JP2018136797A - Embedded graph simplification device, embedded graph simplification method, and computer program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an embedded graph simplification device by which an apex position is not limited to an apex position of an original embedded graph.SOLUTION: An embedded graph simplification device is provided with an embedded graph aligning unit which aligns apex positions. The embedded graph aligning unit is provided with: an optimization problem equation establishment unit which establishes an equation of an optimization problem for alignment to be output; and an optimization problem solution unit which solves the optimization problem to calculate the aligned apex positions. The optimization problem equation establishment unit includes an alignment regularization function establishment unit which generates an alignment regularization function for evaluating a degree that the aligned embedded graphs are close to a straight line by using the apex positions as variables. The alignment regularization function establishment unit performs weighting synthesis so that an influence by size of two sides to be connected with the apexes becomes relatively small, and an influence by an angle to be formed by the two sides becomes relatively large to generate the alignment regularization function.SELECTED DRAWING: Figure 11

Description

  The present invention relates to an embedded graph simplification device, an embedded graph simplification method, and a computer program.

Data structures called embedded graphs exist in various fields such as networks, geographic informatics, and image engineering. An embedded graph is defined by combining a graph, which is a set of edges connecting a plurality of vertices and their vertices, and the vertex positions. For example, in a network, a graph can be constructed by associating network nodes with vertices and network links with edges.
At this time, the embedded graph can be found by associating the physical position of the network node or the node coordinates when visualizing the network with the vertex position. In addition, in the case of geographic informatics, border line data such as borders, prefectural borders, coast lines, lake shore lines, etc. are generally held as embedded graphs. The boundary line is expressed by densely plotting vertices on the boundary line and connecting the edges. In the field of image engineering, an embedded graph is used to hold data on shapes and areas. For example, data such as an object shape in a vector image, a depth boundary line in a depth map, a subject shape of a natural image, and a texture region shape are represented as an embedded graph. By plotting the vertices on the boundary of the shape or area, the boundary line is expressed by an edge.

  In the embedded graph illustrated above, the shape of the embedded graph is maintained as much as possible, and the number of vertices and the number of edges are reduced as much as possible (hereinafter referred to as simplification of the embedded graph or simply simplified). Has a great advantage.

  In a network, more data can be expressed as the number of vertices and the number of sides increases. On the other hand, it becomes difficult to grasp the outline of the network configuration at a glance. Take it. If the embedded graph for this network can be simplified, the network configuration can be presented visually intelligibly, the number of vertices and sides can be reduced, data can be compressed, and the drawing load can be reduced. In geographic informatics, a boundary line is accurately expressed by densely plotting vertices on the boundary line, but the amount of data becomes enormous. By simplifying the boundary line data, it is possible to reduce the amount of data to be transmitted and stored, and to generate visually easy-to-understand geographical information with simplified boundaries. In image engineering, as the number of vertices and sides increases, shape and area data can be retained in detail and expressed accurately, but the amount of data becomes enormous and the amount of code required for transmission and storage increases. End up. If this shape and area can be simplified, the number of vertices and the number of sides can be reduced and the amount of codes can be reduced while maintaining the accuracy of expression as much as possible.

  As an effective method for simplifying a very simple embedded graph such as a line graph or a polygon, there is a Ramer-Doubles-Peucker method (NDP 1 or later, RDP method). Hereinafter, an algorithm when the RDP method is applied to a broken line segment will be described. In this method, (1) First, a line segment connecting the vertices of both ends of a broken line segment is set as an initial broken line segment. (2) Sequentially add vertices farthest from the current line segment to the current line segment. (3) When the distance to the farthest vertex is smaller than a predetermined allowable error ε (> 0), the process is terminated.

  FIG. 13 shows an operation example of the RDP method. In FIG. 13, white circles and broken lines indicate line components of the input embedded graph, and black circles and solid lines indicate line components of the simplified embedded graph. FIG. 13A shows an initial state (state ST0). In the initial state, a line segment connecting the end points of the broken line segment is defined as an initial broken line segment E1. The vertex to be added in the next step is the vertex V1 farthest from the broken line segment E1. As shown in FIG. 13B, in the state ST1, the vertex V1 is added to the polygonal line component E1 to form a new polygonal line component E2. The vertex to be added in the next step is the vertex V2 farthest from the broken line segment E2. As shown in FIG. 13C, in the state ST2, the vertex V2 is added to the polygonal line component E2 to obtain a new polygonal line component E3. The vertex added in the next step is the vertex V3 farthest from the broken line segment E3. As shown in FIG. 13D, in the state ST3, the vertex V3 is added to the polygonal line component E3 to obtain a new polygonal line component E4. In the state ST3, all the vertices are within the allowable error from the broken line segment E4, and the simplification process ends here.

