CN104764712A - Method for detecting quality of inner wall of via hole of PCB - Google Patents

Method for detecting quality of inner wall of via hole of PCB Download PDF

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CN104764712A
CN104764712A CN201510214044.9A CN201510214044A CN104764712A CN 104764712 A CN104764712 A CN 104764712A CN 201510214044 A CN201510214044 A CN 201510214044A CN 104764712 A CN104764712 A CN 104764712A
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CN104764712B (en
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刘恺
杨力帆
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ZHEJIANG OULONG ELECTRIC Co Ltd
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Zhejiang University of Technology ZJUT
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Abstract

The invention provides a method for detecting the quality of the inner wall of a via hole of a PCB. The method includes the steps that step101, the PCB to be detected and a standard template are acquired; step102, an original image of the PCB to be detected and an original image of the standard template are collected through a terahertz wave imaging technology; step103, thresholding and denoising are performed on the original image of the PCB to be detected and the original image of the standard template to obtain a recovered image to be detected and a recovered image of the standard template; step104, characteristic functions of the recovered images are calculated to obtain the circle center and diameter information of the via hole; step105, template matching operation is performed on the characteristic information of the recovered image of the standard template and the characteristic information of the recovered image to be detected; step106, if the matching operation result is within a production permissible error range, it is displayed that the PCB is detected to be qualified; step107, if the matching operation result is beyond the production permissible error range, it is displayed that the PCB is detected to be unqualified. By means of the method, the quality of the inner wall of the via hole is effectively detected, and the product quality of the PCB is guaranteed.

Description

A kind of detection method of PCB via hole inwall quality
Technical field
The present invention relates to printed circuit board (PCB) (Printed Circuit Board, PCB) detection method field, especially a kind of detection method of PCB via hole inwall quality.
Background technology
Printed circuit board (PCB) (Printed Circuit Board, PCB) carries the main platform of various electronic component as one, has undertaken connection electronic component, makes them really become the effect of a complete functional module.It, as the indispensable basic component part of any electronic product, creates tremendous influence to the electronic industry of the world today.Along with the develop rapidly of electronics industry, the requirement of people to product quality, stable performance is more and more higher, and therefore the detection means of PCB becomes the problem being badly in need of research.
In the existing detection technique that PCB produces, general use flying needle or particular manufacturing craft detect the conducting electrical specification of PCB, be placed with automatic optics inspection (Automatic Optic Inspection, AOI) equipment Inspection PCB paste solder printing, element and the surface quality of lines via hole.But at present not for the detection means fast and effectively of PCB via hole (via) inner wall hole quality (comprising the aspects such as the crude degree of hole wall, hole wall hole integrity degree and hole thickness).And along with the develop rapidly of electronic technology, the frequency of transmission and processing signals is more and more higher, PCB via hole inwall quality directly has influence on the consistance of its conduction, amplitude-frequency phase-frequency characteristic and product, thus in the production run of pcb board, detect that the defect of via hole inwall quality is one and significantly works, to be widely used space in PCB manufacturing enterprise.
Existing domestic and international PCB manufacturing enterprise mainly relies on AOI detection system to carry out for the quality testing of PCB via hole [6-8].Usual employing CCD (Charge-coupled Device) video camera or CIS (Contact ImageSensor) absorb detected image, be translated into digital signal, computer hardware and software engineering is adopted to process image digital signal again, thus obtain required target image characteristics value, and the several functions such as implementation pattern identification on this basis, coordinate calculating, intensity profile figure [9-11].According to the difference of measured target, the testing result of output can be size or the Form and position error of testee, also can be the surface integrity etc. of testee.
