JP4775540B2 - Distortion correction method for captured images - Google Patents

Description

  The present invention relates to a distortion aberration correction method for a captured image.

  In recent years, with the improvement in performance of photography equipment such as digital cameras, it has become relatively easy to obtain high-quality images. For example, a structure such as a bridge is photographed with a digital camera, and the structure is obtained from the photographed image. A system has been developed that can measure various dimensions such as the shape and size of the non-contact and with high accuracy.

However, in a system using an optical lens such as a digital camera, a measurement error due to lens distortion is included.
For this reason, a measurement method has been proposed in which a calibration plate on which a target mark is provided in advance is attached to a measurement object, and distortion is corrected by photographing the calibration plate (for example, Patent Document 1).
JP 2001-133225 A

  As described above, when a calibration plate attached to a measurement object, for example, a measurement method that corrects distortion aberration by photographing a lattice pattern, positional shake or rotational shake occurs when taking a picture with a camera. Therefore, the error due to camera shake and rotational shake cannot be separated, so even if distortion is corrected, there is a problem that errors occur in the measurement result and high-precision measurement cannot be performed. It was. Of course, the same problem occurs when the grid pattern is not correctly arranged.

  Therefore, the present invention provides a distortion aberration correction method in a captured image that can correct errors caused by these even when the camera shakes or the position of the lattice pattern is shifted. With the goal.

In order to solve the above-described problem, a distortion aberration correction method for a captured image according to claim 1 of the present invention includes:
Applying Fourier transform to a photographed image showing a luminance value obtained by photographing a two-dimensional lattice pattern through a lens to obtain a frequency spectrum;
Filtering to remove noise from the frequency spectrum, leaving only the first harmonic wave; and
Applying a Fourier inverse transform to the first harmonic wave to obtain a phase value;
Performing phase connection of the phase values determined over a predetermined range in the above steps;
Substituting the phase values (δ _x , δ _y ) obtained by the above phase connection into the following equations (A) and (B ) to determine the deformation amount (β _x , β _y ) of the lattice pattern;
An aberration model quantity (α _x , based on the following formulas (C) and (D) generated in the grid pattern in the shift amount calculation formula expressed by a plane formula considering the rotational movement and parallel movement of the grid pattern: The deformation amount is substituted into a deformation amount calculation formula represented by the following equations (E) and (F) to which α _y ) is added, and at least an aberration model amount of the deformation amount calculation equation is expressed using a least square method. Determining a coefficient;
And correcting the distortion using the coefficient of the determined aberration model quantity.
β _x = −δ _x / 2πω (A)
β _y = −δ _y / 2πω (B)
Where ω is the frequency of the grating before deformation.
α _x = (g ₁ + g ₃ ) x ² + g ₄ xy + g ₁ y ² + k ₁ x (x ² + y ² ) (C)
α _y = g ₂ x ² + g ₃ xy + (g ₂ + g ₄ ) y ² ++ k ₁ y (x ² + y ² ) (D)
β _x = a ₁ x + a ₂ y + a ₃ + α _x (E)
β _y = a ₄ x + a ₅ y + a ₆ + α _y (F)

Further, a distortion aberration correction method for a captured image according to claim 2 is:
Applying Fourier transform to a photographed image showing a luminance value obtained by photographing a two-dimensional lattice pattern through a lens to obtain a frequency spectrum;
Filtering to remove noise from the frequency spectrum, leaving only the first harmonic wave; and
Shifting the primary harmonic wave to the origin side by the frequency spectrum of the lattice-like pattern having no aberration, and then performing a Fourier inverse transform to obtain a phase value;
Performing phase connection of the phase values determined over a predetermined range in the above steps;
Substituting the phase values (δ _x , δ _y ) obtained by the above phase connection into the following equations (A) and (B ) to determine the deformation amount (β _x , β _y ) of the lattice pattern;
An aberration model quantity (α _x , based on the following formulas (C) and (D) generated in the grid pattern in the shift amount calculation formula expressed by a plane formula considering the rotational movement and parallel movement of the grid pattern: The deformation amount is substituted into a deformation amount calculation formula represented by the following equations (E) and (F) to which α _y ) is added, and at least an aberration model amount of the deformation amount calculation equation is expressed using a least square method. Determining a coefficient;
And correcting the distortion using the coefficient of the determined aberration model quantity.
β _x = −δ _x / 2πω (A)
β _y = −δ _y / 2πω (B)
Where ω is the frequency of the grating before deformation.
α _x = (g ₁ + g ₃ ) x ² + g ₄ xy + g ₁ y ² + k ₁ x (x ² + y ² ) (C)
α _y = g ₂ x ² + g ₃ xy + (g ₂ + g ₄ ) y ² ++ k ₁ y (x ² + y ² ) (D)
β _x = a ₁ x + a ₂ y + a ₃ + α _x (E)
β _y = a ₄ x + a ₅ y + a ₆ + α _y (F)