  In the case of a polygon, the RDP method can be applied by appropriately selecting two points from polygon vertices and setting them as the initial broken line instead of the end points. Furthermore, in the RDP method, simplification can be performed so that a specific vertex designated by the user is always included at the same vertex position as that of the input embedded graph. The specified vertex is added when the initial line segment is created, and the process is started.

  The RDP method is most often used because of its high accuracy of shape approximation, ease of mounting, and the like. For example, OpenCV, which is a computer program library for computer vision, employs the RDP method for simplifying a subject shape and approximating a contour.

Hershberger and Snoeyink, "An O (n log n) implementation of the Douglas-Peucker algorithm for line simplification," pp. 383-384, 1994.

  The RDP method described above has the following two problems. (1) It is difficult to apply to complicated embedded graphs excluding broken line segments and polygons. (2) Since the vertex position of the simplified embedded graph strictly protects the vertex position of the original embedded graph, it is not possible to calculate better simplification by slightly shifting the vertex position.

  The above problem (1) will be described. As seen in the network, map / traffic analysis, and image / video coding examples described above, embedded graphs generally have branches, multiple loops, unconnected graphs, etc., and have a more complex structure than line segments and polygons. Have. The RDP method cannot simplify such a complex graph. Alternatively, the RDP method can be applied to complex graphs by performing pre-processing that decomposes into polygonal lines and polygons and post-processing that combines the decomposed graphs. However, since the number of decompositions increases and the number of endpoints increases in a complex graph with a large number of branches, it is more difficult to reduce vertices and edges in a complex graph.

  The above problem (2) will be described. In the RDP method, since a new embedded graph is configured by selecting vertices of the original embedded graph, the vertex positions of the new embedded graph are always limited to the same vertex positions as the original embedded graph. In general, better simplification can be achieved if the vertex position is allowed to be slightly displaced, but this cannot be calculated by the RDP method. For the same reason, the constraint imposed on the specific vertex by the user can only specify that the vertex position is the same as the input.

In view of the above circumstances, an object of the present invention is to provide an embedded graph simplification device, an embedded graph simplification method, and a computer program whose vertex positions are not limited to the vertex positions of the original embedded graph.
Further, a secondary problem of the present invention is to make it possible to reduce vertices and edges even in a complicated embedded graph.

  One aspect of the present invention is an embedding graph simplification device that simplifies an embedding graph having vertex position information, and embedding graph alignment that aligns vertex positions so that an angle formed by edges connected to each vertex is reduced. The embedded graph alignment unit includes an optimization problem expression unit that forms and outputs an optimization problem for alignment, and an optimization for alignment output by the optimization problem expression unit. An optimization problem solving unit that solves the optimization problem and obtains aligned vertex positions; and a vertex information replacement unit that replaces the position information of the vertices of the embedded graph with the vertex positions obtained by the optimization problem solving unit; The optimization problem expression unit generates an alignment regularization function that evaluates the degree to which the aligned embedded graph is close to a straight line using a vertex position of the aligned embedded graph as a variable. A regularization function formula part, a constraint condition formula part for generating a constraint condition relating to the range of the vertex positions after being aligned, and an object based on the alignment regularization function generated by the alignment regularization function formula part An optimization problem generation unit that generates an optimization problem by combining a function and the constraint condition, and the alignment regularization function expression unit includes a direction of an edge connected to each vertex included in the embedded graph, and A side vector representing a size is obtained, and the size of the two sides connected to the vertex and the angle formed by the two sides are determined so that the influence of the size of the two sides is relatively small. This is an embedded graph simplification device that generates the alignment regularization function by weighting and synthesizing so that the influence of the formed angle becomes relatively large.

