KR920010548B1 - Shape measuring method and system of three dimensional curved surface - Google Patents

Abstract

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Description

3D curved shape measuring method and device

1a and b are explanatory views showing the measuring principle of the present invention.

2 is a conceptual diagram of a conventional light cutting method.

3A and 3B are explanatory diagrams showing a change in measurement accuracy due to a slope angle of a conventional light cutting method.

4 is an explanatory diagram showing a configuration of an optical system of the present invention.

5 is a configuration diagram of a three-dimensional shape measuring apparatus according to an embodiment of the present invention.

6 is an explanatory diagram showing a measurement example of a slope shape.

7 is a detailed block diagram of the image synthesis circuit of FIG.

8 is a block diagram showing another example of a shape calculation circuit.

* Explanation of symbols for the main parts of the drawings

1: Reference plane 2: Measured object

3: Slit light source 4: Linear stage

5: power controller 6: motor

7: Position sensor 8: TV camera

9: Shape measuring device 10: Shape calculating circuit

11: Sequential controller 12: Image synthesis calculation circuit

13: Object plane image memory 14: Reference plane image memory

15: difference image calculation circuit 16: height correction circuit

17: 3-D shape memory 18: Maximum luminance image calculating unit

19: synthetic image calculation unit 20: synchronous circuit

21: memory address generation circuit 22: output control circuit

23: Maximum luminance image memory 24: A / D conversion circuit

25: Maximum luminance image memory control circuit 26: In comparison

27: switch circuit 28: synthetic image memory

The present invention relates to a method and apparatus for measuring the shape of a three-dimensional curved surface without contact.

Since the measurement of the three-dimensional curved shape can be considered in a wide range of applications, such as three-dimensional CAD input, robot vision, or human body shape measurement for medical and garment design, various techniques have been proposed in the past.

Among them, a method known in general as a light cutting method is described on pages 398 and 399 of the "Image Processing Guide" issued by Japan, for example, but as shown in FIG. When the light beam pattern formed on the surface of the object when the slit light 53 from the slit light source 52 is irradiated to the object to be measured 51 is observed from a direction different from the irradiation direction, the object to be measured 51 This method focuses on the phenomenon of corresponding to the cross-sectional shape at the slit light irradiation position, and has been widely used in the past for its simplicity, non-contacting and quantitative property.

In measuring the shape of a three-dimensional curved surface using this light cutting method, a video obtained by observing a light beam pattern with a television camera 55 while moving the slit light 53 in the direction of the arrow 54 in FIG. As the signal is processed, the light cutting line (light beam pattern type) in the screen is extracted 56, and a curved shape is constructed 57 by reconstructing it.

As an optical system, an image pickup system using a light spot scanner instead of the slit light source 52 as a light source, and a PSD (Position Sensitive Detector) sensor instead of the television camera 55, for example. There is also a method using a high speed light spot position detection device as is known, but the basic principle is the same as that of FIG.

Although the above-described light cutting method has various advantages, the process of extracting the light cutting line in the screen is indispensable for detecting and specifying each point on the measured object. As shown, problems arise in terms of measurement accuracy or reliability.

(1) deterioration of measurement accuracy and spatial resolution due to the shape of the object to be measured; In the light cutting method, as shown in FIG. 3 (a), when the surface of the object to be measured 51 is a slope at an angle close to the right angle with respect to the optical axis of the slit light 53, the light beam pattern on the object surface Because of the narrow width w, high precision measurement is possible. However, as shown in FIG. 3 (b), when the object to be measured becomes a slope at an angle parallel to the optical axis of the slit light 53, the width w of the light beam pattern on the surface of the object is widened to cut the light. As the uncertainty of the position at the time of line extraction increases, the precision deteriorates, and the amount of movement of the light beam pattern on the object surface when the slit light source 53 is moved increases, thereby degrading the resolution of the spatial measurement at the same time. .

(2) deterioration of measurement reliability due to surface reflectance of the measurement target; In the light cutting method, it is assumed that the light beam pattern must be sufficiently brighter than the surroundings in the process of extracting the light cutting lines in the screen, and therefore, for example, the surface of the object has a large uneven reflectance. In addition, when the inclination angle of the surface of the object is close to the optical axis of the slit light and the reflected light intensity is low, there are some disadvantages when the light cutting line is extracted, or a wrong detection of a completely different point as the light cutting line occurs. Such a phenomenon occurs even when light of a background other than slit light is present at the time of measurement, and in either case, there is a reduction in the reliability of the measurement and a limitation on the application target measurement environment.