  According to each of the above distortion aberration correction methods, when calculating the phase value of the lattice pattern from the photographed image, the deformation amount calculation formula is obtained by adding the plane equation representing the rotation amount and the parallel movement amount of the lattice pattern to the aberration model amount. In addition, the coefficient of at least the aberration model amount in this deformation amount calculation formula is determined by the least square method using the actually obtained deformation amount, so when correcting distortion aberration using a lattice pattern In addition, even when the lattice pattern is shifted, correction can be performed in consideration of the shift amount, so that distortion can be corrected with high accuracy.

[Embodiment]
Hereinafter, a distortion correction method for a captured image according to an embodiment of the present invention will be described with reference to the drawings.

  In the present embodiment, for example, a photographing object such as a digital camera (hereinafter referred to as a camera) is photographed, that is, a photographing object is photographed through a lens, and a photographed image of the photographing object photographed by the camera is used. When measuring various dimensions related to the surface such as the shape and size of the object to be photographed, the displacement due to the distortion of the lens is included as an error, so that the displacement due to the distortion is removed. A distortion aberration correction method will be described.

  The distortion correction method according to the present embodiment captures a reference two-dimensional lattice pattern (which is an example of a lattice pattern and can also be referred to as a reference pattern), and a luminance value signal in a captured image of the lattice pattern When a phase value is obtained by performing Fourier transform on the pattern (signal frequency), a displacement amount (deformation amount) is obtained based on the phase value, and the distortion amount of the lattice pattern is corrected using the displacement amount, the lattice pattern This is a method that can correct, for example, the amount of deviation due to rotational movement and parallel movement due to the fact that it is not correctly arranged.

  In the following description, after describing a method for obtaining the phase value of a grating pattern whose peripheral part has been deformed by distortion by the Fourier transform method, a method for correcting the amount of deviation of the grating pattern on the plane will be explained.

First, a method for obtaining a phase value in a captured image of a lattice pattern using Fourier transform will be described.
Distortion is a phenomenon in which an object that is a straight line is photographed as a curve, for example, and the distance from the center of aberration (in many cases, the center of the image, but not always). Can be used to model.

For example, as shown in FIG. 1, it is assumed that a point that should originally appear at the position of point A is reflected at point B due to distortion.
The distance between the points A and B (the amount of aberration) can be decomposed into a radial direction (radial direction) distortion alpha _r and circumferential distortion alpha _theta.

The circumferential distortion aberration α _θ is sufficiently smaller than the radial distortion aberration α _r . For example, when the circumferential distortion aberration α _θ is ignored, the radial distortion aberration α _r is a distance from the center of the aberration. As a function of r, it is expressed by the following equation (1).

ここで、ｋ_１，ｋ_２，ｋ_３，・・・は収差の量を表す係数である。多くの場合、高次項を無視できるので、（１）式は下記（２）式にて近似することができる。

Here, k ₁ , k ₂ , k ₃ ,... Are coefficients representing the amount of aberration. In many cases, higher-order terms can be ignored, so equation (1) can be approximated by equation (2) below.

すなわち、画像上の歪曲収差の量（幾何学的なずれ量）は収差の中心からの距離ｒの３乗に比例し、その係数ｋ_１を決定することで歪曲収差の量を知ることができ、これに基づき補正を行うことができる。

That is, the amount of distortion of the image (geometric deviation amount) is proportional to the cube of the distance r from the center of the aberration, it is possible to know the amount of distortion by determining the coefficients k ₁ Based on this, correction can be performed.

