[1][2][3] Anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process.
Each of the resulting images in this family are given as a convolution between the image and a 2D isotropic Gaussian filter, where the width of the filter increases with the parameter.
This diffusion process is a linear and space-invariant transformation of the original image.
As a consequence, anisotropic diffusion is a non-linear and space-variant transformation of the original image.
In its original formulation, presented by Perona and Malik in 1987,[1] the space-variant filter is in fact isotropic but depends on the image content such that it approximates an impulse function close to edges and other structures that should be preserved in the image over the different levels of the resulting scale space.
Such methods are referred to as shape-adapted smoothing[6][7] or coherence enhancing diffusion.
Both these cases can be described by a generalization of the usual diffusion equation where the diffusion coefficient, instead of being a constant scalar, is a function of image position and assumes a matrix (or tensor) value (see structure tensor).
Anisotropic diffusion is normally implemented by means of an approximation of the generalized diffusion equation: each new image in the family is computed by applying this equation to the previous image.
Consequently, anisotropic diffusion is an iterative process where a relatively simple set of computation are used to compute each successive image in the family and this process is continued until a sufficient degree of smoothing is obtained.
Pietro Perona and Jitendra Malik pioneered the idea of anisotropic diffusion in 1990 and proposed two functions for the diffusion coefficient: and the constant K controls the sensitivity to edges and is usually chosen experimentally or as a function of the noise in the image.
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
is a real-valued function which is intimately related to the diffusion coefficient.
Then for any compactly supported infinitely differentiable test function
inner product evaluated at I, this gives Therefore, the gradient descent equations on the functional E are given by Thus by letting
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.
[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 where
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.
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.