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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,It is Gaussian image smoothing with variance $2t$ in each coordinate. In Fourier variables, $\hat{u}(\xi ,t)=e^{-t|\xi {|}^2}\hat{g}(\xi )$: 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, andThese 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 $g+\lambda (g-G_t\ast g)$, 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 areFor a smooth solution with the corresponding zero-flux condition, $dE/dt=-\int u_t^2\le 0$. 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 isIts coefficient along an image signal level curve is $c(s)$; across that curve it is $c(s)+s{c}^{\prime }(s)={\Psi }^{\prime \prime }(s)$. 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 $\Psi (s)=\sqrt{{\epsilon }^2+s^2}$, the coefficients areDiffusion across steep image edges is weak but remains forward. As $\epsilon \to 0$, the total variation flow formally becomes $u_t=\mathrm{div}(\nabla u/|\nabla u|)$. 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,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 $s>\kappa /\sqrt{2}$. 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 exampleThe 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:Here $e_n$ estimates the image edge normal and $e_t$ its tangent. A structure tensor $J_{\rho }=G_{\rho }\ast (\nabla u_{\sigma }\nabla u_{\sigma }^T)$ 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 $u_t=-\mathrm{sign}(\Delta u)|\nabla u|$. 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.

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.

A point-sampling model assigns $p_{rc}=f(x_c,y_r)$ to a pixel on a grid, where $f$ is a continuous function representing an image signal. An area-sampling model instead assigns the average value of a function over a specified footprint. These are different models: a narrow feature between point samples can be missed, whereas its contribution to an area sample depends on its width and shade. Neither model says that the recorded value uniquely determines everything inside a grid cell.

The upper panels start with exactly the same $6\times 4$ array. Nearest-neighbor interpolation repeats its values into a $24\times 16$ display; successive linear interpolations in the two coordinate directions produce a smoother $24\times 16$ display. Original sample positions are marked. Both displays use 384 output pixels, but only the same 24 recorded values. The extra values follow the chosen interpolation rule. Between-grid appearance is inferred, not independently recorded.

Spatial resolution needs a stated meaning. A grid over a rectangle of width $L_x$ and height $L_y$ has nominal cell pitches $\Delta x=L_x/W$ and $\Delta y=L_y/H$. Over the same rectangle, doubling both dimensions halves these pitches and quadruples the pixel count. That is a finer grid, not a guarantee of more recoverable detail. Blurring before sampling, quantization and aliasing can make different originals share the same recorded array. Enlarging the rectangle without adding samples instead increases the displayed pitch.

A reproducible computational demonstration makes the ambiguity exact. Take the one-dimensional profiles $f(x)=0.5+0.4\sin (2\pi x)$ and $g(x)=0.5+0.4\sin (10\pi x)$, repeated identically along every row. Sample at $x_j=j/4$ for $j=0,\dots ,8$. Since $10\pi x_j=2\pi x_j+2\pi j$, the fact that the sine is a periodic function gives $f(x_j)=g(x_j)$. Both yield $0.5,0.9,0.5,0.1,0.5,0.9,0.5,0.1,0.5$, although at $x=1/8$ they give approximately $0.78284$ and $0.21716$. The lower panel draws both originals and their common samples: no operation on those samples alone can identify the original profile. This is aliasing in an ideal point-sampling model, without image noise or rounding.

The Nyquist–Shannon sampling theorem describes reconstruction under a bandwidth restriction and ideal sampling assumptions; it does not grant uniqueness for arbitrary image signals from a finite raster image.^[ref] The beautiful lesson is constructive: choose what detail the grid should preserve before sampling, and choose a reconstruction rule suited to the information actually retained.