On the Degree of Ill-Posedness of Multi-Dimensional Magnetic Particle Imaging

Magnetic particle imaging is an imaging modality of relatively recent origin, and it exploits the nonlinear magnetization response for reconstructing the concentration of nanoparticles. Since first invented in 2005, it has received much interest in the literature. In this work, we study one prototypical mathematical model in multi-dimension, i.e., the equilibrium model, which formulates the problem as a linear Fredholm integral equation of the first kind. We analyze the degree of ill-posedness of the associated linear integral operator by means of the singular value decay estimate for Sobolev smooth bivariate functions, and discuss the influence of various experimental parameters. In particular, applied magnetic fields with a field free point and a field free line are distinguished. The study is complemented with extensive numerical experiments.Comment: 20 pages, 6 figure

Intrinsic dimensionality in vision: Nonlinear filter design and applications

Biological vision and computer vision cannot be treated independently anymore. The digital revolution and the emergence of more and more sophisticated technical applications caused a symbiosis between the two communities. Competitive technical devices challenging the human performance rely increasingly on algorithms motivated by the human vision system. On the other hand, computational methods can be used to gain a richer understanding of neural behavior, e.g. the behavior of populations of multiple processing units. The relations between computational approaches and biological findings range from low level vision to cortical areas being responsible for higher cognitive abilities. In early stages of the visual cortex cells have been recorded which could not be explained by the standard approach of orientation- and frequency-selective linear filters anymore. These cells did not respond to straight lines or simple gratings but they fired whenever a more complicated stimulus, like a corner or an end-stopped line, was presented within the receptive field. Using the concept of intrinsic dimensionality, these cells can be classified as intrinsic-two-dimensional systems. The intrinsic dimensionality determines the number of degrees of freedom in the domain which is required to completely determine a signal. A constant image has dimension zero, straight lines and trigonometric functions in one direction have dimension one, and the remaining signals, which require the full number of degrees of freedom, have the dimension two. In this term the reported cells respond to two dimensional signals only. Motivated by the classical approach, which can be realized by orientation- and frequency-selective Gabor-filter functions, a generalized Gabor framework is developed in the context of second-order Volterra systems. The generalized Gabor approach is then used to design intrinsic two-dimensional systems which have the same selectivity properties like the reported cells in early visual cortex. Numerical cognition is commonly assumed to be a higher cognitive ability of humans. The estimation of the number of things from the environment requires a high degree of abstraction. Several studies showed that humans and other species have access to this abstract information. But it is still unclear how this information can be extracted by neural hardware. If one wants to deal with this issue, one has to think about the immense invariance property of number. One can apply a high number of operations to objects which do not change its number. In this work, this problem is considered from a topological perspective. Well known relations between differential geometry and topology are used to develop a computational model. Surprisingly, the resulting operators providing the features which are integrated in the system are intrinsic-two-dimensional operators. This model is used to conduct standard number estimation experiments. The results are then compared to reported human behavior. The last topic of this work is active object recognition. The ability to move the information gathering device, like humans can move their eyes, provides the opportunity to choose the next action. Studies of human saccade behavior suggest that this is not done in a random manner. In order to decrease the time an active object recognition system needs to reach a certain level of performance, several action selection strategies are investigated. The strategies considered within this work are based on information theoretical and probabilistic concepts. These strategies are finally compared to a strategy based on an intrinsic-two-dimensional operator. All three topics are investigated with respect to their relation to the concept of intrinsic dimensionality from a mathematical point of view

L1 data fitting for robust reconstruction in magnetic particle imaging: quantitative evaluation on Open MPI dataset

Magnetic particle imaging is an emerging quantitative imaging modality, exploiting the unique nonlinear magnetization phenomenon of superparamagnetic iron oxide nanoparticles for recovering the concentration. Traditionally the reconstruction is formulated into a penalized least-squares problem with nonnegativity constraint, and then solved using a variant of Kaczmarz method which is often stopped early after a small number of iterations. Besides the phantom signal, measurements additionally include a background signal and a noise signal. In order to obtain good reconstructions, a preprocessing step of frequency selection to remove the deleterious influences of the noise is often adopted. In this work, we propose a complementary pure variational approach to noise treatment, by viewing highly noisy measurements as outliers, and employing the l1 data fitting, one popular approach from robust statistics. When compared with the standard approach, it is easy to implement with a comparable computational complexity. Experiments with a public domain dataset, i.e., Open MPI dataset, show that it can give accurate reconstructions, and is less prone to noisy measurements, which is illustrated by quantitative (PSNR / SSIM) and qualitative comparisons with the Kaczmarz method. We also investigate the performance of the Kaczmarz method for small iteration numbers quantitatively

