CN100410964C - Acquisition and splicing method of rolling fingerprints on three sides - Google Patents

Abstract

A method for collecting and stitching three-sided scrolling fingerprints, comprising: collecting a background image; collecting sequence fingerprints; removing the background from the fingerprint image; mapping the histogram of the obtained image into a histogram of uniform distribution of all gray levels in the image; performing Gaussian processing on the image Smoothing process; calculate the threshold value for the image, and binarize the image according to the threshold value; judge whether there is fingerprint input in the image, if not, re-acquire the next image, and continue if there is; calculate the center and four boundaries of the fingerprint; repeat The above steps are to collect the next sequence fingerprint image; the block with the largest correlation between the sequence fingerprint image obtained and the sequence fingerprint image obtained last time, the center line of the block is the best splicing line; eliminate the dislocation at the splicing line; splicing Two sequence fingerprints; repeat the above steps, continuously collect and splice sequence fingerprints, until it is judged that there is no fingerprint input in a certain image, the three-sided rolling fingerprint collected by the method of the present invention is more complete and has more feature points.

Description

Acquisition and splicing method of rolling fingerprints on three sides

technical field

The invention relates to a method for collecting and splicing fingerprints, in particular to a method for collecting and splicing three-sided rolling fingerprints.

Background technique

Due to the stability and uniqueness of the fingerprint itself, the fingerprint plays a huge role in identification. Fingerprint collection can be divided into flat fingerprint collection and three-sided rolling fingerprint collection.

Planar fingerprints are usually used in civilian applications. It is mostly used in the case of 1:1 comparison, that is, for verification. When verifying, the system compares the fingerprints collected by the current printing with the fingerprints stored in the database during the original registration. The quality of the fingerprints stored in the database at the time of registration is relatively good and the comparison is 1:1, so when used In the verification technology, relatively few fingerprint feature points are required. Figure 13 shows the flat fingerprint collector and the collected flat fingerprints. Its collection process is as follows: put your finger flat on the flat fingerprint collector, collect and generate a flat fingerprint image, and the obtained fingerprint image is generally only the middle of the fingerprint. part, which contains relatively few feature points.

Three-sided rolling fingerprints are usually used in police applications. It is mostly used in the case of 1:N comparison and N is particularly large, from tens of thousands to millions, that is, for identification. Police fingerprints are usually used to compare with fingerprints obtained at the scene of a crime. For example: use a stamped fingerprint to compare 1:N with the on-site fingerprint database to find out which cases the suspect is related to. This process is called reverse investigation in criminal investigation technology; Make a 1:N comparison in the fingerprint database to find out which suspects have appeared at the scene of the case. This process is called positive investigation in criminal investigation technology. It can be seen that in the case of police use, because the on-site fingerprints are often incomplete, the quality is not high, and there is a 1:N comparison, it is required that the fingerprints taken are complete and have many fingerprint feature points. Therefore, the fingerprints used by the police must be three-sided rolling fingerprints. The existing method of collecting three-sided rolling fingerprints is to use ink printing to collect them, and then make them into fingerprint cards and store them. However, the fingerprints collected by this method are of poor quality and are easily damaged. It is also difficult to compare.

Contents of the invention

The purpose of the present invention is to provide a method for collecting and splicing three-sided scrolling fingerprints. Using a scrolling fingerprint collector, the fingers are rolled from left to right or from right to left, and a series of sequence fingerprints are collected during the rolling process, and these sequences are Fingerprint stitching, and finally generate three-sided rolling fingerprints.

In order to achieve the above object, the present invention adopts following technical scheme:

The collection and splicing method of three-sided rolling fingerprint of the present invention, comprises the following steps:

1) Collect the background image, and collect a frame of background image before the finger starts printing;

2) Collect serial fingerprints, scroll your finger on the collection window, and collect a serial fingerprint image during the scrolling process;

3) Remove the background from the fingerprint image, and subtract the sequence fingerprints collected from the background image;

4) map the histogram of the image obtained in step 3) into a histogram of uniform distribution of all gray levels in an image;

5) Gaussian smoothing is performed on the image obtained in step 4);

6) calculate the threshold value to the image obtained in step 5), and carry out binarization to the image according to the threshold value;