As described above, the light cutting method has many application constraints such as the shape of the object to be measured, the state on the surface, or the measurement environment due to some problems in measurement due to the light cutting line extraction process. Compared with the superiority, its use is limited, and it has not yet been assembled until it is widely used as a three-dimensional curved shape measuring device widely used.

SUMMARY OF THE INVENTION The present invention has been made to solve the above-mentioned problems with the light cutting method, and uses the same optical system as the light cutting method to encode the surface to be measured in time using slit light as a medium at a constant speed. With the introduction of a new method, it is aimed at obtaining a method and apparatus for measuring a three-dimensional curved shape based on the new measuring principle without requiring any light cutting line extraction process.

According to the measuring method of the three-dimensional curved shape according to the present invention, linear slit light transmitted from a direction forming an angle (θ) not perpendicular to the reference surface on the surface to be measured is linear at a constant speed over the entire surface of the surface to be measured. And the time at which the slit light passes the point at each position of the surface under measurement corresponding to each pixel in the screen of the video signal obtained by imaging the surface under measurement and the image to be measured. And a step of measuring a three-dimensional curved shape of the object to be measured based on the above-described synthesized image.

In addition, the three-dimensional curved shape measuring device according to the present invention includes slit light transmitting means for transmitting a linear slit light projected from a direction forming an angle θ not perpendicular to the reference plane on the surface to be measured, and a target to be measured. Slit light scanning means for linearly scanning slit light at a constant speed over the entire surface of the surface, a television camera for imaging the surface to be measured from a direction different from the above-mentioned slit light transmitting means, and a video from the television camera. It has an image synthesizing means for synthesizing an image by processing a signal from time to time and for each position of the surface to be measured corresponding to each pixel in the screen, and synthesizing an image whose time the slit light passes as the value of the pixel.

Furthermore, it has image calculation means for calculating the three-dimensional curved shape of the object to be measured by performing arithmetic processing on the basis of the synthesized image obtained by the image synthesizing means.

The television camera captures an object to be measured from a direction perpendicular to the reference plane.

In addition, the three-dimensional curved shape measuring device according to the present invention scans the slit light with respect to the reference image and the composite image u (x, y) obtained by the image synthesizing means when scanning the slit light with respect to the surface to be measured. Slit light projection angle (θ) and slit light scanning in which the three-dimensional formation f (x, y) of the surface to be measured on the reference plane is obtained according to the synthesized image u ₀ (x, y) obtained by the image synthesizing means. Using speed (v)

An image calculating means obtained according to the equation is provided.

Further, the three-dimensional curved shape measuring device according to the present invention scans the slit light with respect to the reference image and the composite image u (x, y) obtained by the image synthesizing means when scanning the slit light with respect to the surface to be measured. And a distance d (parallel to the near side of the television camera, + to the far side), parallel to the reference plane and obtained from the composite image u ₀ (x, y) obtained by the image synthesizing means. The three-dimensional shape f (x, y) of the measurement target surface is determined by using the synthesized image u ₁ (x, y) obtained by the image synthesizing means when scanning the slit light with respect to the second reference plane separated by

An image calculating means obtained according to the equation is provided.

In addition, the three-dimensional curved shape measuring device according to the present invention has a television set for a virtual reference plane and a synthetic image u (x, y) obtained by the image synthesizing means when scanning the slit light onto the surface to be measured. The three-dimensional shape f (x, y) of the surface to be measured is calculated on the basis of the synthesized image u ₀ (x, y) calculated by using the angle (θ) formed by the horizontal axis of the camera screen and the scanning direction of the slit light. Using the slit light transmission angle (θ) and the slit light scanning speed (v) with respect to the reference plane

An image calculating means obtained according to the equation is provided.

In the present invention, the linear reflection pattern of the slit light moves on the surface of the object to be measured, which is captured by a television camera, and the slit light indicates the position of the object to be measured corresponding to the pixel for each pixel in the screen. A synthesized image is created by using the passing time as the pixel value. For example, the three-dimensional curved shape of the object under measurement is measured by obtaining the difference between the values of the pixels corresponding to the two synthetic images for the synthesized image created in the same manner with respect to the above-described synthetic image and the reference plane except the object to be measured. .