In addition, when the radial distortion aberration α _r is decomposed in the x direction and the y direction, that is, the amount of distortion aberration (aberration model amount) in the orthogonal coordinate system is expressed by the following equation (3).

ここで、θ＝arctan（ｘ，ｙ）である。

Here, θ = arctan (x, y).

By the way, when it is also necessary to consider the circumferential distortion aberration α _θ , the distortion aberration model is expressed by the following equation (4) using the coefficients g ₁ , g ₂ , g ₃ , g ₄ , and k _1. Can be expressed. This equation (4) is known (for example, “Weng, J., Cohen, P., and Herniou, M., Camera Calibration with Distortion Models and Accuracy Evaluation, IEEE Trans. Pattern Anal. Mach. Intell., 14 (10), 965-980 (1992) ”). By determining these coefficients g ₁ , g ₂ , g ₃ , g ₄ , and k ₁ , the amount of distortion can be known, and correction can be performed based on this.

Hereinafter, a case in which the circumferential distortion aberration α _θ is considered will be described.

すなわち、（４）式における各係数ｇ_１，ｇ_２，ｇ_３，ｇ_４，ｋ_１を決定する必要がある。

That is, it is necessary to determine the coefficients g ₁ , g ₂ , g ₃ , g ₄ , and k ₁ in the equation (4).

Further, when α _x and α _y are used, the coordinates x ′ and y ′ of the point distorted by the aberration can be expressed by the following equation (5) with respect to the coordinates x and y when the aberration is not included.

ところで、フーリエ変換を用いて、格子パターンにおける撮影画像の位相値を決定し得ることは一般的に知られている。

By the way, it is generally known that the phase value of a captured image in a lattice pattern can be determined using Fourier transform.

For example, when distortion occurs in the undeformed lattice pattern shown in FIG. 2, the image is taken as a deformed image as shown in FIG.
A Fourier transform is performed on the captured lattice pattern of luminance values to obtain a frequency spectrum, and a filtering process (that is, a noise removal process) is performed so as to leave only the first harmonic wave of the frequency spectrum. The spectrum of the primary harmonic wave is shifted to the center side (origin side) by the frequency spectrum in the lattice pattern that is not deformed (when there is no distortion), and then the inverse Fourier transform is performed to Phase distributions ψ _x and ψ _y representing displacement (deformation) in the x direction and y direction are obtained.

The phase distribution obtained in this way is for a pair of lattice lines of the lattice pattern, and these must be represented continuously.
Therefore, phase connection of these phase distributions is performed, and phase values δ _x and δ _y related to deformation in the x direction and the y direction are obtained.

The method described above is shown in the drawings as shown in FIGS.
FIG. 4 shows the first harmonic wave component (x direction and y direction) of the frequency spectrum after the Fourier transform is applied to the lattice pattern recorded in the image, which is deformed by the influence of distortion, and other components are removed. FIG. 5 (a) and FIG. 5 (b) show the result of filtering to be removed, and each inverse harmonic component is shifted to the origin (center of FIG. 4) side, and inverse Fourier transform is performed. The resulting phase distribution is shown. The white portion in FIG. 5 represents the phase value πrad, the black portion represents the phase value −πrad, and the gray in between represents the value in between.

  When phase connection of these phase distributions is performed, a continuous phase distribution can be obtained as shown in FIGS. 6 (a) and 6 (b). In FIGS. 6A and 6B, white represents 2πrad, and black represents −2πrad.

  This phase distribution represents the deformation state of the grating pattern, that is, the distribution state of distortion, and the deformation amount (aberration amount) can be calculated from the phase value based on the following equation (6).

ここで、δ_ｘおよびδ_ｙは位相接続後のｘ方向およびｙ方向における位相値であり、ωは変形前の格子の周波数である。

Here, δ _x and δ _y are phase values in the x direction and y direction after phase connection, and ω is the frequency of the grating before deformation.

By the way, it is actually difficult to install the lattice pattern without tilting it with respect to the horizontal posture and vertical posture (also vertical posture) of the camera, and to accurately shift the first harmonic component to the origin side. Therefore, the obtained deformation amounts β _x and β _y do not represent accurate aberrations, but also include a lattice pattern shift amount (rotation amount and parallel movement amount).

