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 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 ...

, 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 sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, 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 discre ...

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

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 Lapl ...

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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

of the graph Laplacian corresponding to its smallest eigenvalues. 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 mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pr ...

, e.g., the Chebyshev iteration and the conjugate gradient method 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 ...

, 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 un ...

; 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 excessi ...

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 ...

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 the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...

of a

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

over the edge in the Markov chain on the graph. If the graph weight was negative, that would correspond to a negative conductivity in the heat equation, 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 takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...

. While not-physical for the heat equation, 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 (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...

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 resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe a ...

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