JPH0245886A - Method for measuring edge position - Google Patents

Abstract

PURPOSE:To make a measurement error smaller, and simultaneously, to reduce a calculating quantity by making coordinates to give an extreme value in a regression curve obtained in causing the primary differentiation of a density to regress to a normal distribution into an edge position. CONSTITUTION:The primary differentiation of an edge vicinity is made to regress to a normal distribution curve. Namely, f' is made to regress to f'(u)=f'peXp{-(u-un)<2>/(2sigma<2>)}. Here, f'p is the extreme value of the regression curve, sigmais a standard deviation of an expansion, and un is a position to give the extreme value, namely, the edge position. When a minimum square method is applied, the edge position un by the normal distribution can be obtained. Namely, an image (f) is inputted from a TV camera. The primary differentiation f' is calculated from a density (f). The f' is made to regress to the normal distribution curve.

Description

[Detailed Description of the Invention] (Industrial Application Field) This invention samples an analog image signal output from a TV camera into a plurality of pixels using an A/D converter, and The present invention relates to a method of highly accurately measuring the position of an edge in a digital image obtained by quantizing the image and inputting it into a computer's memory.

[Conventional technology]

Edge detection in an image is one of the main feature extraction methods (Image Processing Handbook Editorial Committee: “Image Processing Handbook”, +1p, 277-308. Shokodo, 1
187) (Reference 1), an important application of detected edges is indirect stereo viewing using a monocular (Yoshiaki Shirai:
"Computer Vision", pp. 83-65, Shokodo, 1980) (Reference 2), Binocular Stereo Vision (Y
,5eki:”5tereo matchingbas
ed on edgesegmants”, Trans.
, IEC:E Japan Vol, E69. No, 5
．． pp.

666-674.1986) (Reference 3), or Muqin stereo vision (Xu, Asada, Tsuji; “Moving stereo by hierarchical control”, IEICE Theory, Vol. J69-D,
No, 11. pp.

1765-1733.1986) (Reference 4)
There are position measurements such as Normally, image processing systems that perform these measurements use images sampled into pixels (vixels) of about 256 or 512 square pixels, and the accuracy is equivalent to one pixel.

In measurement technology, the issue of high precision is inevitable in expanding the range of application, and up until now, the tracking system for small animals (P, Machuca and A, L,
rt:Finding edges in noise
y 5cenes”, IEEE Trans.

PAMI, Vol. 3. No, 1. pp, 103-11
1.1981) (Reference 5), Position measurement of objects on the earth's surface from an aircraft (A, J.

Tabatabai and O, R, Mitchell
l:”Edge 1 location.

5ubpixel values in digita
l imagery”, IEEE Tr-ans, PA
Ml, Vol, 6. No, 2. pp, 188-201.
1984) (Reference 6), IC positioning (Suzuki, Yasushi, Fujii, Nakamura: “High-precision visual algorithm for surface mounters”, Proceedings of the 1986 Society for Precision Engineering Spring Conference).

pp, 109-110.1986) (Reference 7)
, Measuring parameters of rigid bodies such as robot hands (J
, Fangand, T., S., Huang: “Some
experiments on estimation
g the 3-D motion parameters
rs of a rigid body from tw
o Consecutive image frame
s”, IEEE Trans-ns, PAMI, Vol. 6
．． No, 5. pp, 545-554.1984) (
Reference 8), Analysis of airplanes on runways (P, J, Ma
cVicar-Whelan and T, 0. Bin
ford:”Intensity discon-ti
nullity 1 location to 5ubp
ixel precision, Pr-oc
, 7th Int, Joint Conf, Artif
, Intel, pp. 752-754 (1981
) ) (Reference 9), research is being conducted to improve the accuracy of edge position measurement to subpixel levels for various purposes. These methods are broadly classified into methods using concentration, methods using first-order differentiation, and methods using second-order differentiation.

The method based on density determines a virtual edge pattern that can be optimally fitted to the original image for the product moments of densities from 1st to 3rd order. The first-order differential method uses the edge position as the center of gravity of the first-order differentials of all pixels forming the edge, and the other uses the peak of the regression curve obtained by fitting the first-order differentials of three consecutive pixels near the peak to a parabola. There are some edge positions. Furthermore, a kind of second-order differential is obtained using a special filter, and the edge position is defined as the zero crossing when the second-order differentials of two adjacent pixels whose polarities are reversed are interpolated into a straight line.

These conventional methods will be explained below.

