Tomographic reconstruction

reconstructedreconstruction algorithmtomographyalgorithmback-projection and reconstructionCT scansreconstructiontomographic image reconstructiontomographic imagingtomographic principle
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.wikipedia
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Tomography

tomographictomogramtomograms
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.
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.

Inverse problem

inverse problemsinversioninverse method
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.
Another example is the inversion of the Radon transform, essential to tomographic reconstruction for X-ray computed tomography.

CT scan

computed tomographyCTcomputerized tomography
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.
Once the scan data has been acquired, the data must be processed using a form of tomographic reconstruction, which produces a series of cross-sectional images.

Johann Radon

RadonRadon, Johann
The mathematical basis for tomographic imaging was laid down by Johann Radon.
the Radon transform, in integral geometry, based on integration over hyperplanes — with application to tomography for scanners (see tomographic reconstruction);

Radon transform

filtered back projectionfiltered backprojectionsinograms
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. These algorithms are designed largely based on the mathematics of the Radon transform, statistical knowledge of the data acquisition process and geometry of the data imaging system. In practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm.
Hence the inverse of the Radon transform can be used to reconstruct the original density from the projection data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as iterative reconstruction.

SAMV (algorithm)

SAMViterative Sparse Asymptotic Minimum Variance
An alternative family of recursive tomographic reconstruction algorithms are the Algebraic Reconstruction Technique s and iterative Sparse Asymptotic Minimum Variance.
SAMV (iterative Sparse Asymptotic Minimum Variance ) is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomographic reconstruction with applications in signal processing, medical imaging and remote sensing.

Operation of computed tomography

spiral CThelical CT scanspiral computed tomography
Operation of computed tomography#Tomographic reconstruction
Once the scan data has been acquired, the data must be processed using a form of tomographic reconstruction, which produces a series of cross-sectional images.

Cone beam reconstruction

cone beamCone-beamreconstruction
Cone beam reconstruction
Tomographic reconstruction

Projection (linear algebra)

orthogonal projectionprojectionprojection operator
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.

Airport security

aviation securitysecuritysecurity checkpoint
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.

X-ray

x-rayssoft x-rayx rays
3). In X-ray CT, the line integral represents the total attenuation of the beam of x-rays as it travels in a straight line through the object.

Attenuation coefficient

absorption coefficientattenuationAbsorption coefficients
As mentioned above, the resulting image is a 2D (or 3D) model of the attenuation coefficient.

Parallel projection

parallel linear projectionprojection
The simplest and easiest way to visualise the method of scanning is the system of parallel projection, as used in the first scanners.

Attenuation

attenuateattenuatedattenuating
Attenuation occurs exponentially in tissue:

Exponential decay

mean lifetimedecay constantlifetime
Attenuation occurs exponentially in tissue:

Fourier transform

Fouriercontinuous Fourier transformuncertainty principle
The Fourier Transform of the projection can be written as

Fourier inversion theorem

inverse Fourier transformFourier inversionFourier integrals
Using the inverse Fourier transform, the inverse Radon transform formula can be easily derived.

Hilbert transform

discrete Hilbert transformTitchmarsh theoremCauchy transform
where is the derivative of the Hilbert transform of

Projection-slice theorem

FHA cyclefourier slice theoremFourier-based back projection
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).

Algorithm

algorithmscomputer algorithmalgorithm design
These algorithms are designed largely based on the mathematics of the Radon transform, statistical knowledge of the data acquisition process and geometry of the data imaging system.

Discrete Fourier transform

DFTcircular convolution theoremFourier transform
The Discrete Fourier transform on each projection will yield sampling in the frequency domain.

Discretization

discretizeddiscretizingdichotomization
In practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm.

DC bias

DCDC offsetDC-balance
The filter used does not contain DC gain, thus adding DC bias may be desirable.

Algebraic reconstruction technique

ART
An alternative family of recursive tomographic reconstruction algorithms are the Algebraic Reconstruction Technique s and iterative Sparse Asymptotic Minimum Variance.

Collimated beam

collimatedcollimationcollimated light
Use of a noncollimated fan beam is common since a collimated beam of radiation is difficult to obtain.