  One aspect of the present invention is the above-described embedded graph simplification device, wherein the optimization problem equation unit uses the vertex position of the aligned embedded graph as a variable, and the variable is the original vertex position. A vertex position change loss function formula part that generates a vertex position change loss function that evaluates how far apart, and a multiplication that multiplies the alignment regularization function output by the alignment regularization function formula part by a trade-off parameter. And an adder that linearly combines the vertex position change loss function and the aligned regularization function multiplied by the trade-off parameter to generate an objective function, and the optimization problem generator includes the optimization problem generator An embedded graph simplification device that generates an optimization problem by combining the objective function output from an adder and the constraint condition.

  One aspect of the present invention is the above-described embedding graph simplification device, wherein a vertex having a small angle between connecting edges is selected from the embedding graph aligned by the embedding graph alignment unit, and the connecting edges are formed. The embedded graph simplification device further includes an unnecessary vertex removal unit that removes a vertex having a small angle.

One aspect of the present invention is the above-described embedded graph simplification device, in which the alignment regularization function expression unit includes vertices included in the embedded graph having vertices with two connected edges, An edge vector generation linear map generation unit for generating an edge vector generation linear map L _m (m is an index of the vertex, m = 1,..., M) representing the size and direction of each edge connected to the vertex; The edge vector generation linear map L _m output from the edge vector generation linear map generation unit and the variable U indicating the vertex information of the embedded graph are input, and the edge vector matrix P _m (P _m = L _m U) and the side vector matrix generating unit that generates, the edge vector matrix generating unit the side vector matrix P _m outputted, the first singular value corresponding to the magnitude of the sides, the second corresponding to the angle at which the two sides forms Singular value and singularity to find Weighting of the first singular value and the second singular value using a value decomposing unit, and a first weight value w ₁ and a second weight value w ₂ respectively corresponding to the first singular value and the second singular value A weighted sum generation unit that generates a sum; and a sum generation unit that generates a sum of all vertices of the weighted sum or the square of the weighted sum output by the weighted sum generation unit as an aligned regularization function. The weighted sum generation unit is an embedded graph simplification device that uses the first weight value w ₁ and the second weight value w ₂ that satisfy the relationship (w ₂ ≧ w ₁ ≧ 0).

  One aspect of the present invention is an embedding graph simplification method for simplifying an embedding graph having vertex position information, and embedding graph alignment for aligning vertex positions so that an angle formed by edges connected to each vertex is reduced. The embedded graph alignment processing step is output in the optimization problem formulation processing step for formulating and outputting an optimization problem for alignment, and the optimization problem formulation processing step. An optimization problem solution processing step for finding an optimization problem for alignment and obtaining aligned vertex positions, and vertex position information of the embedded graph at the vertex positions determined in the optimization problem solution processing step And the optimization problem formula processing step uses the vertex position of the aligned embedded graph as a variable. An alignment regularization function formula processing step for generating an alignment regularization function for evaluating the degree of the aligned embedded graph being close to a straight line, and a constraint condition formula for generating a constraint condition regarding the range of the vertex positions after the alignment And an optimization problem generation processing step for generating an optimization problem by combining the objective function based on the alignment regularization function generated in the alignment regularization function formula processing step and the constraint condition. The alignment regularization function equation processing step obtains an edge vector representing the direction and size of an edge connected to each vertex included in the embedded graph, and calculates the size of the two edges connected to the vertex and the 2 The angle formed by two sides is weighted so that the effect of the size of the two sides is relatively small and the effect of the angle formed by the two sides is relatively large. To generate the alignment regularization function, an embedded graph simplification method.

  One embodiment of the present invention is a computer program for causing a computer to function as the above-described embedded graph simplification apparatus.

  According to the present invention, vertices and edges can be reduced even in a complicated embedded graph, and the vertex position is not limited to the vertex position of the original embedded graph, and the embedded graph can be simplified.

It is a block diagram which shows the structure of the embedded graph simplification apparatus by 1st Embodiment of this invention. FIG. 3 is a block diagram of an embedded graph straight line alignment unit according to the first embodiment of the present invention. FIG. 3 is a block diagram of an optimization problem formula for straight line alignment according to the first embodiment of the present invention. It is a block diagram which shows the detailed function structure of the linear alignment regularization function equation part by 1st Embodiment of this invention. It is a flowchart of the embedded graph simplification process by 1st Embodiment of this invention. It is a flowchart of the embedded graph straight line alignment process by 1st Embodiment of this invention. 6 is a flowchart of optimization problem formulation processing for straight line alignment according to the first embodiment of the present invention; It is a figure which shows the example of the embedding graph used as the input in the embedding graph simplification apparatus by 1st Embodiment of this invention. It is a figure which shows the example of the line-aligned embedding graph in the embedding graph simplification apparatus by 1st Embodiment of this invention. It is a block diagram which shows the function structure of the optimization problem formulation part for the linear alignment by 2nd Embodiment of this invention. It is a block diagram which shows the detailed function structure of the linear alignment regularization function equation part by 2nd Embodiment of this invention. It is the schematic for demonstrating the problem which the embedding graph simplification apparatus by 2nd Embodiment of this invention avoids. It is a figure which shows the outline | summary (prior art) of RDP method which is an embedded graph simplification method.