But, existing AOI equipment realizes the detection of PCB via hole and there is obvious shortcoming, AOI equipment can only detect the place that direct projection light can arrive, thus only can detect aperture shape, diameter, the attributes such as position, but for hole inner wall defect, (such as hole inwall has tiny burr, hole inwall has screw thread, and the holeization brought thus is incomplete, the problems such as hole unqualified thickness lattice) then cannot accomplish effective detection, these defects directly affect the following process quality of pcb board (as pad bubbles, weld not congruent), and the quality of PCB finished product is (as impedance operator, amplitude versus frequency characte etc.).The general hole quality adopting process control to ensure PCB in current PC B production testing process, and do not have specialized equipment equipment can detect hole inwall.
Summary of the invention
In order to overcome the deficiency that cannot realize detecting, cannot ensureing to via hole inwall quality PCB product quality of existing PCB detection mode, the invention provides the detection method of a kind of effective realization to the detection of via hole inwall quality, the PCB via hole inwall quality of guarantee PCB product quality.
The technical solution adopted for the present invention to solve the technical problems is:
A detection method for PCB via hole inwall quality, described detection method comprises the steps:
Step 101: obtain PCB to be measured and standard form;
Step 102: use THz wave imaging technique to gather PCB original image to be measured and standard form original image;
Step 103: carry out thresholding, denoising to PCB original image to be measured and standard form original image, obtains restored image to be measured and standard form restored image;
Step 104: the fundamental function calculating standard form restored image and restored image to be measured respectively, obtain the via hole center of circle and aperture information, process is as follows:
If point on arbitrfary point P (r, θ) and viewing plane on the circular hole of radius a 2 distance L, hole and sightingpiston distance Z, terahertz pulse is Gaussian in the distribution of time domain and spatial domain, and incident wave pulse width is T, center angular frequency ω, wave number k=2 π/λ, then
During plane wave incidence, incident wave light field is described as
Transmitted wave light field is described as consideration light is the travel-time from P to Q, gets when being brought in U by E c is the light velocity, both
Distribution of light intensity I=|U| 2;
The standard form restored image hole circle heart and radius data obtain from the PCB file of standard form; PCB restored image 8-neighborhood contours extract method to be measured obtains bore edges profile, then calculates central coordinate of circle and radius with least square fitting;
Step 105: the standard form restored image of gained and the characteristic information of restored image to be measured are carried out template matches computing, and process is as follows:
First carrying out image registration, both by carrying out coordinate transform to PCB aperture heart coordinate, making restored image coordinate identical with standard form coordinate;
For the point on random two-dimensional image, all transformation matrix can be passed through G = g 11 g 12 g 13 g 21 g 22 g 23 0 0 1 Realize the geometric transformation to it, both x 2 y 2 1 = G x 1 y 1 1 , Wherein, submatrix g 11 g 12 g 21 g 22 Achieve the rotational transform to impact point, submatrix g 13 g 23 Realize the translation transformation to impact point;
Because one group of respective coordinates point determines 2 equations, therefore by minimum three groups of respective coordinates point determination transformation matrix G;
Be analyzed for reference point standard form and each hole transmitted field field strength distribution of recovery template to be measured with each hole heart coordinate, draw the field intensity difference on arbitrary coordinate point;
Step 106: if matching operation result is within the scope of production permissible error, then show detection qualified;
Step 107: if matching operation result exceeds produce permissible error scope, then show detection defective.
Further, in described step 107, export defect area information, the enlarged image in this region, containing the difference of this area coordinate parameter, this coordinate position actual difference, this defect area region transmission coefficient and standard form that field intensity and standard form detected, is presented confession reviewer according to the coordinate parameters provided and does final judgement by described defect area packets of information on a display screen.
Further again, in described step 103, adopt process of iteration to carry out carrying out image threshold segmentation.
Further, in described step 103, described denoising adopts Wavelet Denoising Method function: process is as follows: the first, signal is carried out wavelet transformation, the second, signal and noise different conditions are spatially utilized, carry out signal to wavelet coefficient to cut out, noise is removed from signal, the 3rd, utilize restructing algorithm also original image.
