Feature detection |
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Edge detection |

Corner detection |

Blob detection |

Ridge detection |

Hough transform |

Structure tensor |

Affine invariant feature detection |

Feature description |

Scale space |

**Affine shape adaptation** is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local image structure in neighbourhood region of a specific image point. Equivalently, affine shape adaptation can be accomplished by iteratively warping a local image patch with affine transformations while applying a rotationally symmetric filter to the warped image patches. Provided that this iterative process converges, the resulting fixed point will be *affine invariant*. In the area of computer vision, this idea has been used for defining affine invariant interest point operators as well as affine invariant texture analysis methods.

The interest points obtained from the scale-adapted Laplacian blob detector or the multi-scale Harris corner detector with automatic scale selection are invariant to translations, rotations and uniform rescalings in the spatial domain. The images that constitute the input to a computer vision system are, however, also subject to perspective distortions. To obtain interest points that are more robust to perspective transformations, a natural approach is to devise a feature detector that is *invariant to affine transformations*.

Affine invariance can be accomplished from measurements of the same multi-scale windowed second moment matrix as is used in the multi-scale Harris operator provided that we extend the regular scale space concept obtained by convolution with rotationally symmetric Gaussian kernels to an *affine Gaussian scale-space* obtained by shape-adapted Gaussian kernels (Lindeberg 1994 section 15.3; Lindeberg and Garding 1997). For a two-dimensional image , let and let be a positive definite 2×2 matrix. Then, a non-uniform Gaussian kernel can be defined as

and given any input image the affine Gaussian scale-space is the three-parameter scale-space defined as

Next, introduce an affine transformation where is a 2×2-matrix, and define a transformed image as

- .

Then, the affine scale-space representations and of and , respectively, are related according to

provided that the affine shape matrices and are related according to

- .

Disregarding mathematical details, which unfortunately become somewhat technical if one aims at a precise description of what is going on, the important message is that *the affine Gaussian scale-space is closed under affine transformations*.

If we, given the notation as well as local shape matrix and an integration shape matrix , introduce an *affine-adapted multi-scale second-moment matrix* according to

it can be shown that under any affine transformation the affine-adapted multi-scale second-moment matrix transforms according to

- .

Again, disregarding somewhat messy technical details, the important message here is that *given a correspondence between the image points and , the affine transformation can be estimated from measurements of the multi-scale second-moment matrices and in the two domains.*

An important consequence of this study is that if we can find an affine transformation such that is a constant times the unit matrix, then we obtain a *fixed-point that is invariant to affine transformations* (Lindeberg 1994 section 15.4; Lindeberg and Garding 1997). For the purpose of practical implementation, this property can often be reached by in either of two main ways. The first approach is based on *transformations of the smoothing filters* and consists of:

- estimating the second-moment matrix in the image domain,
- determining a new adapted smoothing kernel with covariance matrix proportional to ,
- smoothing the original image by the shape-adapted smoothing kernel, and
- repeating this operation until the difference between two successive second-moment matrices is sufficiently small.

The second approach is based on *warpings in the image domain* and implies:

- estimating in the image domain,
- estimating a local affine transformation proportional to where denotes the square root matrix of ,
- warping the input image by the affine transformation and
- repeating this operation until is sufficiently close to a constant times the unit matrix.

This overall process is referred to as *affine shape adaptation* (Lindeberg and Garding 1997; Baumberg 2000; Mikolajczyk and Schmid 2004; Tuytelaars and van Gool 2004; Ravela 2004; Lindeberg 2008). In the ideal continuous case, the two approaches are mathematically equivalent. In practical implementations, however, the first filter-based approach is usually more accurate in the presence of noise while the second warping-based approach is usually faster.

In practice, the affine shape adaptation process described here is often combined with interest point detection automatic scale selection as described in the articles on blob detection and corner detection, to obtain interest points that are invariant to the full affine group, including scale changes. Besides the commonly used multi-scale Harris operator, this affine shape adaptation can also be applied to other types of interest point operators such as the Laplacian/Difference of Gaussian blob operator and the determinant of the Hessian (Lindeberg 2008). Affine shape adaptation can also be used for affine invariant texture recognition and affine invariant texture segmentation.

Closely related to the notion of affine shape adaptation is the notion of *affine normalization*, which defines an *affine invariant reference frame* as further described in Lindeberg (2013a,b, 2021:Appendix I.3), such that any image measurement performed in the affine invariant reference frame is affine invariant.

In probability theory and statistics, the **multivariate normal distribution**, **multivariate Gaussian distribution**, or **joint normal distribution** is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

In physics and mathematics, the **Lorentz group** is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In mathematics, a **Gaussian function**, often simply referred to as a **Gaussian**, is a function of the form

In probability theory and statistics, a **Gaussian process** is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.

