Anisotropic Diffusion: Enhancing Image Analysis Through Anisotropic Diffusion

By Fouad Sabry

What is Anisotropic Diffusion

In image processing and computer vision, anisotropic diffusion, also called Perona-Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image. 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. Anisotropic diffusion is a generalization of this diffusion process: it produces a family of parameterized images, but each resulting image is a combination between the original image and a filter that depends on the local content of the original image. As a consequence, anisotropic diffusion is a non-linear and space-variant transformation of the original image.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Anisotropic diffusion

Chapter 2: Fick's laws of diffusion

Chapter 3: Diffusion equation

Chapter 4: Heat equation

Chapter 5: Navier-Stokes equations

Chapter 6: Total variation

Chapter 7: Divergence

Chapter 8: Laplace operator

Chapter 9: Curl (mathematics)

Chapter 10: Divergence theorem

(II) Answering the public top questions about anisotropic diffusion.

(III) Real world examples for the usage of anisotropic diffusion in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Anisotropic Diffusion.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Apr 28, 2024

Book preview

Chapter 1: Anisotropic diffusion

Anisotropic diffusion, also known as Perona-Malik diffusion, is a method used in image processing and computer vision to reduce noise in a picture without sacrificing interpretable image features like edges, lines, and other finer details. In anisotropic diffusion, an image develops a parameterized family of increasingly blurry images through a diffusion process, analogous to the process that builds a scale space. Each output image in this family is represented by the convolution of the original with an isotropic 2-dimensional Gaussian filter whose width scales as the parameter is increased. The image is transformed in a linear and space-invariant fashion by the diffusion process. In anisotropic diffusion, the original image is combined with a filter that is itself dependent on the local content of the original image to get a family of parameterized images. Therefore, anisotropic diffusion is a change of the original image that is both non-linear and space-variant.

Since its inception with Perona and Malik's 1987 presentation, the generated images have been able to maintain linear structures while still being smoothed along these same patterns. In both of these scenarios, the diffusion coefficient is a function of the spatial position of the picture and so takes on a matrix (or tensor) value rather than remaining a constant scalar (see structure tensor).

Although the locally adapted filter and its combination with the picture can be conceptualized as a combination of the original image and space-variant filters, this is not required for the resulting family of images. Each new image in the family is calculated by applying this equation to the preceding image, making anisotropic diffusion possible using an approximation of the generalized diffusion equation. To achieve the desired level of smoothing, anisotropic diffusion is an iterative process where a very simple set of computations are employed to compute each consecutive image in the family.

Formally, let \Omega \subset {\mathbb {R}}^{2} denote a subset of the plane and I(\cdot ,t):\Omega \rightarrow {\mathbb {R}} be a family of gray scale images.

{\displaystyle I(\cdot ,0)} is the input image.

Then, we can characterize anisotropic diffusion 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 \Delta denotes the Laplacian, \nabla denotes the gradient, {\displaystyle \operatorname {div} (\cdots )} is the divergence operator and c(x,y,t) is the diffusion coefficient.

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

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.

Anisotropic diffusion was first proposed in 1990 by Pietro Perona and Jitendra Malik, who also suggested two functions for the diffusion coefficient:

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

and

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

The constant K determines how sensitive the system is to edges; it is typically selected empirically or based on the level of image noise.

Let 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 E:M\rightarrow {\mathbb {R}} defined by

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

where 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 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 is a consequence of multi-part multi-dimensional integration.

Letting \nabla E_{I} denote the gradient of E with respect to the 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)}

This leads to the following equations for the gradient descent of the function E:

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

Thus by letting c=g' the anisotropic diffusion equations are obtained.

Coefficient of diffusion, c(x,y,t) , as proposed by Perona and Malik can lead to instabilities when {\displaystyle \|\nabla I\|^{2}>K^{2}} .

This condition is shown to be equivalent to a negative value for the physical diffusion coefficient (which is distinct from the mathematical diffusion coefficient defined by Perona and Malik) and thus results in backward diffusion that accentuates contrasts of image intensity rather than smoothing them.

Avoiding the issue, People have proved that spatial regularizations lead to a convergent and constant steady-state solution, hence regularization is required.

Regularization of the P-M equation (which will be explained) has another name:.

Using this method, To obtain a modified Perona-Malik equation, the unknown is convolved with a Gaussian inside the non-linearity.

{\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)}

.

This regularization allows the equation to be well-posed, but it also introduces the blurring effect that is typically associated with regularization. Since the regularization parameter must be selected beforehand, knowing the noise level in advance is essential.

