The ideal observer sets an upper limit on the performance of an observer on a detection or classification task. The performance of the ideal observer can be used to optimize hardware components of imaging systems and also to determine another observer's relative performance in comparison with the best possible observer. The ideal observer employs complete knowledge of the statistics of the imaging system, including the noise and object variability. Thus computing the ideal observer for images (large-dimensional vectors) is burdensome without severely restricting the randomness in the imaging system, e.g., assuming a flat object. We present a method for computing the ideal-observer test statistic and performance by using Markov-chain Monte Carlo techniques when we have a well-characterized imaging system, knowledge of the noise statistics, and a stochastic object model. We demonstrate the method by comparing three different parallel-hole collimator imaging systems in simulation.