= Solution
In <diffusion image processing>, the observed grey-value <image signal> $g$ is the initial condition for an evolution $u(x,t)$; time controls the <image smoothing> scale. A useful model must suppress <image noise> while respecting the <image signal>'s geometric boundaries. Unless exterior values are intended, use periodic boundaries or <Neumann boundary conditions>, so the filter does not lose intensity through the <image signal> border.
The basic linear model is the <heat equation>, $u_t=\Delta u$, $u(0)=g$. On the whole plane,
$$
\boxed{u(\cdot,t)=G_t*g,\qquad
G_t(x)=\frac1{4\pi t}e^{-|x|^2/(4t)}.}
$$
It is Gaussian <image smoothing> with variance $2t$ in each coordinate. In Fourier variables, $\widehat u(\xi,t)=e^{-t|\xi|^2}\widehat g(\xi)$: high spatial frequencies are damped most strongly. Under zero-flux or periodic boundaries the mean is conserved, the maximum principle keeps values within the initial range, and
$$
\frac d{dt}\frac12\int_\Omega(u-\bar u)^2dx=-\int_\Omega|\nabla u|^2dx\leq0.
$$
These give stable <image noise> suppression, but a sharp step also contains high frequencies and is blurred across a width of order $\sqrt t$. Constant diffusivity has no mechanism for distinguishing <image noise> from an <image edge>.
Linear sharpening by the backward <heat equation> would multiply Fourier modes by $e^{t|\xi|^2}$ and amplify arbitrarily fine <image noise> without bound. It is an ill-posed initial-value problem. A controlled <unsharp masking> step instead uses $g+\lambda(g-G_t*g)$, with multiplier $1+\lambda(1-e^{-t|\xi|^2})$. This amplifies high frequencies by at most $1+\lambda$; it can improve apparent contrast but also amplifies <image noise> and cannot reliably restore information already lost by <image smoothing>.
Nonlinear models use <image signal> structure to select the diffusion. A gradient-based energy and its formal $L^2$ <gradient flow> are
$$
E(u)=\int_\Omega\Psi(|\nabla u|)dx+\frac\lambda2\int_\Omega(u-g)^2dx,
\qquad
u_t=\operatorname{div}\big(c(|\nabla u|)\nabla u\big)-\lambda(u-g),
\quad c(s)=\frac{\Psi'(s)}s.
$$
For a smooth solution with the corresponding zero-flux condition, $dE/dt=-\int u_t^2\leq0$. The fidelity term prevents indefinite drift towards a constant reconstruction; pure diffusion is usually stopped at a selected finite time.
The local <principal diffusion coefficients> distinguish suppression of diffusion from backward diffusion. Where $s=|\nabla u|>0$, the flux <derivative> is
$$
c(s)I+\frac{c'(s)}s\nabla u\otimes\nabla u.
$$
Its coefficient along an <image signal> level curve is $c(s)$; across that curve it is $c(s)+sc'(s)=\Psi''(s)$. Forward parabolic behavior requires both coefficients nonnegative, with strict positive lower bounds giving uniform parabolicity. Merely choosing $c>0$ and decreasing does not establish well-posedness.
For a convex area-type penalty $\Psi(s)=\sqrt{\varepsilon^2+s^2}$, the coefficients are
$$
c(s)=\frac1{\sqrt{\varepsilon^2+s^2}},\qquad
c(s)+sc'(s)=\frac{\varepsilon^2}{(\varepsilon^2+s^2)^{3/2}}>0.
$$
Diffusion across steep <image edges> is weak but remains forward. As $\varepsilon\to0$, the <total variation flow> formally becomes $u_t=\operatorname{div}(\nabla u/|\nabla u|)$. Its convex <subgradient> formulation handles flat regions and discontinuities. It favors piecewise constant <image signals> and preserves sharp transitions better than Gaussian <image smoothing>, but can produce <staircasing in total variation denoising>, shrink small objects and move boundaries by <curvature>. <image edge> preservation does not mean exact preservation of all <image edge> locations or amplitudes.
The <Perona-Malik equation> takes a decreasing diffusivity such as $c(s)=1/(1+s^2/\kappa^2)$. Small <gradients> are smoothed strongly, while large <gradients> have weak flux. More precisely,
$$
\boxed{c(s)+sc'(s)=\frac{1-s^2/\kappa^2}{(1+s^2/\kappa^2)^2}.}
$$
For $s>\kappa$, diffusion in the <gradient> direction is backward, so a strong transition can steepen. This gives formal <image edge> enhancement but also causes instability and continuum <ill-posedness>. The exponential choice $c(s)=e^{-s^2/\kappa^2}$ similarly has a negative normal coefficient for $s>\kappa/\sqrt2$. Discrete results depend on the stencil, step size and implicit regularization; appealing visual results are not a proof of a well-posed PDE.
A <regularized Perona-Malik diffusion> computes the conductance from a smoothed <image signal>, for example
$$
u_t=\operatorname{div}\big(c(|\nabla(G_\sigma*u)|)\nabla u\big).
$$
The flux still acts on $u$, but the <image edge> detector is less sensitive to raw <image noise>. With a smooth kernel and a positive conductance bounded away from zero on the attained range, the local principal diffusion is forward; appropriate regularity and boundary assumptions give a well-posed nonlocal parabolic model. The exact regularization and fixed <image smoothing> scale are part of the model.
A genuinely directional filter uses a <diffusion tensor for image filtering>:
$$
u_t=\operatorname{div}(D\nabla u),\qquad
D=a_n e_ne_n^T+a_t e_te_t^T,\quad 0<a_n\ll a_t.
$$
Here $e_n$ estimates the <image edge> normal and $e_t$ its tangent. A <structure tensor> $J_\rho=G_\rho*(\nabla u_\sigma\nabla u_\sigma^T)$ provides robust directions at a second averaging scale. Strong tangent diffusion smooths <image noise> along an <image edge>, while weak normal diffusion reduces mixing across it. Related coherence-enhancing choices connect elongated features along their dominant orientation. Positive tensor <eigenvalues> preserve forward parabolicity; this type of enhancement must be distinguished from Perona–Malik's backward normal diffusion.
For more direct sharpening, a <shock filter for image enhancement> formally evolves $u_t=-\operatorname{sign}(\Delta u)|\nabla u|$. It is a Hamilton–Jacobi-type transport mechanism, not a positive diffusion operator, and steepens transitions around inflection boundaries. In practice it can be combined with regularized forward diffusion to control <image noise>. Any enhancement method needs a scale or stopping rule and an honest treatment of <image noise> amplification.
\b[Linear heat flow offers predictable <image smoothing> but blurs <image edges>; convex nonlinear diffusion can preserve them; backward or shock mechanisms sharpen them at a greater stability cost.] <Gradient> thresholds, conductance regularization and positive tensor directions determine which of these behaviors a proposed filter actually has.
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