A limitation in the use of spatially stationary strain variance estimation in the regularization of shear modulus reconstruction

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To yield spatially uniform stability in shear modulus reconstruction, an effective method for setting appropriate spatially varied regularization parameters using the variances of measured strain tensor components was previously reported by us. In the method, the strain variances are evaluated at each position using multiple field measurements or in real time using a single field measurement with the Ziv-Zakai lower bound (ZZLB, i.e., Vz) when using all cross-correlation-based methods. However, a stationary assumption of the displacement or strain measurement error can also be used as an alternative method of real-time reconstruction using an estimated variance (i.e., Vd or Vs). In this study, a comparison of real-time spatially variant regularizations using the above three estimated strain variances performed on an agar phantom reveals that stationary assumptions cannot be used when the target region has a large spatial variation in the shear modulus distribution (e.g., when stress concentration regions are generated in front of and behind a stiff inclusion or tumor), although the ZZLB can be used if sufficient care is taken owing to its low local correlation. When a target has a released tissue boundary, none of the estimated variances are useful. In such regions, none of the estimated variances should be used directly, because the shear modulus is not spatially smooth.
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