7) judging step 6) whether there is fingerprint input in the image of gained, if there is no fingerprint input then to step 2) to collect the next image, if there is fingerprint input then to step 8);

8) calculating the center and four boundaries of the fingerprint;

9) Repeat steps 2)-8), collect the next sequence of fingerprint images, and go to step 10) after collecting one image;

10) just go out step 9) the block of maximum correlation between the sequence fingerprint image that obtains and the sequence fingerprint image that obtains the previous time, the center line of this block is exactly the optimal splicing line;

11) Eliminate the dislocation at the splicing line, and smooth the border of the sequence fingerprint to be spliced;

12) Splicing two serial fingerprints, taking the reverse of the direction of finger scrolling as the front, taking the part in front of the splicing line for the previous sequence fingerprint, and taking the part behind the splicing line for the latter sequence fingerprint;

13) Steps 9)-12) are repeated to continuously collect and stitch serial fingerprints until it is judged that there is no fingerprint input in a certain image.

Due to the adoption of the above technical solution, the three-sided fingerprint collection and splicing method of the present invention can quickly and accurately collect three-sided rolling fingerprints, and the collected fingerprints are complete and have many feature points, which are easy to compare.

Description of drawings

Fig. 1 is the flow chart of the acquisition and splicing method of three-side rolling fingerprint of the present invention;

Fig. 2 is a frame background image collected earlier before fingers start to print in the method of the present invention;

Fig. 3 is a series of fingerprints collected in the method of the present invention—serial fingerprints;

Fig. 4 is the effect figure that fingerprint image removes the background in the method of the present invention;

Fig. 5 is the schematic diagram of doing Gaussian smoothing to image in the method of the present invention;

Fig. 6 is the distribution diagram of image gray scale after normalization processing in the method of the present invention;

Fig. 7 is a schematic diagram of fitting images in the method of the present invention;

Fig. 8 is the schematic diagram of seeking fingerprint image center and boundary in the method of the present invention;

Fig. 9 is a schematic diagram of seeking the maximum correlation block of two fingerprint images in the method of the present invention;

Fig. 10 is a schematic diagram of smoothing the stitching line boundary in the method of the present invention;

Figure 11 is a schematic diagram of sequence fingerprint mosaic in the method of the present invention;

Fig. 12 is the three-sided rolling fingerprint that a series of sequence fingerprints are spliced into in the method of the present invention;

Fig. 13 is a planar fingerprint collector and the planar fingerprints collected;

Figure 14 is a three-sided rolling fingerprint collector and a series of serial fingerprints collected and three-sided rolling fingerprints after splicing.

Detailed ways

The technical solution of the present invention will be further described below in conjunction with the accompanying drawings.

Fig. 1 is the flow chart of the collection and splicing method of three-side rolling fingerprint of the present invention, as shown in Figure 1, the collection of three-side rolling fingerprint of the present invention and splicing method comprise:

1) collect the background image, collect a frame of background image before the finger starts printing (S11);

2) Collecting serial fingerprints, the finger scrolls on the collection window, and a serial fingerprint image is collected during the scrolling process (S12);

3) removing the background from the fingerprint image, and subtracting the collected sequence fingerprint from the background image (S13);

4) mapping the histogram of the image obtained in step 3) into a histogram of uniform distribution of all gray levels in an image (S14);

5) Gaussian smoothing processing (S15) is performed on the image obtained in step 4);

6) Calculate the threshold value for the image obtained in step 5), and carry out binarization (S16) to the image according to the threshold value;

7) judging step 6) whether there is fingerprint input in the image of the gained, if there is no fingerprint input then to step 2) to collect the next image, if there is fingerprint input then to step 8) (S17);

8) calculating the center and four boundaries of the fingerprint (S18);

9) Repeat steps 2)-8), collect the next sequence of fingerprint images, and go to step 10) after collecting one image;

10) just go out step 9) the block of maximum correlation between the sequence fingerprint image that obtains and the sequence fingerprint image that obtains the previous time, the center line of this block is exactly the optimal splicing line (S19);

11) Eliminate the dislocation at the splicing line, and smooth the border of the sequence fingerprint to be spliced (S110);

12) Splicing two sequence fingerprints, with the reverse direction of the finger scrolling as the front, the previous sequence fingerprint takes the part in front of the splicing line, and the latter sequence fingerprint takes the part behind the splicing line (S111);

13) Steps 9)-12) are repeated to continuously collect and stitch serial fingerprints until it is judged that there is no fingerprint input in a certain image.

