Moiré Measurements Using Intersecting Lines

1. Introduction

The moiré effect [1,2] is a physical phenomenon in superimposed periodic or nearly periodic structures (grids, gratings with periodically modulated transmission/reflection) or their projections viewed together through each other. Moiré patterns appear as alternating dark and light large-scale bands with a period exceeding that of the original small-scale structures. Thus, two multi-line grids [1] are required, and in measurements, we typically have one grid on an object and another grid as a static reference.

The definition of the moiré effect [2] states that the moiré effect is an incoherent, nonlinear, point-by-point interaction between the images of grids or their projections with low-pass filtration at corresponding points. There are two fundamental parts in this definition: the interaction (superimposition/overlap, mathematically expressed by multiplication) and the low-pass filtering (averaging, Gaussian filter).

Important practical applications of the moiré effect are measurements of displacement [3,4,5] and angle [6,7]. Other applications of the moiré effect include alignment [8,9], measurement of refraction [10,11], detection of structural cracks [12,13], earthquake detection [14], and much more. Moiré measurements using a digital camera have advantages over direct (non-moiré) measurements, mostly, because of the moiré magnification [15,16].

To reduce the need for two physical grids, we generated a reference grid on a computer [17]. Similarly, in the sampling moiré method (CCD moiré) [18,19,20], the second grid is the grid of camera pixels. However, in that case, exact ratios between the sizes of the image and the CCD pixel pitch must be ensured.

Apparently, it is possible to use existing object content, even if it is not strictly periodic, such as a paragraph of text [21]. In addition, a line of a finite width can be used [22], which avoids the need to install a large optical target on the object.

The width of the line might be reduced. However, the visual contrast of the moiré patterns depends on the opening ratio. Maximum contrast of moiré patterns in coplanar grids with a rectangular profile (lying in the same plane) can be achieved with an aperture ratio of one half [23]. With other opening ratios, the visual contrast decreases, and moiré patterns in grids with low opening ratios (i.e., thin lines) have very low contrast (see Figure 1).

Low-contrast patterns can be equivalently replaced [24], but this requires some special size adjustment. However, as the grid moves, the shifted moiré patterns become noticeable. Therefore, the moiré effect can be observed dynamically even in grids with very low aperture ratios (i.e., in very thin lines).

In this paper, we propose treating the intersection of two lines as a moiré pattern. Moiré detection requires overlapping, followed by filtering. Particularly noticeable moiré patterns appear at large distances, i.e., at low resolution. Averaging effectively reduces resolution.

The intersection itself is uninformative and may depend of particular features of lines. Treating it as a moiré pattern (including a low-pass filter) makes it less dependent on specific characteristics of the line. Simulations in the spatial and spectral domains reveal many similarities between the classic moiré effect, which occurs when multiple lines intersect, and the effect of a single lines intersecting.

Except for simulations, practical examples of measurements in the lab (in fully controllable conditions) and in the field (in unreachable locations) using various lines are provided in Sec. 3.

2. Materials and Methods

According to [1], the period of the moiré patterns T_M in coplanar grids is,

$$T_M={\displaystyle \frac{T_1{T}_2}{\sqrt{T_1{}^2+T_2{}^2-2T_1{T}_2\cos \alpha }}}$$

(1)

where T₁ and T₂ are the periods of the grids, and α is the angle between them.

In particular, when T₁ = T₂, the formula is simplified:

$$T_M={\displaystyle \frac{T}{2\left|\sin \frac{\alpha }{2}\right|}}$$

(2)

And the moiré magnification coefficient is

$$\mu ={\displaystyle \frac{1}{2\left|\sin \frac{\alpha }{2}\right|}}$$

(3)

Note that for the moiré magnification, only the angle is important, not the period, aperture ratio, or line count. For small angles

$$\mu \approx {\displaystyle \frac{1}{\left|\sin \alpha \right|}}\approx {\displaystyle \frac{1}{\left|\tan \alpha \right|}}$$

(4)

It means that a small angle between the intersecting lines (a small value in the denominator) can produce a significant magnification. Compare the grid displacement with the displacement of the moiré patterns in Figure 1.

3. Results

3.1. Simulation of Visual Moiré

In simulation, we compare the moiré patterns in vertical grids/lines overlapped with a slanted reference grid/line. The vertical grid/line is displaced. The intersection of two lines is treated as the moiré pattern. When measuring displacement using moiré patterns, it is necessary to filter the overlapped grids/lines, and measure the phase of the pattern, which is proportional to the initial grid/line displacement.

The simulation additionally includes the 2D FFT. The sinusoidal grids with different phases are shown in Figure 2 and Figure 3; the rectangular pulses (duty cycle 0.5 and 0.2) in Figure 4 and Figure 5.

In these examples (rectangular/sinusoidal profile, single/periodic pulse), the displacement of the pattern’s center in the spatial domain is the same for sine and cosine functions. In particular, the displacement is the same for trigonometric and rectangular functions with any aperture ratio, including a single thin line. Thus, we can consider a single moiré pattern. This means that for measurements, we can consider the intersection of two lines as a moiré pattern.

3.2. Measured Moiré Displacement

Consider a single vertical line oscillating horizontally, as shown in Figure 6.

The amplitude of the oscillation was 10 pixels. The position of the moiré pattern was obtained according to the definition (i.e., superimposed and averaged). The tangent of the slant angle of the reference line was 1/5, and correspondingly (because the magnification coefficient is 5), the amplitude of the moiré pattern is 50 pixels.

The position of the intersection (and the moiré pattern, therefore) depends on the position of the oscillating line. The resulting measured displacement is shown in Figure 7.

