Nuutti Hyvönen

Aalto University · Department of Mathematics and Systems Analysis

The aim of magnetorelaxometry imaging is to determine the distribution of magnetic nanoparticles inside a subject by measuring the relaxation of the superposition magnetic field generated by the nanoparticles after they have first been aligned using an external activation magnetic field that has subsequently been switched off. This work applies techniq...

This work considers Bayesian experimental design for the inverse boundary value problem of linear elasticity in a two-dimensional setting. The aim is to optimize the positions of compactly supported pressure activations on the boundary of the examined body in order to maximize the value of the resulting boundary deformations as data for the inverse...

We investigate a linearised Calderón problem in a two-dimensional bounded simply connected $C^{1,\alpha}$ domain $\Omega$. After extending the linearised problem for $L^2(\Omega)$ perturbations, we orthogonally decompose $L^2(\Omega) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_...

This work extends the results of [Garde and Hyvönen, Math. Comp. 91:1925–1953] on series reversion for Calderón’s problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the i...

The aim of magnetorelaxometry imaging is to determine the distribution of magnetic nanoparticles inside a subject by measuring the relaxation of the superposition magnetic field generated by the nanoparticles after they have first been aligned using an external activation magnetic field that has subsequently been switched off. This work applies tec...

Objective. Diffuse optical tomography (DOT) provides a relatively convenient method for imaging haemodynamic changes related to neuronal activity on the cerebral cortex. Due to practical challenges in obtaining anatomical images of neonates, an anatomical framework is often created from an age-appropriate atlas model, which is individualized to the...

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values 0 and ∞ in some parts of the domain and values bounded away from 0 and ∞...

This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is ∇ · (A∇u) = 0 in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends A to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of...

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from...

This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is ∇ · (A∇u) = 0 in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends A to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of...

This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is ∇ · (A∇u) = 0 in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends A to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of...

This work extends the results of [Garde and Hyvönen, Math. Comp. 91:1925-1953] on series reversion for Calderón's problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the i...

This work considers electrical impedance tomography imaging of the human head, with the ultimate goal of locating and classifying a stroke in emergency care. One of the main difficulties in the envisioned application is that the electrode locations and the shape of the head are not precisely known, leading to significant imaging artifacts due to im...

This work considers electrical impedance tomography imaging of the human head, with the ultimate goal of locating and classifying a stroke in emergency care. One of the main difficulties in the envisioned application is that the electrode locations and the shape of the head are not precisely known, leading to significant imaging artifacts due to im...

This work applies Bayesian experimental design to selecting optimal projection geometries in (discretized) parallel beam X-ray tomography assuming the prior and the additive noise are Gaussian. The introduced greedy exhaustive optimization algorithm proceeds sequentially, with the posterior distribution corresponding to the previous projections ser...

This work considers sequential edge-promoting Bayesian experimental design for (discretized) linear inverse problems, exemplified by X-ray tomography. The process of computing a total variation type reconstruction of the absorption inside the imaged body via lagged diffusivity iteration is interpreted in the Bayesian framework. Assuming a Gaussian...

This paper introduces a constructive method for approximating relative continuum measurements in two-dimensional electrical impedance tomography based on data originating from either the point electrode model or the complete electrode model. The upper bounds for the corresponding approximation errors explicitly depend on the number (and size) of th...

Electrical impedance tomography is an imaging modality for extracting information on the interior structure of a physical body from boundary measurements of current and voltage. This work studies a new robust way of modeling the contact electrodes used for driving current patterns into the examined object and measuring the resulting voltages. The i...

The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insul...

This work applies Bayesian experimental design to selecting optimal projection geometries in (discretized) parallel beam X-ray tomography assuming the prior and the additive noise are Gaussian. The introduced greedy exhaustive optimization algorithm proceeds sequentially, with the posterior distribution corresponding to the previous projections ser...

Iron loss determination in the magnetic core of an electrical machine, such as a motor or a transformer, is formulated as an inverse heat source problem. The sensor positions inside the object are optimized in order to minimize the uncertainty in the reconstruction in the sense of the A-optimality of Bayesian experimental design. This paper focuses...

This paper introduces a constructive method for approximating relative continuum measurements in two-dimensional electrical impedance tomography based on data originating from either the point electrode model or the complete electrode model. The upper bounds for the corresponding approximation errors explicitly depend on the number (and size) of th...

