdst_derivmap() can be made shorter by passing
VirtDim / sqrt(abs(VirtDim.x * VirtDim.y)) as a constant. Furthermore, this vector constant is ±1.0 when VirtDim.x == VirtDim.y. Finally, the scale and offset which is applied to the texture coordinate can be moved to the vertex shader. However, this requires the use of multiple interpolators so this might not be the preferred way.

One relevant thing to notice here is that the numerator VirtDim is a float2 and not part of the bump scale. It serves to convert the height derivatives to the normalized texture domain from which we apply the chain rule. The factor 1 / sqrt(abs(VirtDim.x * VirtDim.y)) on the other hand is part of the bump scale. The distinction is subtle when doing bump mapping but becomes relevant when we do parallax mapping.

If for instance you are using derivative maps made with xNormal or you are just using this way of auto calculating a bump scale then the expression for the bump scale is:

bump_scale = g_fMeshSpecificAutoBumpScale / sqrt(abs(VirtDim.x * VirtDim.y))

where g_fMeshSpecificAutoBumpScale is the square root of the average ratio of the surface to normalized texture area. There is code which shows how to calculate this in the demo. The final value bump_scale is the same as the value which is returned by