[Method using product moment of concentration] (References (5),
(See (6)) When we experimentally investigate the density of pixels on a path that crosses edges, we obtain data like the white circles in Figure 3. On the other hand, consider a virtual edge whose density changes stepwise from fl to f2 at the coordinate u1. Let the density of a pixel at coordinate U be f (u), and for n pixels including an edge, the product moment ml with respect to f (u) from the first order to the third order is defined as follows. On the other hand, in the virtual edge, if the ratio occupied by the density f, , f2 in the above n pixels is P8°P2,
m5 is m+-7f% + P2 f2', i-1,2
, 3"...(2). Solve Pl by combining equations (1) and (2), and get Pl=0.5 + o, ss"7n亙n-・-(3). However, S is S= (m3”21n+3-3m1l112)/(m
2-1nl')'°5...-(4). Therefore, the transition position u, Il of the virtual edge due to the product moments is uIIls u, + PIn
``'' (5) gives sub-pixel precision.

[Method based on center of gravity of first-order differential] If we take the first-order differential, that is, the rate of change in density, for the broken line pattern in FIG. 3, rises and falls occur before and after the edge, as shown by the broken line in FIG. 4. If the edge position is at the center of the weight of the first-order differential, it can be found by calculating the weighted average. That is, 1 of the pixel at coordinate U
If the derivative is f' (u), the edge position uq due to the center of gravity is uq - Σuf'(u)/Σf' (u)
--------(6). However, in order to exclude noise at areas other than edges, it is necessary to add pixels for which f'(u) is equal to or greater than threshold value f1 near edges. f' is 1/1 of the maximum value of fo
0, f' (u) - f', and f' (
Replace with u). Note that fo is a 5obel operator (R, 0, Duda and P, E, Har
t:”Pattern classificatio
"n andscene analysis", Wile
y, New York, pp, 271.1973) (Reference 10) as well as f'(ul, 5) -f(u+1)-f(u)
"" (7) (Reference 7), f'(u) = 0.5 (f(u+1)-f(u
-1)) ... = (8) There is a method (Reference 8).

[Method using parabolic regression of first-order differential] (References (8)
) When looking at the first derivative, we note that several pixels near the peak can be approximated to a parabola like the solid line in Figure 4, and assume that the edge position is at the peak position of the regression parabola.

Note that since the first-order differential in FIG. 4 uses equation (7), it is shifted by 0.5 pixel. If the pixel coordinate with the largest first-order differential in the sample data is Uc, the edge position u9 obtained by parabolic regression is given as follows (9).

[Method using linear interpolation for two orders] (Reference (9)
(See) If we take the quadratic component of the broken line pattern in FIG. 4, a positive maximum, a zero crossing, and a negative maximum occur when passing an edge, as shown by the broken line in FIG. This method is based on the fact that the edge is at the zero crossing of a second order number, and the position of the zero crossing is determined by interpolating the second order number of two pixels whose polarity is reversed into a straight line.

The coordinates of two pixels whose sign is reversed for the second order are ud+
ud”, and each second-order number is f”(Ua),
f'' (ud+1), the edge position ut by linear interpolation is ut = ud+lf''(ua)l/(If''(ud)
DI f”(ud+1)l)...(lO) is given.

Whelan & Binford use the following operator as f''.If 2h◆1 is the length of one side of the square operator, then the center pixel (u6゜ve
), where ulmu, −h, u, −h+1.・畢−・・−, u, +
hV(-V@-h*v,-h+1+...'”'+V@”
It is h. In addition, 2G operators (W, L, Grimson: “Computer
tionalexperiments with a
feature based 5tereo alg-
orithm”, IEEE Trans, PAMI, v
ol, 7. No, 1. pp. 17-34 (1985)
) (Reference document 11) will also be considered using a typical filter size w=4.

This method is mathematically the same as parabolic regression of the first derivative, but different operator patterns yield different results.

[Density at edge portion and its first derivative] In the conventional method described above, even if an object where the density suddenly changes, such as a step edge, is photographed, the pixel is affected by factors such as defocus and response delay specific to the horizontal scanning direction. As a result, the concentration changes gradually. Therefore, typical edges, namely,
Examine the density change pattern for horizontal edges where the density decreases in the vertical scanning direction. The TV camera used is CCD
The image signal from the TV camera is sampled into pixels of 256 degrees horizontally and 240 vertically by an A/D converter, and the density of each pixel is 0, 1, .・・・・・・、
The image is quantized to 127 levels and stored in the image memory.