Step 1) collect the background image, Fig. 2 is a frame background image collected before the fingers start printing in the method of the present invention, the acquisition window of the fingerprint acquisition instrument will inevitably be dirty by dust, hand sweat, oil stains, etc., these factors Both will cause interference and noise when collecting fingerprint images, which will affect the clarity of fingerprint images, especially the fingerprints left when collecting the previous fingerprint will greatly interfere with the fingerprint image. Before the fingers start to print, a frame of background image is first collected, as shown in Figure 2, this frame of background image contains all the original imprints on the acquisition window, so that these imprints can be eliminated from the collected fingerprint image in the subsequent steps, and the obtained Current clear fingerprint image.

Step 2) collect sequence fingerprints, Fig. 3 is a series of fingerprints collected in the method of the present invention---sequence fingerprints, put your finger on the collection window of the fingerprint collection device, from left to right or from right to left Roll your fingers sideways, and collect a series of fingerprint images—serial fingerprints—in turn during the rolling process. When processing, process one sequence of fingerprint images collected each time.

Step 3) Remove the background of the fingerprint image. Fig. 4 is an effect diagram of removing the background of the fingerprint image in the method of the present invention, and subtract each piece of the sequence fingerprints collected from the background to remove the background. The specific calculation method is as follows: as shown in Figure 4, the fingerprint image has a length M and a width N, and the gray value of any point (i, j) of the fingerprint image is O(i, j), and the gray value of any point (i, j) of the sequence fingerprint is The value is O ₁ (i, j), and the gray value of any point (i, j) in the background image is B (i, j), then: O'(i, j)=O ₁ (i, j)-B (i, j), O'(i, j) ∈ [-255, 255]; linearly map the gray value O'(i, j) of all pixels: the maximum value of O'(i, j) is max, the minimum value is min, and the gray value after linear mapping $0(i,j)=255*\frac{o^{\′}(i,j)-\min }{\max -\min },$ O(i, j) is the gray value after removing the background. In this embodiment, the above-mentioned method is used to remove the background of the fingerprint image. The above-mentioned method is not the only method for removing the background, and those skilled in the art can use other methods to remove the background image.

Step 4) Histogram equalization. When comparing two images on a specific basis, the histogram of the image is generally firstly processed so that the histogram after the specification is a "standard histogram". The most commonly used method of prescriptive processing is histogram equalization. It uses the function b=f(a) to map the histogram of the original image into a histogram with uniform distribution of all gray levels in the image, increasing the dynamic range of the pixel gray value, thereby achieving the effect of enhancing the overall contrast of the image. For a suitable function f(*) representing the input probability density and the output probability density, f(*) is defined as follows:

$\mathrm{<span\; class="notranslate">\; pb</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; db</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; pa</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; da</span>}<span\; class="notranslate">\; \&DoubleRightArrow;</span>\mathrm{<span\; class="notranslate">\; df</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; pa</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; da</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; pb</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; )</span>$

f(*) is differentiable and df/da≥0. Define pb(b) as a constant, and P(a) as the gray distribution probability function, namely:

f(a)=(2 ^B -1)*P(a);

Step 5) Gaussian smoothing is performed on the image. Gaussian smoothing uses the Gaussian kernel to perform image smoothing. It is performed in the image space with the help of templates for neighborhood operations, so that the high-frequency components of the image in Fourier space are suppressed without affecting the low-frequency portion. Because the high-frequency components correspond to the parts of the image with large and fast-changing gray values such as the edge of the region, the filter can filter out these components to smooth the image and achieve the purpose of enhancing clarity. Its main steps are:

5.1) Roam the template in the picture, and coincide the center of the template with a certain pixel position in the picture;

5.2) Multiply the coefficient on the template with the corresponding pixel under the template;

5.3) Add all products together

5.4) Assign the sum (the output response of the template) to the pixel corresponding to the central position of the template in the figure.