In step 104, the process of 8-neighborhood contours extract method is as follows:
For a secondary bianry image, if background pixel is black, object is white, for black background pixel a certain in image, if be all background pixel point in its 8 neighborhoods, then this point is set to white, all pixels of traversing graph picture just can complete the extraction to all circular hole profiles;
The Fitting Calculation of central coordinate of circle radius is completed by least square method:
For the N group data (x on edge contour i, y i), if central coordinate of circle P (A, B), radius is R, then equation of a circle is written as (x-A) 2+ (y-B) 2=R 2
Note residuals squares sum functions is
S = Σ i = 1 N [ ( x i - A ) 2 + ( y i - B ) 2 - R 2 ] 2 , Have according to the principle of least square ∂ S ∂ A = ∂ S ∂ B = ∂ S ∂ R = 0 .
Order x n y m ‾ = Σ i = 1 N ( x i n y i m ) / N , Solve central coordinate of circle and radius is:
A = ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 3 ‾ - xy 2 ‾ ) [ ( y ‾ ) 2 - y 2 ‾ ] - ( x 2 ‾ y ‾ + y ‾ y 2 ‾ - x 2 y ‾ - y 3 ‾ ) ( x ‾ y ‾ - xy ‾ ) 2 [ ( x ‾ ) 2 - x 2 ‾ ] [ ( y ‾ ) 2 - y 2 ‾ ] - 2 ( x ‾ y ‾ - xy ‾ ) 2 B = ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 2 y ‾ - y 3 ‾ ) [ ( x ‾ ) 2 - x 2 ‾ ] - ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 3 ‾ - x y 2 ‾ ) ( x ‾ y ‾ - xy ‾ ) 2 [ ( x ‾ ) 2 - x 2 ‾ ] [ ( y ‾ ) 2 - y 2 ‾ ] - 2 ( x ‾ y ‾ - xy ‾ ) 2 R = A 2 - 2 x ‾ A + B 2 - 2 y ‾ B + x 2 ‾ + y 2 ‾
Technical conceive of the present invention is: as shown in Figure 1, and the electromagnetic radiation that frequency is in 0.1THz-10THz is called as THz wave, and it is in the region of microwave to infrared ray transition.Have the characteristic of microwave and optics concurrently.THz wave self has some unique characteristic.
(1) transient state.The typical pulse-widths of THz pulse, can easily to the research of various material line time resolved spectroscopy in picosecond level.
(2) broadband property.Terahertz pulse source only comprises the electromagnetic oscillation in several cycles usually, and individual pulse frequency band can cover the scope of GHz to tens THz, is convenient to inner analysis substance spectra characteristic on a large scale.
(3) coherence.The coherence of THz comes from its relevant generation mechanism.THz coherent measurement technology directly can measure amplitude and the phase place of electric field, thus extracts the optical parametric such as refractive index, absorption coefficient, extinction coefficient, specific inductive capacity of sample easily.
(4) low energy.The energy of THz photon only has milli electron-volt, compared with X ray, can not destroy detected material because of photoionization.
(5) absorption, reflection characteristic.THz ripple has stronger penetrability to a lot of nonmetal, non-polar materials, and pottery, timber, plastic or other material are transparent concerning THz ripple; And water has extremely strong absorbability to THz ripple, in addition, metal material has strong reflection to THz ripple.Absorb reflection characteristic based on THz ripple to the difference of different materials, and the structure and material of pcb board is formed, we think, THz ripple is kind of the wave source be well suited for for the quality testing of PCB via hole.
THz wave can be the same with optical imagery as microwave, as the radiation source of imaging, so the imaging mode that two spectral ranges adopt can be used for reference in THz imaging technology easily, form novel imaging mode, supplement as the effective of other imaging modes.The ultimate principle of THz wave imaging is: utilize the THz wave of known waveform as imaging ray, the space distribution of sample complex permittivity is contained through the Terahertz wave intensity of Imaged samples (or from sample reflection) and phase place, the intensity of terahertz electromagnetic wave of transmission (or reflection) and the two-dimensional signal of phase place are recorded, through suitable digital processing and spectrum analysis, the THz wave image of sample just can be obtained.