Embodiments of the present invention will be described below with reference to the drawings.
[First Embodiment]
FIG. 1 is a block diagram showing a configuration of an embedded graph simplification apparatus 1 according to an embodiment of the present invention. The embedded graph simplification apparatus 1 includes a CPU (Central Processing Unit) connected via a bus, a memory, an auxiliary storage device, and the like, and executes a program. The embedded graph simplification device 1 functions as a device including an embedded graph straight line alignment unit 11 and an unnecessary vertex removal unit 12 by executing a program. All or some of the functions of the embedded graph simplification apparatus 1 are realized by using hardware such as an application specific integrated circuit (ASIC), a programmable logic device (PLD), and a field programmable gate array (FPGA). Also good. The program may be recorded on a computer-readable recording medium. The computer-readable recording medium is, for example, a portable medium such as a flexible disk, a magneto-optical disk, a ROM, a CD-ROM, or a storage device such as a hard disk built in the computer system. The program may be transmitted via a telecommunication line.

In FIG. 1, an embedded graph simplification apparatus 1 inputs an original embedded graph G = (V, E) and a trade-off parameter λ, and a simplified embedded graph G ^* = (V ^* , E ^* ). Is output. Here, as shown in FIG. 1, the embedded graph is two-dimensional data indicating a shape by connecting a plurality of points arranged at different positions and connecting the plurality of points. In a broader concept, it can be said that this is a two-dimensional data simplification device that simplifies such two-dimensional data.

  In the original embedding graph G = (V, E), V is the vertex information of the original embedding graph G, and includes the vertex ID (IDentification) of each vertex and information on the position of the vertex. As an example of the data structure, when N vertices distributed in the D-dimensional coordinate space are considered, an integer vertex ID from “1” to “N” is arbitrarily assigned to each vertex, and the coordinate value of each vertex is assigned to the vertex. If the N-row D-column matrix arranged in the order of ID is V, the above vertex information requirement is satisfied. In the present embodiment, following this example, the original vertex information V is described as an N-row D-column matrix. In addition to the data structure of the N-row D-column matrix, another data structure that can be converted equivalently may be used. E is edge information of the original embedded graph G, and includes information indicating which vertices are connected to form each edge. Examples of the data structure of the edge information E include an adjacency list, an adjacency matrix, a connection matrix, a Laplacian matrix, and a Tute matrix. Any of these may be used because they can be converted equivalently.

In the simplified embedded graph G ^* = (V ^* , E ^* ), V ^* is the vertex information of the simplified embedded graph G ^* , and E ^* is the edge information of the simplified embedded graph G ^*. is there. The data structure of the vertex information V ^* and the edge information E ^* of the simplified embedded graph G ^* is the same data structure as the vertex information V and the edge information E of the original embedded graph G. However, the simplified embedded graph G ^* has a reduced number of vertices and sides compared to the original embedded graph G.

  The trade-off parameter λ is a positive real value. The larger the tradeoff parameter λ, the stronger the degree of simplification, and the smaller the number of vertices of the embedded graph can be adjusted.

  As shown in FIG. 1, the embedded graph simplification apparatus 1 includes an embedded graph straight line alignment unit 11 and an unnecessary vertex removal unit 12. The embedded graph linear alignment unit 11 inputs the original embedded graph G = (V, E) and the trade-off parameter λ, and outputs a linearly aligned embedded graph G ′ = (V ′, E).

  In this embodiment, “straight line alignment” means to change the vertex position so that at least a part of a series of vertices constituting the embedded graph becomes, for example, a zigzag shape as close to a straight line as possible. It is. As a result of this straight line alignment processing, a series of vertexes may be arranged in a truly straight line, or the degree close to a straight line may be increased although it does not become a true straight line. “Linear alignment” is also simply referred to as “alignment”.

