Osteotomy surface contour generation method and device

Abstract

The invention discloses an osteotomy surface contour generation method and device. The method comprises the following steps: determining an osteotomy surface in a three-dimensional model corresponding to an osteotomy part; according to the intersection relation between the osteotomy surface and a plurality of triangular patches, the contour points of the osteotomy surface are determined, and the triangular patches are constituent parts of the three-dimensional model corresponding to the osteotomy part; and sorting the contour points of the osteotomy surface according to a coplanar relationship between any adjacent contour points to obtain ordered contour points. According to the method, the generated contour line is more stable and more accurate.

Description

Technical Field

[0001] The present application belongs to the field of image processing technology, and in particular, relates to a method and device for generating an osteotomy surface contour. Background Art

[0002] During surgery, the contour lines provide surgeons with clear markings that can help them clearly understand the shape and position of the osteotomy site and help develop a reasonable surgical plan.

[0003] Current imaging technologies such as (X-rays, etc.) have difficulty providing images of sufficient resolution, especially in complex anatomical structures. Imaging noise or blur may affect the accuracy of the contour. In addition, the osteotomy site usually involves irregular and complex geometric shapes. Traditional methods have difficulty accurately locating the boundary line, resulting in poor accuracy of the generated contour line, which does not meet surgical requirements. Summary of the invention

[0004] The embodiments of the present application provide a method and device for generating an osteotomy surface contour, which can accurately generate an osteotomy surface contour line.

[0005] In a first aspect, an embodiment of the present application provides a method for generating an osteotomy surface contour, the method comprising: determining the osteotomy surface in a three-dimensional model corresponding to the osteotomy site; determining the contour points of the osteotomy surface based on the intersection relationship between the osteotomy surface and a plurality of triangular facets, wherein the plurality of triangular facets are components of the three-dimensional model corresponding to the osteotomy site; and sorting the contour points of the osteotomy surface based on the coplanar relationship between any adjacent contour points to obtain ordered contour points.

[0006] In a second aspect, an embodiment of the present application provides an osteotomy surface contour generating device, the device comprising: a determination module, used to determine the osteotomy surface in a three-dimensional model corresponding to the osteotomy site; a processing module, used to determine the contour points of the osteotomy surface based on the intersection relationship between the osteotomy surface and a plurality of triangular facets, wherein the plurality of triangular facets are components of the three-dimensional model corresponding to the osteotomy site; a sorting module, used to sort the contour points of the osteotomy surface according to the coplanar relationship between any adjacent contour points to obtain ordered contour points.

[0007] In a third aspect, an embodiment of the present application provides an osteotomy surface contour generation device, the device comprising: a processor and a memory storing computer program instructions; when the processor executes the computer program instructions, the above-mentioned osteotomy surface contour generation method is implemented.

[0008] In a fourth aspect, an embodiment of the present application provides a computer storage medium, on which computer program instructions are stored. When the computer program instructions are executed by a processor, the above-mentioned osteotomy surface contour generation method is implemented.

[0009] In a fifth aspect, an embodiment of the present application provides a computer program product. When the instructions in the computer program product are executed by a processor of an electronic device, the electronic device executes the above-mentioned osteotomy surface contour generation method.

[0010] The osteotomy surface contour generation method and device of the embodiment of the present application, by determining the osteotomy surface from the three-dimensional model of the osteotomy site, can preliminarily judge the range of surgical operation required, which is helpful for the subsequent steps to accurately generate the boundary line of the osteotomy surface, and provides a visual basis for the surgery; the contour points of the osteotomy surface are determined by the intersection relationship between the osteotomy surface and multiple triangular facets, and the spatial position of the contour points can be accurately located; the coplanar relationship between adjacent contour points is used for sorting to prevent contour line fluctuations caused by the random order of points, making the generated contour line more stable and more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] In order to more clearly illustrate the technical solution of the embodiments of the present application, the following is a brief introduction to the drawings required for use in the embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0012] Figure 1 It is a flow chart of a method for generating an osteotomy surface contour provided by an embodiment of the present application;

[0013] Figure 2 It is a schematic diagram of a three-dimensional model of an osteotomy site provided in an embodiment of the present application;

[0014] Figure 3 is a schematic diagram of the intersection of an osteotomy surface and a triangular surface provided in an embodiment of the present application;

[0015] Figure 4 is a schematic diagram of an ordered boundary point provided in an embodiment of the present application;

[0016] Figure 5 is a schematic diagram of calculating an intersection point provided in an embodiment of the present application;

[0017] Figure 6 It is a schematic diagram of a case where two or more osteotomy surfaces exist, provided in an embodiment of the present application;

[0018] Figure 7 This is a rendering of an osteotomy surface contour generation provided in an embodiment of the present application;

[0019] Figure 8 It is a structural schematic diagram of an osteotomy surface contour generating device provided in an embodiment of the present application;

[0020] Fig. 9 It is a structural schematic diagram of the osteotomy surface contour generating device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0021] The features and exemplary embodiments of various aspects of the present application will be described in detail below. In order to make the purpose, technical solutions and advantages of the present application clearer, the present application will be further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only intended to explain the present application, rather than to limit the present application. For those skilled in the art, the present application can be implemented without the need for some of these specific details. The following description of the embodiments is only to provide a better understanding of the present application by illustrating the examples of the present application.

[0022] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, the elements defined by the statement "include..." do not exclude the presence of other identical elements in the process, method, article or device including the elements.

[0023] In robot-assisted total knee replacement surgery, in order to protect the patient's posterior cruciate ligament from being damaged during the operation, the boundary line of the osteotomy plane needs to be marked during the surgical planning stage, so as to indicate the range of osteotomy during the robot-guided osteotomy process.

[0024] The solutions in the prior art are difficult to automatically generate corresponding contour lines (or boundary lines) according to the osteotomy surface, and there are problems such as contour line mismatch and poor accuracy.

[0025] In order to solve the problems of the prior art, the embodiments of the present application provide a method, device, equipment, computer storage medium and computer program product for generating an osteotomy surface contour. The method for generating an osteotomy surface contour provided by the embodiments of the present application is first introduced below.

[0026] Figure 1 FIG. 1 is a flow chart of a method for generating an osteotomy surface contour according to an embodiment of the present application. Figure 1 As shown, Figure 1 The process includes the following steps S101 to S103.

