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We propose a simple telescope with three-dimensional image capability for surface profile measurements. Our method based on the algorithm of reflectivity-height transformation is applied to a traditional commercial telescope with a webcam for determining the third dimension of the test object. It is also useful for thickness, deformation, and surface profile measurements.

In recent decades, telescopes have not undergone any major variations in their structures or applications. A picture from a telescope can be taken by a camera. We can take a picture in two-dimensional (2D) but not in three- dimensional (3D) format. In this paper, we propose a new telescope that has the ability to show and plot a 3D image for long distance surface profile measurements. The method is based on the reflectivity-height transformation, including the internal reflection effect [

When a parallelogram prism made of BK7 with the base angle of 45˚ is used to be an angular sensor to sense the angle deviation of the object light, the light passing through it with twice internal reflections. n_{1} and n_{2} are the indices of refraction of the prism and air respectively. Assume that n_{2} = 1.000 and n_{1} = 1.517 (the average wavelength λ = 550 nm), and the range of external incident angle θ = 0˚ - 10˚. The simulation results of the reflectivities of the s- and p-polarized lights are shown in _{s}_{2} and R_{p}_{2} are represented the reflectivities of twice internal reflection of the s- and p-polarized lights, respectively. Apparently, the change in the reflectivity of the p-polarized light (dotted line) close to the critical angle is more sensitive than that of the s- polarized light (solid line) close to the critical angel.

Regarding objects with a large change in height, we can prefer to select the s-polarized light as the light source because of its larger angle measurement range. Convert the practical measured curve into coordinates, where the x-coordinate is converted to R_{S}_{2}, y-coordinate is converted to θ, and convert the angle θ to be the function of R_{S}_{2}, as shown in

From _{s}_{2}; on the contrary, we can derive the change of _{s}_{2}, as shown in _{s}_{2}. When the interval of the two points of the object is

If we can acquire the reflectivity difference R_{s}_{2} of the two points after imaging and calculate

As

where

where h_{0} is the initial height.

The system structure of the experiment is shown in _{s}_{2} and h, where DR_{s}_{2} ≒ dR_{s}_{2} can be considered the reflectivity difference of the adjacent pixels and dX can be considered the interval of the adjacent pixels. According to Equation (3), the surface height of object can be calculated and the 3D surface profile can be plotted.

In order to measure more accurate 3D surface profiles, we need to use a green filter plate to perform the measurement. As described above, after we perform curve fitting for obtaining the external incident angle

In addition to the above experiment, we also performed experiments on other coins to make sure the proposed method is feasible and to find out the average error range. We compared the average thickness with the measurement results of the vernier caliper. The measurement results of the vernier caliper serve as the reference values. The error of the thickness is about 0.07 mm and the average error percentage is 4%. According to first- order optics approximation, as the measurement distance of the structure is 10 m, the f-number is 5.346, and the diameter of telescope is 52 mm, the corresponding depth of field is 31 mm, the object range that we measured cannot exceed the limit [

The definition of the sensitivity S is written as

∂R_{S} is the reflectivity change corresponding to the micro height change. The highest sensitivity and the lowest sensitivity with a measurement range of 5.8˚ - 8.5˚ When the external angle is 5.8˚, we find that the highest sensitivity S = 0.41 (change/mm); when the external angle is 8.5˚, we find that the lowest sensitivity S = 0.07 (change/mm).

Vertical resolution means the minimal surface height which the system can discriminate. Where S is the sensitivity,

The vertical resolution of the system is shown in

The light source is an indoor fluorescent lamp, so the experiment does not need to be conducted in a dark room. Thus, we not only need to consider the stability of the power of the fluorescent lamp, but we also need to consider whether the system receives light from other sources in order to be aware of any measurement errors. Regarding the stability of the light source, we use the power meter to continuously measure the stability of fluorescent lamp, as shown in

As shown in

Lateral resolution means the minimal discriminable interval. According to the Rayleigh Criterion,

hand, if the effective light collection aperture is smaller, the minimal discriminable distance will also increase and the lateral resolution will be reduced. If we want to improve the lateral resolution, we can use a telescope with a bigger aperture; that is to say, we increase to improve the lateral resolution. With the telescope that we currently use, D = 52 mm, the measurement distance is 10 m, and the lateral resolution is 0.13 mm.

This experiment adopts the angle deviation method by using the parallelogram prism, common telescope and webcam to measure the 3D surface profiles of coins with a thickness of several millimeters at a 10 m distance. The error percentage is about 4%. According to reflectivity-height transformation, this method converts the reflectivity into the height of the object point by point, and then acquires its 3D surface profile in real time. Thus, the device can be used for remotely monitoring or detecting the change of the surface profile. Examples include measuring a change in topography with aerial photos, monitoring dangerous or inaccessible environments, ob-

serving the cracks or deformation of buildings, investigating soil flows or floods, and remote testing for other purposes.

This study was supported in part by the National Science Council of Taiwan with contract number NSC 100- 2221-E-150-067-MY2.