The form producing and detect mechanism according to THz wave can be divided into Pulse Imageing and continuous wave imaging.When THz wave imaging is applied to the detection of object defect, due to the defect of interior of articles or damage to the absorption of THz wave different and edge to the scattering effect of terahertz light, the intensity distributions of Terahertz electric field can be affected, the light and shade Terahertz image being reacted to object being shown as image is different, namely corresponding intensity is different, releases the position at interior of articles defect or damage place accordingly.
By the detection of THz wave imaging technique application with PCB via hole quality, in conjunction with digital image processing techniques, standard component Feature Extraction Technology and corresponding template matching algorithm, build the PCB via hole inwall quality detecting system that is different from existing equipment, make up the defect of existing detection technique.
Beneficial effect of the present invention is mainly manifested in: the absorption characteristic different to THz wave according to metal material on PCB and nonmetallic materials, is incorporated into THz wave imaging technique in the quality testing of PCB via hole.The digital image processing techniques such as combining image Threshold segmentation, Wavelet Denoising Method and characteristic matching method, achieve the detection fast and effectively of PCB being crossed to hole surface and inwall quality, particularly the incomplete PCB via hole of aperturesization detects, and compensate for the deficiency of existing checkout equipment.This patent to ensureing PCB product quality, stability of enhancing product performance, realize the raising of PCB manufacturing enterprise profit and have extremely important meaning.
Accompanying drawing explanation
Fig. 1 is the schematic diagram of THz wave.
Fig. 2 is the block diagram of pcb board via hole detection system structure.
Fig. 3 is the process flow diagram of the detection method of PCB via hole inwall quality.
Embodiment
Below in conjunction with accompanying drawing, the invention will be further described.
With reference to Fig. 1 ~ Fig. 3, a kind of detection method of PCB via hole inwall quality, described detection method comprises the steps:
Step 101: obtain PCB to be measured and standard form;
Step 102: use THz wave imaging technique to gather PCB original image to be measured and standard form original image;
Step 103: carry out thresholding, denoising to PCB original image to be measured and standard form original image, obtains restored image to be measured and standard form restored image;
Step 104: the fundamental function calculating standard form restored image and restored image to be measured respectively, obtain the via hole center of circle and aperture information, process is as follows:
The circular hole of radius a is put point on P (r, θ) and viewing plane 2 distance L, hole and sightingpiston distance Z, if terahertz pulse is Gaussian in the distribution of time domain and spatial domain, incident wave pulse width is T, center angular frequency ω, wave number k=2 π/λ,
During plane wave incidence, incident wave light field describes
Transmitted wave light field describes light is the travel-time from P to Q, gets when being brought in U by E c is the light velocity, both
Distribution of light intensity I=|U| 2;
The standard form restored image hole circle heart and radius data obtain from the PCB file of standard form; PCB restored image 8-neighborhood contours extract method to be measured obtains bore edges profile, then calculates central coordinate of circle and radius with least square fitting;
Step 105: the standard form restored image of gained and the characteristic information of restored image to be measured are carried out template matches computing, and process is as follows:
First carrying out image registration, both by carrying out coordinate transform to PCB aperture heart coordinate, making restored image coordinate identical with standard form coordinate;
For the point on random two-dimensional image, all transformation matrix can be passed through G = g 11 g 12 g 13 g 21 g 22 g 23 0 0 1 Realize the geometric transformation to it, both x 2 y 2 1 = G x 1 y 1 1 . Wherein, submatrix g 11 g 12 g 21 g 22 Achieve the rotational transform to impact point, submatrix g 13 g 23 Realize the translation transformation to impact point.
