In this section, we report and discuss the results obtained by the proposed algorithms on samples of two different materials, which represent typical case tests well.
Numerical computations were carried out using Matlab R2022b on a laptop equipped with an Apple M1 chip with 16 GB of RAM. Please observe that throughout the section, we refer to the frequencies instead of the angular frequencies , i.e., .
4.1. Experimental Setting
The fitted NMRD profiles, computed by Algorithms 1–3 are compared to the
data by means of the
value, which is defined as follows:
where
is the estimated data value, i.e.,
with
being the computed parameters. We quantitatively compare the computed correlation time distributions
given by the three algorithms, determining the peak values and the area below
in the neighborhood of such peaks, and defining a value such as SpecificWeight. Let us assume that
f has
local maxima at the correlation times
,
. Then, we define the neighborhood of interest
using the Full Width at Half Maximum parameter, i.e.,
We compute the SpecificWeight value for each peak
, i.e.,
where
is the number of correlation times belonging to
,
.
The value of the tolerance parameters used in the stopping criteria of all algorithms is . Moreover, a maximum number of 10 iterations k has been set but never been reached. The computational cost is evaluated in terms of execution time.
To test the algorithms’ robustness, we apply them to a set of artificial profiles obtained by adding to uniformly distributed noise within the experimental error intervals. With these tests, referred to as dispersion analysis, we aim to explore the intervals containing the recovered parameter and how the calculated estimates scatter around the average value. Additionally, we want to examine how the position and value of the peaks change in the recovered correlation times distributions.
4.2. Numerical Results from FFC Measures
We present the results obtained by applying all the algorithms to NMRD profiles obtained from two experimental samples, Manganese and Poplar, respectively. Both systems are considered a “gold standard” in relaxometry studies, especially when the involvement of paramagnetic species is necessary. The relaxometric properties of aqueous manganese solutions have been thoroughly investigated [
19,
20] and, as such, these solutions are routinely utilized to assess the performance and stability of instruments. Additionally, the characteristics of Poplar char have been extensively studied [
21], making it an effective model for examining the textural properties and functional mobility of solvents within these porous materials. The NMRD profile for the manganese sample was acquired by the authors, while the data pertaining to Poplar char were taken from [
22].
These two samples show how the algorithms’ results can complement each other to improve the overall quality of the information provided. We evaluate the global quality of the examined methods in terms of , offset , and computation time.
The
data for the Manganese Sample are measured at 26 frequency values
, ranging from
to
MHz. The error intervals for these measurements vary from
s
−1 to
s
−1. These are illustrated by the black error bars in the left panel of
Figure 1. The
data for the Poplar sample are measured at 21 frequency values
, ranging from
to
MHz. The error intervals for these measurements vary from
s
−1 to
s
−1. These are illustrated by the black error bar in the right panel of
Figure 1.
Table 1 presents the estimated parameter
, the goodness-of-fit measure
, and the computation time in seconds obtained by the three algorithms. We observe that MF-L1 achieves a moderate
value and the shortest computation time. In contrast, MF-UPen shows a slightly higher
and takes longer to compute. Conversely, MF-MUPen achieves the best fit, as evidenced by the lowest
. This suggests superior model accuracy, with a modest increase in computation time.
Table 2 outlines the computational results obtained for the Poplar sample by the three algorithms. MF-UPen and MF-L1 both report nearly identical values for
, with minimal
and very short computation times, indicating efficient and effective performance. However, MF-MUPen, while yielding a similar
to the other two algorithms, shows a higher
value, suggesting a slightly poorer fit. Additionally, MF-MUPen requires longer computation time.
The outcomes for the Manganese and Poplar samples represent two scenarios, each indicative of the potential variability in sample analysis. This diversity highlights the importance of utilizing multiple methods to fully understand sample characteristics under varying conditions. The peak analysis for both the Manganese and Poplar samples across the three methods is performed by plotting the correlation times amplitudes
computed by each method in
Figure 2 and reporting the peak position amplitudes, half-width, and SpecificWeights for each sample in
Table 3 and
Table 4.
In
Table 3, we observe a perfect agreement among the three methods in locating the peak at the longest correlation time
. Meanwhile, MF-UPen and MF-L1 have quite good agreement at the intermediate times:
and
, respectively. The distribution pattern in
Figure 2, left panel, shows similarity features between MF-UPen and MF-MUPen, and reveals a tendency of MF-L1 to resolve multiple components at the shortest times.
In the case of the Poplar sample, as shown in the right panel of
Figure 2 and
Table 4, we observe a tighter clustering of peaks across the methods, particularly at the highest amplitude peak around
. This suggests that all three methods are in agreement concerning the main features of the Poplar sample’s distribution.
4.3. Dispersion Analysis
The robustness of the methods is investigated through the dispersion analysis, described in
Section 4.1. The boxplots in
Figure 3 offer a comparative view of algorithmic performance on the two samples. Each boxplot outlines the algorithms’ interquartile range (IQR) and median of
values.
We observe uniformity in medians for the Manganese sample, with outliers highlighted by red plus symbols, suggesting occasional significant deviations for MF-UPen. The symmetry of the data is evident from the whiskers’ lengths.
Conversely, the Poplar sample exhibits a tighter IQR for each algorithm, denoting less variability. Despite the close median values indicating consistent algorithmic performance, outliers for MF-L1 reveal notable deviations in some cases.
Table 5 compares the
confidence intervals [
23], mean
, and medians for both Manganese and Poplar samples across the three algorithms.
The confidence intervals and mean values suggest a wider range of estimates for the Manganese sample, indicating a less uniform agreement among the algorithms. The median values, while closer, still reflect notable variation between the algorithms, suggesting that the model fit depends on the algorithm applied.
Conversely, the Poplar sample demonstrates remarkable consistency, with both confidence intervals and mean values being nearly identical across all three algorithms. The median values also closely align, reinforcing the observation of uniform performance. This indicates that for the Poplar sample, the choice of algorithm does not significantly influence the outcome, and all three algorithms provide equivalent information.
Table 5 represents two distinct scenarios that may emerge when these algorithms are applied to samples with varying characteristics. In the case of the Poplar sample, the outcome from all three algorithms is congruent, implying that the algorithms are robust and interchangeable for this type of sample. Conversely, the Manganese sample demonstrates less consistency across the algorithms, suggesting that additional insights from alternative investigative methods are necessary to supplement the analysis.
Concerning the distribution intensities, we computed the mean distribution obtained by each method and analyzed the peak positions and amplitudes analogously to
Table 1 and
Table 4 for the single samples.
From
Table 6 and
Table 7, we notice that MF-L1 tends to identify a greater number of peaks compared to the other two methods, suggesting a higher sensitivity of the algorithm.
In the case of the Manganese sample, the data reported in
Table 6 show that there is a perfect correspondence in peak position at the longest correlation time
among the three algorithms, while the peaks at shortest and intermediate times are split into multiple components.
Concerning the Poplar sample (
Table 7), we observe that all algorithms show identical peak positioning corresponding to the largest amplitude, which was reached at the shortest correlation time
. At the longest correlation times, MF-UPen finds a single peak around
, while MF-L1 and MF-MUPen split the amplitudes into two peaks at
and
, respectively.
However, despite the differences in the number of peaks identified,
Figure 4 shows that all three algorithms exhibit a fundamental robustness in the localization of the positions of the highest peaks.
From
Table 5, we observe that MF-MUPen has the smallest confidence intervals in both samples and
Figure 3 shows that the number of outliers in MF-MUPen is smaller than in the other methods. This, together with
Figure 4, suggests a higher robustness of MF-MUPen compared to the other methods.