  The edge information E of the embedded graph G ′ is the same information as the edge information E of the original embedded graph G. The data structure and vertex ID of the vertex information V ′ are the same as the original vertex information V, and only the vertex position is different from the vertex information V. Further, the embedded graph straight line aligning unit 11 generates an embedded graph G ′ that is deformed so as to be linearly aligned with respect to the original embedded graph G. For example, the embedded graph straight line alignment unit 11 moves the coordinates of the vertices of the embedded graph G minutely as shown in FIG. 1 and linearly aligns so that many vertices are arranged on one straight line (hereinafter, The embedded graph G ′ is generated. In other words, the embedded graph linear alignment unit 11 linearly aligns the vertex positions so that the angle formed by the edges connected to each vertex is small. Here, “the angle formed by the sides connected to each vertex is reduced” means that the angle formed by the two sides connected to a specific vertex approaches 180 degrees (straight line).

  In addition, the embedded graph straight line alignment unit 11 increases the degree of line alignment as the trade-off parameter λ increases. In other words, as the trade-off parameter λ is larger, the shape of the embedded graph G ′ is represented by a smaller number of straight lines.

The unnecessary vertex removing unit 12 inputs the embedded graph G ′ = (V ′, E) aligned in a straight line, and outputs a simplified embedded graph G ^* = (V ^* , E ^* ). At many vertices of the linearly aligned embedded graph G ′, the angle formed by the connecting edges is small. Therefore, the unnecessary vertex removal unit 12 generates an embedded graph G ^* = (V ^* , E ^* ) simplified by selecting and removing vertices whose angle is equal to or less than a predetermined threshold, that is, decimating. For example, FIG. 1 shows an example in which the black dots of the line-aligned embedded graph G ′ are removed as unnecessary vertices and the line-aligned embedded graph G ′ configured with white points is generated.

  FIG. 2 is a block diagram showing a detailed configuration of the embedded graph linear alignment unit 11 shown in FIG. As shown in FIG. 2, the embedded graph straight line aligning unit 11 includes an optimization problem formulating unit 111 for straight line aligning, an optimization problem solving unit 112, and a vertex information replacing unit 113.

  The optimization problem formulation unit 111 for linear alignment inputs the original embedded graph G = (V, E) and the trade-off parameter λ, and outputs an optimization problem for linear alignment. The optimization problem to be output is a mathematical expression describing an objective function and constraint conditions, and is represented by Expression (1). This optimization problem may be a convex optimization problem or a non-convex optimization problem.

  This equation (1) is defined as the variable indicating the vertex information of the straight line-aligned embedded graph to be obtained, and the objective function value on the right side of “minimize” is minimized among the variables U satisfying the constraints on the right side of “subject to”. The optimization problem of “find the matrix U of N rows and D columns” is expressed.

  The objective function consists of a linear combination of two terms. The first term is a cost for evaluating how far the vertex information V of the original embedded graph and the variable U differ from each other when the vertex information of the linearly aligned embedded graph is a variable U, that is, an evaluation value. is there. That is, the first term can be said to be a fidelity term indicating whether extreme changes in vertex coordinates due to alignment can be suppressed.

  The second term is a cost for evaluating the degree to which the embedded graph (U, E) is aligned in a straight line when the vertex information of the embedded graph aligned in a straight line is a variable U, that is, an evaluation value. In other words, the second term can be said to be a regularization term indicating a penalty for side bending, in other words, a smoothness of straight line alignment.

  The constraint condition represents the movable range of the variable U indicating the vertex information of the embedded graph arranged in a straight line. This optimization problem is designed so that the vertex information V ′ aligned in a straight line is generated by obtaining a solution by the subsequent optimization problem solving unit 112.

In Expression (1), f is a vertex position change loss function, and g is an edge vector angle evaluation function. The vertex position change loss function f and the edge vector angle evaluation function g are differentiable functions or functions that can be calculated by proximity mapping. L _m is an edge vector generation linear mapping. C is a constraint set. The constraint set C is a set whose projection can be calculated.

