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How does the Radon transform work?
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.
Why is Abel transform useful?
The Abel transform is used in calculating the radial mass distribution of galaxies (Binney and Tremaine 1987, p. 651; Arfken and Weber 2005, p. 1014) and inverting planetary radio occultation data to obtain atmospheric information as a function of height.
What are tomographic images?
Tomography is imaging by sections or sectioning through the use of any kind of penetrating wave. 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.
Why do we need image reconstruction?
Image reconstruction has fundamental impacts on image quality and therefore on radiation dose. For a given radiation dose it is desirable to reconstruct images with the lowest possible noise without sacrificing image accuracy and spatial resolution.
Is back projection linear?
Linear back projection (LBP) is the most used ET algorithm due to its advantages of dynamic imaging process, real-time response, and easy realization. In light of the data decomposition method, a new linear back projection algorithm is proposed to improve the spatial resolution of the ET image.
Which is the complex analogue of the Radon transform?
The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.
Why is the Radon transform called a backprojection?
Concretely, for the two-dimensional Radon transform, the dual transform is given by In the context of image processing, the dual transform is commonly called backprojection as it takes a function defined on each line in the plane and ‘smears’ or projects it back over the line to produce an image.
How is a sinogram generated in a Radon transform?
Horizontal projections through the shape result in an accumulated signal (middle bar). The sinogram on the right is generated by collecting many such projections as the shape rotates. Here, color is used to highlight which object is producing which part of the signal.