CN113096147B - MATLAB-based automatic laser marking shadow generation method - Google Patents

Abstract

The invention provides an automatic generation method of laser marking shadow based on MATLAB, comprising the following steps: extraction and interpolation processing of graphic edge data, polygon equidistant offset, establishment of an equidistant contour shadow filling model of multi-connected patterns, A "zigzag" shadow filling model for multi-connected patterns is established. Based on the equidistant contour lines and the "zigzag" shadow filling method, a clear, uniform, and discontinuous multi-connected graphics laser marking shadow filling can be finally obtained. The invention constructs a multi-connected graph isometric contour line and "zigzag" shadow filling method based on MATLAB, which can realize efficient and fast filling of multi-connected graphs, and solves the problem of poor plasticity of laser marking drawing programs, slow calculation speed, and results. Problems such as breakpoints, misconnections, and excessive offsets occur.

Description

An automatic generation method of laser marking shadow based on MATLAB

technical field

The invention belongs to the technical field of laser marking, and relates to a method for automatically generating laser marking shadows based on MATLAB. The line cutting algorithm implements the contour line and "zigzag" shadow filling of any multi-connected domain graph on MATLAB, and makes this function open source.

Background technique

As a modern precision machining method, laser marking is often used to mark text, symbols, images and other information on models or products. Compared with traditional corrosion processing methods and pneumatic processing methods, laser marking has the advantages of no workpiece contact, high-precision trajectory and strong marking. Durability and other advantages, the marking process will not pollute the environment. At present, laser marking technology has been widely used in all walks of life, opening up broad prospects for high-quality, high-efficiency, pollution-free and low-cost modern processing and production.

The basic principle of laser marking is that under the irradiation of a high-energy laser beam, along a specific trajectory, certain physical and chemical traces such as melting, vaporization, and heating discoloration are left on the surface of the workpiece, so as to realize the information marking on the surface of the workpiece. The core system of the laser marking machine is the laser control system, and the drawing system is the key branch system of the control system. The first problem to be solved by laser marking is to identify the two-dimensional contour of the pattern, and then generate a series of equidistant irradiation trajectories within the contour. At present, the commonly used contour filling schemes for laser marking include closed curve filling, diagonal filling, cross-hatching, contour filling and zigzag filling, etc. Among them, contour lines and zigzag filling are the two most common filling methods: the basic principle of zigzag filling is to use equidistant parallel lines to cut contour lines to form a series of parallel line segments, and then connect these line segments in zigzag order. , the discrete laser trajectories are obtained. The basic principle of parallel contour line filling is to use polygons to offset inward equidistantly, and continuously derive new sub-boundaries inward from the outer contour until the equidistant contour lines cover the entire figure. In practical applications, the choice of filling scheme largely depends on the physical and chemical properties of the marking material and the impact of the marking process on the processing task.

There are two common drawing modules for laser marking: one is the drawing software, program, App, etc. that comes with the machine, which automatically recognizes the outline, color and other characteristic information of the object through a certain algorithm. Batch generate different types of fill shadows. The advantage is that the trajectory file can be produced and uploaded to the control center in a short time, which is not only efficient, but also automated. But the biggest disadvantage is that it cannot process the printed object in real time, and the plasticity is weak. For example, when marking complex patterns such as laser engraving LOGO, it still needs external software to reprocess. Once the drawing software fails, the entire laser marking machine will stop working. Equipment maintenance The cost and safety issues caused by dismantling and assembling the marking machine must also be considered. The other is to use external drawing software to assist in the production of shadow files, such as CorelDraw, CAD, etc., which can generate shadows outside the marking machine and plan the laser trajectory, and then output the shadow files (usually in pdf or jpg and other image formats) ) back to the control system and perform subsequent operations. Most of this kind of software is non-open source software developed abroad, which requires the purchase of the right to use, and the internal code of the software cannot or is difficult to change, and the program plasticity is not high. In addition, most software also have certain requirements on the file format of input and output data, and incompatibility often exists. At present, the development of laser marking software and programs in my country is still blank. With the continuous promotion of laser marking technology, automatic shadow generation programs for various programming languages have become a potential demand.