[0027] S101, determining an osteotomy surface in a three-dimensional model corresponding to the osteotomy site.

[0028] The osteotomy site refers to the specific location where osteotomy (i.e. bone removal or trimming) is required, that is, the site that needs to be treated during fracture repair, bone transplantation or orthopedic surgery.

[0029] For example, in total knee replacement surgery, the doctor may need to cut off the distal femur and proximal tibia to prepare the prosthesis installation surface, which is the osteotomy site.

[0030] The three-dimensional model of the osteotomy site is obtained through medical imaging technology. It can represent the three-dimensional structure of the specific part to be treated in the surgical operation in the computer, including information such as the shape, size and relative position of the bone. The three-dimensional model of the osteotomy site can usually be obtained through computed tomography (CT) technology or magnetic resonance imaging (MRI) technology.

[0031] For example, CT scans produce detailed cross-sectional images of bones and surrounding soft tissue, which can be processed to reconstruct a three-dimensional model.

[0032] For example, soft tissue imaging through MRI technology, combined with CT data, provides a more complete anatomical image. Medical imaging data is processed using 3D software to convert 2D slice data into 3D models.

[0033] The osteotomy plane refers to the plane where the doctor wants to cut the bone during the osteotomy operation. The osteotomy plane is usually determined by the doctor based on medical images and surgical requirements. The location of the osteotomy plane varies from case to case. For example, there can be osteotomy planes in the femur, tibia, etc. The specific location is not limited here.

[0034] Combine the following Figure 2 Let's introduce the osteotomy surface in detail.

[0035] Figure 2 It is a schematic diagram of a three-dimensional model of an osteotomy site provided in an embodiment of the present application.

[0036] Upload the CT data required for surgical planning, segment and reconstruct the CT data to obtain the 3D models of the femur and tibia, and display the contour of the osteotomy surface after adjusting the appropriate amount of osteotomy in the surgical planning interface. Figure 2 As shown, Figure 2 The white area in the figure represents the 3D model corresponding to the osteotomy site, and the blue area represents the outline of the osteotomy surface.

[0037] The osteotomy surface can accurately define the specific location and direction of the osteotomy in the 3D model, thereby ensuring that the final surgical plan is consistent with the patient's anatomical structure. Determining the osteotomy surface first helps to accurately generate the boundary line of the osteotomy surface in the subsequent steps, providing a visual basis for the surgery.

[0038] S102: Determine the contour points of the osteotomy surface according to the intersection relationship between the osteotomy surface and a plurality of triangular facets.

[0039] The plurality of triangular facets are components of the three-dimensional model corresponding to the osteotomy site.

[0040] It can be understood that the three-dimensional model is Standard Tessellation Language (STL) data, and STL data is composed of triangular facets. Each triangular facet consists of three vertices and a normal vector, which is used to describe the geometric shape in three-dimensional space. STL data can store the geometric information of the entity, such as vertex coordinates and normal vectors, etc. It can also be understood that the surface of the three-dimensional model is formed by a large number of triangular facets linked together. Each triangular facet consists of three vertices and a plane surrounded by these three vertices. These triangular facets are connected to each other in a specific way, and finally form a complete three-dimensional shape.

[0041] Intersection relationship means that in three-dimensional space, if there is only one common straight line between two planes, then the positional relationship between the two planes is called intersection. Specifically, this means that among all the points on each plane, at least one point also belongs to the other plane, and these common points happen to form a straight line.

[0042] By using the intersection relationship between the osteotomy surface and these triangular facets, the intersection points of the osteotomy surface and the triangular facets can be calculated to obtain the contour points of the osteotomy surface. In other words, by using geometric operations, such as linear algebra, vector operations, etc., the intersection points of two planes can be calculated, and these intersection points are the contour points of the osteotomy surface.

[0043] For example, the plane equations corresponding to the osteotomy surface and the triangular patch can be used to solve the equation group using a mathematical intersection algorithm to obtain the position of the intersection point.

[0044] For another example, the triangle normal vector can be used to determine the relationship between a point and a plane. If the dot product of the vectors is zero, it means that the point is on the plane, and it can be used as the intersection point, and then the coordinates of the intersection point can be calculated.

[0045] Combine the following Figure 3 Let's specifically introduce how to determine the contour points of the osteotomy surface.

[0046] Figure 3 It is a schematic diagram of the intersection of an osteotomy surface and a triangular surface provided in an embodiment of the present application.

[0047] like Figure 3 As shown, face1 and face2 represent two different triangular faces, which intersect with the cross section. The intersection points are A, B, and C, which are the contour points of the osteotomy surface.

[0048] Through intersection calculation, the intersection points of the osteotomy surface and multiple triangular facets can be accurately located. The contour lines formed by these intersection points can accurately reflect the true shape of the osteotomy surface in three-dimensional space, making surgical planning more accurate.

[0049] S103, sorting the contour points of the osteotomy surface according to the coplanar relationship between any adjacent contour points to obtain ordered contour points.

[0050] From the above, it can be understood that the intersection of the osteotomy surface and the triangular patch is the contour point of the osteotomy surface. When these two planes intersect, two adjacent contour points are always located in the same triangular patch, that is, there is a coplanar relationship between any adjacent contour points.

[0051] In principle, when the osteotomy surface intersects with the triangular surface, there are at least two intersection points in theory, that is, there are at least two intersection points on a triangular surface. Figure 3 It can be clearly seen that the two adjacent contour points A and B are both located on the triangle face face1. Since face1 and face2 are connected, the contour points B and C are adjacent to each other, and the adjacent contour points B and C are both located on the triangle face face2.

[0052] Sorting contour points means arranging a set of contour points into an ordered sequence according to a certain sorting method. This ensures that these points can form a coherent path or curve when drawn.

[0053] The above-mentioned sorting method may include sorting by the size of coordinate values, or sorting by searching for adjacent points.

[0054] According to the above principle, the contour points on any triangle patch can be sorted first, and then the next triangle patch connected to the arbitrary triangle patch can be found, and then the contour points on the next triangle patch can be sorted, and so on, until all the contour points are sorted.

[0055] For example, the coordinates of the contour points located on the same triangle are calculated, and they are sorted according to the size of the coordinate values, from large to small or from small to large, to obtain ordered contour points.

