Theoretical Formulation/Development of Signal Sampling with an Equal Arc Length Using the Frame Theorem

Non-uniform sampling with equal arc length intervals can be found in shape measurements with contact sensor arrays. In this study, the conditions of non-uniform spatial sampling with an equal arc length interval are derived from two frame theorems. First, for general non-uniform sampling, the condition is that the equal arc length interval of the sensors should be less than 1 4  . Second, for strictly increasing sampling, the condition is that the equal arc length interval of the sensors should be less than 1 2  . The  is the maximum frequency of the detected object. For the latter, if the sampling frequency is more twice than the sampling frequency required, the reconstruction error (RRMSE and MRE) is less than 5%. If the sampling frequency is more than 2.5 times, the reconstruction error is less than 3%. The simulation and the experiment are carried out and the results show that a sensor array with equal arc length interval can reconstruct the detected object with high accuracy.


Introduction
Sampling plays a critical role in signal analysis and contact sensor shape detection. However, samples cannot be collected uniformly in various applications, such as underwater terrain subsidence monitoring based on a Micro-Electro-Mechanical System (MEMS) 9-axis sensor array [1][2][3], shape monitoring based on the Fiber Bragg Grating (FBG) sensor array [4][5][6][7][8][9][10], and data gloves or motion capture devices based on fiber curvature sensor arrays [11,12]. The sensor array has an implicit characteristic-the arc length interval between the sensors is equal and constant when the sensor array moves with the detected object. Therefore, the sampling of the sensor array becomes non-uniform even though the sensor array is deployed in an equal arc interval at the beginning (uniform sampling at the beginning) when the sensor array moves with the detected object.
Whether the detected object can be completely reconstructed from its nonuniform spatial sampled values with the condition of equal arc interval is a general and import problem which is should be studied. There are two reasons: one is that it cannot be sure if the reconstructed signal is not distorted from the non-uniform sampling set obtained from the contact sensor array. The other is that more sensors are often used to ensure the reconstruction without distortion, resulting in increased costs, more complex systems and larger data volumes if the non-uniform sampling conditions are not known.
Consequently, there have been many studies on randomized non-uniform sampling [13][14][15][16][17][18] and periodic non-uniform sampling [19][20][21][22]. For randomized nonuniform sampling, the non-uniform samples are often regarded as random perturbations from a uniform sampling grid in the analysis. The sampling times are often constructed by adding a Gaussian distribution sampling noise to the previous sampling time [23]. For periodic non-uniform sampling, the offset of sampling time n t from the equal interval sampling time sequence nT has a periodic structure with a period of M, where T is the average sampling period and M is an integer. The sampling time is ( ) n n m t nT r T kM m T r T      , where n rT is the offset of n t from nT, and (mod ) m n M  [22]. This means that the adjacent sampling points are not uniform, but the interval between each sampling point and the subsequent sampling point is equal. The total sampling period is MT [24,25]. It is often applied in analogto-digital (A/D) sampling technology, combining M AD converters with lower sampling rate to obtain one virtual sensor with high sampling rate [22]. There are also several algorithms in the literature that deal with the reconstruction of bandlimited signals from periodic non-uniform samples [19,26,27].
However, non-uniform sampling with the condition of equal arc interval is neither the randomized non-uniform sampling with a known distribution such as Gaussian distribution sampling noise nor periodic non-uniform sampling. There is no research on non-uniform spatial sampling for contact sensor arrays with the condition of equal arc interval. In this article, two theorems of non-uniform spatial sampling with the condition of equal arc interval are studied. The condition of non-uniform spatial sampling is derived from the frame theorem. For general non-uniform spatial sampling, the condition of sampling interval is obtained by the Kadec theorem. For the strictly increasing sampling sequence, the condition of sampling interval is obtained by the Benedetto theorem.
The rest of this paper is organized as follows. Section 2 introduces the conditions of non-uniform spatial sampling which are derived from the frame theorem. A simulation is presented in Sect.
3. An experiment is presented in Sect. 4 to evaluate the performance of the proposed sampling conditions. Finally, the conclusion is given in Sect. 5. Fig. 1 and Fig. 2 show uniform and non-uniform spatial sampling respectively.

Definition of non-uniform spatial sampling with the condition of equal arc interval
The 'o' in Fig. 1 and Fig. 2 is the sampling point. In the uniform sampling, the interval of the x-axis is spaced equally, which is x in Fig. 1. In the non-uniform case, the interval of the x-axis is spaced unequally which is 1 1 ( -) ( -), 0,1, 2,3... f(x) x Fig. 2. Non-uniform spatial sampling.