Fig. 5 is a principle diagram of performing Gaussian smoothing on an image in the method of the present invention, wherein (a) is a part of an image, wherein the gray values of some pixels are marked. A 3*3 template is now assumed, as shown in (b), and the template coefficients are marked in the template. If the position of k ₀ is coincident with the pixel whose gray value is s ₀ in the figure, the output response R of the template is: R=k ₀ s ₀ +k ₁ s ₁ +...k ₈ s ₈

Assign R to the enhanced map as the gray value at point (x, y). see (c)

Do this for each pixel of the original image to get new gray values for all positions in the enhanced image. Different high-pass or low-pass effects can be obtained by assigning different values to k in the template.

Gaussian smoothing is using the Gaussian function $h(x,\mathrm{the\; y})=\frac{1}{\sqrt{2\mathrm{\&pi;}}\mathrm{\&sigma;}}e-((x^2+{\mathrm{the\; y}}^2)/2{\mathrm{\&sigma;}}^2)$ The normalized h'(x, y) is used as the neighborhood operation of the template, and the normalization process is as follows: $h^{,}(x,\mathrm{the\; y})=\frac{k_i}{\underset{i=1}{\overset{9}{\mathrm{\&Sigma;}}}{k}_i}$

In this embodiment, the above-mentioned method is used to perform Gaussian smoothing on the image. The above-mentioned method is not the only method for Gaussian smoothing. Those skilled in the art can use other methods to perform Gaussian smoothing on the image.

阈值thr应该是自适应的。现假设认为灰度值小于10的没有指纹图像输入。根据经验和试验估计б＝20，则Step 6) Calculate the threshold value on the image, and carry out binarization to the image according to the threshold value, and first calculate the threshold value before doing the binarization. A fingerprint image similar to Figure 7, any one of the double peaks is normally distributed,

Threshold thr should be adaptive. Now assume that there is no fingerprint image input with a gray value less than 10. According to experience and experiments, it is estimated that б=20, then

H: h(i), i ∈ [0, 255];

AH: ah(0)=h(0);

ah(i)=ah(i-1)+h(i);

Fig. 7 is a schematic diagram of image fitting in the method of the present invention, and a pair (x ₁ , б ₁ ), (x ₂ , б ₂ ) is obtained through fitting to minimize the fitting error. Then the threshold thr=(x ₁ +x ₂ )/2 is obtained;

The image is binarized, and for any gray value in the image, the gray value greater than or equal to the threshold is equal to 255, and the gray value smaller than the threshold is 0, that is:

I(i,j)=255 when I'(i,j)≥thr

I(i,j)=0 when I'(i,j)<thr;

In this embodiment, the above-mentioned method and parameters are used to carry out binarization processing on the image. The above-mentioned method and selected parameters are not unique, and those skilled in the art can use other methods or select different parameters to process the image. Binary processing.

Step 7) Judging whether the fingerprint is present or not, judging whether the fingerprint is present or not according to the threshold value Thr obtained in the previous step, and judging as having a fingerprint input when the gray scale is greater than the threshold value.

Step 8) Find the center and the four boundaries of the sequence fingerprint, and judge the four boundaries and the center of each frame image in the sequence fingerprint after the fingerprints gathered are binarized. Fig. 8 is the method for asking the fingerprint image center and the boundary Schematic diagram, in which 81 is a scanning line, 82 is a fingerprint image boundary, 83 is a scanning direction, and 84 is a fingerprint boundary point.

Step 10) seek optimal splicing line and splicing width, Fig. 9 is the schematic diagram that seeks the maximum correlation block of two fingerprint images in the method of the present invention, to the continuous two frame fingerprint images that gather, first seek the block of maximum correlation, set The size of the fingerprint image is M*N,

MaxCorr=Max(Corri)i=1,...n

Index=MaxCorr=Corrindex

i＝0~M-1，j＝0~N-1；Seek related:

i=0~M-1, j=0~N-1;

Find the most relevant block, the best stitching line is exactly the center line of the block, 91 is the fingerprint of the latter sequence among the figure, 92 is the largest correlation block, 93 is the largest correlation center line that is the stitching line, and 94 is the fingerprint of the previous sequence.

