Page 169 -

148 3 Image processing

y y y

x’ x’ x’
a x’x
y’ y’ y’
a y’x a x’y
a y’y minor axis major axis

x x x

(a) (b) (c)

Figure 3.48 Anisotropic texture filtering: (a) Jacobian of transform A and the induced horizontal and vertical
resampling rates {a x x ,a x y ,a y x ,a y y }; (b) elliptical footprint of an EWA smoothing kernel; (c) anisotropic
filtering using multiple samples along the major axis. Image pixels lie at line intersections.

One simple solution is to resample the texture from the next higher or lower pyramid level,
depending on whether it is preferable to reduce aliasing or blur. A better solution is to re-
sample both images and blend them linearly using the fractional component of l. Since most
MIP-map implementations use bilinear resampling within each level, this approach is usu-
ally called trilinear MIP-mapping. Computer graphics rendering APIs, such as OpenGL and
Direct3D, have parameters that can be used to select which variant of MIP-mapping (and of
the sampling rate r computation) should be used, depending on the desired tradeoff between
speed and quality. Exercise 3.22 has you examine some of these tradeoffs in more detail.

Elliptical Weighted Average

The Elliptical Weighted Average (EWA) filter invented by Greene and Heckbert (1986)is

based on the observation that the affine mapping x = Ax defines a skewed two-dimensional
coordinate system in the vicinity of each source pixel x (Figure 3.48a). For every destina-

tion pixel x , the ellipsoidal projection of a small pixel grid in x onto x is computed (Fig-

ure 3.48b). This is then used to filter the source image g(x) with a Gaussian whose inverse
covariance matrix is this ellipsoid.
Despite its reputation as a high-quality filter (Akenine-M¨ oller and Haines 2002), we have
found in our work (Szeliski, Winder, and Uyttendaele 2010) that because a Gaussian kernel
is used, the technique suffers simultaneously from both blurring and aliasing, compared to
higher-quality filters. The EWA is also quite slow, although faster variants based on MIP-
mapping have been proposed (Szeliski, Winder, and Uyttendaele (2010) provide some addi-
tional references).

Anisotropic filtering

An alternative approach to filtering oriented textures, which is sometimes implemented in
graphics hardware (GPUs), is to use anisotropic filtering (Barkans 1997; Akenine-M¨ oller and
Haines 2002). In this approach, several samples at different resolutions (fractional levels in
the MIP-map) are combined along the major axis of the EWA Gaussian (Figure 3.48c).