Edge-preserving smoothing or edge-preserving filtering is an

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

technique that smooths away noise or textures while retaining sharp edges. Examples are the

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

bilateral Bilateral may refer to any concept including two sides, in particular: *Bilateria, bilateral animals *Bilateralism, the political and cultural relations between two states *Bilateral, occurring on both sides of an organism ( Anatomical terms of l ...

, guided,

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 t ...

, and Kuwahara filters.

Introduction

In many applications, e.g., medical or satellite imaging, the edges are key features and thus must be preserved sharp and undistorted in smoothing/denoising. Edge-preserving filters are designed to automatically limit the smoothing at “edges” in images measured, e.g., by high gradient magnitudes. For example, the motivation for

(also called nonuniform or variable conductance diffusion) is that a Gaussian smoothed image is a single time slice of the solution to the heat equation, that has the original image as its initial conditions. Anisotropic diffusion includes a variable conductance term that is determined using the differential structure of the image, such that the heat does not propagate over the edges of the image. The edge-preserving filters can conveniently be formulated in a general context of graph-based

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

, where the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...

is first determined using the differential structure of the image, then the

graph Laplacian In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Lap ...

is formulated (analogous to the

operator), and finally the approximate low-pass filter is constructed to amplify the

eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...

of the graph Laplacian corresponding to its smallest

eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

. Since the edges only implicitly appear in constructing the edge-preserving filters, a typical filter uses some parameters, that can be tuned, to balance between aggressive averaging and edge preservation. A common default choice for the parameters of the filter is aimed for natural images and results in strong denoising at the cost of some smoothing of the edges.

Iterative filters

Requirements of the strict edge preservation commonly limit the smoothing power of the filter, such that a single application of the filter still results in unacceptably large noise away from the edges. A repetitive application of the filter may be useful to reduce the noise, leading to the idea of combining the filter with an

iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...

, e.g., the

Chebyshev iteration In numerical linear algebra, the Chebyshev iteration is an iterative method for determining the solutions of a system of linear equations. The method is named after Russian mathematician Pafnuty Chebyshev. Chebyshev iteration avoids the computation ...

and the

conjugate gradient method In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an it ...

are proposed in for graph-based image denoising. Due to the interpretation of the edge-preserving filters as low-pass graph-based filters, iterative eigenvalue solvers, such as

LOBPCG Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem :A x= \lambda B x, for a g ...

, can be used for

denoising Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an u ...

; see, e.g., to accelerate the repeated application of the

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 excess ...

Edge-enhancing smoothing

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 t ...

generates small conductance at the location of the edge of the image to prevent the heat flow over the edge, thus making the

filter edge-preserving. In the graph-based interpretation, the small conductance corresponds to a small weight of an edge of the graph describing a

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

of a

random walk In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a rand ...

over the edge in the

Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...

on the graph. If the graph weight was negative, that would correspond to a negative conductivity in the

heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...

, stimulating the heat concentration at the graph vertices connected by the graph edge, rather than the normal heat

dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...

. While not-physical for the

, this effect results in sharpening corners of one-dimensional signals, when used in graph-based smoothing filters, as shown in reference that also provides an alternative physical interpretation using the

wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...

describing mechanical vibrations of a mass-spring system with some repulsive springs.

Edge-preserving upsampling

Signal

upsampling In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of sample rate conversion, resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'' ...

via the traditional interpolation followed by smoothing for denoising evidently distorts the edges in the original ideal or downsampled signal. The edge-preserving interpolation followed by the edge-preserving filters is proposed in e.g., to upsample a no-flash RGB photo guided using a high resolution flash RGB photo, and a depth image guided using a high resolution RGB photo.

References

{{Reflist Image processing

Introduction

Iterative filters

Edge-enhancing smoothing

Edge-preserving upsampling

See also

References