Because one group of respective coordinates point determines 2 equations, therefore by minimum three groups of respective coordinates point determination transformation matrix G;
Be analyzed for reference point standard form and each hole transmitted field field strength distribution of recovery template to be measured with each hole heart coordinate, draw the field intensity difference on arbitrary coordinate point;
Step 106: if matching operation result is within the scope of production permissible error, then show detection qualified;
Step 107: if matching operation result exceeds produce permissible error scope, then show detection defective.
Further, in described step 107, export defect area information, the enlarged image in this region, containing the difference of this area coordinate parameter, this coordinate position actual difference, this defect area region transmission coefficient and standard form that field intensity and standard form detected, is presented confession reviewer according to the coordinate parameters provided and does final judgement by described defect area packets of information on a display screen.
Further again, in described step 103, adopt process of iteration to carry out carrying out image threshold segmentation.
Further, in described step 103, described denoising adopts Wavelet Denoising Method function: process is as follows: the first, signal is carried out wavelet transformation, the second, signal and noise different conditions are spatially utilized, carry out signal to wavelet coefficient to cut out, noise is removed from signal, the 3rd, utilize restructing algorithm also original image.
In step 104, the process of 8-neighborhood contours extract method is as follows:
For a secondary bianry image, if background pixel is black, object is white, for black background pixel a certain in image, if be all background pixel point in its 8 neighborhoods, then this point is set to white, all pixels of traversing graph picture just can complete the extraction to all circular hole profiles;
The Fitting Calculation of central coordinate of circle radius is completed by least square method:
For the N group data (x on edge contour i, y i), if central coordinate of circle P (A, B), radius is R, then equation of a circle is written as (x-A) 2+ (y-B) 2=R 2
Note residuals squares sum functions is
S = Σ i = 1 N [ ( x i - A ) 2 + ( y i - B ) 2 - R 2 ] 2 , Have according to the principle of least square ∂ S ∂ A = ∂ S ∂ B = ∂ S ∂ R = 0 .
Order x n y m ‾ = Σ i = 1 N ( x i m y i n ) / N , Solve central coordinate of circle and radius is:
A = ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 3 ‾ - xy 2 ‾ ) [ ( y ‾ ) 2 - y 2 ‾ ] - ( x 2 ‾ y ‾ + y ‾ y 2 ‾ - x 2 y ‾ - y 3 ‾ ) ( x ‾ y ‾ - xy ‾ ) 2 [ ( x ‾ ) 2 - x 2 ‾ ] [ ( y ‾ ) 2 - y 2 ‾ ] - 2 ( x ‾ y ‾ - xy ‾ ) 2 B = ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 2 y ‾ - y 3 ‾ ) [ ( x ‾ ) 2 - x 2 ‾ ] - ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 3 ‾ - x y 2 ‾ ) ( x ‾ y ‾ - xy ‾ ) 2 [ ( x ‾ ) 2 - x 2 ‾ ] [ ( y ‾ ) 2 - y 2 ‾ ] - 2 ( x ‾ y ‾ - xy ‾ ) 2 R = A 2 - 2 x ‾ A + B 2 - 2 y ‾ B + x 2 ‾ + y 2 ‾ .
There is the features such as interference fringe, contrast is lower, image is fuzzyyer in the original image that THz wave imaging obtains.This patent processes image from algorithm, reaches stress release treatment, interference fringe, improves the object of picture contrast and enhancing image border etc.Produce in conjunction with PCB actual, enable the image after process meet the needs of PCB via hole detection.
Original image threshold division: imagethresholding is a kind of traditional the most frequently used image partition method, because of its realize simple, calculated amount is little, performance is comparatively stable and to become in Iamge Segmentation the most most widely used cutting techniques of fundamental sum.It is specially adapted to the image that target and background occupies different grey-scale scope, under many circumstances, is the Image semantic classification process of the necessity of carrying out before graphical analysis, feature extraction and pattern-recognition.Common carrying out image threshold segmentation method has Otsu, maximum entropy, process of iteration, Adaptive Thresholding etc., and this patent adopts process of iteration to carry out carrying out image threshold segmentation.
