realization of is obtained by sampling with replacement from the set with probability equal to the corresponding element of . The assumptions that object dynamics form a temporal Markov chain and that observations are independent of each other and the dynamics facilitate the implementation of the condensation algorithm. The first assumption allows the dynamics of the object to be entirely determined by the conditional density . The model of the system dynamics determined by must also be selected for the algorithm, and generally includes both deterministic and stochastic dynamics. The algorithm can be summarized by initialization at time and three steps at each time t: Initialization Form the initial sample set and weights by sampling according to the prior distribution. For example, specify as Gaussian and set the weights equal to each other. Iterative procedure Sample with replacement times from the set with probability to generate a realization of . Apply the learned dynamics to each element of this new set, to generate a new set . To take into account the current observation , set for each element . This algorithm outputs the probability distribution which can be directly used to calculate the mean position of the tracked object, as well as the other moments of the tracked object. Cumulative weights can instead be used to achieve a more efficient sampling. Implementation considerations Since object-tracking can be a real-time objective, consideration of algorithm efficiency becomes important. The condensation algorithm is relatively simple when compared to the computational intensity of the Ricatti equation required for Kalman filtering. The parameter , which determines the number of samples in the sample set, will clearly hold a trade-off in efficiency versus performance. One way to increase efficiency of the algorithm is by selecting a low degree of freedom model for representing the shape of the object. The model used by Isard 1998 is a linear parameterization of B-splines in which the splines are limited to certain configurations. Suitable configurations were found by analytically determining combinations of contours from multiple views, of the object in different poses, and through principal component analysis (PCA) on the deforming object. Isard and Blake model the object dynamics as a second order difference equation with deterministic and stochastic components: where is the mean value of the state, and , are matrices representing the deterministic and stochastic components of the dynamical model respectively. , , and are estimated via