Prior to the description of the embodiments of the present invention, the measuring principle of the present invention will first be conceptually explained according to FIGS. 1A and 1B.

As shown in FIG. 1A, the slit light 3a which is widened from the obliquely upward to the ground perpendicularly to the ground is transmitted to the surface of the measurement target object 2 placed on the reference plane 1, and the slit light 3a is horizontally horizontally grounded. For example, the image is captured by the television camera 8 from directly above the measurement target 2 while moving in the direction. At this time, on the monitor television 8a connected to the television camera 8, it can be observed that the linear reflection pattern of the slit light on the object surface moves in the horizontal direction of the screen.

As described above, the linear shape of the linear reflection pattern of the slit light 3a reflects the uneven information on the surface of the object, and in the conventional light cutting method, the reflection pattern The three-dimensional shape of the object to be measured has been measured by extracting the linear phase of and reconstructing it.

In the present invention, based on the video signal output from the television camera 8 in which the linear reflection pattern of the slit light 3a moves on the object surface, the pixel is applied to each pixel in the screen. An image is synthesized in which the time of the moment when the slit light passes through the position of the corresponding object surface, that is, the time of the moment when the luminance of the position becomes the brightest, is the value of the pixel.

The image synthesized in this way has a height profile of the object surface based on the plane A (hereinafter, referred to as a temporary reference plane) whose value in each pixel is indicated by a dashed-dotted line in FIG. 1A. It shows the right pen. In this way, the height profile based on the provisional reference plane A of the object surface can be measured.

However, the three-dimensional shape measurement of an object is generally not a profile with respect to the provisional reference plane A in FIG. 1A, but a plane position on which the object to be measured 1 is placed (reference plane in FIG. 1A, hereinafter referred to as reference plane). You should measure the profile based on the standard.

In order to satisfy such a request, first, the above-described measurement is performed on the object surface to measure the height profile based on the provisional reference plane A, and then the object to be measured is removed, and then the reference plane 1 is The same measurement is made, and the height profile based on the temporary reference plane A is measured, and then as shown in FIG. 1B, these two height profile images, that is, the correspondence between the object plane composite image and the reference plane composite image Compute the difference between the pixel values. According to this calculation, a height profile image based on the reference plane 1 can be obtained, and the value of each pixel of the height profile image is based on the reference plane 1 of the measurement target surface position corresponding to the pixel. It becomes proportional to the height.

4a, b, and c are schematic views of the optical system of the present invention. FIG. 4a is a perspective view, FIG. 4b is a light side view, and FIG. 4c is a side view.

In FIG. 4A, the xy plane is referred to as the reference plane 1, and the object to be measured 2 having the object surface z = f (x, y) is placed thereon. The television camera 8 observes the object plane at a predetermined angle, for example, directly from the z axis as the optical axis. The slit light source 3 transmits the slit light 3a widening in the y-axis direction at an angle θ with respect to the x-axis and scans in the x-axis direction. At this time, the position of the slit light source 3 is defined as the position x = x ₀ where the slit light 3a is in contact with the reference plane 1. (If the position of the slit light source 3 is a plane parallel to the xy plane) Whatever the definition, the following theory holds true: Here we define the position of the slit light source on the xy plane for simplicity).

Therefore, the object plane and the slit beam plane can be defined by the following equations, respectively.

The equations (1) and (2) hold at the same time on the line where the ray hits the object plane, so the relationship of the following equation holds.

If the composite image u (x, y) corresponding to the (x, y) coordinate is given at that time t, the following equation is established by setting u (x, y) = t.

On the other hand, if the synthesized image on the reference plane (xy plane) from which the measurement object 1 is removed is u ₀ (x, y), the following equation is satisfied by setting f (x, y) = 0 in equation (4). do.

Therefore, the object profile f (x, y) can be calculated | required from following Formula (4) and (5).

Moreover, the following method can be easily considered as an application example of the measuring principle. First, the first application example is a system in which two reference planes are provided. That is, in addition to the reference plane described above, a second reference plane is provided which is parallel to this and separated by a distance d (the side closer to the TV camera and the one farther away). Calculate If the composite image on the second reference plane is u ₁ (x, y), the equation (4) gives f (x, y) = d.

