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Non-separable Wavelet
Non-separable wavelets are multi-dimensional wavelets that are not directly implemented as tensor products of wavelets on some lower-dimensional space. They have been studied since 1992. They offer a few important advantages. Notably, using non-separable filters leads to more parameters in design, and consequently better filters. The main difference, when compared to the one-dimensional wavelets, is that multi-dimensional sampling requires the use of lattices (e.g., the quincunx lattice). The wavelet filters themselves can be separable or non-separable regardless of the sampling lattice. Thus, in some cases, the non-separable wavelets can be implemented in a separable fashion. Unlike separable wavelet, the non-separable wavelets are capable of detecting structures that are not only horizontal, vertical or diagonal (show less anisotropy). Examples * Red-black wavelets * Contourlets * Shearlet In applied mathematical analysis, shearlets are a multiscale framework which allows effic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the Middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications. As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including but not limited to au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for and , a basis of V \otimes W is formed by all tensor products of a basis element of and a basis element of . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (see Universal property). Tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Vetterli
Martin Vetterli is the current president of École polytechnique fédérale de Lausanne (EPFL) in Switzerland, succeeding Patrick Aebischer. He's a professor of engineering and was formerly the president of the National Research Council of the Swiss National Science Foundation. Martin Vetterli has made numerous research contributions in the general area of digital signal processing and is best known for his work on wavelets. He has also contributed to other areas, including sampling (signal processing), computational complexity theory, signal processing for communications, digital video processing and joint source/channel coding. His work has led to over 150 journal publications and to two dozen of patents. Career Martin Vetterli received his Dipl. El.-Ing. degree from the ETH Zurich in 1981, and then completed a Master of Science degree in electrical engineering at Stanford University in 1982. He later pursued his Doctorat ès Sciences at EPFL in 1986. After his disser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multidimensional Sampling
In digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. This article presents the basic result due to Petersen and MiddletonD. P. Petersen and D. Middleton, "Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces", Information and Control, vol. 5, pp. 279–323, 1962. on conditions for perfectly reconstructing a wavenumber-limited function from its measurements on a discrete lattice of points. This result, also known as the Petersen–Middleton theorem, is a generalization of the Nyquist–Shannon sampling theorem for sampling one-dimensional band-limited functions to higher-dimensional Euclidean spaces. In essence, the Petersen–Middleton theorem shows that a wavenumber-limited function can be perfectly reconstructed from its values on an infinite lattice of points, provided the la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anisotropy
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 physical or mechanical properties (absorbance, refractive index, conductivity, tensile strength, etc.). An example of anisotropy is light coming through a polarizer. Another is wood, which is easier to split along its grain than across it. Fields of interest Computer graphics In the field of computer graphics, an anisotropic surface changes in appearance as it rotates about its geometric normal, as is the case with velvet. Anisotropic filtering (AF) is a method of enhancing the image quality of textures on surfaces that are far away and steeply angled with respect to the point of view. Older techniques, such as bilinear and trilinear filtering, do not take into account the angle a surface is viewed from, which can result in aliasing or bl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adhemar Bultheel
Adhemar François Bultheel (born 1948) is a Belgian mathematician and computer scientist, the former president of the Belgian Mathematical Society. He is a prolific book reviewer for the Bulletin of the Belgian Mathematical Society and for the European Mathematical Society. His research concerns approximation theory. Education and career Bultheel was born in Zwijndrecht, Belgium on December 14, 1948. He earned a licenciate in mathematics in 1970 and another in industrial mathematics in 1971, both from KU Leuven KU Leuven (or Katholieke Universiteit Leuven) is a Catholic research university in the city of Leuven, Belgium. It conducts teaching, research, and services in computer science, engineering, natural sciences, theology, humanities, medicine, l .... He remained at KU Leuven for a bachelor's degree in 1975 and a PhD in mathematics in 1979. His dissertation, ''Recursive Rational Approximation'', was jointly supervised by Patrick M. Dewilde and Hugo Van de Vel. Except for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contourlet
Contourlets form a multiresolution directional tight frame designed to efficiently approximate images made of smooth regions separated by smooth boundaries. The contourlet transform has a fast implementation based on a Laplacian pyramid decomposition followed by directional filterbanks applied on each bandpass subband. Contourlet transform Introduction and motivation In the field of geometrical image transforms, there are many 1-D transforms designed for detecting or capturing the geometry of image information, such as the Fourier and wavelet transform. However, the ability of 1-D transform processing of the intrinsic geometrical structures, such as smoothness of curves, is limited in one direction, then more powerful representations are required in higher dimensions. The contourlet transform which was proposed by Do and Vetterli in 2002, is a new two-dimensional transform method for image representations. The contourlet transform has properties of multiresolution, localization, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shearlet
In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions f \in L^2(\R^2). They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena. Shearlets are constructed by parabolic scaling, shearing, and translation applied to a few generating functions. At fine scales, they are essentially supported within skinny and directional ridges following the parabolic scaling law, which reads ''length² ≈ width''. Similar to wavelets, shearlets arise from the affine group and allow a unified treatment of the continuum and digital situation leading to faithful implementations. Although they do not constitute an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gitta Kutyniok
Gitta Kutyniok (born 1972) is a German applied mathematician known for her research in harmonic analysis, deep learning, compressed sensing, and image processing. She has a Bavarian AI Chair for "Mathematical Foundations of Artificial Intelligence" in the institute of mathematics at the Ludwig Maximilian University of Munich. Education and career Kutyniok was educated in Detmold, and in 1996 earned a diploma in mathematics and computer science at University of Paderborn. She then completed her doctorate ( Dr. rer. nat.) at Paderborn in 2000. Her dissertation, ''Time-Frequency Analysis on Locally Compact Groups'', was supervised by Eberhard Kaniuth. From 2000 to 2008 she held short term positions at Paderborn University, the Georgia Institute of Technology, the University of Giessen, Washington University in St. Louis, Princeton University, Stanford University, and Yale University. In 2006 she earned her habilitation in Giessen, in 2008 she became a full professor at Osnabrüc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramid (image Processing)
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. Pyramid generation There are two main types of pyramids: lowpass and bandpass. A lowpass pyramid is made by smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of 2 along each coordinate direction. The resulting image is then subjected to the same procedure, and the cycle is repeated multiple times. Each cycle of this process results in a smaller image with increased smoothing, but with decreased spatial sampling density (that is, decreased image resolution). If illustrated graphically, the entire multi-scale representation will look like a pyramid, with the original i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eero Simoncelli
Eero Simoncelli is an American computational neuroscientist and Silver Professor at New York University. He was a Howard Hughes Medical Institute Investigator from 2000 to 2020. In 2020, he became the inaugural director of the Center for Computational Neuroscience at the Flatiron Institute of the Simons Foundation. Education and early career Simoncelli graduated summa cum laude with a bachelor's degree in physics at Harvard University in 1984. He then attended Cambridge University on a Knox Fellowship to study the Mathematical Tripos, after which he joined the graduate program at the Massachusetts Institute of Technology in electrical engineering and computer science. He received his master's degree in 1988 and his PhD in 1993. He then joined the faculty at the University of Pennsylvania as an assistant professor, and in 1996 he moved to New York University. Awards and professional recognition In 2009, he became an IEEE Fellow. He received an Engineering Emmy Award in 2015 with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
