# 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.