At present, the automatic generation module of laser marking shadow is still mainly based on the graphics processing program that comes with the marking machine. There is no shared open source program for the time being, and the processing speed is not ideal. The Polygon package that comes with MATLAB can complete the equidistant offset of polygons. In theory, the shadow generation of equidistant contour lines can be achieved. However, the offset polygon shadow curve still has an end position opening, and the end point connection is chaotic and excessive offset. and other problems, especially for multi-connected domain graphs, further optimization is still needed. There is currently no MATLAB command that can generate zigzag shadows with one click, which requires manual programming. In general, there is still a lot of room for development of laser marking shadow programming based on various programming languages such as MATLAB.

In order to solve the problem of shortage of open source drawing programs and realize the efficient and rapid generation of laser marking shadows, the present invention introduces two algorithms of polygon equidistant offset and equidistant parallel line cutting, and establishes multi-connectivity according to the graphics processing commands and programming characteristics of MATLAB. Domain graphics parallel lines and contour lines laser mark shading automatically generate models and programs. First, read the multi-connected domain graphic data and interpolate the two-dimensional resolution to 1/5 spacing. Through the maximum number of equidistant offsets, remove overlapping edges, breakpoint identification, reconnection, and delete dislocation connections to achieve contour shadowing The automatic generation of zigzag shadows is realized through five steps of single equidistant offset, equidistant line cutting, intersection sorting, parity edge processing and non-monotonic edge removal.

SUMMARY OF THE INVENTION

The technical problem solved by the present invention is to provide an automatic generation method for laser marking shadows of multi-connected domain graphics parallel lines and contour lines, and to package and implement the algorithm by using the MATLAB open source programming language.

An automatic generation method of laser marking shadow based on MATLAB, including the following steps:

Step 1: Extraction and interpolation of graphic edge data;

Step 2: Polygon equidistant offset;

Step 3: Establish contour line shadow filling model;

Step 4: Create an equidistant parallel line shadow fill model;

Step 5: Package implementation based on MATLAB open source programming language.

The step 1 specifically includes:

Use the Incontour or Edge function that comes with MATLAB, or with the help of other graphics methods, to obtain the contour boundary coordinate set of a given graph, and linearly interpolate the boundary coordinates through the interp function to control the horizontal and vertical distances between each point at the offset. within 1/4 of the distance.

MATLAB is a commercial mathematical software produced by MathWorks in the United States. It is used in data analysis, wireless communication, deep learning, image processing and computer vision, signal processing, quantitative finance and risk management, robotics, control systems and other fields.

The step 2 specifically includes:

First, the curve equation of the target contour is solved according to the contour boundary coordinate data set; the target contour curve is not a smooth curve in essence, but a polygon formed by multiple straight lines connected end to end; in the Cartesian coordinate system, a simple polygon is used as For example, assuming that there are two points on the outer contour whose coordinates are (XA, YA) and (XB, YB), as shown in Figure 1, the slope k of the straight line AB can be expressed as:

Let the equation of straight line AB be: y=kx+b, put the coordinates of point A or point B into the straight line equation, we get:

Therefore, the equation of any line can be expressed as:

Assuming that the coordinates of the two points on the inner contour are P ₁ and P ₂ , then the P ₁ P ₂ line segment is the equidistant inner contour line segment we need to generate; compared with the straight line AB, P ₁ P ₂ generates an increment δb on the Y axis , the P ₁ P ₂ equation is:

where the sign of d is determined by the slope k and the x-axis increment; bringing δb back into the equation, the P ₁ P ₂ equation becomes:

Similarly, for another adjacent inner contour line segment P ₂ P ₃ , its intersection with P ₁ P ₂ is:

Simultaneously combine the above two equations to obtain the coordinates of the intersection points, and connect all the intersection points to obtain the next-order equidistant offset contour; in MATLAB, this method can be automatically implemented using the Polybuffer command, while in Python, C++, Java and other languages, it can be based on The above scheme is implemented step by step.