Time-dependent parameter identification in a Fokker-Planck equation based magnetization model of large ensembles of nanoparticles

In this article, we consider a model motivated by large ensembles of nanoparticles' magnetization dynamics using the Fokker-Planck equation and analyze the underlying parabolic PDE being defined on a smooth, compact manifold without boundary with respect to time-dependent parameter identification using regularization schemes. In the context of magnetic particle imaging, possible fields of application can be found including calibration procedures improved by time-dependent particle parameters and dynamic tracking of nanoparticle orientation. This results in reconstructing different parameters of interest, such as the applied magnetic field and the particles' easy axis. These problems are in particular addressed in the accompanied numerical study

Deep image prior for 3D magnetic particle imaging: A quantitative comparison of regularization techniques on Open MPI dataset

Magnetic particle imaging (MPI) is an imaging modality exploiting the nonlinear magnetization behavior of (super-)paramagnetic nanoparticles to obtain a space- and often also time-dependent concentration of a tracer consisting of these nanoparticles. MPI has a continuously increasing number of potential medical applications. One prerequisite for successful performance in these applications is a proper solution to the image reconstruction problem. More classical methods from inverse problems theory, as well as novel approaches from the field of machine learning, have the potential to deliver high-quality reconstructions in MPI. We investigate a novel reconstruction approach based on a deep image prior, which builds on representing the solution by a deep neural network. Novel approaches, as well as variational and iterative regularization techniques, are compared quantitatively in terms of peak signal-to-noise ratios and structural similarity indices on the publicly available Open MPI dataset

Structural modifications of low-energy heavy-ion irradiated germanium

Heavy-ion irradiation of crystalline germanium (c-Ge) results in the formation of a homogeneous amorphous germanium (a-Ge) layer at the surface. This a-Ge layer undergoes structural modification such as a strong volume expansion accompanied by drastic surface blackening with further ion irradiation. In the present paper we investigate the mechanism of this ion-induced structural modification in a-Ge basically for the irradiation with I ions (3 and 9 MeV) at room and low temperature as a function of ion fluence for the ion incidence angles of Θ=7 and Θ=45. For comparison, Ag- and Au-ion irradiations were performed at room temperature as a function of the ion fluence. At fluences two orders of magnitude above the amorphization threshold, morphological changes were observed for all irradiation conditions used. Over a wide range of ion fluences we demonstrate that the volume expansion is caused by the formation of voids at the surface and in the depth of the projected ion range. At high ion fluences the amorphous layer transforms into a porous structure as established by cross section and plan view electron microscopy investigations. However, the formation depth of the surface and buried voids as well as the shape and the dimension of the final porous structure depend on the ion fluence, ion species, and irradiation temperature and will be discussed in detail. The rate of the volume expansion (i.e., porous layer formation) depends linearly on the value of εn. This clearly demonstrates that the structural changes are determined solely by the nuclear energy deposited within the amorphous phase. In addition, at high ion fluences all perpendicular ion irradiations lead to a formation of a microstructure at the surface, whereas for nonperpendicular ion irradiations a nonsaturating irreversible plastic deformation (ion hammering) without a microstructure formation is observed. For the irradiation with ion energies of several MeV, the effect of plastic deformation shows a linear dependence on the ion fluence. Based on these results, we provide an explanation for the differences in surface morphology observed for different angles of incidence of the ion beam will be discussed in detail

Bayesian view on the training of invertible residual networks for solving linear inverse problems

Learning-based methods for inverse problems, adapting to the data's inherent structure, have become ubiquitous in the last decade. Besides empirical investigations of their often remarkable performance, an increasing number of works addresses the issue of theoretical guarantees. Recently, [3] exploited invertible residual networks (iResNets) to learn provably convergent regularizations given reasonable assumptions. They enforced these guarantees by approximating the linear forward operator with an iResNet. Supervised training on relevant samples introduces data dependency into the approach. An open question in this context is to which extent the data's inherent structure influences the training outcome, i.e., the learned reconstruction scheme. Here we address this delicate interplay of training design and data dependency from a Bayesian perspective and shed light on opportunities and limitations. We resolve these limitations by analyzing reconstruction-based training of the inverses of iResNets, where we show that this optimization strategy introduces a level of data-dependency that cannot be achieved by approximation training. We further provide and discuss a series of numerical experiments underpinning and extending the theoretical findings