[0056] For another example, first determine a point as the starting point, then find the next point on the same triangle, and search and sort in sequence. That is, select any contour point as the starting point (the first point), and the other point on the same triangle as the starting point is the second point. According to the second point, find the third point on the next triangle, and so on, until all points are found, so as to obtain the numbering order of the contour points and obtain the ordered contour points.

[0057] It can be understood that the contour line is connected by a series of points. Only when the order of the points is reasonable can the generated contour line form a closed and continuous shape. If the order of the contour points is incorrect, the generated lines may cross or not meet expectations. Since the coplanar relationship can ensure that the contour points are in the same plane, sorting the coplanar points can prevent the contour line from fluctuating due to the random order of the points, reduce errors, and make the generated contour line more stable and accurate, thereby ensuring the accuracy of the surgical process.

[0058] By determining the osteotomy surface from the three-dimensional model of the osteotomy site, the scope of the surgical operation can be preliminarily determined, which helps to accurately generate the boundary line of the osteotomy surface in the subsequent steps and provides a visual basis for the operation; the contour points of the osteotomy surface are determined by the intersection relationship between the osteotomy surface and multiple triangular facets, and the spatial position of the contour points can be accurately located; sorting by the coplanar relationship between adjacent contour points can prevent contour line fluctuations caused by the random order of points, making the generated contour line more stable and more accurate, thereby ensuring the accuracy of the surgical process.

[0059] In one implementation, the method further includes: determining ordered boundary points of the osteotomy surface according to the ordered contour points; and connecting the ordered boundary points to generate a boundary line of the osteotomy surface.

[0060] Boundary point can be understood as a point that determines the position of the regional boundary. In the present application, boundary point refers to a point at the position of the regional boundary of the osteotomy surface.

[0061] A borderline refers to a dividing line between two regions or objects, and may also refer to an outer contour line or edge line of a region, and is a line used to indicate the boundary or edge between different regions or objects. In the present application, a borderline may be understood as a contour line between an osteotomy surface region and a non-osteotomy surface region.

[0062] In combination with the above, by sorting the contour points, an orderly arranged point set is obtained, namely the ordered contour points. By performing image processing on the ordered contour points according to a preset image processing method, the ordered boundary points are obtained.

[0063] The preset image processing method refers to forming new boundary points on the periphery of the contour line formed by the ordered contour points by transforming the spatial positions of the ordered contour points, that is, forming ordered boundary points.

[0064] For example, the ordered boundary points are obtained by translating the ordered contour points. Assuming there is an ordered contour point A, the contour point A is translated outward (ie, away from the osteotomy surface) by 0.5 cm to obtain the ordered boundary point A'.

[0065] For another example, the ordered boundary points are obtained by amplifying (expanding) the ordered contour points. Assuming there is an ordered contour point A, the distance between the ordered contour point A and the osteotomy surface is amplified to 1.2 times the original distance to obtain the ordered boundary point A'.

[0066] In other words, the numbering order of the ordered boundary points is consistent with that of the ordered contour points, except that the ordered boundary points are farther from the osteotomy surface in spatial position than the ordered contour points, the ordered contour points are closer to the osteotomy surface, and the ordered boundary points are outside the ordered contour points.

[0067] By connecting these ordered boundary points one by one, the boundary line of the osteotomy surface can be formed, for example, connecting from A to B, from B to C, from C to D, ..., and so on, until all the ordered boundary points are connected.

[0068] Combine the following Figure 4 Let me introduce it in detail.

[0069] Figure 4 It is a schematic diagram of an ordered boundary point provided in an embodiment of the present application.

[0070] like Figure 4 As shown, the blue line is the outline of the three-dimensional model boundary line point information of the osteotomy site, the red points are the calculated boundary line annotation points (that is, the ordered boundary points), and the yellow line formed by connecting the boundary line annotation points is the boundary line connection line.

[0071] Combined with the above, it can be understood that by sorting the ordered contour points ( Figure 4 The points on the blue line in the middle) are translated, enlarged and other image processing steps to obtain the ordered boundary points ( Figure 4 By connecting these ordered boundary points one by one, the boundary line of the osteotomy surface can be formed.

[0072] Alternatively, first connect the ordered contour points to obtain Figure 4 The blue line in the image is then processed by translation, amplification and other image processing steps to obtain the blue line directly. Figure 4 The yellow line in the figure, and then select a series of points from the yellow line as ordered boundary points.

[0073] The specific processing method can be selected according to the actual situation and is not limited here.

[0074] It can be understood that the marking of the boundary line helps to clarify the specific area that needs to be osteotomized during surgery, helping doctors to clearly identify the cutting range and avoid accidental damage to surrounding tissues and structures. Compared with directly connecting the ordered contour points to form a boundary line, determining the ordered boundary points of the osteotomy surface based on the ordered contour points can provide more room for surgical operations and avoid damage to surrounding tissues due to factors such as cutting errors; if the ordered contour points are directly connected to form a boundary line, the distance between the boundary line and the surrounding tissues may be too close, which is not conducive to protecting the surrounding tissues.

[0075] In one implementation, the contour points of the osteotomy surface are determined based on the intersection relationship between the osteotomy surface and multiple triangular facets, including: judging whether the spatial position relationship between the osteotomy surface and the multiple triangular facets is an intersection relationship; when the spatial position relationship between the osteotomy surface and the multiple triangular facets is an intersection relationship, identifying the points where the osteotomy surface intersects with the multiple triangular facets as contour points.

[0076] First, it is necessary to determine the spatial position relationship between the osteotomy surface and the multiple triangular facets. Common spatial relationships include intersection, separation and tangency. In this application, the most critical determination is to observe whether the osteotomy surface and the triangular facets intersect.

[0077] If they intersect, the intersection point calculation can be further performed, that is, the intersection points between the osteotomy surface and each intersecting triangular facet need to be calculated. These intersection points are the contour points of the osteotomy surface. For all intersecting triangular facets, their intersection points are collected, and these intersection points constitute the contour point set of the osteotomy surface.

[0078] It can be understood that triangular patches are usually composed of high-resolution mesh models. By taking the intersection of the osteotomy surface and the triangular patch as the contour point, the position of the contour point can be calculated more accurately in the calculation compared to the low-resolution image, thereby improving the positioning accuracy of the contour point and ensuring the accuracy of the subsequently generated contour line.

