Figure 3.

The pretzel: samples of true pretzels (top row) and reconstructed pretzels (bottom row, see Section 5). The two arms have the same shape and range of motion, but for each instance of the synthetic molecule the conformation of each arm is chosen independently from the state of the other arm. For the purpose of this illustration, we present various states of one of the arms (the arm in the green ball in Figure 2), while holding the other arm (the arm in the blue ball in Figure 2) at a fixed state. In the simulation and the recovered object, the arms move independently.

The resulting hyper-molecule has two state variables: one state variable encodes the state of the “arm” in the green ball (see Figure 2) and the other state variable encodes the state of the arm in the blue ball (see Figure 2). The third state variable for the rigid center in the yellow ball is ignored in this figure. Again, for illustration purposes, we present one of the arms at various states, while holding the other arm fixed. The reconstruction captures the conformation found in the synthetic molecule. The reconstruction at a given value of $t$ is similar to the true object at that state, but they do not correspond to the exact same state (see discussion in Section 5), since the choice of parameterization of states is not unique (see discussion in Section 6.2).