The inverse problem of electrical impedance tomography is severely ill-posed. In particular, the resolution of images produced by impedance tomography deteriorates as the distance from the measurement boundary increases. Such depth dependence can be quantified by the concept of distinguishability of inclusions. This paper considers the distinguisha...

This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of...

The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insul...

This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of...

The inverse problem of electrical impedance tomography is severely ill-posed. In particular, the resolution of images produced by impedance tomography deteriorates as the distance from the measurement boundary increases. Such depth dependence can be quantified by the concept of distinguishability of inclusions. This paper considers the distinguisha...

This work introduces a computational framework for applying absolute electrical impedance tomography to head imaging without accurate information on the head shape or the electrode positions. A library of fifty heads is employed to build a principal component model for the typical variations in the shape of the human head, which leads to a relative...

This work tackles an inverse boundary value problem for a p-Laplace type partial differential equation parametrized by a smoothening parameter T ≥ 0. The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. Th...

Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current-voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and...

This work tackles an inverse boundary value problem for a $p$-Laplace type partial differential equation parametrized by a smoothening parameter $\tau \geq 0$. The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve a...

This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the...

This work reformulates the complete electrode model of electrical impedance tomography in order to enable its more efficient numerical solution. The model traditionally assumes constant contact conductances on all electrodes, which leads to a discontinuous Robin boundary condition since the gaps between the electrodes can be described by vanishing...

We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies t...

Thermal tomography is an imaging technique for deducing information about the internal structure of a physical body from temperature measurements on its boundary. This work considers time-dependent thermal tomography modeled by a parabolic initial/boundary value problem without accurate information on the exterior shape of the examined object. The...

Electrical impedance tomography aims at reconstructing the conductivity inside a physical body from boundary measurements of current and voltage at a finite number of contact electrodes. In many practical applications, the shape of the imaged object is subject to considerable uncertainties that render reconstructing the internal conductivity imposs...

The objective of electrical impedance tomography is to reconstruct the internal conductivity of a physical body based on measurements of current and potential at a finite number of electrodes attached to its boundary. Although the conductivity is the quantity of main interest in impedance tomography, a real-world measurement configuration includes...

Quantitative photoacoustic tomography is an emerging imaging technique aimed at estimating the distribution of optical parameters inside tissues from photoacoustic images, which are formed by combining optical information and ultrasonic propagation. This optical parameter estimation problem is ill-posed and needs to be approached within the framewo...

In optical tomography a physical body is illuminated with near-infrared light and the resulting outward photon flux is measured at the object boundary. The goal is to reconstruct internal optical properties of the body, such as absorption and diffusivity. In this work, it is assumed that the imaged object is composed of an approximately homogeneous...

We explain how to build invisible isotropic conductivity perturbations of the unit conductivity in the framework of the point electrode model for two-dimensional electrical impedance tomography. The theoretical approach, based on solving a fixed point problem, is constructive and allows the implementation of an algorithm for approximating the invis...

The objective of electrical impedance tomography is to deduce information about the conductivity inside a physical body from electrode measurements of current and voltage at the object boundary. In this work, the unknown conductivity is modeled as a random field parametrized by its values at a set of pixels. The uncertainty in the pixel values is p...

The aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage. The most accurate forward model for impedance tomography is the complete electrode model, which consists of the conductivity equation coupled with boundary conditions that take in...

Electrical impedance tomography is an imaging modality for extracting information on the conductivity distribution inside a physical body from boundary measurements of current and voltage. In many practical applications, it is a priori known that the conductivity consists of embedded inhomogeneities in an approximately constant background. This wor...

Electrical impedance tomography aims at reconstructing the internal conductivity of a physical body from boundary measurements of current and voltage. In this work, the conductivity is modelled as a log-normal random field with a known (prior) distribution, and the reconstruction task is reformulated as a Bayesian inference problem. Combining the p...

This work considers finding optimal positions for the electrodes within the Bayesian paradigm based on available prior information on the conductivity; the aim is to place the electrodes so that the posterior density of the (discretized) conductivity, i.e., the conditional density of the conductivity given the measurements, is as localized as possi...

We propose a novel numerical method based on a generalized eigenvalue decomposition for solving the diffusion equation governing the correlation diffusion of photons in turbid media. Medical imaging modalities such as diffuse correlation tomography and ultrasound-modulated optical tomography have the (elliptic) diffusion equation parame- terized by...