[0079] In one implementation, determining whether the spatial position relationship between the osteotomy surface and a plurality of triangular facets is an intersection relationship includes: respectively obtaining endpoints corresponding to the plurality of triangular facets, wherein the endpoints of any triangular facet include at least a first endpoint and a second endpoint; for any triangular facet, calculating the distance between the first endpoint and the osteotomy surface to obtain a first distance; calculating the distance between the second endpoint and the osteotomy surface to obtain a second distance; calculating the projection distance of a line segment between the first endpoint and the second endpoint in the normal direction of the osteotomy surface to obtain a third distance; and when the sum of the first distance and the second distance is equal to the third distance, determining that the spatial position relationship between the osteotomy surface and the triangular facet where the first endpoint and the second endpoint are located is an intersection relationship.

[0080] Combine the following Figure 5 To explain in detail.

[0081] Figure 5 It is a schematic diagram of calculating intersection points provided in an embodiment of the present application.

[0082] like Figure 5 As shown, first see Figure 5 Geometric relationship diagram of the blue triangle patch.

[0083] The distance between the first endpoint (point1) and the osteotomy surface is calculated to obtain the first distance (L1), and the distance between the second endpoint (point2) and the osteotomy surface is calculated to obtain the second distance ( Figure 5 The projection of the line segment between point1 and point2 in the normal direction of the osteotomy surface is calculated to obtain the third distance (i.e. Figure 5 L1+L2 in ).

[0084] It can be understood that, according to the geometric relationship that the sum of the first distance plus the second distance is equal to the third distance, it can be seen that the osteotomy surface intersects with the triangular surface patch.

[0085] On the contrary, see Figure 5 Geometric relationship diagram of the yellow triangle patch.

[0086] The distance between the first endpoint (point1) and the osteotomy surface is calculated to obtain the first distance (L1), the distance between the second endpoint (point2') and the osteotomy surface is calculated to obtain the second distance (L2'), and the projection of the line segment between point1 and point2 in the normal direction of the osteotomy surface is calculated to obtain the third distance (L3).

[0087] It can be clearly seen that the sum of L1 and L2' is greater than L3, so it can be determined that the osteotomy surface does not intersect the triangle.

[0088] By calculating the distance between any two endpoints of the triangular patch and the osteotomy surface and comparing the projection distance of the line segment formed by the two endpoints to the normal direction of the osteotomy surface, it is determined whether the osteotomy surface and the triangular patch intersect. The calculation principle is simple and easy to understand, the calculation amount is small, and it is easy to improve the calculation efficiency.

[0089] In one implementation, when the spatial position relationship between the osteotomy surface and multiple triangular facets is an intersection relationship, the point where the osteotomy surface intersects the multiple triangular facets is identified as a contour point, including: calculating the coordinate value of the contour point based on the first distance, the third distance, and the spatial distance between the first endpoint and the second endpoint.

[0090] Combination Figure 5, assuming that the distance from point1 to contour point A is Y, then Y / X=L2 / L1, the Y value can be calculated, and the coordinates of point A can be further solved.

[0091] Similarly, when the coordinates of the two endpoints of two triangles and the ratio of the distances between each endpoint and the contour point are known, the coordinate values ​​of the contour point can be calculated by establishing an equation based on the similarity principle of triangles.

[0092] It can be understood that using the similarity principle of triangles to calculate the coordinates of contour points can transform complex geometric problems into simple proportional relationships. The calculation method is simple and can also reduce the accumulation of errors and improve the accuracy of the calculation results.

[0093] In one implementation, the contour points of the osteotomy surface are sorted to obtain ordered contour points, including: selecting any contour point as a starting point; searching for the next contour point based on the starting point, wherein the search condition is that the starting point and the next contour point are located in the same triangle patch; when the next contour point is found, the next contour point is used as the starting point, and the search for the next contour point based on the starting point is performed again until all contour points of the osteotomy surface are found.

[0094] Combined with the above Figure 5 , assuming that there is a set of contour points of the osteotomy surface as follows:

[0095] A=(1,1,1), B=(2,1,1), C=(2,2,1), D=(1,2,1),….

[0096] Assume that P0 is selected as the starting point, and the next point is found. A and B are in the same triangle patch, and are ranked as the second point. The next point is found from B. B and C are in the same patch, and are ranked as the third point. C and D are in the same patch, and are ranked as the fourth point... until the last point is found. The sorting result is A, B, C, D,...

[0097] It can be understood that searching point by point based on the coplanar relationship can effectively prevent repeated searching of processed points and reduce redundant calculations. At the same time, it can ensure that adjacent points are located in the same plane, thereby ensuring the continuity of the contour line.

[0098] In one implementation, searching for the next contour point based on the starting point includes: calculating the normal vector of the triangle where the starting point is located to obtain a first vector; calculating the vector between the starting point and any contour point to be sorted to obtain a second vector; judging whether the starting point and any contour point to be sorted are located in the same triangle based on the product of the first vector and the second vector; in the case that the starting point and any contour point to be sorted are located in the same triangle, determining any contour point to be sorted as the next contour point.

[0099] Here are some common methods for calculating normal vectors.

[0100] (1) Three-point method

[0101] By calculating the vectors AB and AC formed by any three non-collinear points on the plane, the normal vector of the plane can be obtained by using the cross product operation. The normal vector is AB×AC.

[0102] (2) Coordinate method

[0103] For the plane equation Ax+By+Cz+D=0, (A,B,C) is the normal vector of the plane. That is, take any two points P(x1,y1,z1) and Q(x2,y2,z2) on the plane, the normal vector is the direction vector of vector PQ, that is, (x2-x1,y2-y1,z2-z1).

[0104] (3) Standard equation method

[0105] For a plane with the standard equation Ax+By+Cz+D=0, the normal vector of the plane is (A, B, C).

[0106] (4) Straight-line method

[0107] If the equation of a plane is Ax+By+Cz+D=0, then (A, B, C) is the normal vector of the plane.

[0108] If the conditions are met, Pi is selected as the next contour point and added to the ordered contour point list. The search is repeated using the new next contour point until all contour points have been processed.