The step 3 specifically includes: establishing a contour shadow filling model on the basis of the polygon equidistant offset algorithm;

The contour filling of a single-connected graph only needs to repeat the polygon equidistant offset process several times until the blank area is completely covered. For a multi-connected graph, it can be regarded as the intersection area of multiple simply-connected graphs;

Let the outermost graph be the first layer, and the graphs of the odd-numbered layers are shifted inward, and the graphs of the even-numbered layers are shifted outward. Compared with the single-connected graphs, the multi-connected graphs The equidistant offset of the graph is prone to interruptions, overlaps, and dislocations at intersections, inflection points, etc., which greatly reduces the computational efficiency and even generates algorithm loopholes. optimization;

When the shadows of two simply connected graphs of AB overlap:

First, use the inpolygon function to determine the overlap between the two closed shadow curves of the same layer contour;

Secondly, take the start and end points of the overlapping line segments, and then reconnect them according to the principle of minimum distance to remove the overlapping curve between the two points;

After that, equidistant offset is performed on the basis of the compound graph. If the graph is broken into two or more sub-graphs during the offset process, the equidistant offset will be performed independently in the next step, and there will be no mutual influence. , until the blank area is completely covered;

Finally, set the threshold to 1/3 of the offset spacing, detect curves whose length exceeds the threshold, that is, misconnected line segments, and delete them, and finally form a complete and smooth equidistant contour line filled with shadows.

The step 4 specifically includes:

First, the original set of coordinates is interpolated according to the parallel line spacing requirements, keeping the horizontal and vertical resolution above 1/4 of the parallel line spacing.

Then use the contour line shadow filling model to realize the first-order equidistant offset of the peripheral curve, and finally generate the "zigzag" parallel line shadow on the basis of the first-order sub-contour;

Different from the contour line shadow filling scheme, the parallel line shadow filling algorithm for multi-connected graphs is also suitable for single connected domain graphs: the x direction is the direction of the parallel line, and the y direction is the vertical parallel line direction; the max function and the min function are used. Find the upper boundary Y _max and the lower boundary Y _min of the graph in the y direction respectively. At this time, the graph boundary is appropriately expanded. Specify Y _max +0.2*(Y _max -Y _min ) as the top boundary of the graph, Y _min -0.2*( Y _max -Y _min ) is the bottom boundary of the graph. According to the parallel line interval specified by the user, parallel lines with equal spacing are drawn between the top and bottom boundaries to intersect with the sub-contour, and the intersection point set I _inter is obtained;

Then use the Inpolygon function to judge and remove the line segments outside the contours of the odd-numbered layers and within the contours of the even-numbered layers; the points in the remaining line segments are stored in the point set I _seg , and are arranged in the order of the "zigzag" trend during storage, that is, the odd-numbered lines are from the left. Arrange to the right, and even rows are arranged from right to left;

After obtaining point sets I and J, loop to find all adjacent points in point set J that have displacement in the y-direction to form a jump point set I _jump , and the outer contour points between all adjacent jump points in point set I _jump form point set I _boundary ;

The feature of the "zigzag" shadow is that the left border is reserved in odd-numbered lines, and the right border is removed, and the right border is reserved in even-numbered lines, and the left border is removed;

According to this rule, all points in the I _boundary are firstly classified, and all points that constitute the same edge are classified into one category; at this time, since the graph outline may turn between one or two parallel lines, it is not monotonous. Therefore, it is necessary to check the monotonicity of each type of points, and multiply the difference between the ordinates of the first two points and the difference of the ordinates of the last two points. If If the number is negative, all points of this type are removed to obtain the optimized point set I _boundary-1 ;

Then, cyclic judgment is performed on each point class in I _boundary-1 in turn. If the point of this class belongs to the right boundary of odd-numbered rows or the left boundary of even-numbered rows, all points of this class are removed from I _boundary-1 . ;

Finally, connect each remaining class of points to get the zig-zag shadow.

The step 5 specifically includes: packaging the two filling methods through MATLAB, and inputting the outline data of the graph through a specified file, which can be compatible with tables, text documents, mat files, and data stream files, and can also directly read and convert image files. is the gray value matrix.

The invention realizes the open source of the laser marking shadow generation program, makes the laser marking function open source, enables the user to change the marking algorithm at any time according to the actual situation, improves the algorithm plasticity of the drawing program and the compatibility of data types and programming languages; Contour and parallel line shadow filling based on MATLAB, the algorithm is simple and easy to operate, the calculation speed is fast, and it is suitable for laser marking of all multi-connected domain patterns. There is a shortage of open source programs compatible with laser marking machines. The method is simple and efficient, fully utilizes the program function of MATLAB, and solves the problems of the traditional filling curve end position opening, end point misconnection, excessive offset of the equidistant line and so on. The designed MATLAB open source program is suitable for more complex graphics, and can be rewritten according to other programming languages, realizes the open source of shadow generation, provides more platform choices for the shadow generation module of laser marking, and has wide application prospects.