In this paper, the simultaneous retrieval of the exterior boundary shape and the interior admittivity distribution of an examined body in electrical impedance tomography is considered. The reconstruction method is built for the complete electrode model and it is based on the Fréchet derivative of the corresponding current-to-voltage map with respec...

This work considers the Cauchy problem for a second order elliptic operator in a bounded domain. A new quasi-reversibility approach is introduced for approximating the solution of the ill-posed Cauchy problem in a regularized manner. The method is based on a well-posed mixed variational problem on H 1 ×H div with the corresponding solution pair con...

Electrical impedance tomography is a noninvasive imaging technique based on measurements of currents and voltages on the boundary of the object of interest. The most accurate forward model for impedance tomography is the complete electrode model that takes into account the electrode shapes and the contact impedances at the corresponding interfaces;...

In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation...

This paper considers detection of conductivity inhomogeneities inside an otherwise homogeneous object by electrical impedance tomography using only two electrodes: one of the electrodes is held fixed, while the other moves around the examined object. Unit current is maintained between the electrodes, and the corresponding (relative) potential diffe...

Electrical impedance tomography is a noninvasive imaging technique for recovering the admittance distribution inside a body from boundary measurements of current and voltage. In practice, impedance tomography suffers from inaccurate modelling of the measurement setting: The exact electrode locations and the shape of the imaged object are not necess...

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particul...

This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded and simply connected domain. It is demonstrated that the so-called bisweep data, i.e., the (relative) potential differences between two bound...

This work extends the algorithm for computing the convex source support in the framework of the Poisson equation to a bounded three-dimensional domain. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the boundary of the object. In particular, it belo...

In electric impedance tomography, one tries to recover the spatial admittance distribution inside a body from boundary measurements of current and voltage. In theoretical considerations, it is usually assumed that the available data is the infinite-dimensional Neumann-to-Dirichlet map, i.e. one assumes to be able to use any boundary current and mea...

We consider the inverse problem of determining the position and the shape of a thin tubular object, such as for instance a wire, a thin channel or a curve-like crack, embedded in some three-dimensional homogeneous body from a single measurement of electrostatic currents and potentials on the boundary of the body. Using an asymptotic model describin...

We address a certain inverse problem in ultrasound-modulated optical tomography: the recovery of the amplitude of vibration of scatterers [p(r)] in the ultrasound focal volume in a diffusive object from boundary measurement of the modulation depth (M) of the amplitude autocorrelation of light [φ(r,τ)] traversing through it. Since M is dependent on...

This work considers electrical impedance tomography in the special case that the boundary measurements of current and voltage are carried out with two (infinitely) small electrodes. One of the electrodes lies at a fixed position while the other is moved along the object boundary in a sweeping motion, with the corresponding measurement being the (re...

The most accurate model for real-life electrical impedance tomography is the complete electrode model, which takes into account electrode shapes and (usually unknown) contact impedances at electrode-object interfaces. When the electrodes are small, however, it is tempting to formally replace them by point sources. This simplifies the model consider...

This paper reinvestigates a recently introduced notion of backscattering for the inverse obstacle problem in impedance tomography. Under mild restrictions on the topological properties of the obstacles, it is shown that the corresponding backscatter data are the boundary values of a function that is holomorphic in the exterior of the obstacle(s), w...

We fix an incorrect statement from our paper [SIAM J. Math. Anal. 41, No. 5, 1948–1966 (2009; Zbl 1197.35321)] claiming that two different perfectly conducting inclusions necessarily have different backscatter in impedance tomography. We also present a counterexample to show that this kind of nonuniqueness does indeed occur.

This work extends the concept of convex source support to the framework of inverse source problems for the Poisson equation in an insulated upper half-plane. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the horizontal axis. In particular, it belon...

A novel three-dimensional dental X-ray imaging method is intro-duced. It is based on hybrid data collected with a digital dental panoramic device. Such a device uses a special geometric movement of the X-ray source and detector around the head of a patient to produce a so-called panoramic image, where all teeth are in sharp focus and details at a d...

Electrical impedance tomography is a noninvasive imaging technique for recovering the admittance distribution inside a body from boundary measurements of current and voltage. In this work, we consider application of impedance tomography to nondestructive testing of reinforced concrete: The aim is to estimate the thickness of the concrete layer on t...