[0109] Combined with the above, it can be understood that by determining any point A as the starting point, and knowing the coordinates of the three vertices of the triangular patch, the normal vector of the triangular patch can be calculated; multiply the normal vector of the triangular patch by vectors AB and AC respectively. If they are equal to 0, it means that the line segment is perpendicular to the triangular patch, which proves that the line segment is in the same plane, and the next point of point A is point B. Then find a new point C on the same plane according to the normal vector of face2. Similarly, all contour points are sorted. When the traversal point information is the coordinate information of point A, it means that the contour is closed, and the ordered point information of the intersection surface contour of the osteotomy surface and the corresponding three-dimensional model is generated. Using normal vector calculation to determine whether a contour point is located on a certain triangular patch is simple to calculate and efficient to process, which is convenient for fast calculation in practical applications.

[0110] In one implementation, the ordered boundary points of the osteotomy surface are determined based on the ordered contour points, including: connecting the ordered contour points to obtain an ordered contour line; performing image magnification processing on the ordered contour line to obtain a target contour line; performing thinning calculation based on the target contour line to obtain ordered boundary points.

[0111] Image magnification processing refers to magnifying the size of the ordered contour lines, that is, making the size of the ordered contour lines larger.

[0112] The image enlargement processing methods include proportional enlargement processing and non-proportional enlargement processing.

[0113] Proportional image enlargement means that during the enlargement process, the width and height of the image change in the same proportion, thus keeping the aspect ratio of the image unchanged. This means that the objects in the image still maintain their original shape and proportion in the enlarged image without stretching or compressing.

[0114] On the contrary, non-proportional enlargement means that during the enlargement process, the width and height of the image can change in different proportions, such as stretching the height and width.

[0115] When performing image processing, you can choose to enlarge the image in equal or unequal proportions according to actual needs to achieve the desired visual effect and purpose, which is not limited here.

[0116] For example, according to the steps above, a number of ordered contour points are obtained, and these points are connected in sequence to form an ordered contour line, which represents the boundary of the osteotomy surface. The ordered contour line is enlarged in equal proportion to obtain the target contour line.

[0117] The target contour is thinned to reduce the number of points while retaining the main shape features.

[0118] A simplification algorithm is an algorithm used to simplify or reduce the number of data points. Specific simplification methods include the following.

[0119] Iterative approximation is a classic thinning algorithm that recursively connects data points to approximate line segments, then compares the distances between line segments and deletes points whose distances are less than a set threshold to simplify the data.

[0120] Area method (Visvalingam's algorithm), which is based on the area of ​​triangles for simplification. First, the area of ​​the triangle formed by each data point and its two adjacent points is calculated, and then the data point with the smallest area is deleted until the simplification condition is met.

[0121] Douglas-Peucker algorithm, which constructs simplified line segments by selecting representative points based on the angle and distance of the lines between points, can maintain the turning information of the original line segments.

[0122] The specific thinning algorithm can be selected according to the actual situation and is not limited here. The points obtained after thinning the target contour line are the ordered boundary points.

[0123] For example, for the tibia, the outline of the point is enlarged proportionally, the boundary line is thinned out, and the points are removed at equal intervals, and the remaining ordered points are connected.

[0124] It can be understood that the use of image magnification processing can retain the original boundary features, maintain the basic shape of the entire contour, and avoid obvious loss of the overall contour line appearance; through thinning, the number of points in the contour line can be reduced, and the number of ordered points can be simplified, which is conducive to reducing computational complexity and improving the efficiency of subsequent processing or rendering.

[0125] In one implementation, when there are two or more osteotomy surfaces, the method further includes: calculating the circumscribed polygons corresponding to the two or more osteotomy surfaces using a convex hull algorithm; and performing image magnification processing on the ordered contour lines according to the circumscribed polygons to obtain the target contour lines.

[0126] The convex hull algorithm is an algorithm for calculating the convex hull. The convex hull is a geometric concept that refers to the smallest convex polygon that encloses a given set of points inside a convex polygon. The convex hull algorithm is used to find the outer boundary of a set of points in order to quickly identify and process the geometric features of data points.

[0127] The goal of the convex hull algorithm is to find the smallest convex polygon around a set of data points. The algorithm for calculating the convex hull usually selects an appropriate algorithm for calculation according to the nature and number of input data points. Common convex hull algorithms include Graham Scan, Jarvis March, Quickhull, etc.

[0128] Graham Scan algorithm constructs a convex hull by sorting and traversing the input point set.

[0129] Jarvis March algorithm, also known as "wrapping algorithm" or "corner picking algorithm", is also an algorithm for calculating the convex hull. It gradually constructs the convex hull by finding a starting point from a point set.

[0130] The fast convex hull algorithm (Quickhull) uses a divide-and-conquer strategy to find a convex polygon that encloses a point set. It can quickly calculate the convex hull of a medium-sized data point set.

[0131] The circumscribed polygon of two or more osteotomy surfaces refers to the minimum convex polygon calculated for two or more given osteotomy surfaces. This minimum convex polygon surrounds each osteotomy surface and forms a convex polygon that can enclose the area defined by each osteotomy surface.

[0132] The image magnification process may be the equal-proportional magnification process mentioned above or the non-equal-proportional magnification process, and the specific processing method is not limited here.

[0133] The convex hull algorithm (such as Graham scanning method) is used to calculate the circumscribed polygon of all osteotomy surface point sets. The obtained polygon vertices constitute the circumscribed contour, which can be regarded as a preliminary representation of the contour line. An ordered contour line is constructed according to the vertex sequence of the circumscribed polygon to ensure the continuity of the contour. For example, a closed contour line can be formed by connecting the vertices along the edge of the polygon. For the obtained ordered contour line, the image magnification processing algorithm is used to magnify the ordered contour line to obtain the target contour line.

[0134] Combine the following Figure 6 Let me introduce it in detail.

[0135] Figure 6 This is a schematic diagram of an embodiment of the present application in which two or more osteotomy surfaces exist.

[0136] like Figure 6 As shown, for the femur, since there are two cross sections, the convex hull algorithm is first used to calculate the circumscribed polygon, and then the boundary line is proportionally enlarged. The thinning algorithm is used to calculate the intervals and remove the points, and then the points are connected to generate the boundary line annotation information.

