Anisotropic diffusion

Formally, let ${\displaystyle \Omega \subset \mathbb {R} ^{2}}$ denote a subset of the plane and ${\displaystyle I(\cdot ,t):\Omega \rightarrow \mathbb {R} }$ be a family of gray scale images. ${\displaystyle I(\cdot ,0)}$ is the input image. Then anisotropic diffusion is defined as

${\displaystyle {\frac {\partial I}{\partial t}}=\operatorname {div} \left(c(x,y,t)\nabla I\right)=\nabla c\cdot \nabla I+c(x,y,t)\,\Delta I}$

where ${\displaystyle \Delta }$ denotes the Laplacian, ${\displaystyle \nabla }$ denotes the gradient, ${\displaystyle \operatorname {div} (\cdots )}$ is the divergence operator and ${\displaystyle c(x,y,t)}$ is the diffusion coefficient.

For ${\displaystyle t>0}$, the output image is available as ${\displaystyle I(\cdot ,t)}$, with larger ${\displaystyle t}$ producing blurrier images.

${\displaystyle c(x,y,t)}$ controls the rate of diffusion and is usually chosen as a function of the image gradient so as to preserve edges in the image. Pietro Perona and Jitendra Malik pioneered the idea of anisotropic diffusion in 1990 and proposed two functions for the diffusion coefficient:

${\displaystyle c\left(\|\nabla I\|\right)=e^{-\left(\|\nabla I\|/K\right)^{2}}}$

and

${\displaystyle c\left(\|\nabla I\|\right)={\frac {1}{1+\left({\frac {\|\nabla I\|}{K}}\right)^{2}}}}$

the constant K controls the sensitivity to edges and is usually chosen experimentally or as a function of the noise in the image.

Let ${\displaystyle M}$ denote the manifold of smooth images, then the diffusion equations presented above can be interpreted as the gradient descent equations for the minimization of the energy functional ${\displaystyle E:M\rightarrow \mathbb {R} }$ defined by

${\displaystyle E[I]={\frac {1}{2}}\int _{\Omega }g\left(\|\nabla I(x)\|^{2}\right)\,dx}$

where ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }$ is a real-valued function which is intimately related to the diffusion coefficient. Then for any compactly supported infinitely differentiable test function ${\displaystyle h}$,

${\displaystyle {\begin{aligned}\left.{\frac {d}{dt}}\right|_{t=0}E[I+th]&={\frac {d}{dt}}{\big |}_{t=0}{\frac {1}{2}}\int _{\Omega }g\left(\|\nabla (I+th)(x)\|^{2}\right)\,dx\\[5pt]&=\int _{\Omega }g'\left(\|\nabla I(x)\|^{2}\right)\nabla I\cdot \nabla h\,dx\\[5pt]&=-\int _{\Omega }\operatorname {div} (g'\left(\|\nabla I(x)\|^{2}\right)\nabla I)h\,dx\end{aligned}}}$

where the last line follows from multidimensional integration by parts. Letting ${\displaystyle \nabla E_{I}}$ denote the gradient of E with respect to the ${\displaystyle L^{2}(\Omega ,\mathbb {R} )}$ inner product evaluated at I, this gives

${\displaystyle \nabla E_{I}=-\operatorname {div} (g'\left(\|\nabla I(x)\|^{2}\right)\nabla I)}$

Therefore, the gradient descent equations on the functional E are given by

${\displaystyle {\frac {\partial I}{\partial t}}=-\nabla E_{I}=\operatorname {div} (g'\left(\|\nabla I(x)\|^{2}\right)\nabla I)}$

Thus by letting ${\displaystyle c=g'}$ the anisotropic diffusion equations are obtained.

The diffusion coefficient, ${\displaystyle c(x,y,t)}$, as proposed by Perona and Malik can lead to instabilities when ${\displaystyle \|\nabla I\|^{2}>K^{2}}$. It can be proven that this condition is equivalent to the physical diffusion coefficient (which is different from the mathematical diffusion coefficient defined by Perona and Malik) becoming negative and it leads to backward diffusion that enhances contrasts of image intensity rather than smoothing them. To avoid the problem, regularization is necessary and people have shown that spatial regularizations lead to converged and constant steady-state solution.[9]

To this end one of the modified Perona–Malik models[10] (which is also known as regularization of P-M equation) will be discussed. In this approach, the unknown is convolved with a Gaussian inside the non-linearity to obtain a modified Perona–Malik equation

${\displaystyle {\frac {\partial I}{\partial t}}=\operatorname {div} \left(c(|\nabla (G_{\sigma }*I)|^{2})\nabla I\right)}$

where ${\displaystyle G_{\sigma }=C\sigma ^{-1/2}\exp \left(-|x|^{2}/4\sigma \right)}$.

The well-posedness of the equation can be achieved by this regularization but it also introduces blurring effect, which is the main drawback of regularization. A prior knowledge of noise level is required as the choice of regularization parameter depends on it.

Anisotropic diffusion can be used to remove noise from digital images without blurring edges. With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the heat equation which is equivalent to Gaussian blurring. This is ideal for removing noise but also indiscriminately blurs edges too. When the diffusion coefficient is chosen as an edge avoiding function, such as in Perona–Malik, the resulting equations encourage diffusion (hence smoothing) within regions of smoother image intensity and suppress it across strong edges. Hence the edges are preserved while removing noise from the image.

Along the same lines as noise removal, anisotropic diffusion can be used in edge detection algorithms. By running the diffusion with an edge seeking diffusion coefficient for a certain number of iterations, the image can be evolved towards a piecewise constant image with the boundaries between the constant components being detected as edges.

• Bilateral filter
• Edge detection
• Edge-preserving smoothing
• Heat equation
• Image noise
• Noise reduction
• Scale space
• Total variation denoising
• Bounded variation

• Mathematica PeronaMalikFilter function.
• IDL nonlinear anisotropic diffusion package(edge enhancing and coherence enhancing):