Therefore, the object plane f (x, y) can be obtained from the following equations from the equations (4), (5) and (7).

으로 되는 화상을 각도(ø)만큼 회전한 화상Furthermore, as a second application example, a physical image u ₀ (x, y) for a virtual reference plane is generated by calculation processing using Equation (5) without physically installing a reference plane, and this is expressed in Equation (6). By substituting, the method of obtaining the object plane f (x, y) can be considered. In this case, however, when the horizontal axis (raster direction) of the television camera screen and the scanning direction of the slit light source form an angle ø

Image which is rotated by an angle ø

You must use it.

Next, an embodiment of the present invention will be described with reference to FIGS. 5 to 7.

5 is a configuration diagram of a three-dimensional shape measuring device according to the above-described embodiment. The object to be measured 2 is placed on the reference plane 1 on which the measurement is to be made. The slit light source 3 is mounted on the linear stage 4, and transmits the slit light 3a on the reference plane 1 and the object to be measured 2 at a light transmission angle having an angle θ. The linear stage 4 on which the slit light source 3 is mounted is driven by a motor 6 controlled by the power controller 5 so that the slit light source 3 is fixed at a constant speed in parallel with the reference plane 1. To move accordingly.

At this time, the position of the slit light source 3 is measured by the position sensor 7 assembled to the linear stage 4, and is comprised so that position control by the power controller 5 is possible.

The video signal obtained by photographing by the television camera 8 in which the reference plane 1 and the object to be measured 2 are arranged so that the optical axis is perpendicular to the reference plane 1 is input to the shape measuring apparatus 9.

The shape measuring device 9 is divided into a shape calculation circuit 10 as shape calculation means for performing shape calculation by image synthesis, and a command to the power controller 5 and an operation timing control on the shape calculation circuit 10. A sequential controller 11 is provided.

In shape measurement, the shape measuring device 9 drives the linear stage 4 via a sequence control 11 based on a star art signal supplied from the outside to move the slit light source 3 to an initial position. Set it. Then, scanning of the slit light source 3 is started.

The shape calculating circuit 10 has an image synthesizing circuit 12 which will be described later in its input section, and simultaneously processes the video signal input from the television camera 8 at the same time as the start of scanning of the slit light source 3, thereby allowing the inside of the screen. For each pixel, an image synthesis operation is performed in which the time at which the image of the slit light passes is set to the value of the pixel, and at the same time as the completion of one scan of the slit light source 3, the result is u (x, y). The image data is transferred to the surface composite image memory 13.

Next, after removing the measurement target object 2 from the reference plane 1, the sequential controller 11 returns the slit light source 3 to the initial position, and then starts scanning the slit light source 3 again. . In the image synthesizing circuit 12, the image synthesizing operation performed on the object 2 to be measured is performed on the reference plane 1, and at the same time as the completion of scanning of the slit light source, the resultant u ₀ (x, y) is synthesized by the reference plane. Transfer to image memory 14.

After completion of these image synthesis operations, the shape calculation circuit 10 uses the difference image calculation circuit 15 in accordance with the instruction of the sequential controller 11 to perform the operation of the object plane composite image memory 13. The image {u ₀ (x, y) -u (x, y)} of the difference between the image and the corresponding pixel value of the image in the reference plane composite image memory 14 is calculated, and then the height auxiliary circuit 16 is used. The height profile is calibrated, and the resulting height profile data {u ₀ (x, y) -u (x, y)} v tan θ is accumulated in the three-dimensional shape memory 17.

The height profile data accumulated in the three-dimensional shape memory 17 is appropriately transmitted to the calculator to the CAD system according to instructions from the upper calculator to the CAD system.

In this embodiment, for example, as shown in FIG. 6, when the measurement object 2 has an inclined angle close to the light transmission angle of the slit light, the inclination of the slope is very close to the light transmission angle of the slit light. When the light arrives at the position indicated by " 1 " in the figure, the entire slope becomes equally bright, so that the stored contents of the object plane composite image memory 13 are as indicated by reference numeral 13a. The contents of the reference plane composite image memory 14 are as indicated by the reference numeral 14a. Therefore, the data stored in the three-dimensional shape memory 17 by arithmetic processing of such image data in accordance with the difference image calculating circuit 15 and the height correction circuit 16 becomes as shown by reference numeral 17a. From this fact, it can be seen that a sufficiently high resolution is obtained even for a shape having a surface close to the angle of the slit light.