The **scale-invariant feature transform** (**SIFT**) is a feature detection algorithm in computer vision to detect and describe local features in images. It was published by David Lowe in 1999. Applications include object recognition, robotic mapping and navigation, image stitching, 3D modeling, gesture recognition, video tracking, individual identification of wildlife and match moving.

In probability theory and statistics, the **generalized extreme value** (**GEV**) **distribution** is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution need to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.

**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 theory for handling image structures at different scales, by representing an image as a one-parameter family of smoothed images, the **scale-space representation**, parametrized by the size of the smoothing kernel used for suppressing fine-scale structures. The parameter in this family is referred to as the *scale parameter*, with the interpretation that image structures of spatial size smaller than about have largely been smoothed away in the scale-space level at scale .

In probability theory, the **Rice distribution** or **Rician distribution** is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice.

**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**.

In computer vision, **blob detection** methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some properties are constant or approximately constant; all the points in a blob can be considered in some sense to be similar to each other. The most common method for blob detection is convolution.

In mathematics, the **structure tensor**, also referred to as the **second-moment matrix**, is a matrix derived from the gradient of a function. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent. The structure tensor is often used in image processing and computer vision.

In statistics, the **generalized Pareto distribution** (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location , scale , and shape . Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as .

In mathematics, the spinor concept as specialised to **three dimensions** can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).

In computer vision, **speeded up robust features** (**SURF**) is a patented local feature detector and descriptor. It can be used for tasks such as object recognition, image registration, classification, or 3D reconstruction. It is partly inspired by the scale-invariant feature transform (SIFT) descriptor. The standard version of SURF is several times faster than SIFT and claimed by its authors to be more robust against different image transformations than SIFT.

The **Kadir–Brady saliency detector** extracts features of objects in images that are distinct and representative. It was invented by Timor Kadir and J. Michael Brady in 2001 and an affine invariant version was introduced by Kadir and Brady in 2004 and a robust version was designed by Shao et al. in 2007.

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.

The **Hessian affine region detector** is a feature detector used in the fields of computer vision and image analysis. Like other feature detectors, the Hessian affine detector is typically used as a preprocessing step to algorithms that rely on identifiable, characteristic interest points.

The **shifted log-logistic distribution** is a probability distribution also known as the **generalized log-logistic** or the **three-parameter log-logistic** distribution. It has also been called the **generalized logistic** distribution, but this conflicts with other uses of the term: see generalized logistic distribution.

**Least-squares support-vector machines (LS-SVM)** are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Suykens and Vandewalle. LS-SVMs are a class of kernel-based learning methods.

In computer vision, pattern recognition, and robotics, **point set registration**, also known as **point cloud registration** or **scan matching**, is the process of finding a spatial transformation that aligns two point clouds. The purpose of finding such a transformation includes merging multiple data sets into a globally consistent model, and mapping a new measurement to a known data set to identify features or to estimate its pose. Raw 3D point cloud data are typically obtained from Lidars and RGB-D cameras. 3D point clouds can also be generated from computer vision algorithms such as triangulation, bundle adjustment, and more recently, monocular image depth estimation using deep learning. For 2D point set registration used in image processing and feature-based image registration, a point set may be 2D pixel coordinates obtained by feature extraction from an image, for example corner detection. Point cloud registration has extensive applications in autonomous driving, motion estimation and 3D reconstruction, object detection and pose estimation, robotic manipulation, simultaneous localization and mapping (SLAM), panorama stitching, virtual and augmented reality, and medical imaging.

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*Encyclopedia of Computer Science and Engineering (Benjamin Wah, ed), John Wiley and Sons*.**IV**. pp. 2495–2504. doi:10.1002/9780470050118.ecse609. - T. Lindeberg (2013a). "Invariance of visual operations at the level of receptive fields".
*PLOS ONE*.**8**(7): e66990:1-33. doi: 10.1371/journal.pone.0066990 . - T. Lindeberg (2013b). "Generalized axiomatic scale-space theory".
*Advances in Imaging and Electron Physics*.**178**(7): 1–96. doi:10.1016/B978-0-12-407701-0.00001-7. - T. Lindeberg (2021). "Normative theory of visual receptive fields".
*Heliyon*.**7**(1): e05897. doi: 10.1016/j.heliyon.2021.e05897 . - K. Mikolajczyk, K. and C. Schmid (2004). "Scale and affine invariant interest point detectors" (PDF).
*International Journal of Computer Vision*.**60**(1): 63–86. doi:10.1023/B:VISI.0000027790.02288.f2.Integration of the multi-scale Harris operator with the methodology for automatic scale selection as well as with affine shape adaptation.

- T. Tuytelaars and L. van Gool K. (2004). "Matching Widely Separated Views Based on Affine Invariant Regions" (PDF).
*International Journal of Computer Vision*.**59**(1): 63–86. doi:10.1023/B:VISI.0000020671.28016.e8. Archived from the original (PDF) on 2010-06-12.

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