Noise in digital photographs can be smoothed out using anisotropic diffusion without affecting the sharpness of the image's edges. The anisotropic diffusion equations can be simplified to the heat equation, which is identical to Gaussian blurring, when the diffusion coefficient remains constant. This works wonderfully for suppressing background noise, but it softens edges without discrimination. If the diffusion coefficient is used as a function to prevent sharp edges, as in the Perona-Malik method, then the resulting equations promote diffusion (and consequently smoothing) in the less intense parts of the image while inhibiting it across sharp edges. As a result, the image's boundaries are protected while the noise is reduced.

In a manner analogous to that of noise cancellation, edge detection algorithms can benefit from the application of anisotropic diffusion. After a certain number of iterations of diffusion with an edge-seeking diffusion coefficient, the image will have evolved to become piecewise constant, with the edges indicating the boundaries between the constant components.

{End Chapter 1}

Chapter 2: Fick's laws of diffusion

Adolf Fick originally proposed his laws of diffusion in 1855, describing diffusion on the basis of mostly experimental evidence. D. Fick's first law may be used to obtain his second law, which is equal to the diffusion equation, and both can be used to solve for the diffusion coefficient.

Normal or Fickian diffusion refers to a diffusion process that follows Fick's laws; anomalous or non-Fickian diffusion refers to a process that deviates from these rules.

The now-famous rules of mass diffusion were first described by scientist Adolf Fick in 1855. Inspiring Fick's work were Thomas Graham's previous investigations, which, while interesting, did not provide the essential laws for which Fick would become famous. Fick's law is comparable to other laws discovered at the same time by other luminaries, such as Darcy's law (hydraulic flow), Ohm's law (charge transport), and Fourier's Law (frequency analysis) (heat transport).

Based on Graham's work, Fick conducted experiments in which he measured the concentrations and fluxes of salt as it diffused through tubes of water from one reservoir to the other. Diffusion in solids was not generally thought to be conceivable at the time, therefore Fick's study focused exclusively on diffusion in fluids. non-Fickian is a term used to describe it.

According to Fick's first law, the diffusion flux is proportional to the concentration gradient. In its simplest form, it is the idea that a solute would migrate from a high-concentration to a low-concentration region across a concentration gradient, with the amount of the flux being proportional to the concentration gradient (spatial derivative). Different variants of the law can be expressed in a single (spatial) dimension, with the molar foundation being the most prevalent:

{\displaystyle J=-D{\frac {d\varphi }{dx}}}

where

J is the diffusion flux, of which the dimension is the amount of substance per unit area per unit time. J measures the amount of substance that will flow through a unit area during a unit time interval.

D is the diffusion coefficient or diffusivity. Its dimension is area per unit time.

φ (for ideal mixtures) is the concentration, which is measured in terms of the mass per unit volume.

x is position, the dimension of which is length.

D is proportional to the squared velocity of the diffusing particles, that is temperature dependent, accordance to the Stokes-Einstein relation, the viscosity of the fluid and the particle size.

In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of (0.6–2)×10−9 m²/s.

For biological molecules the diffusion coefficients normally range from 10−10 to 10−11 m²/s.

In two or more dimensions we must use ∇, gradient or del operator, It encompasses the first derivative in its whole, obtaining

{\displaystyle \mathbf {J} =-D\nabla \varphi }

The diffusion flux vector is denoted by the letter J.

The driving force for the one-dimensional diffusion is the quantity −∂φ/

∂x

, This is the concentration gradient for homogeneous mixtures.

Another form for the first law is to write it with the primary variable as mass fraction (yi, presented, for illustration, in kg/kg, Afterwards, the equation shifts to:

{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}}

where

The ith species is indicated by the index i, Ji is the diffusion flux vector of the ith species (for example in mol/m²-s), Mi is the molar mass of the ith species, and

ρ is the mixture density (for example in kg/m³).

Note that the \rho is outside the gradient operator.

For this reason:

{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }}}

where ρsi is the partial density of the ith species.

Furthermore, the chemical potential gradient of a species is the driving force for the diffusion of that species in chemical systems other than perfect solutions or mixtures. Then the first law of Fick's theory (in a single dimension) can be formulated.

J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}

where

The ith species is indicated by the index i.

c is the concentration (mol/m³).

R is the universal gas constant (J/K/mol).

T is the absolute temperature (K).

μ is the chemical potential (J/mol).

The fugacity differential is the driving factor behind Fick's law:

{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}}}

Fugacity f_{i} has Pa units.

f_{i} is a partial pressure of component i in a vapor {\displaystyle f_{i}^{G}} or liquid {\displaystyle f_{i}^{L}} phase.

At vapor liquid equilibrium the evaporation flux is zero because {\displaystyle f_{i}^{G}=f_{i}^{L}} .

You'll find four different formulations of Fick's law for binary gas mixtures here. These presuppose either constant pressure or that the molar masses of the two species are equal, that thermal diffusion is minimal, and that the body force per unit mass is