Conventionally, it is difficult to extract the cut line from the image 13a as described above, and when the light cutting method is applied to such a slope, both measurement accuracy and spatial resolution can be expected, but this embodiment In such a slope, the measurement accuracy and spatial resolution of the non-width or sampling pitch of the slit light can be measured, and a sufficiently high resolution can be obtained even with a shape having an approximation of the angle of the slit light.

7 is a configuration diagram showing an example of the image synthesis circuit 12 that is one component of the shape measuring device 9.

The image synthesizing circuit 12 processes the video signal input from the television camera 8 to calculate the luminance at the brightest moment for each pixel, and the maximum luminance image calculating unit 18 calculates the luminance. And an image synthesizing operation unit 19 for performing image synthesizing operation which sets the time at the moment of capture as the value of the pixel. The synchronizing circuit 20, the memory address generation circuit 21, and the output control circuit are used for controlling these. (22) is provided.

The maximum luminance calculation unit 18 performs A / A video signal on the basis of the timing signal output from the synchronization circuit 20 centering on the maximum luminance image memory 23, which is a buffer memory of the maximum luminance image operation. An A / D change circuit 24 for converting D and digitizing, a maximum luminance image memory 25 for controlling the reading and writing of the address data of the maximum luminance image memory designated by the memory address generation circuit 21, Furthermore, it is comprised by the comparison circuit 26 and the switch circuit 27 which select and output the larger value by comparing the value of the corresponding pixel of the image input from a television camera with the image of a maximum luminance memory.

On the other hand, the synthesized image calculating unit 19 is composed mainly of the synthesized image memory 28 which stores the result of the synthesized image calculation, and is in accordance with the output signal of the comparison circuit 26 in the maximum luminance image calculating unit 18. Synthetic image memory control circuit 29 having a function of writing the time to the synthesized image memory 28 when the signal level input to the robot becomes larger than the pixel value of the address of the corresponding maximum luminance image memory 23. Equipped with.

This circuit uses the A / D conversion circuit 24 to output a video signal input from a television camera by star art with the maximum luminance image memory 13 and the composite image memory 28 cleared at the start of computation. By comparing the value of the video signal with the value of the pixel of the maximum luminance image memory 13 corresponding to the position of the pixel while digitizing the image, and only when the value of the video signal is larger, the pixel of the maximum luminance image memory 13 The value of is updated to the value of the video signal, and the time is written in the corresponding pixel of the composite image.

In this manner, the above-described calculation is performed in accordance with the operation control signal from the outside, and as a result of the above calculation, the constant image described above is generated in the synthesized image memory 28 at the end of the calculation. The synthesized image thus calculated is transmitted to the next calculation circuit via the output control circuit 22.

Moreover, in the above embodiment, a composite image of the reference plane 1 is produced at the time of measurement. Since the synthesized image of the reference plane 1 may be created once, it is better to use the synthesized image of the first reference plane 1 as it is during the second and subsequent measurements. In addition, since the composite image of the reference plane 1 has a simple structure, a composite image is created by adding a calculation function to the shape calculation circuit 10 and calculating and calculating the virtual reference plane by the equation (5). It may be stored in the image memory 14.

In addition, in the above-described embodiment, an example in which one reference plane is provided is illustrated, but a second reference plane may be provided in addition to the reference plane 1. The second reference plane (not shown) is provided at a distance d with respect to the reference plane 1 between the reference plane 1 and the slit light source 3. In this case, the shape calculation circuit 10a creates a composite image u ₁ (x, y) by the second reference plane in accordance with the image synthesis circuit 12 as described above, and stores it as shown in FIG. The synthesized image memory 14a is provided in addition to the object synthesized image memory 13 and the reference plane synthesized image memory 14, and furthermore, in place of the difference image calculation circuit 15 and the height correction circuit 16 described above (8). An operation circuit 30 for calculating an expression is provided.

As described above, according to the present invention, even when using the optical system similar to the light cutting method, even if the object to be measured has a shape of the slope at an angle close to the transmission angle of the slit light, for example, the composite image of the object to be measured and the reference plane Since the difference of the corresponding pixel value of the composite image is obtained to obtain a three-dimensional shape, the measurement accuracy and spatial resolution of the slit light and the sampling width of the slit light can be measured even on such a slope. A sufficiently high resolution is also obtained for shapes with angles and approximations.