[0137] It can be understood that the convex hull algorithm can be used to determine the minimum external boundary enclosed by all osteotomy surfaces. The contour line can be further determined based on the outer polygon calculated by the convex hull algorithm, so that the generated contour line can be the minimum external boundary, avoiding overlap with other areas and preventing damage to the surrounding tissues of the osteotomy site.

[0138] Combine the following Figure 7 To specifically demonstrate the effect of generating an osteotomy surface contour.

[0139] Figure 7 This is a rendering of an osteotomy surface contour generation provided in an embodiment of the present application.

[0140] like Figure 7 As shown in the figure, in robot-assisted total knee osteotomy, in order to protect the patient's posterior cruciate ligament from being damaged during the operation, the boundary line of the osteotomy plane needs to be marked in the surgical planning stage. Figure 7 The blue line in the figure is the final generated boundary line.

[0141] Based on the osteotomy surface contour generation method provided in the above embodiment, the present application also provides a specific implementation of an osteotomy surface contour generation device. Please refer to the following embodiment.

[0142] Figure 8 It is a schematic diagram of an osteotomy surface contour generating device provided in an embodiment of the present application.

[0143] See first Figure 8 The osteotomy surface contour generating device 1000 provided in the embodiment of the present application includes the following modules:

[0144] The determination module 1001 is used to determine the osteotomy surface in the three-dimensional model corresponding to the osteotomy site.

[0145] The processing module 1002 is used to determine the contour points of the osteotomy surface according to the intersection relationship between the osteotomy surface and a plurality of triangular facets, where the plurality of triangular facets are components of the three-dimensional model corresponding to the osteotomy site.

[0146] The sorting module 1003 is used to sort the contour points of the osteotomy surface according to the coplanar relationship between any adjacent contour points to obtain ordered contour points.

[0147] As an implementation of the present application, the processing module 1002 may further include a first processing submodule, which is used to determine the ordered boundary points of the osteotomy surface according to the ordered contour points; and connect the ordered boundary points to generate the osteotomy surface boundary line.

[0148] It can be understood that the marking of the boundary line helps to clarify the specific area that needs to be osteotomized during surgery, helping doctors to clearly identify the cutting range and avoid accidental damage to surrounding tissues and structures. Compared with directly connecting the ordered contour points to form a boundary line, determining the ordered boundary points of the osteotomy surface based on the ordered contour points can provide more room for surgical operations and avoid damage to surrounding tissues due to factors such as cutting errors; if the ordered contour points are directly connected to form a boundary line, the distance between the boundary line and the surrounding tissues may be too close, which is not conducive to protecting the surrounding tissues.

[0149] As an implementation method of the present application, the above-mentioned determination module 1001 may also include a first determination submodule, which is used to determine the contour points of the osteotomy surface based on the intersection relationship between the osteotomy surface and multiple triangular facets, including: judging whether the spatial position relationship between the osteotomy surface and the multiple triangular facets is an intersection relationship; when the spatial position relationship between the osteotomy surface and the multiple triangular facets is an intersection relationship, identifying the points where the osteotomy surface intersects with the multiple triangular facets as contour points.

[0150] It can be understood that triangular patches are usually composed of high-resolution mesh models. By taking the intersection of the osteotomy surface and the triangular patch as the contour point, the position of the contour point can be calculated more accurately in the calculation compared to the low-resolution image, thereby improving the positioning accuracy of the contour point and ensuring the accuracy of the subsequently generated contour line.

[0151] As an implementation method of the present application, the above-mentioned device may also include a judgment module, which is used to judge whether the spatial position relationship between the osteotomy surface and multiple triangular facets is an intersection relationship, including: respectively obtaining the endpoints corresponding to multiple triangular facets, wherein the endpoints of any triangular facet include at least a first endpoint and a second endpoint; for any triangular facet, calculating the distance between the first endpoint and the osteotomy surface to obtain a first distance; calculating the distance between the second endpoint and the osteotomy surface to obtain a second distance; calculating the projection distance of the line segment between the first endpoint and the second endpoint in the normal direction of the osteotomy surface to obtain a third distance; when the sum of the first distance and the second distance is equal to the third distance, it is determined that the spatial position relationship between the osteotomy surface and the triangular facet where the first endpoint and the second endpoint are located is an intersection relationship.

[0152] By respectively calculating the distances between any two endpoints of the triangular patch and the osteotomy surface, it is determined whether the osteotomy surface and the triangular patch intersect. The calculation principle is simple and easy to understand, the amount of calculation is small, and the calculation efficiency is improved.

[0153] As an implementation method of the present application, the above-mentioned device may also include a recognition module, which is used to identify the points where the osteotomy surface intersects with multiple triangular facets as contour points when the spatial position relationship between the osteotomy surface and the multiple triangular facets is an intersection relationship, including: calculating the coordinate value of the contour point using the principle of similar triangles based on the first distance, the third distance and the spatial distance between the first endpoint and the second endpoint.

[0154] It can be understood that using the similarity principle of triangles to calculate the coordinates of contour points can transform complex geometric problems into simple proportional relationships. The calculation method is simple and can also reduce the accumulation of errors and improve the accuracy of the calculation results.

[0155] As an implementation method of the present application, the above-mentioned sorting module 1003 may also include a first sorting submodule, which is used to sort the contour points of the osteotomy surface to obtain ordered contour points, including: selecting any contour point as the starting point; searching for the next contour point according to the starting point, wherein the search condition is that the starting point and the next contour point are located in the same triangular facet; when the next contour point is found, the next contour point is used as the starting point, and the search for the next contour point according to the starting point is executed again until all contour points of the osteotomy surface are found.

[0156] It can be understood that searching point by point based on the coplanar relationship can effectively prevent repeated searching of processed points and reduce redundant calculations. At the same time, it can ensure that adjacent points are located in the same plane, thereby ensuring the continuity of the contour line.

[0157] As an implementation method of the present application, the above-mentioned device may also include a search module, which is used to search for the next contour point based on the starting point, including: calculating the normal vector of the triangle where the starting point is located to obtain a first vector; calculating the vector between the starting point and any contour point to be sorted to obtain a second vector; judging whether the starting point and any contour point to be sorted are located in the same triangle based on the product of the first vector and the second vector; when the starting point and any contour point to be sorted are located in the same triangle, determining any contour point to be sorted as the next contour point.

