Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

In this paper, we aim to provide a statistical theory for object matching based on the Gromov-Wasserstein distance. To this end, we model general objects as metric measure spaces. Based on this, we propose a simple and efficiently computable asymptotic statistical test for pose invariant object discrimination. This is based on an empirical version of a $\beta$-trimmed lower bound of the Gromov-Wasserstein distance. We derive for $\beta\in[0,1/2)$ distributional limits of this test statistic. To this end, we introduce a novel $U$-type process indexed in $\beta$ and show its weak convergence. Finally, the theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons.Comment: For a version with the complete supplement see [v2

Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

In this paper, we aim to provide a statistical theory for object matching based on the Gromov-Wasserstein distance. To this end, we model general objects as metric measure spaces. Based on this, we propose a simple and efficiently computable asymptotic statistical test for pose invariant object discrimination. This is based on an empirical version of a $\beta$-trimmed lower bound of the Gromov-Wasserstein distance. We derive for $\beta\in[0,1/2)$ distributional limits of this test statistic. To this end, we introduce a novel $U$-type process indexed in $\beta$ and show its weak convergence. Finally, the theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons

Distribution of Distances based Object Matching: Asymptotic Inference

In this paper, we aim to provide a statistical theory for object matching based on a lower bound of the Gromov-Wasserstein distance related to the distribution of (pairwise) distances of the considered objects. To this end, we model general objects as metric measure spaces. Based on this, we propose a simple and efficiently computable asymptotic statistical test for pose invariant object discrimination. This is based on a (β-trimmed) empirical version of the afore-mentioned lower bound. We derive the distributional limits of this test statistic for the trimmed and untrimmed case. For this purpose, we introduce a novel U-type process indexed in β and show its weak convergence. The theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons.</p

Sensitivity comparison of absorption and grating-based phase tomography of paraffin-embedded human brain tissue

Advances in high-resolution hard X-ray computed tomography have led to the field of virtual histology to complement histopathological analyses. Phase-contrast modalities have been favored because, for soft tissues, the real part of the refractive index is orders of magnitude greater than the imaginary part. Nevertheless, absorption-contrast measurements of paraffin-embedded tissues have provided exceptionally high contrast combined with a submicron resolution. In this work, we present a quantitative comparison of phase tomography using synchrotron radiation-based X-ray double grating interferometry and conventional synchrotron radiation-based computed tomography in the context of histopathologically relevant paraffin-embedded human brain tissue. We determine the complex refractive index and compare the contrast-to-noise ratio (CNR) of each modality, accounting for the spatial resolution and optimizing the photon energy for absorption tomography. We demonstrate that the CNR in the phase modality is 1.6 times higher than the photon-energy optimized and spatial resolution-matched absorption measurements. We predict, however, that a further optimized phase tomography will provide a CNR gain of 4. This study seeks to boost the discussion of the relative merits of phase and absorption modalities in the context of paraffin-embedded tissues for virtual histology, highlighting the importance of optimization procedures for the two complementary modes and the trade-off between spatial and density resolution, not to mention the disparity in data acquisition and processing