Complex filtering depends on knowing just how much of the texture (or shading) we need to filter. Modern GPUs such as the GeForce FX provide partial derivative functions to help us. For any value used in shading, we can ask the GPU: "How much does this value change from pixel to pixel, in either the screen-x or the screen-y direction?"
These functions are ddx() and ddy(). Although they are little used, they can be very helpful for filtering and antialiasing operations. These derivative functions provide us with the necessary information to perform procedural filtering or to adroitly modify the filtering inherent in most texture sampling.
For GPUs that do not directly support ddx() and ddy(), you can use the method outlined in Chapter 25 of this book, "Fast Filter-Width Estimates with Texture Maps."
The values returned by the GPU for ddx() and ddy() are numerically iterated values. That is, if you evaluateddx(myVar), the GPU will give you the difference between the value of myVar at the current pixel and its value at the pixel next door. It's a straight linear difference, made efficient by the nature of GPU SIMD architectures (neighboring pixels will be calculated simultaneously). Of course, it should apply to values that interpolate across a polygon, passed from the vertex shader—the derivative of any uniform value will always be zero.
Because these derivatives are always linear, the second derivatives—for example, ddx(ddx(myVar))—will alwaysbe zero. If your shader contains some clever function whose higher-order derivatives could instead be accurately calculated analytically, use that formulation if the value is important (say, for scientific calculation or film-level rendering).
Once we know the amount of change in a given pixel, we're able to determine the appropriate filtering. For a texture, the correct filtering will be to integrate the texture not just at a given u-v coordinate, but across a quadrilateral-shaped window into that texture, whose texture coordinates will be ddx(UV) across and ddy(UV) in height. When we call functions such as tex2D(), this is in fact automatically calculated for us. Cg, for advanced profiles, allows us to optionally specify the size of the filter we want to apply. For example, by specifying:
float2 nilUV = float2(0, 0); float4 pt = tex2D(myTextureSampler, IN.UV, nilUV, nilUV);
we can force the filter size to be zero—in effect, forcing the sampling of this texture always to use the "nearest-neighbor" method of filtering, regardless of the mode set by the API or the presence of mipmaps.