 Diffusion tensor for image filtering

A diffusion tensor for image filtering selects directional transport in an image signal throughHere $e_n,e_t$ are estimated normal and tangent directions to an image edge. Taking $0<a_n\ll a_t$ promotes image smoothing along the image edge while reducing mixing across it. A structure tensor provides orientation estimates.

For a prescribed sufficiently regular $D(x,t)$, positive eigenvalues give a forward parabolic partial differential equation; fixed finite upper and strictly positive lower bounds on the eigenvalues give uniform ellipticity. This statement concerns the principal coefficient of the equation actually being solved.

If $D$ depends directly on the evolving gradient, its positivity alone is insufficient. In one dimension, $D(q)=1/(1+q^2)>0$ gives flux $\Phi (q)=q/(1+q^2)$, butis negative when $|q|>1$. Linearizing then gives backward diffusion in that range. Distinguishing a prescribed tensor from a gradient-dependent nonlinear flux prevents a false general parabolicity claim.