|
Reconstruction Algorithm
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security. This article applies in general to reconstruction methods for all kinds of tomography, but some of the terms and physical descriptions refer directly to the reconstruction of X-ray computed tomography. Introducing formula The projection of an object, resulting from the tomographic measurement process at a given angle \theta, is made up of a set of line integrals (see Fig. 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomographic Reconstruction- Projection, Back Projection And Filtered Back Projection
Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, cosmochemistry, astrophysics, quantum information, and other areas of science. The word ''tomography'' is derived from Ancient Greek τόμος ''tomos'', "slice, section" and γράφω ''graphō'', "to write" or, in this context as well, "to describe." A device used in tomography is called a tomograph, while the image produced is a tomogram. In many cases, the production of these images is based on the mathematical procedure tomographic reconstruction, such as X-ray computed tomography technically being produced from multiple projectional radiographs. Many different reconstruction algorithms exist. Most algorithms fall into one of two categories: filtered back projection (FBP) and iterative reconstruction (IR). These procedures give inexact results: they re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Reconstruction Technique
frame, Animated sequence of reconstruction steps, one iteration. The algebraic reconstruction technique (ART) is an iterative reconstruction technique used in computed tomography. It reconstructs an image from a series of angular projections (a sinogram). Gordon, Bender and Herman first showed its use in image reconstruction; whereas the method is known as Kaczmarz method in numerical linear algebra. An advantage of ART over other reconstruction methods (such as filtered backprojection) is that it is relatively easy to incorporate prior knowledge into the reconstruction process. ART can be considered as an iterative solver of a system of linear equations A x = b , where: : A is a sparse m \times n matrix whose values represent the relative contribution of each output pixel to different points in the sinogram ( m being the number of individual values in the sinogram, and n being the number of output pixels); : x represents the pixels in the generated (output) imag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DC Bias
In signal processing, when describing a periodic function in the time domain, the DC bias, DC component, DC offset, or DC coefficient is the mean value of the waveform. A waveform with zero mean or no DC bias is known as a ''DC balanced'' or ''DC free'' waveform. Origin The term originates in electronics, where ''DC'' refers to a direct current voltage. In contrast, various other non-DC frequencies are analogous to superimposed alternating current (AC) voltages or currents, hence called AC component or AC coefficients. Applications In the design of electronic amplifier circuits, every active device has biasing to set its ''operating point'', the steady state current and voltage on the device when no signal is applied. In bipolar transistor biasing, for example, a network of resistors is used to apply a small amount of DC to the base terminal of the transistor. The AC signal is applied at the same terminal and is amplified. The bias network is designed to preserve the applied AC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification). Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, ''discretization'' may also refer to modification of variable or category ''granularity'', as when multiple discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is used in the Fou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Transform
In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over line (geometry), lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ''ƒ'' represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ''ƒ''. Inverse problem, Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ''ƒ'' from its known attenuation data. In detail, if ''ƒ'' is a compact support, compactly supported continuous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection-slice Theorem
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * Take a two-dimensional function ''f''(r), Projection (mathematics), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. * Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line. In operator terms, if * ''F''1 and ''F''2 are the 1- and 2-dimensional Fourier transform operators mentioned above, * ''P''1 is the projection operator (which projects a 2-D function onto a 1-D line), * ''S''1 is a slice operator (which extracts a 1-D central slice from a function), then : F_1 P_1 = S_1 F_2. This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical computed axial tomography, CT scans where a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