3.3. Practical Examples of Moiré Measurements Using a Single Line

In this section, we describe measurements taken without attaching a physical grid to the object, because oscillations of a small/thin object could be distorted. The measurements were made using the new approach.

3.3.1. Examples with Known Parameters

a. Flower

Oscillations of plastic flowers were measured under "windy” conditions in the laboratory; the “wind” was produced by an electric fan. The flowers with a stem diameter of 1.5-2 mm were 24-29 cm long, see Figure 8.

In the first test, we provided a constant horizontal airflow and recorded a 1-minute video. The measured displacement is shown in Figure 9(a). A 512-point FFT was applied to it. The averaged power spectrum (based on 4 measurements) is shown in Figure 9(b); it shows a center frequency of individual measurements at the spectral intervals (bins) 224-228, i.e., 6.61 ± 0.06 Hz (0.8%).

In the second test, we arranged a periodical airflow. The fan automatically rotated around the vertical axis to the left and right, as shown in Figure 10. The airflow at the flower's location was effectively “turned on” and “off” according to the fan’s rotation.

The measured oscillations of a flower are shown in Figure 11.

A continuous wavelet transform (with Ricker wavelet) was applied to the signal shown in Figure 11. The result (2D array of CWT coefficients and its cross-section) is shown in Figure 12.

The moments when the air "touches" the flower in Figure 11 correspond to those in Figure 12(b). The average period of fan rotations is 9.1 seconds.

b. Antenna

We measured the vibrations of a telescopic antenna on a car navigation device placed on a printer mounted on a high table (as in [22]). The antenna length was 22 cm, and its diameter varied from 2 mm at the base to 0.5 mm at the top. In this test, the navigation device was turned off, and only its housing served as a support for the antenna, as shown in Figure 13.

The measured oscillations are shown in Figure 14 for printouts of different lengths.

The 512-point FFT of the long printout (1.5 pages) contained less noise and had a sharper peak than the FFT of the short printout (14 lines), as shown in Figure 15.

Two peaks can be clearly observed in the power spectrum Figure 15: the fundamental frequency of supporting table at the 101^st spectral bin (6.04 Hz) shown by the solid vertical line across two panels (a) and (b) in Figure 15, and the antenna frequency at 188^th bin (10.9 Hz) shown by the dashed line. The previously reported table's frequency was 6.26 Hz [22]. The relative difference of frequency with that measurement is 6.4%.

c. Wires in lab

In this test, steel wires were secured between two tables with clamps. The camera was mounted above the wires, mid-span. A horizontal air flow (perpendicular to the wire) was constant. A diagram of the experimental setup is shown in Figure 16.

According to [17], the fundamental frequency is proportional to the diameter of the wire and inversely proportional to the square root of its length,

$$f~{\displaystyle \frac{\sqrt{S}}{\sqrt{L}}}={\displaystyle \frac{D}{\sqrt{L}}}$$

(5)

Measured frequencies obtained in the FFT and estimated ratios of frequencies are given in Table 1.

The measured frequency ratio for two distances (100 and 200 cm) differs from the calculated value of 1/√2 = 0.71 by 1.9% for a wire diameter of 1 mm and by 6.1% for a diameter of 0.6 mm. The ratio corresponding to two diameters (0.6 and 1 mm) differs from the calculated value of 0.6 by 18.2% at a distance of 100 cm and by 14.7% at a distance of 200 cm. The latter are less accurate, probably because this test used uncalibrated wire.

3.3.2. Examples with Unknown Parameters

Unlike the previous examples, exact parameters were unknown in these tests and could only be estimated indirectly.

a. Street wire

The street was observed from a distance of 30 m. The wire diameter was unknown. Wind speed was approximately 10 m/s. The two wires (one along the street, one across) visually intersected, so a reference line was not required for these illustrative measurements; see Figure 17.

The measured signal and its power spectrum are shown in Figure 18.

The power spectrum exhibits a small but noticeable peak at the 46^th spectral bin (0.67 Hz). It should be noted that the measurements were taken without a reference line; instead, the reference was found in the video.

b. High voltage line

The distance between the high-voltage poles was several hundred meters. The height of the wire above the ground at the lowest point (where the measurements were taken) was approximately 10 meters. The diameter of the wire was unknown.

The camera tripod was mounted on a solid base (stone); the camera was pointed vertically upward, as shown in Figure 19. The video recording lasted up to 10 minutes.

Figure 20 shows the averaged power spectra measured over several videos (4096-point FFT) for different weather conditions (no wind and wind speed up to 5 m/s).

In the presence of wind, the center of the first harmonic was located at the 26^th spectral bin (0.190 Hz, period 5.25 s), and the center of the second harmonic was at the 52^nd spectral bin (0.381 Hz). Without wind, the amplitude of the first harmonic was significantly smaller, and its wider peak was located between the 25^th and 26^th bins (0.186 Hz), but the second harmonic was not detected.

4. Discussion

We examined thin lines with in-camera images fewer than 7-10 pixels.

It should be noted that printer/antenna vibrations were not detected on a hard cement floor.

The limitations of this method are as follows: the line must be straight. The background must be uniform.

Using an inclined line is possible, whereas for cloned lines [22], only a vertical/horizontal line is suitable.

5. Conclusions

Simulations and tests have shown that the intersection of two lines can be considered a moiré pattern, which has the useful property of the moiré magnification. This approach essentially reduces the effective size of the optical target to the intersection of two thin lines. The laboratory and field measurement examples are provided. In the laboratory, the length measurement accuracy is 2-6%, and the diameter measurement accuracy is 14-18%. In the field, measurements were conducted in a hard-to-reach location. The difference with previous measurements is 6.4%. This ensures that single-line moiré measurements can be performed without a special optical target or complex setup.