According to the present invention, the image of the linear reflection pattern of the slit light moving on the measurement target surface is captured by a television camera, and the position of the object surface corresponding to the pixel is slit for each pixel in the screen. The image synthesis operation is performed using the time at which the light passes through as the value of the pixel. However, the requirement for the image synthesis operation to be obtained accurately to obtain the shape information is that the brightness of each position on the measurement target surface corresponding to each pixel is different. The only condition is that the slit light is maximized at the moment it passes through the position.

Therefore, the unevenness of the spatial surface reflectivity of the object to be measured does not affect the measurement, and even if there is background light, the amount of light is constant in time and the signal of the television camera is not saturated. The brightness of each point on the surface of the object is maximized at the moment when the slit light passes, so that the measurement can be performed without being affected by the surface reflectance of the object or the background light.

Furthermore, according to the present invention, by providing two reference planes and measuring them, the measurement can be performed without depending on the projection angle and the scanning speed of the slit light, resulting in high measurement accuracy.

Furthermore, according to the present invention, since the calculation was performed to obtain a composite image of the reference plane, the work at the time of measurement was simplified.

Claims (6)

The linear slit light projected from the direction of the pseudo-theta (θ) not perpendicular to the measurement reference plane on the surface to be measured can be linearly scanned over the entire surface of the surface to be measured, and the image to be obtained can be obtained by imaging the surface to be measured. A step of synthesizing an image having the time when the slit light passes the point as the value of the pixel at each position of the target object surface corresponding to each pixel in the screen of the video signal; The three-dimensional surface shape measuring method comprising the step of measuring a three-dimensional curved shape of the. Slit light transmitting means for transmitting a linear slit light from a direction forming a drawing θ not perpendicular to the reference plane 1 on the surface to be measured, and a slit for linearly scanning the slit light over the entire surface of the surface under measurement A television camera 8 which captures an optical scanning means, a surface to be measured in a direction different from the slit light transmitting means, and a video signal from the television camera 8 are processed at different times to correspond to each pixel in the screen. And an image calculating means for calculating a three-dimensional curved shape of the object to be measured by performing arithmetic processing on the basis of the compoundability obtained by the image synthesizing means on the surface to be measured. The three-dimensional curved measuring device according to claim 2, wherein the television camera (8) captures an object to be measured from a direction perpendicular to the reference plane (1). The synthetic image u (x, y) obtained by the image synthesizing means when scanning the slit light on the surface to be measured and the slit light on the reference plane 1 according to claim 2 The slit light transmission angle θ and the slit light of the three-dimensional shape f (x, y) of the surface under measurement according to the synthesized image u ₀ (x, y) obtained by the image synthesizing means with respect to the reference plane 1 Using scan rate (v) f (x, y) = {u ₀ (x, y) -u (x, y)} v tanθ An apparatus for measuring a three-dimensional curved shape comprising an image calculating means obtained according to the equation. The synthetic image u (x, y) obtained by the image synthesizing means when scanning the slit light on the surface to be measured and the slit light on the reference plane 1 according to claim 2 The synthesized image u ₀ (x, y) obtained by the image synthesizing means, and further, parallel to the reference plane 1 and at a distance d (+ near the TV camera 8; 3D shape f (x, y) of the measurement target surface by using the synthesized image u ₁ (x, y) obtained by the image synthesizing means when scanning the slit light with respect to the second reference plane separated by of

An apparatus for measuring a three-dimensional curved shape comprising an image calculating means obtained according to the equation. 3. The television camera (8) according to claim 2, wherein the composite image u (x, y) obtained by the image synthesizing means when scanning the slit light on the surface to be measured and the virtual reference plane (1). ) The reference plane is based on the three-dimensional shape f (x, y) of the surface to be measured according to the composite image u ₀ (x, y) obtained by calculating using the angle (ø) formed by the horizontal axis of the screen and the scanning direction of the slit light. Using the slit light transmission angle (θ) and the slit light scanning speed (v) for (1) f (x, y) = {u ₀ (x, y) -u (x, y)} v tanθ An apparatus for measuring a three-dimensional curved shape comprising an image calculating means obtained according to the equation.

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