In this embodiment, the above method is used to find the maximum correlation block between two consecutive frames of fingerprint images. The above method is not unique, and those skilled in the art can use other methods to find the maximum correlation between two consecutive frames of fingerprint images. related blocks.

Step 11) eliminate the dislocation at the stitching line, and smooth the border of the sequence fingerprint to be stitched. Fig. 10 is a schematic diagram of smoothing the border of the stitching line in the method of the present invention. The stitching line of the sequence fingerprint is as shown in Fig. 10 The acupuncture-like shape shown in Fig. 1, the sequence fingerprint boundary is smoothed before splicing: if the point ( _xi , y _i ) is any point on the boundary of a certain fingerprint, then, $x_i^{\&prime;}=\underset{k=i-\mathrm{the\; s}}{\overset{i+\mathrm{the\; s}}{\mathrm{\&Sigma;}}}{x}_k/(2\mathrm{the\; s}+1)$ ${\mathrm{the\; y}}_i^{\&prime;}=\underset{k=i-\mathrm{the\; s}}{\overset{i+\mathrm{the\; s}}{\mathrm{\&Sigma;}}}{\mathrm{the\; y}}_k/(2\mathrm{the\; s}+1)$ s is the smoothing factor, (x' _i , y' _i ) is the smoothed boundary.

In this embodiment, the above-mentioned method is used to eliminate the dislocation at the splicing line and smooth the border of the sequence fingerprint to be spliced. The above-mentioned method is not unique, and those skilled in the art can use other methods to eliminate the splicing line. The dislocation of the sequence fingerprints to be spliced is smoothed.

Step 12) splicing two sequence fingerprints, taking the reverse of the direction of finger scrolling as the front, the front frame of the sequence fingerprint image takes the part in front of the center line (i.e. splicing line) of the largest relevant block, and the rear frame takes the part of the largest relevant block The part behind the central line (ie, splicing line), FIG. 11 is a schematic diagram of sequence fingerprint splicing in the method of the present invention.

Do continuous splicing until it is judged that the finger has left. During the entire collection process, the first fingerprint takes all the fingerprints before the center line of the largest correlation block (that is, the splicing line) of the first and second sequence fingerprints, except for the last fingerprint. In addition, for other sequence fingerprints, the partial fingerprints of the largest relevant block after the center line of the largest relevant block (that is, the splicing line) are spliced on the previous fingerprint, and the above steps are repeated until it is judged that there is no fingerprint. All the fingerprints behind the central line of the relevant block are spliced on the previous sequence fingerprint to generate a spliced three-sided rolling fingerprint. Figure 12 is a three-sided rolling fingerprint spliced by a series of sequence fingerprints in the method of the present invention.

Figure 14 shows the three-sided rolling fingerprint collector and a series of serial fingerprints collected and the spliced three-sided rolling fingerprints. It can be seen from the figure that the spliced three-sided rolling fingerprints are more complete and have more feature points than flat fingerprints .

Claims (7)