By moving the TV camera in a direction perpendicular to the edge 8 times by 0° and 25 pixels, and observing the change in density of a specific pixel, the edge image at a 0.25 pixel value can be determined. FIG. 6 shows the results of experiments conducted on the above edges in this manner. ■ indicates when the image is in focus and stopped down appropriately. On the other hand, the circle mark indicates that the subject is in focus, but the aperture is fully open and the focus is largely out of focus. Since the density continuously changes depending on the sub-vixel position of the edge, it is thought that defocus is reflected. In order to make the concentration change more understandable, the first derivative of FIG. 6 was calculated using equation (7), and the results are shown in FIG. From this figure, it can be seen that the horizontal edge has a symmetrical normal distribution shape, and in the case of the black mark, the width of the spread is extremely narrow at 2 pixel diameter.

[Comparison of various edge position measurement methods] For each type of step edge, move the position 6 times by approximately 0.1 pixel, determine the edge position each time using the various measurement methods described above, and compare it with the set value. Examine the systematic errors that occur. FIG. 8 shows experimental results regarding horizontal edges where the reflectance decreases stepwise in the vertical scanning direction. Note that the characteristics were exactly the same for horizontal edges that increase in the vertical scanning direction. FIGS. 8(a) to 8(e) show the method using the product moment of concentration and the method using the center of gravity for the first order, respectively.

Method by parabolic regression of first order, and Whelan
This is a method using a linear regression of two orders using the &Binford operator and the 2G operator. In the steep edge image shown by the mark 1 in Fig. 8, systematic errors that change trigonometrically are noticeable, with a maximum of 0.2 pixels for the parabolic regression method and a maximum of 0.2 pixels for the moment method and the linear regression method using the Whelan & Binford operator. It becomes 0.15 pixels. The remaining centroid method and linear regression method using the 2G operator have a maximum of 0.
1 pixel and standard deviation of 0°06 pixels, which is relatively small and excellent.

On the other hand, for the circle marked with large defocus, the systematic error is reduced by about half in both methods.

Next, we compare the performance of each method from the viewpoint of computational complexity.

In the product moment method, it is necessary to calculate first-order, second-order, and third-order product moments for a pixel range that sufficiently includes the edge image, that is, for a range of 15 pixels in the case of the above experiment. In addition, it is necessary to calculate the irrational function, that is, calculate the square root, twice, resulting in a large amount of calculation.

Similar to the product-moment method, the centroid method requires subtraction using a threshold value for the first order of pixels in a wide range that includes enough edge images, and it is necessary to judge whether or not to use it in the centroid calculation, which requires a large amount of calculation. .

The parabolic regression method requires only the least amount of calculation, as it simply substitutes the first order obtained for three pixels into one rational function.

In the linear regression method for the second order using Whelan &Binford's operator, unless the length of one side of the operator is increased to about 15 pixels, the process will be the same as the parabolic regression method for the first order. In operator calculation for 15 pixels, it is necessary to perform a huge amount of product-sum calculations as many as 255 (=15×15) times per pixel.

The linear regression method for the second order using the 2G operator is also
Similar to the lan & Binford operator, it requires a huge amount of product-sum operations.

[Problem to be solved by the invention]

As mentioned above, all conventional methods have the following drawbacks. That is, the product-moment method has a large systematic error in edge position measurement values and requires a large amount of calculation. The center of gravity method requires a large amount of calculation. The parabolic regression method has large systematic errors. Whelan &amp;
The method using Binford's operator has a large systematic error and requires a large amount of calculation.

The method using 2G operators requires a large amount of calculation.

SUMMARY OF THE INVENTION An object of the present invention is to provide an edge position measurement method that eliminates the above-mentioned drawbacks, has small measurement errors, and requires a small amount of calculation.

[Means to solve the problem]

The edge position measuring method according to the present invention is based on the density f of a pixel at a coordinate U in a specific direction with respect to a digital image.
From (u), the differential step of calculating the first-order molecule '(u) of the concentration and the distribution of the first-order molecule '(u) are calculated by f'P
This process consists of a regression step in which a constant un is determined by calculating regression to a normal distribution function with + un + O as a constant.

[Effect]

In this invention, the first-order density is regressed to a normal distribution, and the coordinates that give the extreme value in the obtained regression curve are defined as edge positions.

〔Example〕

This invention focuses on the fact that the first-order portion forms a normal distribution near the edge, and regresses the first-order portion near the edge to a normal distribution curve. In other words, fo is f'(u) = f'pexp (-(u-un)2/
(2Q2))...Return to (12). Here, f' is the maximum value of the regression curve, σ is the standard deviation of the spread, and un is the position giving the maximum value, that is, the edge position. If we take the natural logarithm of equation (12) and apply the least squares method, we can find the edge position un according to the normal distribution.
is calculated as un... (13). Here, 8, ■Σu2Σu3- ΣU Σu', A2-rΣu
4-(Σu2)2A3-ΣUΣu2-rΣu3. B,
−ΣUΣu3−(Σu2)2B2−ΣUΣu2−rΣu
3. B, , -rΣu2-(Σu)2. r is the number of pixels on which regression calculation is performed, and can be arbitrarily selected. Σ is calculated for r pixels before and after the pixel uc with the largest first-order component. In particular, if r==3, un in equation (13) simply becomes un-UC (14).