  By the way, it has been found that the phase distribution caused by the deviation of the lattice pattern, that is, the error of the inclination (rotation amount) of the lattice pattern and the shift (parallel movement amount) of the frequency spectrum can be approximated by an expression representing a plane. It was.

  That is, when the reference two-dimensional lattice pattern shown in (a) of FIG. 7 is slightly tilted counterclockwise as shown in (b), the displacement of the lattice pattern obtained by Fourier transformation is changed. The represented phase value includes the rotation amount of the lattice pattern.

The coordinates of the point in the plane before the rotation, that is, the coordinates of the point on FIG. 7A are (x, y), and the coordinates after the rotation, that is, the point corresponding to (x, y) on FIG. _Let the coordinates be (x _r , y _r ) {assuming that the point (x, y) has moved to (x _r , y _r ) due to the rotational motion of the rigid body}}, the point (x, y) and the point (x _r , The relationship with y _r ) can be expressed by the following equation (7) using a rotation matrix.

ここで、θは剛体回転角度である。この場合の各点のｘ方向変位は、ｕ_ｘ＝ｘ_２−ｘ_１、ｙ方向変位は、ｕ_ｙ＝ｙ_２−ｙ_１であるので、下記（８）式が得られる。

Here, θ is a rigid body rotation angle. In this case, the x-direction displacement of each point is u _x = x ₂ −x ₁ , and the y-direction displacement is u _y = y ₂ −y ₁ , so the following equation (8) is obtained.

θは回転角度であるので、撮影した特定の基準となる格子パターンの画像に対して定数となる。

Since θ is a rotation angle, it is a constant with respect to the image of the lattice pattern that is a specific reference that is taken.

  By the way, when an error is included in the shift amount when the primary harmonic wave component of the frequency spectrum is shifted to the origin side, the frequency of the lattice pattern before deformation as a reference is different from the actual frequency.

  It means that the interval (pitch) of the lattice pattern before deformation is different. For example, when the first harmonic component of the frequency spectrum in the x direction cannot be accurately shifted to the origin side despite the fact that the lattice pattern of FIG. As shown in FIG. 4, a lattice pattern extending (or contracting) in the x direction is used. In this case, a phase distribution equivalent to x-direction uniaxial deformation is obtained.

  That is, the displacement in the x direction is expressed by the following equation (9).

ここで、ε_ｘはｘ方向の歪であり、この場合には画像上で一様な値となる。ｙ方向についても同様（ｕ_ｙ＝ε_ｙｙ）である。

Here, ε _x is a distortion in the x direction, and in this case, it is a uniform value on the image. The same applies to the y direction (u _y = ε _y y).

  Therefore, in consideration of the rigid body rotational displacement {Equation (8)} and the shift error {Equation (9)} of the first harmonic component, both the x direction and the y direction are expressed as planar equations as in the following Equation (10). Can be expressed.

ここで、定数ｃは理論的には必ずしも必要ではないが、計測データを取り扱うということで組み込んでいる。例えば、フーリエ変換による位相解析の場合の座標の原点と、その後のデータ解析（最小二乗法など）の原点（収差の中心）とが異なる場合に、定数項ｃが必要となるからである。

Here, the constant c is not necessarily required theoretically, but is incorporated by handling measurement data. For example, the constant term c is required when the origin of coordinates in the case of phase analysis by Fourier transform is different from the origin (center of aberration) of subsequent data analysis (such as the least square method).

Accordingly, the deformation amounts β _x and β _y considering the aberration amount (α _x , α _y ) and the shift amount (u _x , u _y ) of the grating pattern should be approximated by the following equations (11) and (12). I'm going.

なお、半径方向成分（ｒ方向）を考えた場合、その変形量β_ｒについては、下記（１３）式にて近似することができるが、ここでは、β_ｘおよびβ_ｙを用いて説明する。

When the radial component (r direction) is considered, the deformation amount β _r can be approximated by the following equation (13), but will be described using β _x and β _y here.

そして、（６）式により得られた値の分布から、（１１）〜（１２）式を用いて、各係数を決定することができる。

Then, each coefficient can be determined from the distribution of values obtained from the equation (6) using the equations (11) to (12).

At this time, the values of β _x and β _y obtained by the equation (6) are obtained as a function of the points x ′ and y ′ including the aberration.
On the other hand, the aberration models of the equations (4) and (11) to (12) are given as a function of the point (x, y) not including the aberration. Therefore, each coefficient cannot be determined directly. Therefore, each coefficient is determined by combining the least square method and the iterative calculation.

  First, coordinate values (x ′, y ′) including aberrations are substituted into (x, y) as initial values, and approximate values of the respective coefficients are determined by a least square method. That is, it can be calculated by the following equation (14) using the phase distribution of the grating in the x direction and the y direction in consideration of the circumferential distortion aberration. Of course, when the least square method is used, calculation is performed for a plurality of pixels.

ここで、

here,

以上の計算により得られた各係数の近似値を用い、収差を含まない座標（ｘ，ｙ）の近似値を計算する。その座標値（ｘ，ｙ）を用い、再び、同様の最小二乗法により各係数を計算する。これを各係数の値が収束するまで繰り返すことにより、各係数を決定することができる。

Using the approximate value of each coefficient obtained by the above calculation, the approximate value of coordinates (x, y) not including aberration is calculated. Using the coordinate value (x, y), each coefficient is calculated again by the same least square method. By repeating this until the value of each coefficient converges, each coefficient can be determined.

In many cases, the center of the aberration is considered to be the center of the image. However, if the center of the aberration is not known, the nonlinear least square method may be used with the coordinates of the center of the aberration as an unknown. At this time, the relationship between r and θ and the central coordinates (x ₀ , y ₀ ) of the aberration is expressed by the following equation (18).

以上の方法により、収差モデル量すなわち収差量を決定した後、その値を用いて撮影画像が補正される。撮影画像の歪みを補正する場合、例えば画像の輝度値補間法などが用いられる。

After determining the aberration model amount, that is, the aberration amount by the above method, the captured image is corrected using the value. When correcting the distortion of the captured image, for example, an image luminance value interpolation method or the like is used.

Here, the result of verifying the effectiveness of applying the aberration correction method using the Fourier transform of the present invention to the displacement measurement result using the digital image correlation method will be described.
FIG. 9 shows an equal displacement diagram (displacement distribution) in which the displacement when the flat plate is moved in the x direction (rigid body translational displacement) is obtained by the digital image correlation method. Since it is a rigid translational displacement, the displacement should not be different from place to place and should be uniform. However, a distribution appears in the displacement as shown in the figure due to the influence of distortion. The unit of legend in these figures is a picture element (pixel). As shown in FIG. 9A, the displacement at the center of the image is 91.5 pixels, whereas a difference of 96.5 pixels and 5 pixels occurs around the image. FIG. 9B shows displacement in the y direction.

  FIG. 10 is a diagram showing a comparison between the displacement distribution (a) corrected using the aberration correction method according to the present invention and the displacement distribution (b) before correction, and it can be seen that the aberration is removed. In this example, equation (4) is used as the aberration model, but similar results were obtained when equation (3) was used.

  FIG. 11 shows the displacement distribution in the vicinity of the center of the plate when a uniaxial tensile load is applied to the flat plate. (A) shows the displacement in the x direction, and (b) shows the displacement in the y direction. In the distribution before aberration correction in this figure, a rounded equal displacement line appears. On the other hand, in FIG. 12 after aberration correction, a substantially straight equal displacement line appears ({a) shows displacement in the x direction and (b) shows displacement in the y direction}, and an appropriate result is obtained. It can be said. In this figure, the equal displacement lines appear obliquely because the test piece to which a tensile load is applied rotates (rigid body rotation) after the load.

Here, the distortion aberration correction method described above is described in a step format.
That is, this distortion correction method includes a step of performing Fourier transform on a photographed image showing a luminance value obtained by photographing a two-dimensional lattice pattern through a lens to obtain a frequency spectrum, A step of removing noise by leaving only the first harmonic wave; and the first harmonic wave is shifted to the origin side by the frequency spectrum of the lattice-like pattern having no aberration, and then subjected to inverse Fourier transform to obtain a phase A step of obtaining a value, a step of performing phase connection of the phase values obtained over a predetermined range in the step, a step of obtaining a deformation amount of the lattice pattern based on the phase value obtained by the phase connection, The aberration model amount generated in the lattice pattern is added to the displacement amount calculation expression expressed by the plane equation considering the rotational movement and parallel movement of the lattice pattern. Substituting the deformation amount into the deformation amount calculation formula, determining a coefficient representing at least the aberration model amount of the deformation amount calculation formula using the least square method, and using the coefficient of the determined aberration model amount And a step of correcting distortion.

In addition, the distortion model in the radial direction is considered as the aberration model quantity, or the distortion aberration in the radial direction and the circumferential direction is considered as the aberration model quantity.
More specifically, a step of obtaining a frequency spectrum by subjecting a photographed image showing a luminance value obtained by photographing a two-dimensional lattice pattern through a lens to obtain a frequency spectrum, A step of removing noise by leaving only a wave, and a step of obtaining a phase value by performing inverse Fourier transform after shifting the primary harmonic wave to the origin side by the frequency spectrum of the lattice pattern without aberration A phase connection of the phase values obtained over the predetermined range in the above steps, and a lattice pattern deformation amount (β _x , β _y ) is obtained based on the phase values obtained by the phase connection. steps and, aberration model amount generated to the grid pattern of the rotational movement and translational movement to the shift amount calculating formula represented by the formula considering the plane of the grid pattern (alpha _x alpha _y) formed by addition of the following equation (A) and (B) by substituting the deformation amount to the deformation value calculation formula of the formula represents at least aberration model of the deformation amount calculation equation using the least square method A step of determining a coefficient, and a step of correcting distortion by using the coefficient of the determined aberration model quantity.

β _x = a ₁ x + a ₂ y + a ₃ + α _x (A)
β _y = a ₄ x + a ₅ y + a ₆ + α _y (B)
As described above, when calculating the phase value in the captured image of the lattice pattern, a deformation amount calculation formula obtained by adding a plane expression representing the rotation amount and the parallel movement amount of the lattice pattern to the aberration model amount is used. Since the coefficients of the plane equation representing the calculation formula are determined by the least square method using the actual deformation, the lattice pattern is shifted when correcting the distortion using the lattice pattern. Even in such a case, the correction can be performed in consideration of the amount of deviation, so that the distortion aberration can be corrected with high accuracy.

  By the way, in the above-described embodiment, a Fourier spectrum is applied to the captured image of the lattice pattern to obtain a frequency spectrum, and only the first harmonic wave is removed from the frequency spectrum, and the noise component is removed. It has been described that the harmonic wave is shifted to the origin side by the frequency spectrum of the lattice-like pattern having no aberration, and then the inverse Fourier transform is performed to obtain the phase value, but the deformation amount calculation formula {(11) and ( 12)} itself includes the amount of rotation and the amount of parallel movement of the lattice pattern, so that the primary harmonic wave need not be shifted to the origin side.

  Moreover, although the case where the circumferential direction distortion aberration was considered in the said embodiment was demonstrated, only radial direction distortion aberration can also be considered. In this case, equation (3) is used instead of equation (4).

  Furthermore, in the above-described embodiment, the lattice lines provided in parallel to each other may be inclined at an angle other than 90 degrees other than those orthogonal to each other as the lattice pattern.

It is a figure which shows the distortion aberration for demonstrating the distortion correction method which concerns on embodiment of this invention. It is a figure of the grating | lattice pattern used as the reference | standard in the same distortion correction method. It is a figure of the lattice pattern after the deformation. The frequency spectrum in the lattice pattern after the deformation is shown. The phase distribution in the lattice pattern after the deformation is shown, (a) is the x-direction phase distribution, and (b) is the y-direction phase distribution. The phase distribution after phase connection in the lattice pattern after the deformation is shown, (a) is the x-direction phase distribution, and (b) is the y-direction phase distribution. It is a figure which shows the lattice pattern in the same distortion correction method, (a) shows what becomes a reference | standard, (b) shows what rotated. The lattice pattern in the same distortion correction method that is deformed in the x direction is shown. It is a figure which shows the displacement distribution at the time of making a rigid body translational displacement, (a) shows x direction displacement, (b) shows y direction displacement. It is a figure which compares the displacement distribution after correction | amendment by the same distortion aberration correction method, and the displacement distribution before correction | amendment, (a) shows x direction displacement, (b) shows y direction displacement. The displacement distribution before the aberration correction in the uniaxial tension | tensile_strength of a flat plate is shown, (a) shows x direction displacement, (b) shows y direction displacement. The displacement distribution after the aberration correction in the uniaxial tension | tensile_strength of the same plate is shown, (a) shows displacement in the x direction, and (b) shows displacement in the y direction.

Claims (2)

Applying Fourier transform to a photographed image showing a luminance value obtained by photographing a two-dimensional lattice pattern through a lens to obtain a frequency spectrum;
Filtering to remove noise from the frequency spectrum, leaving only the first harmonic wave; and
Applying a Fourier inverse transform to the first harmonic wave to obtain a phase value;
Performing phase connection of the phase values determined over a predetermined range in the above steps;
Substituting the phase values (δ _x , δ _y ) obtained by the above phase connection into the following equations (A) and (B ) to determine the deformation amount (β _x , β _y ) of the lattice pattern;
An aberration model quantity (α _x , based on the following formulas (C) and (D) generated in the grid pattern in the shift amount calculation formula expressed by a plane formula considering the rotational movement and parallel movement of the grid pattern: The deformation amount is substituted into a deformation amount calculation formula represented by the following equations (E) and (F) to which α _y ) is added, and at least an aberration model amount of the deformation amount calculation equation is expressed using a least square method. Determining a coefficient;
And correcting the distortion using the coefficient of the determined aberration model quantity. A method for correcting distortion in a photographed image.
β _x = −δ _x / 2πω (A)
β _y = −δ _y / 2πω (B)
Where ω is the frequency of the grating before deformation.
α _x = (g ₁ + g ₃ ) x ² + g ₄ xy + g ₁ y ² + k ₁ x (x ² + y ² ) (C)
α _y = g ₂ x ² + g ₃ xy + (g ₂ + g ₄ ) y ² ++ k ₁ y (x ² + y ² ) (D)
β _x = a ₁ x + a ₂ y + a ₃ + α _x (E)
β _y = a ₄ x + a ₅ y + a ₆ + α _y (F) Applying Fourier transform to a photographed image showing a luminance value obtained by photographing a two-dimensional lattice pattern through a lens to obtain a frequency spectrum;
Filtering to remove noise from the frequency spectrum, leaving only the first harmonic wave; and
Shifting the primary harmonic wave to the origin side by the frequency spectrum of the lattice-like pattern having no aberration, and then performing a Fourier inverse transform to obtain a phase value;
Performing phase connection of the phase values determined over a predetermined range in the above steps;
Substituting the phase values (δ _x , δ _y ) obtained by the above phase connection into the following equations (A) and (B ) to determine the deformation amount (β _x , β _y ) of the lattice pattern;
An aberration model quantity (α _x , based on the following formulas (C) and (D) generated in the grid pattern in the shift amount calculation formula expressed by a plane formula considering the rotational movement and parallel movement of the grid pattern: The deformation amount is substituted into a deformation amount calculation formula represented by the following equations (E) and (F) to which α _y ) is added, and at least an aberration model amount of the deformation amount calculation equation is expressed using a least square method. Determining a coefficient;
And correcting the distortion using the coefficient of the determined aberration model quantity. A method for correcting distortion in a photographed image.
β _x = −δ _x / 2πω (A)
β _y = −δ _y / 2πω (B)
Where ω is the frequency of the grating before deformation.
α _x = (g ₁ + g ₃ ) x ² + g ₄ xy + g ₁ y ² + k ₁ x (x ² + y ² ) (C)
α _y = g ₂ x ² + g ₃ xy + (g ₂ + g ₄ ) y ² ++ k ₁ y (x ² + y ² ) (D)
β _x = a ₁ x + a ₂ y + a ₃ + α _x (E)
β _y = a ₄ x + a ₅ y + a ₆ + α _y (F)

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