Texture Mapping - Perspective Correctness

Perspective Correctness

Texture coordinates are specified at each vertex of a given triangle, and these coordinates are interpolated using an extended Bresenham's line algorithm. If these texture coordinates are linearly interpolated across the screen, the result is affine texture mapping. This is a fast calculation, but there can be a noticeable discontinuity between adjacent triangles when these triangles are at an angle to the plane of the screen (see figure at right – textures (the checker boxes) appear bent).

Perspective correct texturing accounts for the vertices' positions in 3D space, rather than simply interpolating a 2D triangle. This achieves the correct visual effect, but it is slower to calculate. Instead of interpolating the texture coordinates directly, the coordinates are divided by their depth (relative to the viewer), and the reciprocal of the depth value is also interpolated and used to recover the perspective-correct coordinate. This correction makes it so that in parts of the polygon that are closer to the viewer the difference from pixel to pixel between texture coordinates is smaller (stretching the texture wider), and in parts that are farther away this difference is larger (compressing the texture).

Affine texture mapping directly interpolates a texture coordinate between two endpoints and :

where

Perspective correct mapping interpolates after dividing by depth, then uses its interpolated reciprocal to recover the correct coordinate:

All modern 3D graphics hardware implements perspective correct texturing.

Classic texture mappers generally did only simple mapping with at most one lighting effect, and the perspective correctness was about 16 times more expensive. To achieve two goals - faster arithmetic results, and keeping the arithmetic mill busy at all times - every triangle is further subdivided into groups of about 16 pixels. For perspective texture mapping without hardware support, a triangle is broken down into smaller triangles for rendering, which improves details in non-architectural applications. Software renderers generally preferred screen subdivision because it has less overhead. Additionally they try to do linear interpolation along a line of pixels to simplify the set-up (compared to 2d affine interpolation) and thus again the overhead (also affine texture-mapping does not fit into the low number of registers of the x86 CPU; the 68000 or any RISC is much more suited). For instance, Doom restricted the world to vertical walls and horizontal floors/ceilings. This meant the walls would be a constant distance along a vertical line and the floors/ceilings would be a constant distance along a horizontal line. A fast affine mapping could be used along those lines because it would be correct. A different approach was taken for Quake, which would calculate perspective correct coordinates only once every 16 pixels of a scanline and linearly interpolate between them, effectively running at the speed of linear interpolation because the perspective correct calculation runs in parallel on the co-processor. The polygons are rendered independently, hence it may be possible to switch between spans and columns or diagonal directions depending on the orientation of the polygon normal to achieve a more constant z, but the effort seems not to be worth it.

Another technique was subdividing the polygons into smaller polygons, like triangles in 3d-space or squares in screen space, and using an affine mapping on them. The distortion of affine mapping becomes much less noticeable on smaller polygons. Yet another technique was approximating the perspective with a faster calculation, such as a polynomial. Still another technique uses 1/z value of the last two drawn pixels to linearly extrapolate the next value. The division is then done starting from those values so that only a small remainder has to be divided, but the amount of bookkeeping makes this method too slow on most systems. Finally, some programmers extended the constant distance trick used for Doom by finding the line of constant distance for arbitrary polygons and rendering along it.