[0158] Combined with the above, it can be understood that by determining any point A as the starting point, and knowing the coordinates of the three vertices of the triangular patch, the normal vector of the triangular patch can be calculated; multiply the normal vector of the triangular patch by vectors AB and AC respectively. If they are equal to 0, it means that the line segment is perpendicular to the triangular patch, which proves that the line segment is in the same plane, and the next point of point A is point B. Then find a new point C on the same plane according to the normal vector of face2. Similarly, all contour points are sorted. When the traversal point information is the coordinate information of point A, it means that the contour is closed, and the ordered point information of the intersection surface contour of the osteotomy surface and the corresponding three-dimensional model is generated. Using normal vector calculation to determine whether a contour point is located on a certain triangular patch is simple to calculate and efficient to process, which is convenient for fast calculation in practical applications.

[0159] As an implementation method of the present application, the above-mentioned device may also include a thinning module, which is used to determine the ordered boundary points of the osteotomy surface based on the ordered contour points, including: connecting the ordered contour points to obtain an ordered contour line; performing image magnification processing on the ordered contour line to obtain a target contour line; performing thinning calculation based on the target contour line to obtain ordered boundary points.

[0160] It can be understood that the use of image magnification processing can retain the original boundary features, maintain the basic shape of the entire contour, and avoid obvious loss of the overall contour line appearance; through thinning, the number of points in the contour line can be reduced, and the number of ordered points can be simplified, which is conducive to reducing computational complexity and improving the efficiency of subsequent processing or rendering.

[0161] As an implementation method of the present application, the above-mentioned device may also include a convex hull module, which is used to calculate the circumscribed polygons corresponding to two or more osteotomy surfaces using a convex hull algorithm; and perform image magnification processing on the ordered contour lines according to the circumscribed polygons to obtain the target contour lines.

[0162] It can be understood that the convex hull algorithm can be used to determine the minimum external boundary enclosed by all osteotomy surfaces. The contour line can be further determined based on the outer polygon calculated by the convex hull algorithm, so that the generated contour line can be the minimum external boundary, avoiding overlap with other areas and preventing damage to the surrounding tissues of the osteotomy site.

[0163] Fig. 9 It is a schematic diagram of the hardware structure of the osteotomy surface contour generation device provided in an embodiment of the present application.

[0164] The osteotomy surface contour generating device may include a processor 2001 and a memory 2002 storing computer program instructions.

[0165] Specifically, the processor 2001 may include a central processing unit (CPU), or an application specific integrated circuit (ASIC), or may be configured to implement one or more integrated circuits of the embodiments of the present application.

[0166] The memory 2002 may include a large capacity memory for data or instructions. By way of example and not limitation, the memory 2002 may include a hard disk drive (HDD), a floppy disk drive, a flash memory, an optical disk, a magneto-optical disk, a magnetic tape, or a universal serial bus (USB) drive or a combination of two or more of these. In appropriate cases, the memory 2002 may include a removable or non-removable (or fixed) medium. In appropriate cases, the memory 2002 may be inside or outside the integrated gateway disaster recovery device. In a specific embodiment, the memory 2002 is a non-volatile solid-state memory.

[0167] In certain embodiments, the memory may include a read-only memory (ROM), a random access memory (RAM), a magnetic disk storage medium device, an optical storage medium device, a flash memory device, an electrical, optical or other physical / tangible memory storage device. Thus, typically, the memory includes one or more tangible (non-transitory) computer-readable storage media (e.g., a memory device) encoded with software including computer-executable instructions, and when the software is executed (e.g., by one or more processors), it is operable to perform the operations described with reference to a method according to an aspect of the present disclosure.

[0168] The processor 2001 reads and executes the computer program instructions stored in the memory 2002 to implement any one of the osteotomy surface contour generation methods in the above embodiments.

[0169] In one example, the osteotomy surface contour generating device may further include a communication interface 2003 and a bus 2000. Fig. 9 As shown, the processor 2001, the memory 2002, and the communication interface 2003 are connected via a bus 2000 and communicate with each other.

[0170] The communication interface 2003 is mainly used to implement communication between various modules, devices, units and / or equipment in the embodiments of the present application.

[0171] Bus 2000 includes hardware, software or both, and the parts of online data flow billing equipment are coupled to each other. For example, but not limitation, bus may include accelerated graphics port (AGP) or other graphics bus, enhanced industrial standard architecture (EISA) bus, front-end bus (FSB), hypertransport (HT) interconnection, industrial standard architecture (ISA) bus, infinite bandwidth interconnection, low pin count (LPC) bus, memory bus, micro channel architecture (MCA) bus, peripheral component interconnection (PCI) bus, PCI-Express (PCI-X) bus, serial advanced technology attachment (SATA) bus, video electronics standard association local (VLB) bus or other suitable bus or two or more of these combinations. In appropriate cases, bus 2000 may include one or more buses. Although the present application embodiment describes and shows a specific bus, the present application considers any suitable bus or interconnection.

[0172] In addition, in combination with the method for generating the osteotomy surface contour in the above embodiments, the embodiment of the present application can provide a computer storage medium for implementation. The computer storage medium stores computer program instructions; when the computer program instructions are executed by the processor, any method for generating the osteotomy surface contour in the above embodiments is implemented.

[0173] An embodiment of the present application also provides a computer program product, including a computer program, which, when executed, implements any one of the methods for generating an osteotomy surface contour in the above embodiments.

[0174] It should be clear that the present application is not limited to the specific configuration and processing described above and shown in the figures. For the sake of simplicity, a detailed description of the known method is omitted here. In the above embodiments, several specific steps are described and shown as examples. However, the method process of the present application is not limited to the specific steps described and shown, and those skilled in the art can make various changes, modifications and additions, or change the order between the steps after understanding the spirit of the present application.

[0175] The functional blocks shown in the above-described block diagram can be implemented as hardware, software, firmware or a combination thereof. When implemented in hardware, it can be, for example, an electronic circuit, an application specific integrated circuit (ASIC), appropriate firmware, a plug-in, a function card, etc. When implemented in software, the elements of the present application are programs or code segments that are used to perform the required tasks. The program or code segment can be stored in a machine-readable medium, or transmitted on a transmission medium or a communication link by a data signal carried in a carrier wave. "Machine-readable medium" can include any medium capable of storing or transmitting information. Examples of machine-readable media include electronic circuits, semiconductor memory devices, ROM, flash memory, erasable ROM (EROM), floppy disks, CD-ROMs, optical disks, hard disks, optical fiber media, radio frequency (RF) links, etc. The code segment can be downloaded via a computer network such as the Internet, an intranet, etc.