Wavelet Denoising Method function: wavelet function is as one effective time (space)/dimensional analysis algorithm, it applies the multiple research fields throughout signal and image analysis processing, three steps are divided: the first, signal is carried out wavelet transformation to PCB imaging processing, the second, signal and noise different conditions are spatially utilized, carry out signal to wavelet coefficient to cut out, noise is removed from signal, the 3rd, utilize restructing algorithm also original image.
The calculating of standard form and collection image eigenfunction, comprises following process:
First, the center of circle in hole is located and radius derivation algorithm: the core of detection system is exactly solve the quality problems detecting PCB via hole, will carry out the determination of the center of circle and radius, calculate the fundamental function of through hole on this basis for the through hole after each imaging.
Then, the via structure of field strength distribution and symmetric fundamental function: Standard PC B exists symmetry, the result of thus its THz wave imaging also has corresponding symmetry.And the PCB via hole that there is mass defect structurally must have asymmetry, also just necessarily uneven to the absorption characteristic of THz wave in via hole zones of different, this finally result in the asymmetry of the imaging results to defect PCB via hole.
Finally. template matching algorithm: template matching algorithm is ingredient important in PCB detection system, which determines accuracy rate and the degree of accuracy of detection system.This project, according to the feature of this detection system, is intended adopting traditional essential characteristic matching method to combine with relational structure matching method, to improve the accuracy rate of systems axiol-ogy.
Detection system is primarily of the generating means of (1) Terahertz light source as shown in Figure 2, (2) board under test article carrying platform, and (3) terahertz wave detector part and (4) are formed for the computing machine of Digital Image Processing.Objective table is for adjusting the relative position of pcb board and light source.Common THz wave detecting devices is pyroelectric detector.The information input computing machines such as the THz wave field strength distribution obtain checkout equipment, just can obtain the THz wave imaging results of carrying out matching detection for same standard picture after certain Digital Image Processing.

Claims (5)

1. a detection method for PCB via hole inwall quality, is characterized in that: described detection method comprises the steps:
Step 101: obtain PCB to be measured and standard form;
Step 102: use THz wave imaging technique to gather PCB original image to be measured and standard form original image;
Step 103: carry out thresholding, denoising to PCB original image to be measured and standard form original image, obtains restored image to be measured and standard form restored image;
Step 104: the fundamental function calculating standard form restored image and restored image to be measured respectively, obtain the via hole center of circle and aperture information, process is as follows:
If point on arbitrfary point P (r, θ) and viewing plane on the circular hole of radius a 2 distance L, hole and sightingpiston distance Z, terahertz pulse is Gaussian in the distribution of time domain and spatial domain, and incident wave pulse width is T, center angular frequency ω, wave number k=2 π/λ, then
During plane wave incidence, incident wave light field is described as
Transmitted wave light field is described as consideration light is the travel-time from P to Q, gets when being brought in U by E c is the light velocity, both
Distribution of light intensity I=|U| 2;
The standard form restored image hole circle heart and radius data obtain from the PCB file of standard form; PCB restored image 8-neighborhood contours extract method to be measured obtains bore edges profile, then calculates central coordinate of circle and radius with least square fitting;
Step 105: the standard form restored image of gained and the characteristic information of restored image to be measured are carried out template matches computing, and process is as follows:
First carrying out image registration, both by carrying out coordinate transform to PCB aperture heart coordinate, making restored image coordinate identical with standard form coordinate;
For the point on random two-dimensional image, all transformation matrix can be passed through G = g 11 g 12 g 13 g 21 g 22 g 23 0 0 1 Realize the geometric transformation to it, both x 2 y 2 1 = G x 1 y 1 1 , Wherein, submatrix g 11 g 12 g 21 g 22 Achieve the rotational transform to impact point, submatrix g 13 g 23 Realize the translation transformation to impact point;
Because one group of respective coordinates point determines 2 equations, therefore by minimum three groups of respective coordinates point determination transformation matrix G;
Be analyzed for reference point standard form and each hole transmitted field field strength distribution of recovery template to be measured with each hole heart coordinate, draw the field intensity difference on arbitrary coordinate point;
Step 106: if matching operation result is within the scope of production permissible error, then show detection qualified;
Step 107: if matching operation result exceeds produce permissible error scope, then show detection defective.
