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Radon transform. Maps f on the (x, y)-domain to Rf on the (α, s)-domain.. In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line.
In theory, the inverse Radon transformation would yield the original image. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object, f ( x , y ) {\displaystyle f(x,y)} .
The way in which a Fourier transform changes x-t data into x-ω (ω is angular frequency) data shows why phase velocity dominates surface wave inversion theory. Phase velocity is the velocity of each wave with a given frequency. The modified wavefield transform is executed by doing a Fourier transform first before a slant stack.
A real-valued Radon measure is defined to be any continuous linear form on K (X); they are precisely the differences of two Radon measures. This gives an identification of real-valued Radon measures with the dual space of the locally convex space K (X). These real-valued Radon measures need not be signed measures.
A water level device showing both ends at the same height. A water level (Greek: Aλφαδολάστιχο or (υδροστάθμη) [Alfadolasticho]) is a siphon utilizing two or more parts of the liquid water surface to establish a local horizontal line or plane of reference.
Conversely, the obvert level is the highest interior level, and can be considered the "ceiling" level, being the highest level of that sewer. The bottom of the sewer is called the invert from a general resemblance in construction to an "inverted" arch. [2] An inverted arch is a rounded structure with its crown facing in the downward position.
Thus, the formula can be read from left to right or from right to left in order to simplify a given integral. When used in the former manner, it is sometimes known as u -substitution or w -substitution in which a new variable is defined to be a function of the original variable found inside the composite function multiplied by the derivative of ...
In higher dimensions, the X-ray transform of a function is defined by integrating over 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 ...