In this method, if the logarithm calculation values are tabulated in advance for the values that may be given as fo, the processing time of the logarithm calculation can be ignored, and the first-order component is added or subtracted for an appropriate number of pixels before and after the maximum pixel. All you need is multiplication and division,
Can be processed at high speed.

An embodiment of the present invention shown in FIG. 1 is an arrangement of this method according to processing steps. 1 to 4 respectively from 1st to 4th
This is called the fourth step.

As a preliminary step, in the first step, a logarithmic calculation is performed on a value X that may be given as a first order number, and the name of the calculated value is arranged as, for example, LN. Array L
For the number N, X is a natural number 1.2. ......, 127
In that case, just use X itself. Furthermore, if X is given in steps of 0.1, 10x may be used. This processing is to avoid the problem that logarithmic calculations require significantly more calculation time than addition, subtraction, multiplication, and division.

Next, the actual process of determining the edge position is performed.

In the second step, the image f from the TV left camera is input. Originally, an image is created manually in a two-dimensional array, but it is written as f (u) to mean the density of a pixel at coordinates U in a specific direction in which the edge position is to be determined.

In the third step (differentiation step in the claims), the first-order molecule ° is calculated from the concentration f. In this example, the first order calculation formula of equation (7) is used. In this case, the coordinate U° giving the first order is 0.5 different from the sampling coordinate U of the original image.
Pixels are misaligned.

In the fourth step (regression step), fo is regressed to the normal distribution curve. In this example, the number r of pixels targeted for regression calculation is 3.
Equation (14) corresponding to the case of pixels Equation % Regarding this measurement method, experiments similar to those of the conventional method were conducted. The results are shown in FIG. From this figure, it can be seen that the systematic error of this method is comparable to that of the centroid method, which has the smallest error among the conventional methods.

In addition, the amount of calculation of this method in the actual process is comparable to that of the parabolic regression method, which was the least of the conventional methods.
This method suffers from both measurement accuracy and computational complexity.

Note that the systematic error changes trigonometrically, and its maximum value decreases in proportion to the width of the density change region of the edge image. and,
When the edge position is at the center of each pixel or at the boundary between two adjacent pixels; By taking these characteristics into consideration, systematic errors can be corrected. When this correction is made for the normal distribution regression method shown in Figure 2, the maximum value of the systematic error is 0.0.
The number of pixels decreases to about 5 pixels.

(Effects of the Invention) As explained in detail above, the present invention provides a digital image with the density f (u
) to calculate the first-order molecule '(u) of the concentration, and the distribution of the first-order molecule '(u) as f'
Since the regression process consists of calculating the regression to a normal distribution function with p + u n and σ as constants to obtain the constant un, the edge position in the digital image can be measured with high precision, and the corresponding edge can be calculated with high accuracy. A three-dimensional position in real space can be determined with high precision.

[Brief explanation of the drawing]

Figure 1 is a flowchart for explaining the invention in detail;
The figure is a diagram showing the edge position measurement results in the method of this invention, FIG. 3 is a schematic diagram showing the change in concentration at the edge part,
Figure 4 is a diagram showing the first-order density, Figure 5 is a diagram showing the second-order density, Figure 6 is a diagram showing experimental results of density changes at actual horizontal edges, and Figure 7 is a diagram showing the density changes in the horizontal edge. 6 About Figure 1
FIG. 8 is a diagram showing the result of calculating the order, and FIG. 8 is a diagram showing the edge position measurement result in the conventional method. In the figures, 1 to 4 are the first to fourth process stories (pixels), respectively. Figure (CI) Travel distance (mm) (e) Travel distance (Tm)

Claims (1)

[Scope of Claims] Regarding a digital image, a differentiation step of calculating a first-order differential of density f'(u) from the density f(u) of a pixel at a coordinate u in a specific direction; u) distribution, f'
Let _p, U_n, σ be constants (u-u_n)^2 f'(u)=f'_pexp[-(u-u_n)^2/
2σ^2] A method for measuring an edge position comprising a regression step of calculating a regression to a normal distribution function of [2σ^2] to obtain a constant u_n.