This method was implemented in mental ray in order to provide a very high quality texture filtering, far superior to the pyramid filtering explained above. It eliminates most if not all of the aliasing in high texture frequencies. When using checkerboard textures mapped onto a rectangle, for example, there is much less blurring at the horizon where the texture compression is severe. With mip-map textures as explained above, the blurring at such extreme compressions is sometimes still visible.

The main cause for the excessively blurry-looking images using mip-maps is the approximation of the pixel projection area by a square. With elliptical filtering a circle around the current sampling location is projected to texture space and will give either a circle or an ellipse as a projection shape. Instead of approximating this curve by simple shapes like squares, a direct convolution (averaging) of all texels which are inside the ellipse area is done. Averaging all texels in this area can take quite long, so mental ray uses pyramids of prefiltered textures to accelerate this. There are various parameters explained below which control modification of ellipse shape and level selection in the pyramid.

The most difficult part when elliptical projections are used is
that a screen to texture space transformation matrix has to be
provided. This matrix is used in the filtering code to transform
the circle around the current sampling location to texture space.
mental ray provides two helper functions for constructing this
matrix when UV texture coordinates are available; see *mi_texture_filter_project* in the
Writing Shaders chapter. If those are not available and (for
example) direct cylinder projective mappings are used, it is much
easier to calculate this matrix.

The following filtering algorithm is applied: first, a circle in the current sampling location is transformed to the ellipse using the provided transformation matrix. Then the eccentricity of the ellipse is calculated (major radius divided by minor radius). If the eccentricity is larger than a specified maximum, the minor radius is adjusted (made larger) to make sure that this eccentricity maximum always holds. The reason for this enlargement is that the direct convolution is done in the pyramid level based on the minor axis length of the ellipse. There is another parameter which specifies the maximum allowed number of texels the minor radius may cover. If that number is exceeded in the finest level (zero), a higher level is used. In the second level, for example, the minor radius has half the size etc. Enlarging the minor radius when the eccentricity is exceeded, basically means that we are going up in the pyramid. So, for very large ellipses, mental ray is making them "fatter" and uses a higher level in the pyramid. Referring to the checkerboard-mapped plane example above, the circle is projected to very large thin ellipses near the horizon, covering thousands of texels, and using the technique above, mental ray just makes a few texture lookups in the higher pyramid levels.

There is another parameter which modifies the size of the circle to be projected, usually the radius is 0.5, making it larger introduces more blurring, making it less gives more aliasing.

The projection helper functions expect another parameter which is the maximum offset of the central sampling location to the two other points which have to be selected. The other two points should be inside the pixel, but since mental ray is using the current intersection primitive (the triangle) also for these points to determine the UV texture coordinates, a smaller value than 0.5 (pixel corners) is appropriate since mental ray might hit the triangle plane outside the triangle area. Usually 0.3 gives good results. When the UV coordinates are calculated using cylinder projections, it is possible to obtain the UV coordinates much faster and also much more accurately.

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