2. the detection method of a kind of PCB via hole inwall quality as claimed in claim 1, it is characterized in that: in described step 107, export defect area information, the enlarged image in this region, containing the difference of this area coordinate parameter, this coordinate position actual difference, this defect area region transmission coefficient and standard form that field intensity and standard form detected, is presented confession reviewer according to the coordinate parameters provided and does final judgement by described defect area packets of information on a display screen.
3. the detection method of a kind of PCB via hole inwall quality as claimed in claim 1 or 2, is characterized in that: in described step 103, adopts process of iteration to carry out carrying out image threshold segmentation.
4. the detection method of a kind of PCB via hole inwall quality as claimed in claim 1 or 2, it is characterized in that: in described step 103, described denoising adopts Wavelet Denoising Method function: process is as follows: the first, signal is carried out wavelet transformation, the second, signal and noise different conditions are spatially utilized, carry out signal to wavelet coefficient to cut out, noise is removed from signal, the 3rd, utilize restructing algorithm also original image.
5. the detection method of a kind of PCB via hole inwall quality as claimed in claim 1 or 2, it is characterized in that: in step 104, the process of 8-neighborhood contours extract method is as follows:
For a secondary bianry image, if background pixel is black, object is white, for black background pixel a certain in image, if be all background pixel point in its 8 neighborhoods, then this point is set to white, all pixels of traversing graph picture just can complete the extraction to all circular hole profiles;
The Fitting Calculation of central coordinate of circle radius is completed by least square method:
For the N group data (x on edge contour i, y i), if central coordinate of circle P (A, B), radius is R, then equation of a circle is written as (x-A) 2+ (y-B) 2=R 2
Note residuals squares sum functions is
S = Σ i = 1 N [ ( x i - A ) 2 + ( y i - B ) 2 - R 2 ] 2 , Have according to the principle of least square ∂ S ∂ A = ∂ S ∂ B = ∂ S ∂ R = 0 ;
Order x n y m ‾ = Σ i = 1 N ( x i n y i m ) / N , Solve central coordinate of circle and radius is:
A = ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 3 ‾ - xy 2 ‾ ) [ ( y ‾ ) 2 - y 2 ‾ ] - ( x 2 ‾ y ‾ + y ‾ y 2 ‾ - x 2 y ‾ - y 3 ‾ ) ( x ‾ y ‾ - xy ‾ ) 2 [ ( x ‾ ) 2 - x 2 ‾ ] [ ( y ‾ ) 2 - y 2 ‾ ] - 2 ( x ‾ y ‾ - xy ‾ ) 2 B = ( x 2 ‾ y ‾ + y ‾ y 2 ‾ - x 2 y ‾ - y 3 ‾ ) [ ( x ‾ ) 2 - x 2 ‾ ] - ( x 2 ‾ x ‾ + x ‾ y 2 ‾ - x 3 ‾ - xy 2 ‾ ) ( x ‾ y ‾ - xy ‾ ) 2 [ ( x ‾ ) 2 - x 2 ‾ ] [ ( y ‾ ) 2 - y 2 ‾ ] - 2 ( x ‾ y ‾ - xy ‾ ) 2 R = A 2 - 2 x ‾ A + B 2 - 2 y ‾ B + x 2 ‾ + y 2 ‾ .
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