rus
Issue #4/2017
V.A.Solomatin, D.A.Balabanova
Measuring Method Of Relief-Frequency Response Of Laser Locators In The Relief Scanning
Measuring Method Of Relief-Frequency Response Of Laser Locators In The Relief Scanning
The concept of "relief-frequency response" has been introduced, representing a laser scanner as a filter of spatial frequencies when probing the surface relief. A method for measuring the laser locator relief-frequency response has been proposed using test objects in the form of a step change in range (a spatial analogue of measuring the transient response). The article describes the installation design to determine the relief-frequency response, the method and the results of measurements.
Теги: laser locators laser probe of the relief relief-frequency characteristic лазерное зондирование рельефа лазерные локаторы рельефно-частотная характеристика
INTRODUCTION
In the framework of the theory of linear filtration, the following generalized characteristics of optical and optoelectronic systems are used, such as pulse response (point scattering function, line scattering function), spatial frequency characteristic, contrast-frequency response (modulation transfer function), transition response (edge function) [1, 2]. These characteristics are known to be interrelated, representing the system as a whole or its individual links as a spatial frequency filter that transmits the spectral (by spatial frequency) distribution of the energy components of the signal (radiation flux, brightness, irradiance) and allowing us to estimate the quality of the optical or optoelectronic system, which creates an image of the object. Thus, the scanning of the field of view is described by the convolution of the distribution function of the radiation flux in the field of view with the pulse response of the scanning system. By the spatial coordinate (along the scanning axis), the energy (light, brightness) processes are convolved, allowing to model a two-dimensional (flat) spatial structure.
With laser scanning for the purpose of constructing a spatial model of the relief (laser location), an informative parameter is the range as a function of the spatial coordinates determining the direction in the field of view, i. e., the relief of the surface being scanned, and not any energy parameter of the signal. The pulse response of the system and the convolution procedure acquire a different meaning in this case. The pulse response is treated as a function describing the output signal at the input action represented by the δ-function. The model of the δ-function in the energy description of the signal can be a point source of radiation or, for a one-dimensional representation of the signal, a luminous line. When describing a signal by the range distribution function with respect to spatial coordinates, i. e., when describing the surface relief, a model of the δ-function can be, e. g., a narrow protrusion on a flat surface representing a pulse change in the relief (range) structure. When such an object is scanned with a laser locating system, a signal appears that represents the pulse response of the scanning system.
RELIEF SCANNING
Let us assume that the scanning with a laser beam is performed along the x axis, the scanned relief is given by the S(x) function in the given coordinate system (Fig. 1). The S(x) function is a section of the two-dimensional S(x, y) function describing the relief for a fixed value of y and is defined as
,
where b is the width of the line. Fig. 1a shows the divergent laser beams from two points of the scanner position, x1 and x2. On a flat part of the surface, the laser spot has a dimension a along the x axis, where a = b for a symmetric laser beam. When using pulse range finders, the range is determined by the time interval between the transmitted pulse and the reflected pulse from the surface (echo signal) shown in Fig. 1b and 1c. In the general case, for a discrete range measurement with a certain time frequency, a spatial sample of the S(x) function is performed with its averaging inside the size of the laser spot [2, 3]. However, at sufficiently high sampling frequency (dozens of counts within the laser spot), the scanning procedure is reduced to the convolution of the S(x) function with the pulse response of the R(x) system:
,
And the convolution spectrum is the product of the corresponding spectra S(fx)R(fx), where fx is the spatial frequency, • is the convolution sign, and Δx is the x axis shift.
Fig. 2 shows that, as a result of convolution, the spectrum of the S(x) function loses its high spatial frequencies, and the relief model is smoothed out. Ideally, the pulse response of the system is described by a rectangular function, a width equal to the size a of the laser spot along the x axis. The spectrum of the rectangular R(x) function is the function of counts
,
where
.
The R(fx) function decreases to a level of 0.6 when fx = 1/2a, i. e. at a frequency half the frequency of the first zero of the count function, therefore, for the relief modelling, it is convenient to consider the spatial frequency (fx=fm=1/2a) as the maximum frequency characteristic by a level of 0.6.
EXPERIMENTAL DETERMINATION OF THE PULSE RESPONSE
In practice, the pulse response of the scanner may differ from the ideal rectangular shape. Theoretically, it is not possible to calculate its shape, therefore, to describe the scanner as a filter of spatial frequencies, it is expedient to experimentally determine the pulse response for certain test objects. The most convenient practically is a stepped signal model used in determining the transient response of the system (edge function). It is relatively easy to implement such model in the form of a rectangular ledge creating a stepwise range difference when the model plane surfaces are located perpendicular to the axis of the laser beam. The above-mentioned model of the δ-function in the form of a narrow projection on a flat surface proved to be less convenient in practice. The pulse response can be obtained by differentiating the transient response obtained from the stepped surface model.