1. A collection and splicing method of rolling fingerprints on three sides, comprising the steps of: 1) Collect the background image, and collect a frame of background image before the finger starts printing; 2) Collect serial fingerprints, scroll your finger on the collection window, and collect a serial fingerprint image during the scrolling process; 3) Remove the background from the fingerprint image, and subtract the sequence fingerprints collected from the background image; 4) map the histogram of the image obtained in step 3) into a histogram of uniform distribution of all gray levels in an image; 5) Gaussian smoothing is performed on the image obtained in step 4); 6) calculate the threshold value to the image obtained in step 5), and carry out binarization to the image according to the threshold value; 7) judging step 6) whether there is fingerprint input in the image of gained, if there is no fingerprint input then to step 2) to collect the next image, if there is fingerprint input then to step 8); 8) calculating the center and four boundaries of the fingerprint; 9) Repeat steps 2)-8), collect the next sequence of fingerprint images, and go to step 10) after collecting one image; 10) find the block of maximum correlation between the sequence fingerprint image that step 9) obtains and the sequence fingerprint image that obtains the previous time, and the center line of this block is exactly the optimal splicing line; 11) Eliminate the dislocation at the splicing line, and smooth the border of the sequence fingerprint to be spliced; 12) Splicing two serial fingerprints, taking the reverse of the direction of finger scrolling as the front, taking the part in front of the splicing line for the previous sequence fingerprint, and taking the part behind the splicing line for the latter sequence fingerprint; 13) Steps 9)-12) are repeated to continuously collect and stitch serial fingerprints until it is judged that there is no fingerprint input in a certain image. 2. the acquisition of three-side rolling fingerprint as claimed in claim 1 and splicing method, it is characterized in that, described 3) the method for removing background of fingerprint image is as follows: The length of the collected original sequence fingerprint image is M and the width is N. The gray value of any point (i, j) is O(i, j), and the gray value of any point (i, j) of the actual sequence fingerprint is O ₁ (i , j), the gray value of any point (i, j) in the background image is B(i, j), then: O'(i, j)=O ₁ (i, j)-B(i, j), O'(i, j) ∈ [-255, 255], the gray value O'(i, j) of all pixels is linearly mapped: the maximum value of O'(i, j) is max, and the minimum value is min, gray value after linear mapping $0(i,j)=255*\frac{o^{\&prime;}(i,j)-\min }{\max -\min },$ O(i, j) is the gray value of the serial fingerprint after removing the background. 3. the collection of three-side rolling fingerprint as claimed in claim 1 and splicing method, it is characterized in that, described 4) histogram is mapped into the histogram that all gray scales are uniformly distributed in an image and refers to by function b=f( a), f(a)=(2 ^B -1)*P(a) to map the histogram into a histogram with uniform distribution of all gray levels in an image to increase the dynamic range of pixel gray values and enhance the overall contrast of the image . 4. the collection of three-side rolling fingerprint as claimed in claim 1 and splicing method, it is characterized in that, described 5) doing Gaussian smoothing process to image comprises the following steps: 5.1) Roam the template in the picture, and coincide the center of the template with a certain pixel position in the picture; 5.2) Multiply the coefficient on the template with the corresponding pixel under the template; 5.3) Add all products together; 5.4) Assign the sum (the output response of the template) to the pixel corresponding to the central position of the template in the figure. 5. the collection of three-side rolling fingerprint as claimed in claim 1 and splicing method, it is characterized in that, described 6) threshold value is calculated to image, and this image is carried out binarization according to threshold value and comprises: The gray value of the fingerprint image processed by Gaussian smoothing is normally distributed, using the formula ah(0) = h(0); ah(i)=ah(i-1)+h(i); h(i) is a Gaussian function, i∈[0, 255]; Fitting, find a pair of (x ₁ , δ ₁ )(x ₂ , δ ₂ ), so that the fitting error is the smallest, Calculate threshold thr=(x ₁ +x ₂ )/2; The image is binarized according to the threshold thr, and the grayscale is set to 255 for the grayscale greater than the threshold thr, and the grayscale is set to 0 for the grayscale smaller than the threshold thr. 6. the collection of three-sided rolling fingerprint as claimed in claim 1 and splicing method, it is characterized in that, described 10) the sequence fingerprint image that obtains and the block of maximum correlation between the sequence fingerprint image that obtains last time refers to: set fingerprint The size of the image is M*N, MaxCorr=Max(Corr ₁ )i=1,...n Index＝MaxCorr＝Corr _index Seek related:

i=0~M-1, j=0~N-1; The block with the largest correlation is found, and the best stitching line is the center line of the block. 7. the collection of three-sided rolling fingerprint as claimed in claim 1 and splicing method, it is characterized in that, described 11) eliminates the dislocation at splicing line place, and smoothing the sequence fingerprint boundary to be spliced refers to: setting point (x _i , y _i ) is any point on the boundary of a certain fingerprint, then $x_i^{\&prime;}=\underset{k=i-\mathrm{the\; s}}{\overset{i+\mathrm{the\; s}}{\mathrm{\&Sigma;}}}{x}_k/(2\mathrm{the\; s}+1)$ ${\mathrm{the\; y}}_i^{\&prime;}=\underset{k=i-\mathrm{the\; s}}{\overset{i+\mathrm{the\; s}}{\mathrm{\&Sigma;}}}{\mathrm{the\; y}}_k/(2\mathrm{the\; s}+1)$ s is the smoothing factor, (x' _i , y' _i ) is the smoothed boundary.

Images