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 distribution. Crucially, the weights depend not only on Euclidean distance of pixels, but also on the radiometric differences (e.g., range differences, such as color intensity, depth distance, etc.). This preserves sharp edges.
The bilateral filter is defined as [1] [2]
and normalization term, , is defined as
where
The weight is assigned using the spatial closeness (using the spatial kernel ) and the intensity difference (using the range kernel ). [2] Consider a pixel located at that needs to be denoised in image using its neighbouring pixels and one of its neighbouring pixels is located at . Then, assuming the range and spatial kernels to be Gaussian kernels, the weight assigned for pixel to denoise the pixel is given by
where σd and σr are smoothing parameters, and I(i, j) and I(k, l) are the intensity of pixels and respectively.
After calculating the weights, normalize them:
where is the denoised intensity of pixel .
The bilateral filter in its direct form can introduce several types of image artifacts:
There exist several extensions to the filter that deal with these artifacts, like the scaled bilateral filter that uses downscaled image for computing the weights. [5] Alternative filters, like the guided filter, [6] have also been proposed as an efficient alternative without these limitations.
Adobe Photoshop implements a bilateral filter in its surface blur tool. GIMP implements a bilateral filter in its Filters → Blur tools; and it is called Selective Gaussian Blur. The free G'MIC plugin Repair → Smooth [bilateral] for GIMP adds more control. [7] A simple trick to efficiently implement a bilateral filter is to exploit Poisson-disk subsampling. [1]
The bilateral filter has been shown to be an application of the short time kernel of the Beltrami flow [8] [9] [10] that was introduced as an edge preserving selective smoothing mechanism before the bilateral filter.
Other edge-preserving smoothing filters include: anisotropic diffusion, [11] weighted least squares, [12] edge-avoiding wavelets, [13] geodesic editing, [14] guided filtering, [15] and domain transforms. [16]
In digital signal processing, spatial anti-aliasing is a technique for minimizing the distortion artifacts (aliasing) when representing a high-resolution image at a lower resolution. Anti-aliasing is used in digital photography, computer graphics, digital audio, and many other applications.
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The Canny edge detector is an edge detection operator that uses a multi-stage algorithm to detect a wide range of edges in images. It was developed by John F. Canny in 1986. Canny also produced a computational theory of edge detection explaining why the technique works.
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 undesired signal component from the desired signal component, as with common-mode rejection ratio.
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph, the discrete Laplace operator is more commonly called the Laplacian matrix.
In image processing, a Gaussian blur is the result of blurring an image by a Gaussian function.
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In imaging science, difference of Gaussians (DoG) is a feature enhancement algorithm that involves the subtraction of one Gaussian blurred version of an original image from another, less blurred version of the original. In the simple case of grayscale images, the blurred images are obtained by convolving the original grayscale images with Gaussian kernels having differing width. Blurring an image using a Gaussian kernel suppresses only high-frequency spatial information. Subtracting one image from the other preserves spatial information that lies between the range of frequencies that are preserved in the two blurred images. Thus, the DoG is a spatial band-pass filter that attenuates frequencies in the original grayscale image that are far from the band center.
In optics, a caustic or caustic network is the envelope of light rays which have been reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. In some cases caustics can be seen as patches of light or their bright edges, shapes which often have cusp singularities.
Corner detection is an approach used within computer vision systems to extract certain kinds of features and infer the contents of an image. Corner detection is frequently used in motion detection, image registration, video tracking, image mosaicing, panorama stitching, 3D reconstruction and object recognition. Corner detection overlaps with the topic of interest point detection.
Mean shift is a non-parametric feature-space mathematical analysis technique for locating the maxima of a density function, a so-called mode-seeking algorithm. Application domains include cluster analysis in computer vision and image processing.
In the fields of computer vision and image analysis, the Harris affine region detector belongs to the category of feature detection. Feature detection is a preprocessing step of several algorithms that rely on identifying characteristic points or interest points so to make correspondences between images, recognize textures, categorize objects or build panoramas.
Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling. Pyramid representation is a predecessor to scale-space representation and multiresolution analysis.
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.
The random walker algorithm is an algorithm for image segmentation. In the first description of the algorithm, a user interactively labels a small number of pixels with known labels, e.g., "object" and "background". The unlabeled pixels are each imagined to release a random walker, and the probability is computed that each pixel's random walker first arrives at a seed bearing each label, i.e., if a user places K seeds, each with a different label, then it is necessary to compute, for each pixel, the probability that a random walker leaving the pixel will first arrive at each seed. These probabilities may be determined analytically by solving a system of linear equations. After computing these probabilities for each pixel, the pixel is assigned to the label for which it is most likely to send a random walker. The image is modeled as a graph, in which each pixel corresponds to a node which is connected to neighboring pixels by edges, and the edges are weighted to reflect the similarity between the pixels. Therefore, the random walk occurs on the weighted graph.
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In image analysis, the average with limited data validity is an image filter for feature-preserving noise removal, consisting in a smoothing filter that only involves pixels satisfying some validity criterion. If some feature of noise elements is known, it is possible to use it to define a criterion to detect invalid pixels, and selectively smooth only invalid pixels using data coming only from valid pixels, thus avoiding to affect other features of the image.
Shrinkage fields is a random field-based machine learning technique that aims to perform high quality image restoration using low computational overhead.
A guided filter is an edge-preserving smoothing image filter. As with a bilateral filter, it can filter out noise or texture while retaining sharp edges.
