Analysis of Dynamic Magnetic Resonance Elastography Data - Advance Image Processing in Magnetic Resonance Imaging - page 452

which allows estimation of the shear modulus from a single polarization of
motion. A separate assumption path can be used for 2-D imaging by assuming
that all derivatives in the out-of-plane direction are negligible. The shear mode
then decouples and can also be solved by 2-D Helmholtz inversion [42].
In practice, such direct inversion techniques require data smoothing and the
calculation of accurate second derivatives from the noisy data. The resolution is
essentially limited only by the noise level in the data. In a stiffer material, the
shear wave has a longer wavelength, making the derivatives smaller and the effects
of noise more serious. The relative performance of different filtering approaches
for smoothing was studied in detail by Oliphant [43].
These techniques do not depend on planar shear-wave propagation but simply
on the presence of motion (that satisfies the assumed physical model) in the region
of interest. In particular, complex interference patterns from reflection, diffraction,
etc., do not pose difficulties except that these patterns may contain areas of low
amplitude and, hence, low signal-to-noise ratio (SNR). This is also true for the
LFE algorithm, which, despite its origin as an image processing method, actually
involves inverting the Helmholtz equation (with the additional assumption of no
attenuation) and correctly handles superimposed waves.

14.5.6 V ARIATIONAL M ETHOD

Romano et al. [47,48] have suggested using the weak (variational) integral form
of Equation 14.4 and test functions to estimate the Lame constants. The test
functions are chosen such that they and their first derivatives vanish at the local
window boundaries, removing all effects of surface forces. Integration by parts
is used to shift the derivative operations from the noisy data to the analytic
derivatives of the smooth test functions and integrating these over local windows
in product with the data. In practice, this is similar to calculating derivatives by
filtering with the derivative of a smooth function. Their assumption of constant
µ/ρ also is essentially equivalent to the local homogeneity assumption. In the

incompressible case, this is equivalent to direct inversion with the specific con-
ditions described earlier imposed on the smoothing filter.

14.5.7 M ATCHED F ILTER

The matched-filter algorithm uses an adaptive smoothed matched filter (i.e., a
smoothed version of the data itself) and its Laplacian to perform the same division
as direct inversion. It is motivated by theoretical considerations to minimize the
uncertainty in the estimate of

in the face of random noise [43,49]. The processing
is computationally more intensive than direct inversion because a different filter
is calculated and applied at each voxel.

µ

14.5.8 R EMOVING THE L OCAL H OMOGENEITY A SSUMPTION

The assumption of local homogeneity is used in all these techniques to simplify the
equation of motion to an algebraic equation that can be solved locally (Equation 14.5).