Description of drawings

Fig. 1 is the schematic diagram of the equidistant offset method of extraction and interpolation processing of graphic edge data in step 1 of the present invention;

Fig. 2 is the optimization scheme of the equidistant contour line filling method of the present invention in the application of multi-connected graphs;

Figure 3 shows the filling results of the present invention at 1mm and 0.1mm pitches.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and embodiments, but this is not intended to limit the protection scope of the present application.

The automatic generation method of laser marking shadows for multi-connected domain graphics based on MATLAB is implemented through the following steps:

Step 1: Extract the outer contour dataset I _outline of the target pattern through MATLAB's own graphics and data reading functions, or through other graphics edge detection methods

Step 2: Control the x-direction and y-direction resolutions of the outer contour data set to be more than 0.2 times the line spacing through the linear interpolation method, and obtain the interpolated contour data set I _interp

Step 3: Polygon Offset Offset

Solve the curve equation of the target contour based on the contour boundary coordinate data set. In the Cartesian coordinate system, for any figure, it is regarded as a polygon composed of a series of straight lines. When the coordinates of two adjacent points (X _A , Y _A ) and (X _B , Y _B ) are known, calculate The slope k of the line connecting the two points:

Let the equation of AB be: y=kx+b, put the coordinates of point A or point B into the equation of the straight line, we get:

So the equation of any straight line can be expressed as:

The coordinates of the two points on the inner contour are assumed to be P ₁ and P ₂ , then the P ₁ P ₂ line segment is the equidistant inner contour line segment that we need to generate. Compared with the AB line, P ₁ P ₂ produces an increment δb on the Y axis, and the equation of P ₁ P ₂ is:

The sign of d is determined by the slope k and the x-axis increment. Bringing δb back into the equation, the P ₁ P ₂ equation becomes:

For another adjacent inner contour line segment P ₂ P ₃ , its intersection with P ₁ P ₂ is:

Simultaneously combine the above two equations to obtain the coordinates of the intersection points, and connect all the intersection points to obtain the next-order equidistant offset contour. In MATLAB, the Polybuffer command can be used to implement the algorithm, and on other programming platforms such as Python, C++, Java, etc., it only needs to be programmed according to the above steps.

Step 4: Create an equidistant parallel line shadow fill model of a multi-connected pattern

Based on the polygon equidistant offset algorithm, the contour line shadow filling model is established. The contour filling of a simply connected graph only needs to repeat the polygon equidistant offset process many times until the blank area is completely covered. Let the outermost layer of graphics be the first layer, and the inward layers are the second and third layers. Compared with single-connected graphs, the equidistant offsets of multi-connected graphs are prone to interruptions, overlaps, and dislocations at intersections and inflection points, which greatly reduces computational efficiency and even generates algorithm loopholes. Therefore, the shadow generation algorithm of multi-connected graphs needs to be further optimized on the basis of single-connectivity.

When the shadows of the two single-connected graphs of AB overlap, first, use the inpolygon function to determine the overlap between the two closed shadow curves of the contours of the same layer. Second, take the start and end points of the overlapping line segments, and then reconnect them according to the principle of minimum distance to remove the overlapping curve between the two points. After that, equidistant offset is performed on the basis of the compound graph. If the graph is broken into two or more sub-graphs during the offset process, the equidistant offset will be performed independently in the next step, and there will be no mutual influence. , until the white space is completely covered. Finally, the threshold is set to 1/3 of the offset spacing, and the curves whose length exceeds the threshold (misconnected line segments) are detected and deleted, and finally a complete and smooth equidistant contour line is filled with shadows.

