For numerical differentiation, dimensionality can be a blessing!

Anderssen, R. S.; Hegland, Markus

doi:10.1090/s0025-5718-99-01033-9

Cited by 74 publications

(69 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…∂φ m/ ∂x k was scaled to ∂u m/ ∂x k (i. e., spatial derivatives of the physical displacement field u) by the factor given in equation (4b) of [28]. For reducing noise, numerical derivatives were calculated by 3D gradients according to Anderssen and Hegland [29] using a twopixel symmetric window in three dimensions [30]. The resulting strain components of the wave field (derivatives of the three Cartesian components of the displacement field) were used to calculate the curl field, as has been established in 3D MRE for suppressing volumetric strain [31].…”

Section: Data Processingmentioning

confidence: 99%

Patient-Activated Three-Dimensional Multifrequency Magnetic Resonance Elastography for High-Resolution Mechanical Imaging of the Liver and Spleen

Guo

Hirsch

Streitberger

et al. 2013

Fortschr Röntgenstr

View full text Add to dashboard Cite

Purpose: To introduce a novel in-vivo three-dimensional multifrequency magnetic resonance elastography (3D-MMRE) method for high-resolution mechanical characterization of the liver and spleen. Materials and Methods: Ten healthy volunteers were examined by abdominal single-shot 3D-MMRE using a novel patient-activated trigger system with respiratory control. 10 contiguous slices with 2.5?mm cubic voxel resolution, 3 wave components, 8 time steps, and 2 averages were acquired at 7 mechanical excitation frequencies from 30 to 60?Hz. The total imaging time was approximately 15?min. For postprocessing, multifrequency dual elasto-visco (MDEV) inversion was used to calculate high-resolution mechanical parameter maps of the abdomen including the liver and spleen. Results: Two parameters maps were generated from each image slice to capture the magnitude and the phase angle of the complex shear modulus. Both parameters depicted the mechanical structures of the abdomen with unprecedented high spatial resolution. Spatially averaged group mean values of the liver and spleen are 1.27???0.17 kPa and 2.01???0.69 kPa, indicating a significantly higher asymptomatic stiffness of the spleen compared to the liver. Conclusion: Patient-activated respiratory-gated 3D-MMRE combined with MDEV inversion provides highly resolved mechanical maps of the liver and spleen that are superior to previous elastograms measured by abdominal MRE. Citation Format: ??Guo J, Hirsch S, Streitberger KJ et?al. Patient-Activated Three-Dimensional Multifrequency Magnetic Resonance Elastography for High-Resolution Mechanical Imaging of the Liver and Spleen. Fortschr R?ntgenstr 2014; 186: 260???266

show abstract

Section: Data Processingmentioning

confidence: 99%

Patient-Activated Three-Dimensional Multifrequency Magnetic Resonance Elastography for High-Resolution Mechanical Imaging of the Liver and Spleen

Guo

Hirsch

Streitberger

et al. 2013

Fortschr Röntgenstr

View full text Add to dashboard Cite

show abstract

“…In this case, the window size used for computing the average derivatives was taken to be 7 by 7 pixels. It should be mentioned that McLaughlin et al use a hybrid method that combines the averaging scheme of Anderssen and Hegland (1999) with a median scheme to compute derivatives for reconstructing shear stiffness from the same experiment. They use a mathematically rigorous method for adaptively choosing a method and for computing the derivatives based on the local signal-to-noise ratio.…”

Section: Shear Modulus Reconstruction From Mr Measured Datamentioning

confidence: 99%

“…In this work, a simple filter is used to smooth the potentially noisy displacement data and project it onto a volume preserving space prior to applying the inverse algorithm. We also present one example where a method designed, based on statistical analysis (Anderssen and Hegland, 1999), for differentiating noisy data is used. Another disadvantage, as compared to the methods that solve the approximating Helmholtz equation, is that all three components of the displacement field are required as well as their gradients.…”

Section: Introductionmentioning

confidence: 99%

Shear modulus reconstruction in dynamic elastography: time harmonic case

Park

Maniatty²

2006

Phys. Med. Biol.

