4.6 Reconstruction of the sparse PR: the peeling algorithm
4.6.3 Analysis of the peeling algorithm
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Figure 4.7: (A) The ACF of a discrete signal, that is the input of the PA. (B) Reconstructed sparse signal (green) obtained with the PA compared with the original discrete signal (blue).
Note that the PA recovers almost perfectly the original signal.
the first and last delta as
c(1) =
d(i)d(L)
d(j) and c(N )=
d(j)d(L) d(i) ,
where L is the index of the delta located further away, that is L = arg maxny(n). Note that the (unknown) labellings of the elements in the first twin fixes the time direction, that is one of the free parameters we have, see (4.1). This procedure has a major flaw: we need a real twin and without collisions, two conditions that are hard to check. We solve this problem using a voting strategy similar to the one described in Section 4.6.1: we pick the available twins, compute the amplitudesc(1)andc(N )according to the two possible orderings, and we pick the estimate which is confirmed by the majority of twins. If there are at least two real twins without collisions, there are at least two equal estimates. This estimate is correct unless there are 2 or more fake twins and/or with collision that agree on a wrong estimate. Again, if the probability distribution of the{c(n)}Nn=1is continuous, this event has probability zero.
A flowchart of the algorithm is given in Figure 4.6, while the Matlab implementation is available [124].
4.6.3 Analysis of the peeling algorithm
In this section, we analyze the performance of the PA and its fundamental aspects. We show reconstruction examples for both discrete and continuous PR, the impact of collisions in the discrete case, and the impact of noise on the ACF deltas locations in the continuous case. Note
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1 Success Rate of PA
Collisions Rate
Sparsity
%
Success Rate of CF
Figure 4.8: Performance of the PA as a function of the sparsity. For each value of the sparsity, we solved 1000 random discrete PR using the peeling algorithm and we compare it with a state-of-the-art algorithm for discrete PR, known as charge flipping (CF) [116]. The reconstruction is considered to be successful when the MSE is smaller than 10−4. Note that when the signal is sufficiently sparse, the peeling algorithm outperforms charge flipping. We also measure the average number of collisions in the ACF; note that even if the algorithm can manage the presence of some collisions, it starts to fail when there are too many of them.
that we do not consider the impact of collisions for continuous PR because they have probability zero. According to the same philosophy, we do not consider noise in the deltas locations for the discrete problem because noise generally affects the amplitudes.
First, we present an example of input and output of the PA when applied to a discrete sparse PR problem. Figure 4.7 shows the recovery of a 6-sparse discrete signal from its ACF using the PA. This is compared with the original signal, showing perfect recovery. Note that the reconstructed signal has been shifted and properly time-reversed to match the original signal, according to (4.1).
Second, we analyze the impact of collisions on the reconstruction performance for the discrete PR problem. In general, according to the theory presented in Section 4.5, the reconstruction is unique whenever there are no collision. As we previously mentioned, we designed the iterative peeling action of the PA to reduce the negative impact of collisions in the reconstruction and to exploit the sparsity of the signal. Therefore, we are interested to see the success rate of PA as a function of collisions and we consider the following simulation setup: we pick discrete signals of length l = 128 with a sparsity N ∈ [6, 34]. For each N, we consider 1000 randomly generated sparse signals and we measure the percentage of successful reconstructions and the percentage of elements of the ACF with collisions. As a reference, we compare the reconstruction performance to the one obtained by charge flipping (CF) [115, 116], the reference algorithm for the reconstruction of X-ray crystallographic data. The results are shown in Figure 4.8, where we first note that PA outperforms CF for very sparse signals, that is N ≤ 16. Moreover, we underline that for low sparsities, that is N ≤ 16, and consequently a small number of collisions, the reconstruction is often successful. On the other hand, as the number of collisions grows, the PA fails to reconstruct properly most of the sequences.
4.6 Reconstruction of the sparse PR: the peeling algorithm 167
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Figure 4.9: (A) The sparse ACF of a signal, which is the input of the PA. (B) The original sparse (blue) signal and the reconstructed one (green) by the PA. Note that both signals are defined on a continuous domain and a proper constant phase term has been chosen to show the equivalence of the two signals. The reconstruction is exact up to the machine precision.
Reconstruction Error
Measurement Noise 0
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Figure 4.10: For each plot, we show the reconstruction error of PA as a function of the noise in the measured ACF and for a specific relaxation parameter . Each point in the scatter plot is an instance of PA for a random continuous signal. The point location is given by the noise in the measured deltas of the ACF (x axis) and the reconstruction error (y axis). Both axes are in a log10scale and the red line is located at the value of that we considered. The red dots represent those signals whose reconstruction failed due to the negative effects of the relaxation. Note that, the ratio of red-to-blue dots grows with the measurement noise, showing the instability of the PA to such a noise.
Next, we analyze the behavior of the PA when reconstructing sparse signals defined on a continuous domain.
First, we consider an example of reconstruction of a 10-sparse signal and we show the results in Figure 4.9. Note that the locations of the deltas and their amplitudes are perfectly recovered,
up to the constant phase term that has been found a-posteriori.
Second, we analyze the reconstruction performance of the PA as a function of noise. More precisely, we consider to measure an ACF with errors in the delta’s locations,{y(n)+w(n)}, where w(n)are independent random variables uniformly distributed on±ξ. We vary the maximum noise amplitude ξ ={10−7, 10−5, 10−3, 10−1} and the relaxation parameter = [10−14, 1] as defined in (4.8), and we observe the error of the reconstructed sparse signal. The sparsity of the signal is fixed to N = 20 and the deltas are uniformly distributed as x(n)∈ [0, 1]. The results are shown in Figure 4.10: four scatter plots (one for each ξ) show the relationship between the noise in the deltas of the ACF and the respective noise in the reconstructed signal. More precisely, each point represents a randomly generated signal; its horizontal position is given by ξ and its vertical position is given by the error in the reconstruction of the locations of the deltas, defined as
n
|x(n)− x(n)|2.
Due to the absence of denoising, we would expect reconstruction error to be of the same order of magnitude as the measurement noise. Nonetheless, as soon as ξ > , the relaxation is not sufficiently strong to counterbalance the measurement noise w(n), and the PA fails. Another case of failure is introduced by the relaxation itself: if there are deltas closer than to each other, the estimator defined by (4.8) is incorrect. The red dots in Figure 4.10 are examples of the described negative effect introduced by the relaxation.