[0176] It should also be noted that the exemplary embodiments mentioned in this application describe some methods or systems based on a series of steps or devices. However, this application is not limited to the order of the above steps, that is, the steps can be performed in the order mentioned in the embodiment, or in a different order from the embodiment, or several steps can be performed simultaneously.

[0177] Aspects of the present disclosure are described above with reference to the flowchart and / or block diagram of the method, device (system) and computer program product according to the embodiment of the present disclosure. It should be understood that each box in the flowchart and / or block diagram and the combination of each box in the flowchart and / or block diagram can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing device to produce a machine so that these instructions executed by the processor of the computer or other programmable data processing device enable the implementation of the function / action specified in one or more boxes of the flowchart and / or block diagram. Such a processor can be, but is not limited to, a general-purpose processor, a special-purpose processor, a special application processor, or a field programmable logic circuit. It can also be understood that each box in the block diagram and / or flowchart and the combination of boxes in the block diagram and / or flowchart can also be implemented by dedicated hardware that performs a specified function or action, or can be implemented by a combination of dedicated hardware and computer instructions.

[0178] The above is only a specific implementation of the present application. Those skilled in the art can clearly understand that for the convenience and simplicity of description, the specific working processes of the systems, modules and units described above can refer to the corresponding processes in the aforementioned method embodiments, and will not be repeated here. It should be understood that the protection scope of the present application is not limited to this. Any technician familiar with the technical field can easily think of various equivalent modifications or replacements within the technical scope disclosed in this application, and these modifications or replacements should be included in the protection scope of this application.

Claims

1. A method for generating an osteotomy surface contour, characterized in that: include: Determine the osteotomy surface in the three-dimensional model corresponding to the osteotomy site; Determining contour points of the osteotomy surface according to an intersection relationship between the osteotomy surface and a plurality of triangular facets, wherein the plurality of triangular facets are components of a three-dimensional model corresponding to the osteotomy site; The contour points of the osteotomy surface are sorted according to the coplanar relationship between any adjacent contour points to obtain ordered contour points.

2. The method according to claim 1, characterized in that The method further comprises: Determining the ordered boundary points of the osteotomy surface according to the ordered contour points; The ordered boundary points are connected to generate the osteotomy surface boundary line.

3. The method according to claim 2, characterized in that Determining the contour points of the osteotomy surface according to the intersection relationship between the osteotomy surface and a plurality of triangular facets comprises: Determine whether the spatial position relationship between the osteotomy surface and the plurality of triangular facets is an intersection relationship; When the spatial position relationship between the osteotomy surface and the plurality of triangular facets is an intersection relationship, the points where the osteotomy surface and the plurality of triangular facets intersect are identified as contour points.

4. The method according to claim 3, characterized in that The determining whether the spatial position relationship between the osteotomy surface and the plurality of triangular facets is an intersection relationship comprises: Respectively obtain endpoints corresponding to the plurality of triangular facets, wherein the endpoints of any of the triangular facets include at least a first endpoint and a second endpoint; For any of the triangular facets, calculating the distance between the first endpoint and the osteotomy surface to obtain a first distance; Calculating the distance between the second endpoint and the osteotomy surface to obtain a second distance; Calculate the projection distance of the line segment between the first endpoint and the second endpoint in the normal direction of the osteotomy surface to obtain a third distance; When the sum of the first distance and the second distance is equal to the third distance, it is determined that the spatial position relationship between the osteotomy surface and the triangular surface where the first endpoint and the second endpoint are located is an intersection relationship.

5. The method according to claim 4, characterized in that When the spatial position relationship between the osteotomy surface and the plurality of triangular facets is an intersection relationship, identifying the points where the osteotomy surface intersects with the plurality of triangular facets as contour points comprises: The coordinate value of the contour point is calculated according to the first distance, the third distance, and the spatial distance between the first endpoint and the second endpoint.

6. The method according to any one of claims 1 to 5, characterized in that The step of sorting the contour points of the osteotomy surface to obtain ordered contour points includes: Select any one of the contour points as the starting point; Searching for a next contour point according to the starting point, wherein a search condition is that the starting point and the next contour point are located in the same triangle patch; When the next contour point is found, the next contour point is used as the starting point, and the step of searching for the next contour point according to the starting point is performed again until all contour points of the osteotomy surface are found.

7. The method according to claim 6, characterized in that The step of searching for the next contour point according to the starting point comprises: Calculate the normal vector of the triangle where the starting point is located to obtain a first vector; Calculate the vector between the starting point and any contour point to be sorted to obtain a second vector; Determining whether the starting point and any one of the contour points to be sorted are located in the same triangle patch according to the product of the first vector and the second vector; In the case where the starting point and any one of the contour points to be sorted are located in the same triangle patch, the any one of the contour points to be sorted is determined as the next contour point.

8. The method according to any one of claims 2 to 5 or 7, characterized in that: Determining the ordered boundary points of the osteotomy surface according to the ordered contour points comprises: Connecting the ordered contour points to obtain ordered contour lines; Performing image magnification processing on the ordered contour lines to obtain target contour lines; A thinning calculation is performed according to the target contour line to obtain ordered boundary points.

9. The method according to claim 8, characterized in that In the case where there are two or more osteotomy surfaces, the method further comprises: Calculating the circumscribed polygons corresponding to the two or more osteotomy surfaces using a convex hull algorithm; The ordered contour lines are subjected to image magnification processing according to the circumscribed polygon to obtain target contour lines.

10. A device for generating an osteotomy surface contour, characterized in that: include: A determination module, used to determine the osteotomy surface in the three-dimensional model corresponding to the osteotomy site; A processing module, for determining the contour points of the osteotomy surface according to the intersection relationship between the osteotomy surface and a plurality of triangular facets, wherein the plurality of triangular facets are components of the three-dimensional model corresponding to the osteotomy site; The sorting module is used to sort the contour points of the osteotomy surface according to the coplanar relationship between any adjacent contour points to obtain ordered contour points.

Images