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Gabor Filter
In image processing, a Gabor filter, named after Dennis Gabor, is a linear filter used for texture analysis, which essentially means that it analyzes whether there is any specific frequency content in the image in specific directions in a localized region around the point or region of analysis. Frequency and orientation representations of Gabor filters are claimed by many contemporary vision scientists to be similar to those of the human visual system. They have been found to be particularly appropriate for texture representation and discrimination. In the spatial domain, a 2-D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane wave (see Gabor transform). Some authors claim that simple cells in the visual cortex of mammalian brains can be modeled by Gabor functions. Thus, image analysis with Gabor filters is thought by some to be similar to perception in the human visual system. Definition Its impulse response is defined by a sinusoidal wave (a plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution Theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. Functions of a continuous variable Consider two functions g(x) and h(x) with Fourier transforms G and H: \begin G(f) &\triangleq \mathcal\(f) = \int_^g(x) e^ \, dx, \quad f \in \mathbb\\ H(f) &\triangleq \mathcal\(f) = \int_^h(x) e^ \, dx, \quad f \in \mathbb \end where \mathcal denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically 2\pi or \sqrt) will appear in the convolution theorem below. The convolution of g and h is defined by: r(x) = \(x) \triangleq \int_^ g( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bone
A bone is a rigid organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red and white blood cells, store minerals, provide structure and support for the body, and enable mobility. Bones come in a variety of shapes and sizes and have complex internal and external structures. They are lightweight yet strong and hard and serve multiple functions. Bone tissue (osseous tissue), which is also called bone in the uncountable sense of that word, is hard tissue, a type of specialized connective tissue. It has a honeycomb-like matrix internally, which helps to give the bone rigidity. Bone tissue is made up of different types of bone cells. Osteoblasts and osteocytes are involved in the formation and mineralization of bone; osteoclasts are involved in the resorption of bone tissue. Modified (flattened) osteoblasts become the lining cells that form a protective layer on the bone surface. The mineralize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trabecular
A trabecula (plural trabeculae, from Latin for "small beam") is a small, often microscopic, tissue element in the form of a small beam, strut or rod that supports or anchors a framework of parts within a body or organ. A trabecula generally has a mechanical function, and is usually composed of dense collagenous tissue (such as the trabecula of the spleen). It can be composed of other material such as muscle and bone. In the heart, muscles form trabeculae carneae and septomarginal trabeculae. Cancellous bone is formed from groupings of trabeculated bone tissue. In cross section, trabeculae of a cancellous bone can look like septa, but in three dimensions they are topologically distinct, with trabeculae being roughly rod or pillar-shaped and septa being sheet-like. When crossing fluid-filled spaces, trabeculae may offer the function of resisting tension (as in the penis, see for example trabeculae of corpora cavernosa and trabeculae of corpus spongiosum) or providing a cell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feature Extraction
In machine learning, pattern recognition, and image processing, feature extraction starts from an initial set of measured data and builds derived values (features) intended to be informative and non-redundant, facilitating the subsequent learning and generalization steps, and in some cases leading to better human interpretations. Feature extraction is related to dimensionality reduction. When the input data to an algorithm is too large to be processed and it is suspected to be redundant (e.g. the same measurement in both feet and meters, or the repetitiveness of images presented as pixels), then it can be transformed into a reduced set of features (also named a feature vector). Determining a subset of the initial features is called feature selection. The selected features are expected to contain the relevant information from the input data, so that the desired task can be performed by using this reduced representation instead of the complete initial data. General Feature extracti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabor Wavelet
Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters. The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized. However they have the downside of being non-orthogonal, so efficient decomposition into the basis is difficult. Since their inception, various applications have appeared, from image processing to analyzing neurons in the human visual system. Minimal uncertainty property The motivation for Gabor wavelets comes from finding some function f(x) which minimizes its standard deviation in the time and frequency domains. More formally, the variance in the position domain is: : (\Delta x)^2 = \frac where f^(x) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabor Wavelet
Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters. The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized. However they have the downside of being non-orthogonal, so efficient decomposition into the basis is difficult. Since their inception, various applications have appeared, from image processing to analyzing neurons in the human visual system. Minimal uncertainty property The motivation for Gabor wavelets comes from finding some function f(x) which minimizes its standard deviation in the time and frequency domains. More formally, the variance in the position domain is: : (\Delta x)^2 = \frac where f^(x) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabor Function
In applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function. Overview In 1946, Dennis Gabor suggested the idea of using a granular system to produce sound. In his work, Gabor discussed the problems with Fourier analysis. Although he found the mathematics to be correct, it did not reflect the behaviour of sound in the world, because sounds, such as the sound of a siren, have variable frequencies over time. Another problem was the underlying supposition, as we use sine waves analysis, that the signal under concern has infinite duration even though sounds in real life have limited duration – see time–frequency analysis. Gabor applied ideas from quantum physics to sound, allowing an analogy between sound and quanta. He proposed a mathematical method to reduce Fourier analysis into cells. His research aimed at the infor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . For example, is an imaginary number, and its square is . By definition, zero is considered to be both real and imaginary. Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century). An imaginary number can be added to a real number to form a complex number of the form , where the real numbers and are called, respectively, the ''real part'' and the ''imaginary part'' of the complex number. History Although the Greek mathematician and engineer Hero of Alexandria is noted as the first to present a calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