The mathematical expression of the non-uniform spatial sampling with the condition of equal arc interval
Assume that is a mathematical expression of a detected object (such as a ribbon of terrain) which is a band limited function. The terrain is detected by a MEMS nine-axis sensor array [1][2][3]. In other words, () fx is sampled with the condition of equal arc length. N is the sample number which can be obtained by N L X  (L is the arc length of the () fx, and x is the arc length of sensors which have an equal spatial interval at the beginning). Set ( ) ( ) is the sampling value. Like the process of uniform sampling, non-uniform sampling is the product of a series of impulse series and the original function.  In contrast to uniform sampling, the impulse series does not have periodicity (equal intervals), which can be expressed as: Where n x is the projection of the sampling point on the x-axis. Sampled signal which shows as: As seen from formula (1), the sampled signal () s fx is an impulse string weighted by the values () fx.

The condition of non
where is inner product, which is defined as , In the frame theorem, the sampling point set is {} nn x ¢ and sampling value set is In order to completely reconstruct the band-limited function () fx from the sampling value set { ( )} nn fx ¢ , the following inequality (4) should be met.
Then inequality (7) can be expressed as: From inequality (7), it can be concluded that when the spatial arc length interval of the sensor array is not greater than  (7). The derivation process is as follows: Fig. 4 shows the derivation schematic diagram of the spatial sampling interval. Taking n=1 for example, assume that there is a sampling point in the range   , which can be transformed as: As the above analysis shows, there must exist a sampling point in the range . Therefore, in order to satisfy inequality (9), inequality (10) should be satisfied: Add the two inequalities in (10)  It can be seen from the above analysis that when the equal arc length interval between the sampling points is half of the Nyquist sampling interval or one quarter of the period corresponding to the highest frequency of the detected object, the original function () fx can be reconstructed completely.
However, the sampling interval based on the non-uniform sampling theorem is larger. The more sampling points, the more sensors, the more complicated is the sensor network and the more cost. In applications such as underwater seabed monitoring based on sensors, the sampling points are a strictly increasing sequence.
Therefore, non-uniform sampling theorem for increasing sampling is introduced which can obtain a less sampling interval.
(2)The non-uniform sampling of a strictly increasing sequence with equal arc length In the frame theory, Marvasti [27] and Benedetto [28]

Simulation of general non-uniform spatial sampling theorem
A simulation for the non-uniform sampling condition and reconstruction method is carried out. Any signal can be superimposed by sinusoidal signals and cosine signals.    Table 1, Fig. 5 and Fig. 6, first, the reconstructing error of both is mainly at the extremum points (cyan areas in Fig. 5 and Fig. 6), where the non-uniform sampling points are far away from the extremum points. Therefore, the reconstructing error is related to the distribution of non-uniform sampling points. Second, the sampling numbers with equal arc length are more than random sampling, and the reconstructing error of sampling with equal arc length is much less than random sampling. This is because the arc length (about 22) is much longer than the x-axis range (from 0 to 10). The sampling number with equal arc length is more than twice the random sampling. According to the general non-uniform sampling theorem, there must be one sampling point in the range [ when the arc length interval is 0.5. In other words, the sampling frequency is more than twice the sampling frequency of the general non-uniform sampling theorem. Moreover, when the signal is a line, the sampling points are uniform, the signal can also be reconstructed completely.

Simulation of strictly increasing non-uniform sampling theorem
The same simulation signals are selected. As mentioned above, there must be interval is shorter, sampling points are more, and the reconstruction accuracy is higher.
The different sampling intervals are tested which progressively decrease from 1.02 to 0.82 at an interval of 0.05 (0.05*1.02). Fig. 7 shows the cosine signal, the sampling points with different arc length interval, reconstructing signals respectively, and Fig. 8 shows the ones of the sinusoidal complex signal. Table 3 shows the reconstruction error of non-uniform sampling with different equal arc length interval. The RRMSE and MRE between the original signal and the reconstructing signal are also calculated.   From Fig. 7 Fig. 8 and Table 2, first, it can be seen that the arc length of the sinusoidal signal and sinusoidal complex signal is about the same, the same highest frequency, the same sampling points are needed when the arc length interval is the same. Second, the smaller the arc length interval, the more sensors are needed, and the less the reconstruction error. Three, the RRMSE and MRE of the reconstruction error are less than 5% when the arc length interval is 1.02 (the condition of the strictly increasing non-uniform sampling theorem). Those are less than 3% when the arc length interval is 0.87 (0.85*1.02). This is because the condition of the strictly increasing non-uniform sampling theorem is that there must have more than one . In other words, the sampling frequency is more twice than the sampling frequency of the strictly increasing non-uniform sampling theorem. When the arc length interval is 0.87 (0.85*1.02), the number of sampling points is 27 and the sampling frequency is more than 2.5 times the sampling frequency of the strictly increasing non-uniform sampling theorem. Fourth, compared to simulation of the general non-uniform spatial sampling (Table 1), the RRMSE and MRE is much bigger when the arc length interval is 1.02 which satisfies the condition of the strictly increasing non-uniform sampling theorem.
This is because the signal in spatial domain is limited. The frequency spectrum is infinite. There is some energy missing when the highest frequency is regarded as 3 2  . The arc length interval in the simulation of the general non-uniform spatial sampling is much less than the one in this case, and the accuracy is much higher.
Moreover, when the signal is a line, the sampling points are uniform, the signal can also be reconstructed completely.

