Iterative reconstruction
Encyclopedia
Iterative reconstruction refers to iterative
algorithms used to reconstruct 2D and 3D images in certain imaging
techniques.
For example, in computed tomography
an image must be reconstructed from projections of an object. Here, iterative reconstruction techniques are a
better, but computationally more expensive, alternative to the common filtered back projection (FBP) method, which directly calculates the image in
a single reconstruction step.
. Often, it is not possible to exactly solve the inverse
problem directly. In this case, a direct algorithm has to approximate the solution, which might cause visible reconstruction artifacts
in the image. Iterative algorithms approach the correct solution using multiple iteration steps, which allows to obtain a better
reconstruction at the cost of a higher computation time.
In computed tomography
, this approach was the one first used by Hounsfield
. There are a large variety of algorithms, but each starts with an assumed image, computes projections from the image, compares the original projection data and updates the image based upon the difference between the calculated and the actual projections.
There are typically five components to iterative image reconstruction algorithms,
e.g.
.
image in the case of incomplete data. The method has been applied in emission tomography modalities like SPECT and PET
, where there is significant attenuation along ray paths and noise statistics are relatively poor.
As another example, it is considered superior when one does not have a large set of projections
available, when the projections are not distributed uniformly in angle, or when the projections are sparse or missing at certain orientations. These scenarios may occur in intraoperative CT, in cardiac CT, or when metal artifacts
.
require the exclusion of some portions of the projection data.
In Magnetic Resonance Imaging
it can be used to reconstruct images from data acquired with multiple receive coils and with sampling patterns different from the conventional Cartesian grid and allows the use of improved regularization techniques (e.g. total variation
) or an extended modeling of physical processes to improve the reconstruction. For example, with iterative algorithms it is possible to
reconstruct images from data acquired in a very short time as required for Real-time MRI
.
Here is an example that illustrates the benefits of iterative image reconstruction for cardiac MRI.
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
algorithms used to reconstruct 2D and 3D images in certain imaging
Digital imaging
Digital imaging or digital image acquisition is the creation of digital images, typically from a physical scene. The term is often assumed to imply or include the processing, compression, storage, printing, and display of such images...
techniques.
For example, in computed tomography
Computed tomography
X-ray computed tomography or Computer tomography , is a medical imaging method employing tomography created by computer processing...
an image must be reconstructed from projections of an object. Here, iterative reconstruction techniques are a
better, but computationally more expensive, alternative to the common filtered back projection (FBP) method, which directly calculates the image in
a single reconstruction step.
Basic concepts
The reconstruction of an image from the acquired data is an inverse problemInverse problem
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...
. Often, it is not possible to exactly solve the inverse
problem directly. In this case, a direct algorithm has to approximate the solution, which might cause visible reconstruction artifacts
Digital artifact
A digital artifact is any undesired alteration in data introduced in a digital process by an involved technique and/or technology.-Possible causes:...
in the image. Iterative algorithms approach the correct solution using multiple iteration steps, which allows to obtain a better
reconstruction at the cost of a higher computation time.
In computed tomography
Computed tomography
X-ray computed tomography or Computer tomography , is a medical imaging method employing tomography created by computer processing...
, this approach was the one first used by Hounsfield
Godfrey Hounsfield
Sir Godfrey Newbold Hounsfield CBE, FRS, was an English electrical engineer who shared the 1979 Nobel Prize for Physiology or Medicine with Allan McLeod Cormack for his part in developing the diagnostic technique of X-ray computed tomography .His name is immortalised in the Hounsfield scale, a...
. There are a large variety of algorithms, but each starts with an assumed image, computes projections from the image, compares the original projection data and updates the image based upon the difference between the calculated and the actual projections.
There are typically five components to iterative image reconstruction algorithms,
e.g.
.
- An object model that expresses the unknown continuous-space function
that is to be reconstructed in terms of a finite series with unknown coefficients that must be estimated from the data. - A system model that relates the unknown object to the "ideal" measurements that would be recorded in the absence of measurement noise. Often this is a linear model of the form
. - A statistical modelStatistical modelA statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...
that describes how the noisy measurements vary around their ideal values. Often Gaussian noiseGaussian noiseGaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise, in which...
or Poisson statistics are assumed. - A cost function that is to be minimized to estimate the image coefficient vector. Often this cost function includes some form of regularizationRegularizationRegularization may refer to:* Regularization ** Regularization * Regularization * Regularization * Regularization...
. - An algorithmAlgorithmIn mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
, usually iterative, for minimizing the cost function, including some initial estimate of the image and some stopping criterion for terminating the iterations.
Advantages
The advantages of the iterative approach include improved insensitivity to noise and capability of reconstructing an optimalOptimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
image in the case of incomplete data. The method has been applied in emission tomography modalities like SPECT and PET
Positron emission tomography
Positron emission tomography is nuclear medicine imaging technique that produces a three-dimensional image or picture of functional processes in the body. The system detects pairs of gamma rays emitted indirectly by a positron-emitting radionuclide , which is introduced into the body on a...
, where there is significant attenuation along ray paths and noise statistics are relatively poor.
As another example, it is considered superior when one does not have a large set of projections
available, when the projections are not distributed uniformly in angle, or when the projections are sparse or missing at certain orientations. These scenarios may occur in intraoperative CT, in cardiac CT, or when metal artifacts
.
require the exclusion of some portions of the projection data.
In Magnetic Resonance Imaging
Magnetic resonance imaging
Magnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...
it can be used to reconstruct images from data acquired with multiple receive coils and with sampling patterns different from the conventional Cartesian grid and allows the use of improved regularization techniques (e.g. total variation
Total variation
In mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure...
) or an extended modeling of physical processes to improve the reconstruction. For example, with iterative algorithms it is possible to
reconstruct images from data acquired in a very short time as required for Real-time MRI
Real-time MRI
Real-time magnetic resonance imaging refers to the continuous monitoring of moving objects in real time. Because MRIis based on time-consuming scanning of k-space, real-time MRI was possible only with low image quality or low temporal resolution...
.
Here is an example that illustrates the benefits of iterative image reconstruction for cardiac MRI.
See also
- Tomographic reconstructionTomographic reconstructionThe mathematical basis for tomographic imaging was laid down by Johann Radon. It is applied in computed tomography to obtain cross-sectional images of patients...
- Tomogram
- Computed TomographyComputed tomographyX-ray computed tomography or Computer tomography , is a medical imaging method employing tomography created by computer processing...
- Magnetic Resonance ImagingMagnetic resonance imagingMagnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...
- Inverse problemInverse problemAn inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...
- OsemOsem (mathematics)In mathematical optimization, the ordered subset expectation maximization method is an iterative method that is used in computed tomography....
- DeconvolutionDeconvolutionIn mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing...