The problem of tracking membranes of cells as randomly evolving contours has been studied in collaboration with Gregory D. Hager and his students at the Center for Computational Vision and Control (CCVC) at the Yale University, New Haven, CT. We have implemented three methods for tracking cell membranes as active contours in images: a spline-based parameter fitting method (Blake and Izard 1998), a classical non-parametric method based on energy minimization (Kass et al.), and a recently developed variation on the classical method (Tagare 1997, Ma 1998). The first method uses a statistical sampling technique to acquire data about the location of the cell boundary followed by standard spline-fitting techniques. The second method determines the membrane location by optimizing an energy functional combining an external potential derived from the image contrast and an internal energy which places constraints on the shape of the contour representation. The external potential pulls the contour toward the image features representing the cell boundary while the internal energy tends to regularize the shape of the outline. The third method is a technical variant of the second. Essentially, it constrains the evolution of the membrane such that the motion is in the direction locally perpendicular to the contour representation. These methods were tested by tracking membranes in single and four cellblastomeres of a developing mouse embryo. The results of the contour extraction were evaluated in terms of accuracy, stability and the amount of parameter tuning required to achieve a stable boundary representation. The spline-fitting method is designed to minimize the amount of data sampling required to compute a contour, whereas in our case, we prefer to sample densely to achieve high accuracy. Furthermore, it decouples the image processing from the fitting of the spline parameters. Such a two-step procedure has advantages in terms of computational complexity, however, shows deficiencies in contour localization since the estimation of the contour geometry is only indirectly controlled by the image information. For these reasons, we discarded spline-based parameter fitting in favor of energy-based methods. The results achievable by energy-based minimization are more promising. We found that, by suitable tuning of the parameters in the energy functional and by choosing appropriate filters for the computation of the external potential, sufficiently accurate boundary representations are obtainable. The original energy-based formulation has the advantage of being quick to converge, but suffers from a bias introduced by the form of the optimization. Also, we found that the computed contour rotates along the membrane making it difficult to keep the point of probe-target interaction stable over time. The modified optimization technique does not have these problems but is unfortunately slow to converge. While the results in membrane extraction suggest, in principle, that energy-based contours are generally appropriate to solve the task of membrane tracking and that they achieve the required accuracy, our preliminary investigations made clear that significant work is needed to improve the methods. In particular, the image filtering defining the external potential field must be adapted to the image data common to microscopy. A novel filtering scheme has to be devised capable of dealing with the complex and fuzzy contrast generated by semi-transparent biological objects. Also, the energy minimization approach is quite sensitive to parameter tuning. Only a minimal over-tuning of the parameter weighing internal versus external energy can lead to nonsensical results. To be routinely applicable to a machine vision based controller, which is not operated by CV experts, the contour extraction scheme must be parameter free or self-tuning. In consideration of all these aspects as well as of our needs to not only locate the membrane but also to sensitively detect changes we propose to design and test a novel membrane tracker. Initial investigations in this direction are currently made at the CCVC in Yale. Blake, A. and Isard, M. 1998. Active Contours. Springer. Kass, M. Witkin, A. and Terzopoulos, D. 1987. Snakes: Active contour models. Proc. 1st Int'l Conf. Computer Vision ICCV'87, pages 259 - 268. Tagare, H. D. 1997. Non-rigid curve correspondence for estimating heart motion. In Proc. Int'l Conf. Information Processing in Medical Imaging IPMI'97, pages 489 - 494. Ma, T. 1998. Active Contour Models: Consistency, Stability, and ParameterEstimation. PhD thesis, Yale University.
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