Experimental setup
The experimental setup mainly includes terrain deformation simulation platform, three-dimensional (3D) laser scanner, sensor array and computer for data processing and terrain display. The experimental setup is shown in Fig. 9. The deformation simulation platform is 2.1 m × 1.8 m (the length along the sensor array is 2.1 m, and the length in the direction perpendicular to the sensor array is 1.8 m). Sixteen (4 × 4) screw slides were placed in the area. The distances between the screw slides were 70 cm and 45 cm. The heights of the screw slides are controlled by the motors. The shapes are constructed by the different heights of the screw slides. Six sensor arrays ( Fig. 9) are arranged on the net and one end is fixed. Each sensor array is 2.1 m in length. The sensor array #1, #3, #4 and #6 is located directly above the screws, and their status can be changed directly by the screws. The shape change of sensor array has little effect on other sensor arrays. Therefore, the sampling of the sensor array is regarded as one-dimensional non-uniform sampling. The experimental setup includes a 3D laser scanner, and the data obtained by the 3D laser scanner will be used as the real value of the shape. The accuracy of the 3D laser scanner is 3 mm. The 3D laser scanner scans the shapes, obtaining the point cloud data, which are processed by the software Cyclone (shown in Fig. 10), including point cloud data simplification, segmentation and coordinate extraction [32]. In this study, we constructed three different shapes which are shown in Figs. 11 -13. In the experiment of the terrain deformation monitoring, the sensors are placed with equal arc length. The samples are strictly increasing non-uniform. Therefore, we carried out the experiment for strictly increasing non-uniform sampling theorem.

Results and discussion
The one-dimensional amplitude spectrum of the shapes is calculated. The data are   infinite. The lowest frequency of the terrain is zero, and the highest frequency is defined as effective bandwidth in which the frequency range contains 95% energy of the shape. Table 3 to Table 5 are the highest frequency of these three shapes when the y is fixed respectively. As can be seen from Fig. 14 and Table 3 to Table 5, the frequencies of the three shapes are all concentrated in the low frequency and the highest frequency  is 1.428 m -1 . According to the non-uniform sampling theorem with equal arc length, the arc length interval between sensors should be less than  The sensor array senses the tilt angle of the shapes and conducts data collection, processing, modeling and display on a computer. The coordinates of the sensor array are used to analyse the reconstruction error. Table 6 shows the reconstruction error statistics. The RRMSE and MRE between the original signal and the reconstructing signal are also calculated. In other words, the signal is limited in the spatial domain. Though the arc length interval satisfies the strictly increasing non-uniform sampling theorem, the arc length interval accounts to reconstruction error. According to the above simulation, the error caused by the reconstruction error is less than 5%. Three, the measurement errors may increase because the sensors could not fit well with the terrain surface when the terrain moves.

Conclusion
In this article, non-uniform spatial sampling with an equal arc length interval is presented. The definition of the non-uniform spatial sampling and associated mathematical expression is given. The condition of non-uniform spatial sampling is derived based on the frame theorem. Two simulations and an application (the terrain deformation test) of the strict increasing non-uniform sampling theorem are carried out. There are four conclusions: (1) Though the condition of the strictly increasing non-uniform sampling theorem is satisfied, the RRMSE and MRE of the reconstruction signal is not close to zero.
The reason is that the frequency spectrum is infinite when the signal in the spatial domain is limited, and there is some energy missing for the highest frequency which often is defined as effective bandwidth of signal. The effective bandwidth contains 95% of the energy of the signal.
(2) In practical application, sensors are deployed on the detected object. If the sampling point set is strictly increasing, the arc length interval between sensors can be based on the strictly increasing non-uniform sampling theorem, which 1 2 l   . If the sampling frequency is more twice than the sampling frequency of the strictly increasing non-uniform sampling theorem, the reconstruction error (RRMSE and MRE) is less than 5%. If the sampling frequency is more 2.5 times, the reconstruction error (RRMSE and MRE) is less than 3%. If the sampling point set is not strictly increasing, the arc length interval between sensors should satisfy 1 4 l   .
(3) Assume that the arc length of the detected object is L, the arc length interval of the sensor array is l. The sensor array with equal arc length can be deployed on which the highest frequency of the detected object is less than 1 4l  or 1 2l  . For later, the x-axis coordinates of the detected object should be strict increasing.
(4) The smaller the arc length interval, the more sensors are needed, and the less the reconstruction error.
The results can guide technicians on the application of contact sensor array for shape monitoring with less sensors and high accuracy.

Availability of data and materials
All data of simulation and experiment can be obtained by contacting the authors of the paper.

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