INSTALLATION SCHEME
To carry out the experiment to measure the transient response of the laser rangefinder, an installation was assembled according to the scheme shown in Fig. 3. The laser rangefinder (using the laser measuring tape BOSCH GLM 40 Professional) was mounted on a movable base moving perpendicular to the direction of radiation with a screw with a counting mechanism (a table with micrometer from the set of optical bench OSK‑2 was used). The measurements were carried out at a distance (range li) of about 8.5 m with a carriage increment Δ=0.3 mm using a test object in the form of a ledge with a range difference of h=30 mm.
Specifications of laser range finder:
• Measuring range is 0.15–40 m;
• Measurement accuracy (limit value of error) is ±1.5 mm;
• The smallest displayed value is 1 mm;
• Laser class is 2;
• Working wavelength is 635 nm;
• Weight is 0.1 kg;
• Dimensions are 105x41x24 mm.
To obtain reliable values of the counts of the transition response, 10 series of measurements (realizations of the transient response) were made. The carriage increment of Δ=0.3 mm provided 11 counts of the range li within the cross section of the laser beam. The results of measurements and an estimation of their accuracy are shown in the table, where the standard deviation of one measurement is
,
and the standard deviation of the arithmetic mean is
,
where lcp is the average distance in a series of n=10 measurements.
Obviously, the values of σ and M are negligibly small.
EXPERIMENT RESULTS
The H(x) transient response (edge function) was determined from the measurement results, the G(x) pulse response was calculated by differentiating H(x) followed by determination of the G(fx) spatial frequency (relief-frequency) response as the Fourier transform of the G(x) function. The H(x), G(x), G(fx) normalized functions are shown in Fig. 4.5.6. A comparison of the obtained G(fx) relief-frequency response with the R(fx) spectrum of a rectangular pulse of the same width of a (a is the spot width, in the case under consideration a=3 mm) (Fig. 6) allows us to determine the scale of the spatial frequencies during the Fourier transform, since the first zero in the spectrum of a rectangular function occurs at a completely determined frequency fx=1/a. It can be seen that the real G(fx) response covers a much larger range of spatial frequencies, i. e., the approximation of the pulse response by a rectangular function with a width of a (spot size), yields an underestimation of the accuracy of the relief reproduction as a result of scanning.
CONCLUSION
The relief-frequency response, representing the laser scanning location system as a filter of spatial frequencies, makes it possible to consider this system as a linear link in the general scheme of signal transformation when scanning the surface relief. The relief-frequency response of the laser locating system determines the accuracy of the relief reproduction, characterized by the range of transmitted frequencies. In the experimental determination of the relief-frequency response as a test object, it is expedient to use a model with a stepped range change ("ledge"), followed by direct determination of the transition response (edge function), which makes it possible to calculate the relief-frequency response. The developed installation and the measurement method support the possibility of experimental determination of the transient response and subsequent calculation of the relief-frequency response. It is advisable to indicate the relief-frequency response determined at this distance to the object in the list of parameters and characteristics of laser locators used to scan the surface relief, along with the dimensions of the laser beam section at a certain distance.
In the framework of the theory of linear filtration, the following generalized characteristics of optical and optoelectronic systems are used, such as pulse response (point scattering function, line scattering function), spatial frequency characteristic, contrast-frequency response (modulation transfer function), transition response (edge function) [1, 2]. These characteristics are known to be interrelated, representing the system as a whole or its individual links as a spatial frequency filter that transmits the spectral (by spatial frequency) distribution of the energy components of the signal (radiation flux, brightness, irradiance) and allowing us to estimate the quality of the optical or optoelectronic system, which creates an image of the object. Thus, the scanning of the field of view is described by the convolution of the distribution function of the radiation flux in the field of view with the pulse response of the scanning system. By the spatial coordinate (along the scanning axis), the energy (light, brightness) processes are convolved, allowing to model a two-dimensional (flat) spatial structure.
With laser scanning for the purpose of constructing a spatial model of the relief (laser location), an informative parameter is the range as a function of the spatial coordinates determining the direction in the field of view, i. e., the relief of the surface being scanned, and not any energy parameter of the signal. The pulse response of the system and the convolution procedure acquire a different meaning in this case. The pulse response is treated as a function describing the output signal at the input action represented by the δ-function. The model of the δ-function in the energy description of the signal can be a point source of radiation or, for a one-dimensional representation of the signal, a luminous line. When describing a signal by the range distribution function with respect to spatial coordinates, i. e., when describing the surface relief, a model of the δ-function can be, e. g., a narrow protrusion on a flat surface representing a pulse change in the relief (range) structure. When such an object is scanned with a laser locating system, a signal appears that represents the pulse response of the scanning system.
