Multiobject tracking provides situational awareness that enables new applications for modern convenience, applied ocean sciences, public safety, and homeland security. In many multiobject tracking applications, including radar and sonar tracking, after coherent prefiltering of the received signal, measurement data is typically structured in cells, where each cell represent, e.g., a different range and bearing value. While conventional detect-then-track (DTT) multiobject tracking approaches convert the cell-structured data within a detection phase into so-called point measurements in order to reduce the amount of data, track-before-detect (TBD) methods process the cell-structured data directly, avoiding a potential information loss. However, many TBD tracking methods are computationally intensive and achieve a reduced tracking accuracy when objects interact, i.e., when they come into close proximity. We here counteract these difficulties by introducing the concept of probabilistic object-to-cell contributions. As many conventional DTT methods, our approach uses a probabilistic association of objects with data cells, and a new object contribution model with corresponding object contribution probabilities to further associate cell contributions to objects that occupy the same data cell. Furthermore, to keep the computational complexity and filter runtimes low, we here use an efficient Poisson multi-Bernoulli filtering approach in combination with the application of belief propagation for fast probabilistic data association. We demonstrate numerically that our method achieves significantly increased tracking performance compared to state-of-the-art TBD tracking approaches, where performance differences are particularly pronounced when multiple objects interact.