In this paper, we present new results on the Riemannian geometry of symmetric positive semi-definite (SPSD) matrices. First, based on an existing approximation of the geodesic path, we introduce approximations of the logarithmic map, the exponential maps, and Parallel Transport (PT). Second, we derive a canonical representation for a set of SPSD matrices. Based on these results, we propose an algorithm for Domain Adaptation (DA) and demonstrate its performance in two applications: fusion of hyper-spectral images and motion recognition.
A fundamental question in data analysis, machine learning and signal processing is how to compare between data points. The choice of the distance metric is specifically challenging for high-dimensional data sets, where the problem of meaningfulness is more prominent (e.g. the Euclidean distance between images). In this paper, we propose to exploit a property of highdimensional data that is usually ignored -which is the structure stemming from the relationships between the coordinates. Specifically we show that organizing similar coordinates in clusters can be exploited for the construction of the Mahalanobis distance between samples. When the observable samples are generated by a nonlinear transformation of hidden variables, the Mahalanobis distance allows the recovery of the Euclidean distances in the hidden space. We illustrate the advantage of our approach on a synthetic example where the discovery of clusters of correlated coordinates improves the estimation of the principal directions of the samples. Our method was applied to real data of gene expression for lung adenocarcinomas (lung cancer). By using the proposed metric we found a partition of subjects to risk groups with a good separation between their Kaplan-Meier survival plot. metric learning, geometric analysis, manifold learning, intrinsic modeling, biclustering, gene expression
Abstract-Modern imaging systems typically use single-carrier short pulses for transducer excitation. Coded signals together with pulse compression are successfully used in radar and communication to increase the amount of transmitted energy. Previous research verified significant improvement in SNR and imaging depth for ultrasound imaging with coded signals. Since pulse compression needs to be applied at each transducer element, the implementation of coded excitation (CE) in array imaging is computationally complex. Applying pulse compression on the beamformer output reduces the computational load but also degrades both the axial and lateral point spread function (PSF) compromising image quality. In this work we present an approach for efficient implementation of pulse compression by integrating it into frequency domain beamforming. This method leads to significant reduction in the amount of computations without affecting axial resolution. The lateral resolution is dictated by the factor of savings in computational load. We verify the performance of our method on a Verasonics imaging system and compare the resulting images to time-domain processing. We show that up to 77 fold reduction in computational complexity can be achieved in a typical imaging setups. The efficient implementation makes CE a feasible approach in array imaging paving the way to enhanced SNR as well as improved imaging depth and frame-rate.
Modern imaging systems use single-carrier short pulses for transducer excitation. The usage of coded signals allowing for pulse compression is known to improve signal-to-noise ratio (SNR), for example in radar and communication. One of the main challenges in applying coded excitation (CE) to medical imaging is frequency dependent attenuation in biological tissues. Previous work overcame this challenge and verified significant improvement in SNR and imaging depth by using an array of transducer elements and applying pulse compression at each element. However, this approach results in a large computational load. A common way of reducing the cost is to apply pulse compression after beamforming, which reduces image quality. In this work we propose a high-quality low cost method for CE imaging by integrating pulse compression into the recently developed frequency do-main beamforming framework. This approach yields a 26-fold reduction in computational complexity without compromising image quality. This reduction enables efficient implementation of CE in array imaging paving the way to enhanced SNR, improved imaging depth and higher frame-rate.
In contemporary high-dimensional data analysis, intrinsically similar and related data sets are often significantly different due to various undesired factors that could arise from different acquisition equipment, calibration, environmental conditions, and many other sources of batch effects. Therefore, the task of aligning such data sets has become ubiquitous. In this work, we present a method for the alignment of different, but related, sets of Symmetric Positive Semidefinite (SPSD) matrices, which constitute a commonly-used family of features, e.g., covariance and correlation matrices, various kernels, and prototypical graph and network representations. Our method does not require any a-priori correspondence, and it is based on non-Euclidean Procrustes Analysis (PA) using a particular Riemannian geometry of SPSD matrices. While the derivation is focused on the manifold of SPSD matrices, we show that our alignment method can be applied directly in the original high-dimensional data space, when considering SPSD features that are sample covariance matrices. We demonstrate the advantage of our approach over competing methods in simulations and in an application to Brain-Computer Interface (BCI) with electroencephalographic (EEG) recordings.
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