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Medical imaging

imagingdiagnostic imagingdiagnostic radiology
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.
In this restricted sense, medical imaging can be seen as the solution of mathematical inverse problems.

Regularization (mathematics)

regularizationregularizedregularize
This type of constraint is known as regularization.
In mathematics, statistics, and computer science, particularly in machine learning and inverse problems, regularization is the process of adding information in order to solve an ill-posed problem or to prevent overfitting.

Tomographic reconstruction

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

SAMV (algorithm)

SAMViterative Sparse Asymptotic Minimum Variance
Solutions explored include Algebraic Reconstruction Technique, filtered backprojection, and as computing power has increased, iterative reconstruction methods such as 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.

Iterative reconstruction

image reconstructioniterativeiterative CT reconstruction algorithms
Solutions explored include Algebraic Reconstruction Technique, filtered backprojection, and as computing power has increased, iterative reconstruction methods such as iterative Sparse Asymptotic Minimum Variance.
The reconstruction of an image from the acquired data is an inverse problem.

Inverse scattering problem

inverse scattering
Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century, only one class of nonlinear inverse problems was so before 1970, that of inverse spectral and (one space dimension) inverse scattering problems, after the seminal work of the Russian mathematical school (Krein, Gelfand, Levitan, Marchenko).
It is the inverse problem to the direct scattering problem, which is to determine how radiation or particles are scattered based on the properties of the scatterer.

Well-posed problem

ill-posedill-posed problemwell-posed
(However, by the bounded inverse theorem, if the mapping is bijective, then the inverse will be bounded (i.e. continuous).) Thus small errors in the data d are greatly amplified in the solution m. In this sense the inverse problem of inferring m from measured d is ill-posed.
Inverse problems are often ill-posed.

Radon transform

filtered back projectionfiltered backprojectionsinograms
Another example is the inversion of the Radon transform, essential to tomographic reconstruction for X-ray computed tomography. Solutions explored include Algebraic Reconstruction Technique, filtered backprojection, and as computing power has increased, iterative reconstruction methods such as iterative Sparse Asymptotic Minimum Variance.
Reconstruction is an inverse problem.

Fredholm integral equation

Fredholm equationFredholm integralFredholm integral operators
One central example of a linear inverse problem is provided by a Fredholm integral equation of the first kind:
They also commonly arise in linear forward modeling and inverse problems.

Mathematical geophysics

mathematical geophysicistnonlinear geophysics
Mathematical geophysics
Geophysical inverse theory is concerned with analyzing geophysical data to get model parameters.

Optimal estimation

Optimal estimation
In applied statistics, optimal estimation is a regularized matrix inverse method based on Bayes' theorem.

Inverse Problems

Inverse Problems
It combines theoretical, experimental and mathematical papers on inverse problems with numerical and practical approaches to their solution.

Backus–Gilbert method

Backus-Gilbert inverse
Backus–Gilbert method
It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems.

Atmospheric sounding

aerologyaerologistsounding
Atmospheric sounding
It is usually easy to go from the state space to the measurement space—for instance with Beer's law or radiative transfer—but not the other way around, therefore we need some method of inverting \vec f or of finding the inverse model, \vec f^{-1}.

Grey box model

grey-box
Grey box model
A third method is model inversion, which converts the non-linear m(f,p,Ac) into an approximate linear form in the elements of A, that can be examined using efficient term selection and evaluation of the linear regression.

Engineering optimization

biological optimizationOptimizationOptimized
Engineering optimization
inverse optimization (inverse problem)

Tikhonov regularization

ridge regressionregularizeda squared regularizing function
Tikhonov regularization
The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix C_M representing the a priori uncertainties on the model parameters, and a covariance matrix C_D representing the uncertainties on the observed parameters.

Data assimilation

assimilationassimilating3D-Var data assimilation scheme
Data assimilation
This increasing non-linearity in the models and observation operators poses a new problem in the data assimilation.

Physical geodesy

gravity fieldGeodesygravity
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field.

Science

scientificsciencesscientific knowledge
Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe.

Mathematics

mathematicalmathmathematician
Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe.

System identification

identificationmodel identificationmodel order reduction
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.

Optics

opticalopticoptical device
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.

Radar

radar stationradarsradar system
They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields.