TWI788838B - Method for coordinate transformation from spherical to polar - Google Patents

Abstract

This invention is a method for coordinate conversion performed by a processor or a device, possibly a computer-based device. At least one great orientation and one great position individually represented by spherical coordinates (θ_o,Φ_o) and by polar coordinates (r_o,Φ_o) on two-dimensional plane, and the effective focal length f of a lens set are used to determine a first-order coefficient k₁ and a third-order coefficient k₂ which make the radius r in polar coordinates be converted by the colatitude θ and its cube. The first-order coefficient k₁ is proved as the effective focal length f and the third-order coefficient k₂ is proved between -0.166f and 0.410f.

Description

Spherical coordinates to polar coordinates conversion method

This creation is about a coordinate conversion method; especially about a coordinate conversion method using simple mathematical formulas, especially because of its high adaptability.

With the vigorous development of consumer electronics products, related camera devices are becoming increasingly miniaturized. For example, the camera equipment that originally required multiple lenses to complete a panoramic (panoramic) collage is also used in fisheye cameras and digital signal processing. With the combination of Digital Signal Processor (DSP), the panoramic camera device has been successfully reduced to the size of a palm.

Theoretically, through the geometric relationship between the solid angle of the lens and the pixel coordinates of the two-dimensional image sensor, the images at various angles can be converted through the operation of trigonometric functions. In 2006, there was a fisheye lens conversion formula disclosed by J.Kannala & S.S.Brandt (DOI: 10.1109/TPAMI.2006.153). As shown in "Figure 1", the direction of the solid angle is represented by the colatitude θ and the azimuth Φ in spherical coordinates; the incident light in this direction passes through the lens group LS at a refraction angle β is focused on the two-dimensional image sensor PX, and its radius on the polar coordinate (polar coordinate) r=f. tan(β), where f is the effective focal length of the lens set LS.

According to different shooting requirements, the relationship between the colatitude angle θ and the radius r can be roughly divided into three models, among which, type I is an orthogonal projection (orthogonal projection) r=f. sinθ; Type II is stereographic projection (stereographic projection) r=f. 2. tan(θ/2); and type III is isosteric Angle projection (equisolid angle projection) r=f. 2. sin(θ/2).

If a square of graph paper is placed twice f in front of the lens, and there are 41×41 orthogonal straight lines on it (the interval is f), after sequentially applying the above-mentioned projection relationships of types I, II, and III, the two-dimensional The image sensor PX presents the graphics from left to right as shown in "Fig. 2". Most of the fisheye lenses attached to mobile phones are type I (the leftmost picture). Although they can present orthogonal images from different angles, the edge quality is severely compressed and cannot be restored with normal resolution; it is commonly used for video conferencing or panoramic images. Most of the fisheye lenses are Type II (middle picture). Through the combination of high-variance lenses, although the image quality directly in front of the lens is sacrificed, the resolution of the edge can be improved; in addition, there are also lenses within ±60° in front of the lens ( i.e. field angle 120°) as much as possible to present isometric type III (far right image).

The two-dimensional images captured by the fisheye lens are generally processed by DSP into a panoramic image format that can be read by subsequent equipment; angled images. Whether it is the hardware processing in the early stage or the software processing in the later stage, it is necessary to convert the two-dimensional image coordinates into the direction of the incident light, or vice versa, to cut out the two-dimensional image corresponding to the externally required solid angle, and the most What is important and contains the largest error is the conversion between the radius r and the co-latitude angle θ.

k₁．θ+k₂．θ³+k₃．θ⁵+k₄．θ⁷+k₅．θ⁹。這五個轉換參數k₁,k₂,...,k₅，加上光軸偏移與球面偏差的兩個校正參數，一共有七個變數要與實際結果作比對，以找出二維影像座標至所述餘緯角θ以及所述方位角Φ的轉換矩陣(matrix)。 According to the polynomial conversion method revealed by J.Kannala & SSBrandt, it is shown that at least nine orders are required to completely convert the radius r from a small angle to a large angle r(θ)

k ₁ . θ+k ₂ . θ ³ +k ₃ . θ ⁵ +k ₄ . θ ⁷ +k ₅ . θ ⁹ . The five transformation parameters k ₁ , k ₂ ,...,k ₅ , plus the two correction parameters of optical axis offset and spherical deviation, have a total of seven variables to be compared with the actual results to find out the two A transformation matrix (matrix) from the dimensional image coordinates to the colatitude angle θ and the azimuth angle Φ.

k₁．θ+ k₂．θ³，降低不確定性。雖然三階近似法較能得到穩定的數值解，但高階項次造成的過度配適(overfitting)，如何影響座標轉換的還原情形，J.Kannala & S.S.Brandt並沒有多加詳述。 However, in practical applications, it is extremely difficult to obtain a stable numerical solution (numerical solution) for seven variables, so J.Kannala & SSBrandt finally suggested using the third-order approximation method, r(θ)

k ₁ . θ+ k ₂ . θ ³ , reducing uncertainty. Although the third-order approximation method can obtain a stable numerical solution, J.Kannala & SSBrandt did not elaborate on how the overfitting caused by high-order terms affects the restoration of the coordinate transformation.

A coordinate conversion method implemented by a processor or a device equipped with a computer to determine a first-order coefficient k ₁ at least by means of a large orientation of spherical coordinates, a large orientation of two-dimensional coordinates, and an effective focal length f of a lens group And a third-order coefficient k ₂ , so that the radius r of the two-dimensional coordinate can be converted from the first-order item and the third-order item of the colatitude θ of the spherical coordinate, which includes: Determine the latitude angle θ _o of the large orientation and the corresponding radius r _o of the large orientation; set the first-order coefficient k ₁ =f, and calculate the third-order coefficient k ₂ according to f, θ _o and r _o ; and use The radius of the two-dimensional coordinate r=k ₁ . θ+k ₂ . θ ³ , and r _o falls between 0.7 and 1.2 times the maximum radius in the two-dimensional coordinate transformation space. Wherein, the third-order coefficient k ₂ is between -0.166f and 0.410f.

[(s．f．θ_o ³)/(r_o-f．θ_o)]之正整數，其中，在r_o>f．θ_o時，s得從0.334至0.410中選擇，而在r_o<f．θ_o時，s得從-0.166至-0.150中選擇，接著決定k₂=f．s/n²。 In one embodiment, the meaning of calculating the third-order coefficient k ₂ refers to k ₂ =(r _o -f.θ _o )/θ _o ³ . In another embodiment, the meaning of calculating the third-order coefficient k ₂ refers to r _o =f. When θ _o , determine that k ₂ is a value from -0.166/4 ² to 0.334/15 ² ; otherwise, first determine an angle factor n as the closest

A positive integer of [(s.f.θ _o ³ )/(r _o -f.θ _o )], where r _o >f. When θ _o , s has to be selected from 0.334 to 0.410, and in r _o <f. When θ _o , s has to be selected from -0.166 to -0.150, and then determine k ₂ =f. s/n ² .

In order to adapt to the difference actually caused by installing the lens group, there is a corresponding third-order coefficient k ₂ for each azimuth Φ of the spherical coordinates, and the latitude angle θ that determines the large orientation The significance of _o and the corresponding radius r _o of the large orientation refers to the use of the radius r _o of the large orientation of two or more azimuth angles to find out that the remaining latitude angle θ _o of the large orientation is the cone angle The conic section (conic section), such as: elliptic equation; and according to the conic section, the radius r _o (Φ) of the large location on each of the azimuth angles is deduced.

In addition, a coordinate conversion method is implemented by a processor or a device equipped with a computer, by means of the small orientation and large orientation of spherical coordinates and the small orientation and large orientation of two-dimensional coordinates to determine a first-order coefficient k ₁ and a third-order coefficient k ₂ , so that the radius r of the two-dimensional coordinate can be converted from the first-order term and the third-order term of the colatitude θ of the spherical coordinate, which includes : Determine the residual latitude angle θ _f and θ _o between the small orientation and the large orientation and the corresponding radius r _f and r _o between the small orientation and the large orientation, wherein the residual latitude angle θ _f of the small orientation is less than 0.75 (unit: radian); calculate k ₁ =r _f /θ _f and k ₂ =(r _o -k ₁ .θ _o )/θ _o ³ ; and make the radius of the two-dimensional coordinate r=k ₁ . θ+k ₂ . θ ³ , and r _o falls between 0.7 and 1.2 times the maximum radius in the two-dimensional coordinate transformation space.

In order to improve the accuracy of the first-order coefficient k ₁ , the residual latitude angle θ _f of the small orientation is smaller than 0.23 (unit: radian). It is especially suitable for the unknown effective focal length or for correcting the effective focal length.

Furthermore, a coordinate conversion method is implemented by a processor or a device equipped with a computer, and determines a first-order coefficient k ₁ and a third-order coefficient by means of a plurality of orientations of spherical coordinates and a plurality of positioning of two-dimensional coordinates k ₂ , so that the radius r of the two-dimensional coordinate can be converted from the first-order item and the third-order item of the colatitude θ of the spherical coordinate, which includes: determining the plurality of orientations The remaining latitude angle θ _i and the corresponding radius r _i of the plurality of locations; according to θ _i and r _i , determine the first-order coefficient k ₁ and the third-order coefficient k ₂ ; and make the radius r of the two-dimensional coordinates = k ₁ . θ+k ₂ . θ ³ , and at least one of r _i falls between 0.7 times and 1.2 times the maximum radius in the two-dimensional coordinate transformation space.

The meaning of determining the first-order coefficient k ₁ and the third-order coefficient k ₂ refers to initializing the first-order coefficient k ₁ and the third-order coefficient k ₂ to perform a double search procedure by iterative or recursive way, to provide a complex number of parameter values to replace the first-order coefficient k ₁ and the third-order coefficient k ₂ , execute a scoring procedure to obtain a degree of difference as the relationship [r _i -(k ₁ .θ _i +k ₂ . θ _i ³ )] ² and compare the smallest degree of difference with its representative k ₁ and k ₂ .

The meaning of determining the first-order coefficient k ₁ and the third-order coefficient k ₂ refers to setting the first-order coefficient k ₁ as the effective focal length f of a lens group and initializing the third-order coefficient k ₂ to perform a single search The program provides a complex number of parameter values to replace the third-order coefficient k ₂ by an iterative or recursive method, and executes a scoring procedure to obtain a difference degree as a relation [r _i -(k ₁ .θ _i +k _2. The sum of θ _i ³ )] ² and compare the minimum degree of difference with its representative k ₂ .

The meaning of determining the first-order coefficient k ₁ and the third-order coefficient k ₂ refers to calculating a plurality of first-axis data X _i =θ _i ² and a plurality of second-axis data Y _i =r _i /θ _i , And determine the second-order coefficient k ₂ =Σ _i [(X _i -<X _i >). (Y _i -<Y _i >)]/Σ _i (X _i -<X _i >) ² and the first-order coefficient k ₁ =<Y _i >-k ₂ . <X _i >, wherein <X _i > and <Y _i > are the arithmetic mean of the plurality of first-axis data and the plurality of second-axis data respectively.

D: deformation factor

DN: horizontal line below

E: Rising rate

f: effective focal length of the lens group

k ₁ : first-order item coefficient

k ₂ : Third-order term coefficient

k ₃ : fifth-order item coefficient

k ₄ : Seventh-order item coefficient

k ₅ : Ninth order term coefficient

LS: lens group

n: angle factor

OP: optical axis

PX: Two-dimensional image sensor

q: half coefficient

R: normalized radius

r: radius

r _f : Radius of small positioning

r _i : the radius of multiple positioning

r(θ _max ): maximum value

r(θ _min ): minimum value

r _o : the radius of the largest positioning

ST: Tester

S: Optimizing results

s: linear factor

T: Normalized co-latitude angle

UP: upper horizontal line

X _i : first axis data

Y _i : second axis data

<X _i >: Arithmetic mean of the data on the first axis

<Y _i >: Arithmetic mean of the second axis data

α: the co-latitude angle divided by the angle factor

β: Refraction angle

Φ: azimuth

θ: co-latitude angle

θ _f : Minor orientation residual latitude angle

θ _i : Complementary latitude angles of multiple orientations

θ _max : maximum angle

θ _min : minimum angle

θ _o : the latitude angle of the maximum orientation

4: (3-1) The curve on the right side of the equal sign

63: (3-2) The curve on the right side of the equal sign

490: (3-3) The curve on the right side of the equal sign

(θ²/16²)之曲線 11: sin ² (θ/16)

(θ ² /16 ² ) curve

(θ²/8²)之曲線 168: sin ² (θ/8)

(θ ² /8 ² ) curve

0.96(θ²/4²)之曲線 749: sin ² (θ/4)

0.96(θ ² /4 ² ) curve

163:2f. The fitting curve of sin(θ/2) in response to formula (11)

165:3f. The fitting curve of sin(θ/3) in response to formula (11)

166:4f. The fitting curve of sin(θ/4) in response to formula (11)

41:2f. Fitting curve of tan(θ/2) in response to formula (11)

36:3f. Fitting curve of tan(θ/3) in response to formula (11)

35:4f. Fitting curve of tan(θ/4) in response to formula (11)

831:3f. Fitting curve of tan(θ/3) in response to formula (11)

5138:0.5． f. sin(θ)+0.5． f. Fitting curve of tan(θ/2) in response to formula (11)

2992:0.5． f. sin(θ)+0.71． f. The fitting curve of sin(θ/2) in response to formula (11)

Figure 1 is a three-dimensional schematic diagram of coordinate conversion

Figure 2 is a schematic diagram of three projection models from three-dimensional coordinates to two-dimensional coordinates

Figure 3 is the relationship diagram of formula (3) in this creation in θ/4, θ/8 and θ/16

Figure 4 is the relationship between θ/2, θ/3 and θ/4 of the formula (11) in this creation

Fig. 5 is a schematic diagram of the second embodiment of this creation

Figure 6 is a demonstration figure of the results of the third embodiment of this creation

This manual uses the third-order approximation formula disclosed by J.Kannala & S.S.Brandt (2006) as an example, and obtains the coefficients of the third-order approximation with a unique method; however, it is not limited to the third-order approximation order approximation, using the same rules can also be deduced to a higher order approximation.

90°的範圍內θ/16<0.1，故8．sin(θ/16)可近似於θ/2(θ的單位在此說明書中為弧度)。利用此一近似項，將(1)改寫為sin(θ/2)

(θ/2)．cos(θ/4)．cos(θ/8)．cos(θ/16)...(2)。接著，因為cos²α+sin²α=1，且在α<20°的範圍內，(1/4)．sin⁴α<<1-sin²α，故可延伸出一通式(general formula)cos²α

1-sin²α+(1/4)sin⁴α=(1-0.5sin²α)²...(3)。 First, use the double-angle formula to get sin(θ/2)=8. sin(θ/16). cos(θ/4). cos(θ/8). cos(θ/16)...(1); since in θ

θ/16<0.1 in the range of 90°, so 8. sin(θ/16) can be approximated to θ/2 (the unit of θ is radian in this specification). Using this approximation, rewrite (1) as sin(θ/2)

(θ/2). cos(θ/4). cos(θ/8). cos(θ/16)...(2). Then, because cos ² α+sin ² α=1, and in the range of α<20°, (1/4). sin ⁴ α<<1-sin ² α, so a general formula (general formula) cos ² α can be extended

1-sin ² α+(1/4)sin ⁴ α=(1-0.5sin ² α) ² ... (3).

1-0.5sin²(θ/16)...(3-1)、cos(θ/8)

1-0.5sin²(θ/8)...(3-2)以及cos(θ/4)

1-0.52sin²(θ/4)...(3-3)；其中，cos(θ/4)函數有部分角度大於20°，故其係數由0.5調整為0.52。「第3圖」中點線與虛線間變異數(variance)開根號後，為百萬分之4、63與490，顯見其近似程度相當高。 Please refer to "Figure 3", the dotted lines from top to bottom are the relationship of cos(θ/16), cos(θ/8) and cos(θ/4) with the co-latitude angle θ, and the dotted lines 4, 63 and 490 are the curves obtained according to (3). That is, cos(θ/16)

1-0.5sin ² (θ/16)...(3-1), cos(θ/8)

1-0.5sin ² (θ/8)...(3-2) and cos(θ/4)

1-0.52sin ² (θ/4)...(3-3); Among them, the cos(θ/4) function has some angles greater than 20°, so its coefficient is adjusted from 0.5 to 0.52. After taking the square root of the variance between the dotted line and the dotted line in "Figure 3", it is 4, 63, and 490 parts per million, which shows that the degree of approximation is quite high.

(θ/2)．[1-0.52sin²(θ/4)]．[1-0.5sin²(θ/8)]．[1-0.5sin²(θ/16)]...(4)，或是sin(θ/2)

(θ/2)．{1-θ²．[(0.5/4²)+(0.5/8²)+(0.5/16²)]}...(5)；而後者，係(4)代入正弦函數的平方近似其角度的平方之關係。請參閱「第3圖」插圖，點線為正弦函數的平方而虛線為其角度的平方11、168、749，也就是sin²(θ/16)

(θ²/16²)、sin²(θ/8)

(θ²/8²)與sin²(θ/4)

0.96(θ²/4²)。「第3圖」插圖中點線與虛線間變異數開根號，為百萬分之11、168與749。最後，將(5)所有項次乘開即得到sin(θ/2)

(θ/2)．[1-0.16．(θ/2)²]...(6)。雖然sin²(θ/4)

0.96(θ²/4²) 間的變異數開根號為百萬分之749；但因合併cos(θ/4)．cos(θ/8)．cos(θ/16)的關係，最後(6)之正弦函數與其三階近似值間變異數開根號後，為百萬分之413。 Therefore, we substitute (3) and rewrite (2) as sin(θ/2)

(θ/2). [1-0.52sin ² (θ/4)]. [1-0.5sin ² (θ/8)]. [1-0.5sin ² (θ/16)]...(4), or sin(θ/2)

(θ/2). {1-θ ² . [(0.5/4 ² )+(0.5/8 ² )+(0.5/16 ² )]}...(5); and the latter is the relationship between (4) substituting the square of the sine function to approximate the square of its angle. Please refer to the illustration of "Figure 3", the dotted line is the square of the sine function and the dashed line is the square of the angle 11, 168, 749, which is sin ² (θ/16)

(θ ² /16 ² ), sin ² (θ/8)

(θ ² /8 ² ) and sin ² (θ/4)

0.96(θ ² /4 ² ). The square root of the variance between the dotted line and the dashed line in the illustration of "Figure 3" is 11, 168, and 749 parts per million. Finally, multiply all terms in (5) to get sin(θ/2)

(θ/2). [1-0.16. (θ/2) ² ]...(6). Although sin ² (θ/4)

The root sign of the variance between 0.96(θ ² /4 ² ) is 749 parts per million; but due to the combination of cos(θ/4). cos(θ/8). The relationship between cos(θ/16), the variance between the last (6) sine function and its third-order approximation is 413 parts per million.

Different from the traditional use of linear algebra to solve the transformation matrix, this creation can directly obtain the transformation coefficient of the type III projection relationship through (1) to (6). Although the projection relationship between type I and type II obviously depends on higher-order items, we still try to analyze it as follows.

{1-θ²．[(q/2²)+(0.5/4²)+(0.5/8²)+(0.5/16²)]}...(7)，以及對半係數q作0.5附近的掃描，從最小變異數得到該半係數q應為0.44。最後，第I型投影關係的三階近似sin(θ)

θ．(1-0.15．θ²)...(8)，且正弦函數與其變異數開根號，為百萬分之5038，屬於可接受的近似結果。至於tan(θ/2)，則利用恆等式tan(θ/2)=0.5．sinθ．[1+tan²(θ/2)]，套用(8)得到tan(θ/2)

(θ/2)．[1+0.39．(θ/2)²]...(9)，且正切函數與其變異數開根號為百萬分之5510，也屬於可接受的近似結果。 Also use the double angle formula to get sin(θ)=16sin(θ/16). cos(θ/2). cos(θ/4). cos(θ/8). cos(θ/16), and then assume cos(θ/2). cos(θ/4). cos(θ/8). cos(θ/16)

{1-θ ² . [(q/2 ² )+(0.5/4 ² )+(0.5/8 ² )+(0.5/16 ² )]}...(7), and scan the half coefficient q around 0.5, from the minimum The coefficient of variation obtained by this semi-coefficient q should be 0.44. Finally, the third-order approximation sin(θ) of the type I projection relation

θ. (1-0.15.θ ² )...(8), and the square root of the sine function and its variance is 5038 parts per million, which is an acceptable approximate result. As for tan(θ/2), use the identity tan(θ/2)=0.5. sin θ. [1+tan ² (θ/2)], apply (8) to get tan(θ/2)

(θ/2). [1+0.39． (θ/2) ² ]...(9), and the root sign of the tangent function and its variance is 5510 parts per million, which is also an acceptable approximate result.

Accordingly, we set a projection relation general formula r(θ/n)=f. θ. [1+s. (θ/n) ² ]...(10), or after dividing the co-latitude angle θ and the radius r(θ) respectively by a maximum angle θ _max and a maximum value r(θ _max ), the normalized The normalized co-latitude angle T and the normalized radius R(T) represent (10) again, that is, R(T)/T=D. [1+s. (θ _max /n) ² . T ² ]=D． (1+E.T ² )...(11), wherein, deformation factor D=f. θ _max /r(θ _max ), the angle factor n is a positive integer 1, 2, 3, ..., and the linear factor s from -0.17 to 0.41 and the rate of rise E. In the following example, the maximum angle and the farthest distance that the two-dimensional image sensor PX can sense are the maximum angle θ _max and the maximum value r(θ _max ), for example: the maximum angle θ _max = 90° and the maximum r(θ _max )=1000 pixels. The maximum angle θ _max does not represent the maximum angle at which the lens group LS can collect light, and the maximum value r(θ _max ) does not represent the maximum distance that the two-dimensional image sensor PX can sense. The examples in this specification limit the implementation scope of this creation.

1且該角度因子n

4，該線性因子s趨近於一固定值-0.166；反之，當該變形因子D<1且該角度因子n

15，該線性因子s趨近於一固定值0.334。 Please refer to "Figure 4", the dotted lines in the upper three figures from left to right are r(θ)=2f. sin(θ/2), 3f. sin(θ/3) and 4f. When sin(θ/4), the relationship between R(T)/T and the square T ² of the normalized colatitude angle; the dotted lines in the three figures below are r(θ)=2f from left to right. tan(θ/2), 3f. tan(θ/3) and 4f. When tan(θ/4), the relationship between R(T)/T and the square T ² of the normalized colatitude angle. Through the general formula of normalized projection relation in (11), we can use the fitting method of minimum variation to find out the linear factor, which corresponds to r(θ)=2f. sin(θ/2), 3f. sin(θ/3) and 4f. When sin(θ/4), s are -0.163, -0.165 and -0.166 respectively, and the fitting results are the dotted lines 163, 165 and 166 in the three figures above; and correspond to r(θ)=2f. tan(θ/2), 3f. tan(θ/3) and 4f. When tan(θ/4), s are 0.41, 0.36, and 0.35 respectively, and the fitting results are the dotted lines 41, 36, and 35 in the three figures below. When the deformation factor D

1 and the angle factor n

4. The linear factor s tends to a fixed value -0.166; on the contrary, when the deformation factor D<1 and the angle factor n

15, the linear factor s tends to a fixed value of 0.334.

Therefore, the first embodiment of the present invention first finds the maximum value r(θ _max ) of the two-dimensional image sensor PX and the corresponding maximum angle θ _max , according to the effective Focal length f, calculate the deformation factor D = f. θ _max /r(θ _max ); then through the relationship of (11) at T=R=1, s. (θ _max /n) ² =E=(1-D)/D, get the coefficient in (10) or (11). Since the first embodiment uses a relatively large radius, the rate of increase E may be relatively high. However, if the requirements for the converted coordinate position are not too high, this is still a good method to quickly obtain the conversion matrix.

Please refer to "Fig. 5", the second embodiment of the invention is to first determine the minimum colatitude angle other than zero and its corresponding radius as a minimum angle θ _min and a minimum value r(θ _min ), And the maximum angle θ _max and the maximum value r(θ _max ). During implementation, a tester ST is placed in front of the lens group LS, for example: it can project a horizontal or grid-shaped light source or a horizontal or grid-shaped pattern plate, and move up and down near the optical axis OP to test the two-dimensional image The sensor PX produces a symmetrical upper horizontal line UP and a lower horizontal line DN; from the moving distance near the optical axis OP and the distance between the tester ST and the lens group LS, θ _min and r can be calculated (θ _min ). In addition, one tester ST can also be placed on both symmetrical sides of the lens group LS, so that θ _max can be set to 90° and r(θ _max ) can also be obtained from the two-dimensional image sensor PX. Based on this, f=r(θ _min )/θ _min and D=f are calculated. θ _max /r(θ _max ) and E=(1-D)/D. Since the second embodiment utilizes θ _min and r(θ _min ) which are closest to the optical axis, the rate of increase E may be low, but if the requirements for the converted coordinate position are not too high, and the effective focal length f Unavailable, this is still a good way to quickly obtain the transformation matrix.

The third embodiment of the present creation is to fit (11) with a plurality of the normalized colatitude angles T and the corresponding normalized radius R(T), and the fitting result can choose the coefficient of the minimum variation Or select the coefficient when the variation is zero for the differential of the rate E; the former is to test the coefficient by iterative or recursive methods, and the latter is to use the slope and cutoff of the least square method (least square) According to the distance formula, both are commonly used linear regression methods, so this manual will not repeat them. Besides, since D=f in (11). The difference of θ _max /r(θ _max ) is usually not large, so the data of the effective focal length f provided by the lens manufacturer can be used, and the first-order coefficient k ₁ can be fixed directly with θ _max and r(θ _max ); thus, It is only necessary to scan and test the third-order coefficient k ₂ .

Please refer to "Figure 6", from left to right, r(θ)=3f. tan(θ/3), r(θ)=0.5. f. sin(θ)+0.5． f. tan(θ/2) and r(θ)=0.5. f. sin(θ)+0.71． f. The normalized projection relationship (dotted line) of sin(θ/2), and the fitting results (dotted line) 831, 5138, 2992 obtained by the formula of the least square method in the third embodiment. It can be seen that even if the projection relationship is arbitrarily matched, it is still applicable In the method of this creation, and the root sign of the variation between the dotted line and the dotted line in "Figure 6" is from left to right, which are 831, 5138, and 2992 parts per million, and the approximate effect is good.

[s．D/(1-D)]的正整數，但D=1時則令n=15。如此，藉由該有效焦距f以及θ_max與其對應的r(θ_max)。實施時，亦可在D<1時，藉由疊代或遞迴方式在0.334至0.410間選擇s，最後找出能得到最接近正整數之該角度因子n以及其對應的該線性因子s；同理，在D>1時，藉由疊代或遞迴方式在-0.166至-0.150間選擇s，最後找出能得到最接近正整數之該角度因子n以及其對應的該線性因子s。 In the fourth embodiment of the present invention, the maximum value r(θ _max ) and the maximum angle θ _max of the two-dimensional image sensor PX are found first, and then the deformation factor D= is calculated according to the effective focal length f f. θ _max /r(θ _max ). Because (11) when T=R=1, s. D/(1-D)=(n/θ _max ) ² ; so set s=0.334 when D<1 and s=-0.166 when D>1, so as to determine n is the closest to θ _max .

[s. D/(1-D)] is a positive integer, but when D=1, n=15. Thus, by the effective focal length f and θ _max and its corresponding r(θ _max ). During implementation, it is also possible to select s between 0.334 and 0.410 by iterative or recursive methods when D<1, and finally find out the angle factor n and the corresponding linear factor s that can obtain the closest positive integer; Similarly, when D>1, select s between -0.166 and -0.150 by iterative or recursive methods, and finally find out the angle factor n and the corresponding linear factor s that can get the closest positive integer.

[(s．f．θ_o ³)/(r_o-f．θ_o)]|並在該分數小於或等於該目標值時做更新。所述更新係重設該最佳化結果S為當下的該線性因子s；以及重設該目標值為該分數。在該最佳化程序結束後，計算f．s/n²為該三階係數k₂。反之，在r_o<f．θ_o時，初始化該目標值、該最佳化結果S以及該線性因子s，以執行一最佳化程序，藉由疊代或遞迴方式，去掃描介於-0.166至-0.150之參數值為該線性因子s、執行該計分程序以得到該角度因子n以及該分數為 |n-

[(s．f．θ_o ³)/(r_o-f．θ_o)]|並在該分數小於或等於該目標值時做更新。所述更新係重設該最佳化結果S為當下的該線性因子s；以及重設該目標值為該分數。在該最佳化程序結束後，計算f．S/n²為該三階係數k₂。 That is to say, by adjusting the value of the linear factor s, it is found out which positive integer the angle factor n corresponding to the lens set LS is. At r _o =f. When θ _o , the third-order coefficient k ₂ is determined from -0.166/4 ² to 0.334/15 ² . At r _o >f. When θ _o , initialize a target value, an optimization result S and the linear factor s to execute an optimization procedure, and scan the parameter values from 0.334 to 0.410 for the linear factor by iterative or recursive methods s, execute a scoring procedure to obtain the angle factor n and a score of |n-

[(s.f.θ _o ³ )/(r _o -f.θ _o )]| and update when the score is less than or equal to the target value. The updating is resetting the optimization result S to the current linear factor s; and resetting the target value to the score. After the optimization procedure is finished, calculate f. s/n ² is the third-order coefficient k ₂ . On the contrary, in r _o <f. When θ _o , initialize the target value, the optimization result S and the linear factor s to execute an optimization procedure, and scan the parameter values between -0.166 and -0.150 by iterative or recursive methods For the linear factor s, execute the scoring procedure to obtain the angular factor n and the score is |n-

[(s.f.θ _o ³ )/(r _o -f.θ _o )]| and update when the score is less than or equal to the target value. The updating is resetting the optimization result S to the current linear factor s; and resetting the target value to the score. After the optimization procedure is finished, calculate f. S/n ² is the third-order coefficient k ₂ .