The aim in electric impedance tomography is to recover the spatial admittance distribution inside a body from the boundary measurements of current and voltage. In theoretical considerations it is usually assumed that one can use any boundary current and measure the corresponding potential everywhere on the object boundary. On the other hand, a real...

In electrical impedance tomography, one tries to recover the spatial conductivity distribution inside a body from boundary measurements of current and voltage. In many important situations, the examined object has known background conductivity but is contaminated by inhomogeneities. The factorization method of Kirsch provides a tool for locating su...

We consider a variant of (two dimensional) electric impedance tomography with very sparse data that resemble so-called backscatter data in inverse scattering. Such data arise in practice if the same single pair of electrodes is used to drive currents and measure voltage differences, subsequently at various neighboring locations on the boundary of t...

The aim of electric impedance tomography is to gather information on the conductivity inside a physical body from boundary measurements of current and voltage. In many situations of practical importance, the investigated object has known constant background conductivity, but is contaminated by embedded inhomogeneities. In this work, we test numeric...

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the co-efficients of the corresponding partial differential equation. Often the exam-ined object has known background properties but is contaminated by inhomo-geneities that cause perturbations of the coefficient func...

We investigate the inverse source problem of electrostatics in a bounded and convex domain with compactly supported source. We try to extract all information about the unknown source support from the given Cauchy data of the associated potential, adopting by this previous work of Kusiak and Sylvester to the case of electrostatics. We introduce, and...

The aim in electric impedance tomography is to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many situations of practical importance, the investigated object has known background conductivity but is contaminated by inhomogeneities. In this work, we try to extract all possible information about...

The factorization method is a tool for recovering inclusions inside a body when the Neumann-to-Dirichlet operator, which maps applied currents to measured voltages, is known. In practice this information is never at hands due to the discreteness and physical properties of the measurement devices. The complete electrode model of impedance tomography...

The aim of optical tomography is to reconstruct the optical properties inside a physical body, e.g. a neonatal head, by illuminating it with near-infrared light and measuring the outward flux of photons on the object boundary. Because a brain consists of strongly scattering tissue with imbedded cavities filled by weakly scattering cerebrospinal flu...

In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the investigated object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool...

In electrical impedance tomography, one tries to recover the conductivity inside a body from boundary measurements of current and voltage. In many practically important situations, the object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such incl...

In electrical impedance tomography, one tries to recover the spa- tial conductivity distribution inside a body from boundary,measurements,of current and voltage. In many practically important situations, the object has known,background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool...

The aim of optical absorption and scattering tomography is to reconstruct the optical properties inside a physical body, e.g., a neonatal head, by illuminating it with near-infrared light and measuring the outward flux of photons on the object boundary. Because brain consists of strongly scattering tissue with imbedded cavities filled by weakly sca...

In the framework of diffuse tomography, i.e. optical tomography and electrical impedance tomography, the factorization method of Andreas Kirsch provides a tool for locating inhomogeneities inside an object with known background material parameters. In earlier work, it has been shown that inhomogeneities can be characterized with the factorization t...

In optical tomography, one tries to determine the spatial absorption and scattering distributions inside a body by using measured pairs of inward and outward fluxes of near-infrared light on the object boundary. In many practically important situations, the scatter and the absorption inside the object are smooth apart from inclusions where at least...

In electrical impedance tomography one tries to recover the spatial admittance distribution inside a body from boundary measurements. In theoretical considerations it is usually assumed that the boundary data consists of the Neumann-to-Dirichlet map; when conducting real-world measurements, the obtainable data is a linear finite-dimensional operato...

This paper provides mathematical analysis of optical tomography in a situation when the examined object, for example the human brain, is strongly scattering with non-scattering inclusions. Light propagation in biological tissue is often modelled by the diffusion approximation of the radiative transfer equation. To be justified, the diffusion approx...

In this paper we propose a novel computational method for localization of metallic objects with Electrical Impedance Tomography (EIT). The problem of metal localization arises from non-destructive testing (NDT) of reinforced concrete (1). In concrete structures the thickness of the protective concrete layer on top of reinforcing bars is one of the...

This thesis presents mathematical analysis of optical and electrical impedance tomography. We introduce papers [I-III], which study these diffusive tomography methods in the situation where the examined object is contaminated with inclusions that have physical properties differing from the background. Research reports / Helsinki University of Techn...