A.1 Derivation of error resulting from signal recovery procedure

In the following we prove the theorem for average value of the recovery error (σ 2x ) introduced in section 4.2, in Eq. (4.20). We assume, for the sake of simplicity, that the matrix A, transforming the sparse expantion x into the signal y, has normally distributed elements with zero means and 1/N variances.These values of the parameters of normal distribution ensure that the matrix A is orthonormal. Hence, based on Eq. (4.17), the covariance matrix Sr is given by:

where M is the number of acquired samples, N is the number of samples in complete signal, σ is standard deviation of noise in signal and P denotes the covariance matrix of a prior distribution of sparse expantion x. The σ2x is equal to the trace of the matrix Sr and hence:

The sum in the last term in Eq. (A.2) may be approximated by a definite integral. In the following we will use for the calculations a basic rectangle rule, and:

where h = 1/2. At the very beginning, see Eq. (4.19), we assumed that the function P(k) has the form:

We will perform the integration using the substitution t = e−Tk. Without any significant loss of precision, we change the integration limits from [1 − h, N + h] to [0, N]. The calculations of the integral I will be as follows:

Finally, substituting the integral I in Eq. (A.5) into formula in Eq. (A.2), gives the average value of the recovery error:

A.2 Derivation of error resulting from limited number of photoelectrons

As mentioned in section 4.4.1, the function yk, describing the kth signal from a single photoelectron is assumed to be a Gaussian function with standard deviation σp. The function ˜ yk, given in Eq. (4.37), may be approximated with:

where tkr is a random variable with ftr distribution, which denotes the kth λ photon’s registration time, λ contributes to the signal width and n = 1, 2, …,N. The probability that the random variable ˜yk(n) is equal to the specified value may be calculated based on the previously introduced function Ф see Eq. (4.46).