RELIEF SCANNING
Let us assume that the scanning with a laser beam is performed along the x axis, the scanned relief is given by the S(x) function in the given coordinate system (Fig. 1). The S(x) function is a section of the two-dimensional S(x, y) function describing the relief for a fixed value of y and is defined as
,
where b is the width of the line. Fig. 1a shows the divergent laser beams from two points of the scanner position, x1 and x2. On a flat part of the surface, the laser spot has a dimension a along the x axis, where a = b for a symmetric laser beam. When using pulse range finders, the range is determined by the time interval between the transmitted pulse and the reflected pulse from the surface (echo signal) shown in Fig. 1b and 1c. In the general case, for a discrete range measurement with a certain time frequency, a spatial sample of the S(x) function is performed with its averaging inside the size of the laser spot [2, 3]. However, at sufficiently high sampling frequency (dozens of counts within the laser spot), the scanning procedure is reduced to the convolution of the S(x) function with the pulse response of the R(x) system:
,
And the convolution spectrum is the product of the corresponding spectra S(fx)R(fx), where fx is the spatial frequency, • is the convolution sign, and Δx is the x axis shift.
Fig. 2 shows that, as a result of convolution, the spectrum of the S(x) function loses its high spatial frequencies, and the relief model is smoothed out. Ideally, the pulse response of the system is described by a rectangular function, a width equal to the size a of the laser spot along the x axis. The spectrum of the rectangular R(x) function is the function of counts
,
where
.
The R(fx) function decreases to a level of 0.6 when fx = 1/2a, i. e. at a frequency half the frequency of the first zero of the count function, therefore, for the relief modelling, it is convenient to consider the spatial frequency (fx=fm=1/2a) as the maximum frequency characteristic by a level of 0.6.
EXPERIMENTAL DETERMINATION OF THE PULSE RESPONSE
In practice, the pulse response of the scanner may differ from the ideal rectangular shape. Theoretically, it is not possible to calculate its shape, therefore, to describe the scanner as a filter of spatial frequencies, it is expedient to experimentally determine the pulse response for certain test objects. The most convenient practically is a stepped signal model used in determining the transient response of the system (edge function). It is relatively easy to implement such model in the form of a rectangular ledge creating a stepwise range difference when the model plane surfaces are located perpendicular to the axis of the laser beam. The above-mentioned model of the δ-function in the form of a narrow projection on a flat surface proved to be less convenient in practice. The pulse response can be obtained by differentiating the transient response obtained from the stepped surface model.
INSTALLATION SCHEME
To carry out the experiment to measure the transient response of the laser rangefinder, an installation was assembled according to the scheme shown in Fig. 3. The laser rangefinder (using the laser measuring tape BOSCH GLM 40 Professional) was mounted on a movable base moving perpendicular to the direction of radiation with a screw with a counting mechanism (a table with micrometer from the set of optical bench OSK‑2 was used). The measurements were carried out at a distance (range li) of about 8.5 m with a carriage increment Δ=0.3 mm using a test object in the form of a ledge with a range difference of h=30 mm.
Specifications of laser range finder:
• Measuring range is 0.15–40 m;
• Measurement accuracy (limit value of error) is ±1.5 mm;
• The smallest displayed value is 1 mm;
• Laser class is 2;
• Working wavelength is 635 nm;
• Weight is 0.1 kg;
• Dimensions are 105x41x24 mm.
To obtain reliable values of the counts of the transition response, 10 series of measurements (realizations of the transient response) were made. The carriage increment of Δ=0.3 mm provided 11 counts of the range li within the cross section of the laser beam. The results of measurements and an estimation of their accuracy are shown in the table, where the standard deviation of one measurement is
,
and the standard deviation of the arithmetic mean is
,
where lcp is the average distance in a series of n=10 measurements.
Obviously, the values of σ and M are negligibly small.
EXPERIMENT RESULTS
The H(x) transient response (edge function) was determined from the measurement results, the G(x) pulse response was calculated by differentiating H(x) followed by determination of the G(fx) spatial frequency (relief-frequency) response as the Fourier transform of the G(x) function. The H(x), G(x), G(fx) normalized functions are shown in Fig. 4.5.6. A comparison of the obtained G(fx) relief-frequency response with the R(fx) spectrum of a rectangular pulse of the same width of a (a is the spot width, in the case under consideration a=3 mm) (Fig. 6) allows us to determine the scale of the spatial frequencies during the Fourier transform, since the first zero in the spectrum of a rectangular function occurs at a completely determined frequency fx=1/a. It can be seen that the real G(fx) response covers a much larger range of spatial frequencies, i. e., the approximation of the pulse response by a rectangular function with a width of a (spot size), yields an underestimation of the accuracy of the relief reproduction as a result of scanning.
CONCLUSION
The relief-frequency response, representing the laser scanning location system as a filter of spatial frequencies, makes it possible to consider this system as a linear link in the general scheme of signal transformation when scanning the surface relief. The relief-frequency response of the laser locating system determines the accuracy of the relief reproduction, characterized by the range of transmitted frequencies. In the experimental determination of the relief-frequency response as a test object, it is expedient to use a model with a stepped range change ("ledge"), followed by direct determination of the transition response (edge function), which makes it possible to calculate the relief-frequency response. The developed installation and the measurement method support the possibility of experimental determination of the transient response and subsequent calculation of the relief-frequency response. It is advisable to indicate the relief-frequency response determined at this distance to the object in the list of parameters and characteristics of laser locators used to scan the surface relief, along with the dimensions of the laser beam section at a certain distance.
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