101

View full text Add to dashboard Cite

Abstract. This paper presents a direct inversion approach for reconstructing the elastic shear modulus in soft tissue from dynamic measurements of the interior displacement field during time harmonic excitation. The tissue is assumed to obey the equations of nearly incompressible, linear, isotropic elasto-dynamics in harmonic motion. A finite element discretization of the governing equations is used as a basis, and a procedure is outlined to eliminate the need for boundary conditions in the inverse problem. The hydrostatic stress (pressure) is also reconstructed in the process, and the effect of neglecting this term in the governing equations, which is common practice, is considered. The approach does not require iterations and can be performed on subregions of the domain resulting in a computationally efficient method. A sensitivity study is performed to investigate the detectability of abnormal regions of different size and shear modulus contrast from the background. The algorithm is tested on simulated data on a 2D domain, where the data is generated on a very fine mesh to get a near exact solution, then down-sampled to a coarser mesh that is similar to the spatial discretization of actual data, and noise is added. Results showing the effect of the hydrostatic stress term and noise are presented. A reconstruction using MR measured experimental data involving a tissue-mimicking phantom is also shown to demonstrate the algorithm.

show abstract

“…As it happens, however, previous contributions have not demonstrated a capability to produce derivatives, even of the first order, without order-of-accuracy deterioration. In the recent literature we find, for example, differentiation errors of order h 2/9 for mesh spacings of order h in the method [2], errors of the order of several percent even for machine accurate data in the contribution [9], and errors of order δ 1/2 for data containing errors of order δ in the contributions [12,13].…”

mentioning

confidence: 85%

“…with limited loss in the order of accuracy. For example, for (possibly nonsmooth) O(h r ) errors in the values of an underlying infinitely differentiable function, the LDC loss in the order of accuracy is "vanishingly small": derivatives of smooth functions are approximated by the LDC method with an accuracy of order O(h r ) for all r < r. Previous work on reconstruction of numerical derivatives from scattered noisy data [2,9,12,13,19,20,21,22] has focused on two main approaches: use of finite differences on one hand and regularization on the other. Much of the work in this area has been concerned with stability, seeking mainly to eliminate large derivative errors that arise as two function values f (x 1 ) and f (x 2 ) with large errors occur at points x 1 and x 2 that lie very close to each other.…”

mentioning

confidence: 99%

Numerical Differentiation of Approximated Functions with Limited Order-of-Accuracy Deterioration

Bruno¹,

Hoch²

2012

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. We consider the problem of numerical differentiation of a function f from approximate or noisy values of f on a discrete set of points; such discrete approximate data may result from a numerical calculation (such as a finite element or finite difference solution of a partial differential equation), from experimental measurements, or, generally, from an estimate of some sort. In some such cases it is useful to guarantee that orders of accuracy are not degraded: assuming the approximating values of the function are known with an accuracy of order O(h r ), where h is the mesh size, an accuracy of O(h r ) is desired in the value of the derivatives of f . Differentiation of interpolating polynomials does not achieve this goal since, as shown in this text, n-fold differentiation of an interpolating polynomial of any degree ≥ (r − 1) obtained from function values containing errors of order O(h r ) generally gives rise to derivative errors of order O(h r−n ); other existing differentiation algorithms suffer from similar degradations in the order of accuracy. In this paper we present a new algorithm, the LDC method (low degree Chebyshev), which, using noisy function values of a function f on a (possibly irregular) grid, produces approximate values of derivatives f (n) (n = 1, 2 . . . ) with limited loss in the order of accuracy. For example, for (possibly nonsmooth) O(h r ) errors in the values of an underlying infinitely differentiable function, the LDC loss in the order of accuracy is "vanishingly small": derivatives of smooth functions are approximated by the LDC algorithm with an accuracy of order O(h r ) for all r < r. The algorithm is very fast and simple; a variety of numerical results we present illustrate the theory and demonstrate the efficiency of the proposed methodology.

show abstract

For numerical differentiation, dimensionality can be a blessing!

Cited by 74 publications

References 11 publications

Patient-Activated Three-Dimensional Multifrequency Magnetic Resonance Elastography for High-Resolution Mechanical Imaging of the Liver and Spleen

Patient-Activated Three-Dimensional Multifrequency Magnetic Resonance Elastography for High-Resolution Mechanical Imaging of the Liver and Spleen

Shear modulus reconstruction in dynamic elastography: time harmonic case

Numerical Differentiation of Approximated Functions with Limited Order-of-Accuracy Deterioration

Contact Info

Product

Resources

About