This creation uses mathematical methods to calculate the projection relationship between the co-latitude angle and the radius through a simple formula, without the need for time-consuming and laborious scanning tests, and will not fall into the quagmire of over-fitting; especially, the conversion from spherical coordinates to polar coordinates The angle factor can be obtained by the linear factor within a certain range. Therefore, even if there is an error in the assembly of the lens set LS and the two-dimensional image sensor PX, the coordinate transformation method of the present invention can be used as the initialization parameter of the transformation matrix, saving search and calculation time.

To sum up, the coordinate conversion method of this creation has indeed met the requirements for patent application, and a patent application is filed according to law. However, the above is only a preferred embodiment of this creation, and should not limit the scope of implementation of this creation; therefore, all simple equivalent changes and modifications made according to the patent scope of this creation and the content of the specification are all It should still be within the scope covered by this creation patent.

DN: horizontal line below

f: effective focal length of the lens group

LS: lens group

OP: optical axis

PX: Two-dimensional image sensor

r(θ _max ): maximum value

ST: Tester

UP: upper horizontal line

θ _min : minimum angle

Claims (3)

A coordinate conversion method implemented by a processor or a device equipped with a computer, which includes the following steps: receiving a two-dimensional image and image parameters provided by a camera processor, wherein the camera processor is based on a two-dimensional image The electrical signal generated by the sensor is processed into the two-dimensional image and the image parameters; according to the image parameters, the corresponding projection relationship of a lens group is determined, wherein the lens group collects light in a fisheye angle of view to focus on the two A three-dimensional image sensor, the two-dimensional image sensor is located on the effective focal length f of the lens group, the half-angle of the fisheye viewing angle can be expressed as a large directional latitude angle θ _o , and the image parameters are used to determine A large positioning radius r _o ; set the first-order coefficient k ₁ =f, and calculate the third-order coefficient k ₂ according to f, θ _o and r _o ; and convert the two-dimensional image to display pixel values in spherical coordinates, And the colatitude, azimuth and polar coordinate radius on the two-dimensional image sensor are expressed as θ, Φ and r respectively, wherein the conversion is based on r=k ₁ . θ+k ₂ . θ ³ . The projection relationship for determining a lens group is based on r _o and f. The ratio of θ _o determines the projection relationship, r _o is related to the maximum range where the light is focused on the two-dimensional image sensor and falls in f. 0.7 times to 1.2 times of θ _o and the third-order coefficient k ₂ is between -0.166f to 0.410f. The coordinate conversion method as described in Claim 1, wherein the meaning of calculating the third-order coefficient k ₂ according to f, θ _o and r _o includes: when r _o =f. When θ _o , the third-order coefficient k ₂ is determined from -0.01 to 0.001; in r _o >f. When θ _o , scan a linear factor s within 0.334 to 0.410 to find the positive integer closest to the square root of the value of (s.f.θ _o ³ )/(r _o -f.θ _o ) as an angle factor n, and finally calculate f. s/n ² is the third-order coefficient k ₂ ; and in r _o <f. When θ _o , scan the linear factor s from -0.166 to -0.150 to find the positive integer closest to the root sign of the value of (s.f.θ _o ³ )/(r _o -f.θ _o ) Angle factor n, and finally calculate f. s/n ² is the third-order coefficient k ₂ . The coordinate transformation method as described in Claim 1, wherein the meaning of calculating the third-order coefficient k ₂ according to f, θ _o and r _o includes: when r _o =f. When θ _o , the third-order coefficient k ₂ is determined from -0.01 to 0.001; in r _o >f. When θ _o , initialize an objective value, an optimization result S and a linear factor s to execute an optimization procedure, and provide a plurality of parameter values ranging from 0.334 to 0.410 by iterative or recursive methods To replace the linear factor s, perform a scoring procedure to obtain an angle factor n and a score that is n minus (s.f.θ _o ³ )/(r _o -f.θ _o ) and taking the square root of the value Absolute value and when the score is less than or equal to the target value, update the optimization result S to the current linear factor s and reset the target value to the score, and finally calculate f. S/n ² is the third-order coefficient k ₂ ; and when r _o <f. When θ _o , initialize the target value, the optimization result S and the linear factor s to perform an optimization procedure, and provide a plurality of values between -0.166 and -0.150 by iterative or recursive methods parameter value to replace the linear factor s, execute the scoring procedure to obtain the angular factor n and the score is the square root of the value of n minus (s.f.θ _o ³ )/(r _o -f.θ _o ) Then take the absolute value and update the optimization result S as the current linear factor s and reset the target value to the score when the score is less than or equal to the target value, and finally calculate f. S/n ² is the third-order coefficient k ₂ .

Images