Exact solutions for 2N-dimensional phase space reconstruction from N-dimensional projections
This is a draft of a note that I might put on arXiv. I don’t know if it deserves to be published in a journal because it’s kind of obvious from previous work, but it still might be worth sharing.
Let \(f_t(x)\) be a probability density function defined over positions \(x = (x_1, \dots , x_n)\) and velocities \(v = (v_1, \dots , v_n)\) in an \(n\)-dimensional space at time \(t\). Suppose we know the marginal distributions \(\{ f_{t_1}(x) , f_{t_2}(x) , \dots, f_{t_k}(x) \}\) at times \(t_1 < t_2, \dots, < t_k\). Can we use this information to reconstruct the initial distribution \(f_{t_0}(x, v)\)?
If the distribution evolves in an uncoupled harmonic oscillator, the coordinates simply rotate in each \(x_i\)-\(v_i\) plane. We will consider a continuous set of projections \(f_{\theta}(x)\) indexed by the rotation angles \(\theta = (\theta_1, \dots, \theta_n)\), where \(\theta_i \in [0, \pi]\), and write the initial phase space distribution as \(f(x, v)\).
When \(n = 1\), we can define the transformed coordinates as
\[\begin{align} x'(\theta) &= x \cos\theta + v \sin\theta, \\ v'(\theta) &= v \cos\theta - x \sin\theta. \end{align}\]
The Radon Transform (RT) is:
\[\begin{equation} \mathcal{R} f = \int_{-\infty}^{\infty}{f(x'(\theta), v'(\theta)) dv'}. \end{equation}\]
Since the RT is invertible, there is a one-to-one map between the projections \(f_\theta(x)\) and the distribution \(f(x, v)\).
When \(n = 2\), we can write the projections as \(f_{\theta}(x) = f_{\theta_1, \theta_2}(x_1, x_2)\). Hock and Wolski [1] derived an exact solution for this case. The solution is to first apply the Inverse Radon Transform (IRT) to the slices \(f_{\theta_1, \theta_2}(x_1 | x_2)\) for fixed \(\theta_2\), giving \(f_{\theta_2}(x_1, x_2, v_1)\). Then, for fixed \(x_1\) and \(x_2\), apply the IRT to reconstruct the \(x_2\)-\(v_2\) distribution, giving \(f(x_1, x_2, v_1, v_2)\). I’ll call this the Hock-Wolski (HW) method.
Jaster-Merz et al. [2] showed that if one measures the three-dimensional projections \(f_{\theta_1, \theta_2}(x_1, x_2, x_3)\), one can reconstruct the five-dimensional phase space distribution \(f(x_1, x_2, x_3, v_1, v_2)\) by running the HW method for each fixed \(x_3\), i.e., on each two-dimensional slice \(f_{\theta_1, \theta_2}(x_1, x_2 | x_3)\) of the three-dimensional projections. This idea can be extended to reconstruct the six-dimensional phase space distribution \(f(x_1, x_2, x_3, v_1, v_2, v_3)\) by considering three rotation angles \(\{\theta_1, \theta_2, \theta_3\}\).1 One now has a set of five-dimensional distributions \(f_{\theta_3}(x_1, x_2, x_3, v_1, v_2)\) indexed by \(\theta_3\). Applying the IRT to reconstruct the \(x_3\)-\(v_3\) distribution (while fixing the other coordinates) gives the six-dimensional phase space distribution \(f(x_1, x_2, x_3, v_1, v_2, v_3)\).
This idea extends to \(n\) dimensions, where projections are indexed by \(n\) rotation angles.
References
Footnotes
In particle accelerators, one can vary the phase of an accelerating cavity to approximate rotations in the longitudinal phase space.↩︎