[26] G. Taubin, "algebraic Nonplanar Curve and Surface Estimation in 3-space with Applications to 5 Concluding Remarks

In this paper, we apply the previously proposed continuous distance transform neural network (CDTNN) to represent 3-D endocardial (inner) and epicardial (outer) contours and quantitatively estimate the motion of left ventricles of human hearts from ultrasound images acquired using transesophageal echo-cardiography. This CDTNN has many good properties as the conventional distance transforms, which are suitable for 3-D object representation and deformation estimation. We have successfully represented the 3-D epicardia and endocardia of left ventricles using CDTNNs trained by as few as 7.5% of the manually traced data. The mean absolute error in the testing for one patient over the 27 testing planes were (1:4 1:2 mm) for the endocardium, (1:4 1:2 mm) for the epicardium (1:3 1:0 mm) at end diastole and (1:4 1:2 mm) for the endocardium vs. 1:2 1:0 mm for the epicardium at end systole. The absolute error measured compares favorably with the human inter-observer variability reported for analyzing distances. With this unique distance transform representation that is continuous and di erentiable, we are also able to systematically and e ectively measure the amount of 3-D heart motion in terms of a ne transform.