Anisotropic diffusion
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In
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
and
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing
image noise Image noise is random variation of brightness or color information in images, and is usually an aspect of electronic noise. It can be produced by the image sensor and circuitry of a scanner or digital camera. Image noise can also originate in ...
without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image.
Anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
diffusion resembles the process that creates a
scale space Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
, where an image generates a parameterized family of successively more and more blurred images based on a
diffusion process In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
. Each of the resulting images in this family are given as a
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
between the image and a 2D
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
Gaussian filter In electronics and signal processing mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would have infinite impulse respons ...
, 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. In its original formulation, presented by Perona and
Malik Malik, Mallik, Melik, Malka, Malek, Maleek, Malick, Mallick, or Melekh ( phn, 𐤌𐤋𐤊; ar, ملك; he, מֶלֶךְ) is the Semitic term translating to "king", recorded in East Semitic and Arabic, and as mlk in Northwest Semitic duri ...
in 1987, the space-variant filter is in fact isotropic but depends on the image content such that it approximates an
impulse function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
close to edges and other structures that should be preserved in the image over the different levels of the resulting
scale space Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
. This formulation was referred to as ''anisotropic diffusion'' by Perona and Malik even though the locally adapted filter is isotropic, but it has also been referred to as ''inhomogeneous and nonlinear diffusion'' or ''Perona–Malik diffusion'' by other authors. A more general formulation allows the locally adapted filter to be truly anisotropic close to linear structures such as edges or lines: it has an orientation given by the structure such that it is elongated along the structure and narrow across. Such methods are referred to as '' shape-adapted smoothing'' or ''coherence enhancing diffusion''. As a consequence, the resulting images preserve linear structures while at the same time smoothing is made along these structures. 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(or
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
) value (see
structure tensor In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the inf ...
). Although the resulting family of images can be described as a combination between the original image and space-variant filters, the locally adapted filter and its combination with the image do not have to be realized in practice. 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 Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
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.


Formal definition

Formally, let \Omega \subset \mathbb^2 denote a subset of the plane and I(\cdot,t): \Omega \rightarrow \mathbb be a family of gray scale images. I(\cdot, 0) is the input image. Then anisotropic diffusion is defined as : \frac = \operatorname \left( c(x,y,t) \nabla I \right)= \nabla c \cdot \nabla I + c(x,y,t) \, \Delta I where \Delta denotes the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, \nabla denotes the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
, \operatorname(\cdots) is the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
operator and c(x,y,t) is the diffusion coefficient. For t > 0 , the output image is available as 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. Pietro Perona and
Jitendra Malik Jitendra Malik is an Indian-American academic who is the Arthur J. Chick Professor of Electrical Engineering and Computer Sciences at the University of California, Berkeley. He is known for his research in computer vision. Academic biography ...
pioneered the idea of anisotropic diffusion in 1990 and proposed two functions for the diffusion coefficient: : c\left(\, \nabla I\, \right) = e^ and : c\left(\, \nabla I\, \right) = \frac the constant ''K'' controls the sensitivity to edges and is usually chosen experimentally or as a function of the noise in the image.


Motivation

Let M denote the manifold of smooth images, then the diffusion equations presented above can be interpreted as the
gradient descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
equations for the minimization of the energy functional E: M \rightarrow \mathbb defined by : E = \frac \int_ g\left( \, \nabla I(x)\, ^2 \right)\, dx where g:\mathbb \rightarrow \mathbb is a real-valued function which is intimately related to the diffusion coefficient. Then for any compactly supported infinitely differentiable test function h , : \begin \left.\frac \_ E + th&= \frac \big, _\frac \int_\Omega g\left( \, \nabla (I+th)(x)\, ^2 \right)\, dx \\ pt &= \int_\Omega g'\left(\, \nabla I(x)\, ^2 \right) \nabla I \cdot \nabla h\, dx \\ pt &= -\int_\Omega \operatorname(g'\left( \, \nabla I(x)\, ^2 \right) \nabla I) h\, dx \end where the last line follows from multidimensional integration by parts. Letting \nabla E_I denote the gradient of E with respect to the L^2(\Omega, \mathbb)
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
evaluated at I, this gives : \nabla E_I = - \operatorname(g'\left( \, \nabla I(x)\, ^2 \right) \nabla I) Therefore, the
gradient descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
equations on the functional ''E'' are given by : \frac = - \nabla E_I = \operatorname(g'\left( \, \nabla I(x)\, ^2 \right) \nabla I) Thus by letting c = g' the anisotropic diffusion equations are obtained.


Regularization

The diffusion coefficient, c(x,y,t) , as proposed by Perona and Malik can lead to instabilities when \, \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. To this end one of the ''modified Perona–Malik models'' (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 : \frac=\operatorname \left(c(, \nabla(G_\sigma * I), ^2)\nabla I \right) where G_\sigma=C\sigma^\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.


Applications

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 In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
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.


See also

*
Bilateral filter A bilateral filter is a non-linear, edge-preserving, and noise-reducing smoothing filter for images. It replaces the intensity of each pixel with a weighted average of intensity values from nearby pixels. This weight can be based on a Gaussian ...
*
Edge detection Edge detection includes a variety of mathematical methods that aim at identifying edges, curves in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The same problem of finding discontinuitie ...
*
Edge-preserving smoothing Edge-preserving smoothing or edge-preserving filtering is an image processing technique that smooths away noise or textures while retaining sharp edges. Examples are the median, bilateral, guided, anisotropic diffusion, and Kuwahara filters. I ...
*
Heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
*
Image noise Image noise is random variation of brightness or color information in images, and is usually an aspect of electronic noise. It can be produced by the image sensor and circuitry of a scanner or digital camera. Image noise can also originate in ...
* Noise reduction *
Scale space Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
*
Total variation denoising In signal processing, particularly image processing, total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process (filter). It is based on the principle that signals with excessi ...
*
Bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...


References


External links

*Mathematic
PeronaMalikFilter
function. * IDL nonlinear anisotropic diffusion package(edge enhancing and coherence enhancing)

{{Noise, state=uncollapsed Image processing Image noise reduction techniques