The fifth step: establish a "zigzag" shadow fill model;

First, the contour line shadow filling model is used to realize the first-order equidistant offset of the peripheral curve. Then, the x direction is the direction of the parallel line, and the y direction is the vertical parallel line direction. In MATLAB, use the max function and the min function to obtain the upper boundary Y _max and the lower boundary Y _min of the graph in the y direction respectively, and specify Y _max +0.2*(Y _max -Y _min ) as the top boundary of the graph, Y _min -0.2*( Y _max -Y _min ) is the bottom boundary of the graph. According to the parallel line interval specified by the user, parallel lines with equal spacing are drawn between the top and bottom boundaries to intersect with the sub-contour, and the intersection point set I _inter is obtained. Then use the Inpolygon function to judge and remove the line segments outside the contours of odd-numbered layers and within the contours of even-numbered layers. The points in the remaining line segments are stored in the point set I _seg , and are arranged in the order of the "zigzag" trend during storage, that is, the odd-numbered lines are arranged from left to right, and the even-numbered lines are arranged from right to left. After obtaining point sets I and J, loop to find all adjacent points in point set J with displacement in the y direction to form a jump point set I _jump , and the outer contour points between all adjacent jump points in point set I _jump form point set I _boundary . Finally, all the points in the I _boundary are classified, and all the points that constitute the same edge are classified into one class. Multiply the difference between the ordinates of the first two points and the difference of the ordinates of the last two points. If it is a negative number, remove all points of this type to obtain the optimized point set I _boundary . Finally, cyclic judgment is performed on each point class in the I _boundary in turn. If the point of this type belongs to the right boundary of the odd-numbered row or the left boundary of the even-numbered row, all points of this type are removed from the I _boundary . Finally, connect each remaining class of points to get a zig-zag shadow, as shown in Figure 3.

Step 6: Package implementation based on MATLAB open source programming language

The above two methods are implemented step by step through the MATLAB programming language according to the calculation sequence. Before each cycle, the spatial resolution of the contour coordinate set obtained in this round is sparsed to 0.2 times of the equidistant line spacing to reduce unnecessary operations. The graphic outline data is input through the specified file type, which is compatible with various file forms such as Excel table, text document, mat matrix data file, dat data stream file, etc. It can also directly read the picture file and convert it into a gray value matrix. The output data is also compatible with the above formats.

The final multi-connected graphics are filled with equidistant contour lines and "zigzag" equidistant lines at 1mm and 0.1mm resolutions, as shown in Figure 3. The pattern is clear, the curve distance is equal, and there are no breakpoints.

The invention constructs a multi-connected graph isometric contour line and "zigzag" shadow filling method based on MATLAB, which can realize efficient and fast filling of multi-connected graphs, and solves the problem of poor plasticity of laser marking drawing programs, slow calculation speed, and results. Problems such as breakpoints, misconnections, and excessive offsets occur.

The foregoing has shown and described the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above-mentioned embodiments, and the above-mentioned embodiments and descriptions are only preferred examples of the present invention, and are not intended to limit the present invention, without departing from the spirit and scope of the present invention. Under the premise, the present invention will also have various changes and improvements, and these changes and improvements all fall within the scope of the claimed invention. The claimed scope of the present invention is defined by the appended claims and their equivalents.

Claims (1)

1. A method for automatically generating laser marking shadows based on MATLAB, comprising the following steps: Step 1: Extraction and interpolation processing of graphic edge data; specifically, using the Incontour or Edge function that comes with MATLAB to obtain the contour boundary coordinate set of a given graphic, and linearly interpolate the boundary coordinates through the interp function. And the vertical distance is controlled within 1/4 of the offset spacing; Step 2: Polygon equidistant offset; specifically includes: first, solving the curve equation of the target contour according to the contour boundary coordinate data set; the target contour curve is not a smooth curve in essence, but a polygon formed by a plurality of straight lines connected end to end; In the Cartesian coordinate system, taking a simple polygon as an example, set the coordinates of any two adjacent points on the polygon to be A(XA,YA), B(XB,YB), the slope k of the straight line AB can be expressed as:

Let the equation of straight line AB be: y=kx+b, put the coordinates of point A or point B into the straight line equation, we get:

Therefore, the equation of any line can be expressed as:

Assuming that the coordinates of the two points on the inner contour are P ₁ and P ₂ , then the P ₁ P ₂ line segment is the equidistant inner contour line segment we need to generate; compared with the straight line AB, P ₁ P ₂ generates an increment δb on the Y axis , the P ₁ P ₂ equation is:

where the sign of d is determined by the slope k and the x-axis increment; bringing δb back into the equation, the P ₁ P ₂ equation becomes:

Similarly, for another adjacent inner contour line segment P ₂ P ₃ , its intersection with P ₁ P ₂ is:

Simultaneously combine the above two equations to obtain the coordinates of the intersection point, and connect all the intersection points to obtain the next-order equidistant offset contour Step 3: establish a contour line shadow filling model; specifically, it includes: establishing a contour line shadow filling model on the basis of the polygon equidistant offset algorithm; The contour filling of a single-connected graph only needs to repeat the polygon equidistant offset process several times until the blank area is completely covered. For a multi-connected graph, it can be regarded as the intersection area of multiple simply-connected graphs; Let the outermost layer of graphics be the first layer, and the second and third layers inward, wherein the graphics of the odd-numbered layers are shifted inward, and the graphics of the even-numbered layers are shifted outwards; When the shadows of two simply connected graphs of AB overlap: First, use the inpolygon function to determine the overlap between the two closed shadow curves of the same layer contour; Secondly, take the start and end points of the overlapping line segments, and then reconnect them according to the principle of minimum distance to remove the overlapping curve between the two points; After that, equidistant offset is performed on the basis of the compound graph. If the graph is broken into two or more sub-graphs during the offset process, the equidistant offset will be performed independently in the next step, and there will be no mutual influence. , until the blank area is completely covered; Finally, set the threshold to 1/3 of the offset spacing, detect the curve whose length exceeds the threshold, that is, the misconnected line segment, and delete it, and finally form a complete and smooth equidistant contour line filled with shadows Step 4: Establish an equidistant parallel line shadow filling model; specifically, first, interpolate the original coordinate set according to the parallel line spacing requirements, and keep the horizontal and vertical resolutions above 1/4 of the parallel line spacing; Then use the contour line shadow filling model to realize the first-order equidistant offset of the peripheral curve, and finally generate the "zigzag" parallel line shadow on the basis of the first-order sub-contour; Different from the contour line shadow filling scheme, the parallel line shadow filling algorithm for multi-connected graphs is also suitable for single connected domain graphs: the x direction is the direction of the parallel line, and the y direction is the vertical parallel line direction; the max function and the min function are used. Find the upper boundary Y _max and the lower boundary Y _min of the graph in the y direction respectively. At this time, the graph boundary is appropriately expanded. Specify Y _max +0.2*(Y _max -Y _min ) as the top boundary of the graph, Y _min -0.2*( Y _max -Y _min ) is the bottom boundary of the graph. According to the parallel line interval specified by the user, parallel lines with equal spacing are drawn between the top and bottom boundaries to intersect with the sub-contour, and the intersection point set I _inter is obtained; Then use the Inpolygon function to judge and remove the line segments outside the contours of the odd-numbered layers and within the contours of the even-numbered layers; the points in the remaining line segments are stored in the point set I _seg , and are arranged in the order of the "zigzag" trend during storage, that is, the odd-numbered lines are from the left. Arrange to the right, and even rows are arranged from right to left; After obtaining point sets I and J, loop to find all adjacent points in point set J that have displacement in the y-direction to form a jump point set I _jump , and the outer contour points between all adjacent jump points in point set I _jump form point set I _boundary ; The feature of the "zigzag" shadow is that the left border is reserved in odd-numbered lines, and the right border is removed, and the right border is reserved in even-numbered lines, and the left border is removed; According to this rule, first classify all points in the I _boundary , and all points that constitute the same edge are classified into one category; perform monotonicity test on each category of points, and compare the difference between the ordinates of the first two points and the last two points. Multiply the difference between the ordinates of the points, and if it is a negative number, remove all points of this type to obtain the optimized point set I _boundary-1 ; Then, cyclic judgment is performed on each point class in I _boundary-1 in turn. If the point of this class belongs to the right boundary of odd-numbered rows or the left boundary of even-numbered rows, all points of this class are removed from I _boundary-1 . ; Finally, connect each remaining class of points to get a "zigzag" shadow; Step 5: Packaging implementation based on MATLAB open source programming language; specifically including: packaging and implementing two filling models through MATLAB, inputting graphic outline data through a specified file, and two filling models are contour line shadow filling model and equidistant parallel The line shadow filling model is compatible with tables, text documents, mat files, data flow files, and can also directly read image files and convert them into gray value matrices.

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