Towards Analyzing the Influence of Measurement Errors in Magnetic Resonance Imaging of Fluid Flows

Abstract

Magnetic Resonance Imaging (MRI) provides an insight into opaque structures and does not only have a large number of applications in the field of medical examinations but also in the field of engineering. In technical applications, MRI enables a contactless measurement of the two-or threedimensional velocity field within minutes. However, various measurement methods would benefit from an acceleration of the measurement procedure. Compressed Sensing is a promising method to fit this need. A random undersampling of the sampled data points enables a significant reduction of acquisition time. As this method requires a nonlinear iterative reconstruction of unmeasured data to obtain the same data quality as for a conventional fully sampled measurement, it is essential to estimate the influence of uncertainty on the quantitative result. This paper investigates the implementation of interval arithmetic approaches with a focus on the applicability in the frame of compressed sensing techniques. These approaches are able to handle bounded uncertainty not only in the case of linear relationships between measured data and the computed outputs but also allow for solving the necessary optimality criteria for the fluid velocity reconstruction in an iterative manner under the assumption of set-valued measurement errors and bounded representations of noise.

Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications

“…Remark 4. An interval extension of the atan2 function is described in [30]. A straightforward extension of a phase unwrap operation applied individually to all interval bounds of the generalized atan2 function in the case of angles leaving the interval [−π ; π] allows for determining the depicted enclosures in Figure 5.…”

Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System

“…To solve the Bézout identity (14) in the design procedure suggested in this paper, it has to be checked firstly whether a solution for this polynomial equation exists. This can be done using a reduced Gröbner basis.…”

“…Here, a selection of useful techniques ranges from the extension of fundamental arithmetic operations to interval valued expressions [1][2][3], which were recently standardized in the IEEE standard 1788 [11], over the development of set valued counterparts for zero finding techniques for sets of algebraic equations such as the Krawczyk operator [12][13][14], interval based techniques for reachability analysis [15], the verified global optimization [16][17][18], to the solution of identification tasks by means of set inversion techniques via interval analysis (e.g., the algorithms set inversion via interval analysis (SIVIA) and problem specific generalizations) [2,19]. If control applications are concerned, interval analysis can be applied both to an offline design and verification of control procedures under consideration of feasibility and safety requirements such as input and state constraints and to an online interval evaluation in terms of real-time capable robust control strategies generalizing the ideas of variable-structure control techniques and backstepping [20][21][22][23][24][25].…”

An unscented transformation approach to stochastic analysis of measurement uncertainty in magnet resonance imaging with applications in engineering

“…Our previous work has dealt with a first approach to quantify the effect of measurement uncertainty by representing the possible ranges of consistently reconstructed data in MRI-based signal processing with the help of purely set-valued, non-probabilistic approaches. John et al (2020b) and Rauh et al (2020) showed that interval analysis (as a set-valued approach) provides a helpful tool to detect those domains in the reconstructed images that are influenced most by the assumed bounded measurement uncertainty. Quantification of these effects became possible with the help of computing the worst-case deviations between the estimated suprema and infima of reconstructed phase angles.…”

“…However, those interval approaches require application-specific insight concerning the derivation of meaningful bounds for the expected measurement errors. As discussed by John et al (2020b) and Rauh et al (2020), suitable options for such models are the assumption of independent additive bounds for each measured point in the frequency domain or uncertainty models that are related to the power spectral density of the acquired data. The validity of such assumptions, however, needs to be checked for each measurement scenario.…”