Fredrik Orderud

Empirical Bayes estimator for endocardial edge detection in 3D+T echocardiography

This paper presents an empirical Bayes (EB) estimator for detection of endocardial edges in 3D+T echocardiography recordings. A maximum likelihood (ML) edge detector, proposed in a previous study, combines the responses of multiple edge detectors to improve the detection accuracy. We aim to further extend this approach with the use of contextual priors, that gives the probabilistic distribution of correct (yet unknown) endocardial edge positions. For training, a ML model that gives an optimal linear combination of multiple endocardial edge detectors is learned from a pre-segmented dataset. For a given test data, (1) ML edges are estimated using the learned ML model, (2) a conceptual prior is derived using the ML edge estimations in an empirical fashion, and (3) ML estimates and the conceptual prior are fused to produce empirical Bayes endocardial edge estimates. Comparative analyses show that EB reduces the mean square endocardial surface error with respect to ML estimations. This is due to the Stein effect that briefly asserts that the expected mean square error of the ML estimations should be reduced with the use of empirically-derived prior information.

Doo-Sabin Surface Models with Biomechanical Constraints for Kalman Filter Based Endocardial Wall Tracking in 3D+T Echocardiography.

3D+T echocardiography is a valuable tool for assessing cardiac function, as it enables real-time, non-invasive and low cost acquisition of volumetric images of the heart. The automated tracking of heart chambers in 3D+T echocardiography remains a challenging task due to reasons including speckle noise, shadowing, and the existence of intra-cavity structures [6]. Furthermore, the real-time detection of endocardial borders might be desirable for the invasive procedures and intensive care applications. State-space analysis using Kalman filtering can be employed for the detection of left ventricle (LV) structures in time-dependent recordings. Orderud et al. proposed a Kalman tracking framework for the real-time detection of LV structures in 3D+T echocardiography [5]. The study took advantage of compact Doo-Sabin model representations for rapid tracking, but it did not utilize physical properties to constrain model deformations. Liu et al. introduced a biomechanical-model constrained statespace analysis framework for the tracking of short-axis 2D+T echocardiography recordings [4]. Their study used dense Delaunay triangulated models and employed basic tri-nodal linear elements during the finite element analysis (FEA). Due to triangulated high resolution model representations, it offered a computationally expensive solution. This paper proposes an approach to combine the compact model representations with biomechanical constraints for rapid and accurate tracking. We extend the real-time Kalman tracking framework described in [5] by employing biomechanically constrained state transitions. First, FEA for the tracked Doo-Sabin surface model is performed using the isoparametric method introduced in [3]. This step enables the computation of a stiffness matrix K for a given endocardial model using shell elements without changing the model geometry. However, the computed model might lead to inaccurate deformation modes due to hypothesized model shape and FEA parameters (e.g. Young’s modulus, Poisson’s ratio). Accordingly, we improve the model shape and stiffness matrix using statistical information collected from a training data via Control Point Distribution Models (CPDM) [2]. During the improvement stage, (1) the model shape is updated to the population mean, (2) the stiffness matrix for the updated model shape is computed as K′ (see Figure 1), and (3) K′ is further modified to Kopt to produce similar modes of deformation as the statistically observed ones using Baruch and Bar-Itzhack direct matrix modifications (BBDMM) [1]. Finally, the state prediction stage of the Kalman tracking framework is formulated to perform biomechanically constrained tracking. In the results section, endocardial surface tracking quality is compared among (1) Doo-Sabin surface models with different control node resolutions, (2) biomechanically constrained and non-constrained state transitions, and (3) the systems employing statistically improved and not improved Doo-Sabin models (see Figure 2). Our analyses showed that