The optimization problem solving unit 112 receives the optimization problem, obtains a solution thereof, and outputs the vertex information V ′ aligned in a straight line. An algorithm for obtaining a solution can be arbitrarily selected. For example, in the PDS method (Primal Dual Splitting Method), the gradient calculation of the vertex position change loss function f, or the proximity mapping, the gradient calculation of the edge vector angle evaluation function g, or the proximity mapping, the edge vector generation linear mapping L _m and its conjugate. The optimization problem can be solved by an iteration combining the mapping L _m ^T (m = 1,..., M), the projection onto the constraint set C, and the four arithmetic operations. The PDS method is described in Reference Document 1 below.

[Reference 1]: Condat, “A Generic Proximal Algorithm for Convex Optimization-Application to Total Variation Minimization”, Signal Process IEEE, vol. 21, no. 8, pp. 985-989, 2014.

  The vertex information replacement unit 113 receives the vertex information V ′ aligned in a straight line and the original embedded graph G = (V, E), and replaces the original vertex information V with the vertex information V ′ aligned in a straight line. The embedded graph G ′ = (V ′, E) that is linearly aligned is output. That is, the vertex information replacement unit 113 replaces the vertex position information of the embedded graph with the vertex position obtained by the optimization problem solving unit 112.

  FIG. 3 is a block diagram showing a detailed configuration of the optimization problem formula unit 111 for linear alignment shown in FIG. As shown in FIG. 3, the optimization problem equation unit 111 for line alignment includes a vertex side information decomposition unit 1111, a vertex position change loss function equation unit 1112, a line alignment regularization function equation unit 1113, and , A multiplier 1114, an adder 1115, a constraint condition expression unit 1116, and an optimization problem generation unit 1117.

  The vertex edge information decomposition unit 1111 receives the original embedded graph G = (V, E), decomposes it into the original vertex information V and the original edge information E, and outputs the result.

  The vertex position change loss function equation unit 1112 receives the original vertex information V and outputs a vertex position change loss function f. The vertex position change loss function f is a cost for evaluating how far the original vertex information V is deviated from the variable U when the vertex information of the line-aligned embedded graph is the variable U. The vertex position change loss function f evaluates an absolute value residual or a square residual between the variable U and the original vertex information V. As the vertex position change loss function f, functions exemplified in the equations (2) to (4) can be used.

In equations (2) to (4), v _{n, d} and u _{n, d} are the n-row and d-column components of the matrix of vertex information V and the matrix of variable U, respectively, and vertex information whose vertex ID is n Represents the value of the d-th axis component. Note that “vec” in the equations (3) and (4) is a function for rearranging the elements of the matrix into column vectors. The function used as the vertex position change loss function f may be a convex function or a non-convex function.

  The straight line alignment regularization function equation unit 1113 receives the edge information E, and outputs a straight line alignment regularization function (alignment regularization function) for evaluating the angle formed by the edges connected to each vertex. The straight line alignment regular function is a function for evaluating the degree to which a series of vertices is close to a straight line. As the linear alignment regularization function, the one exemplified in the equation (5) can be used.

The function shown in equation (5) is a cost for evaluating the degree to which the embedded graph (U, E) using the variable U is linearly aligned when the vertex information of the embedded graph aligned in a straight line is the variable U. is there. This cost is composed of an edge vector generation linear map L _m (m = 1,..., M), an edge vector angle evaluation function g, and a cumulative operation Σ. The edge vector generation linear map L _m (m = 1,..., M) is an edge connected to every vertex whose degree (number of edges to be connected) is 2 from the variable U indicating the vertex information of the line-aligned embedded graph. An operation for generating a vector (side vector) representing the size and orientation of the. The edge vector angle evaluation function g evaluates the angle formed by the extracted edge vectors using a nuclear norm. The cumulative operation Σ adds these evaluation values. Hereinafter, this calculation method will be described.

  First, the integer M is the number of vertices whose degree is 2 among the vertices of the embedded graph G, and is calculated from the edge information E. Then, a number of (m = 1,..., M) is given to M degree 2 vertices, the vertex ID is curr_id (m), and the vertex IDs of two adjacent vertices are prev_id (m) and next_id, respectively. (M).

The edge vector generation linear map L _m (m = 1,..., M) includes an edge vector of (curr_id (m), prev_id (m)) and an edge vector of (curr_id (m), next_id (m)). This is an operation of calculating and storing these calculated side vectors in a 2 × D matrix. A matrix of two lines D column is called a side vector matrix P _m.

The edge vector generation linear map L _m receives a variable U indicating vertex information of the line-aligned embedded graph and outputs an edge vector matrix P _m = L _m U. The side vector generation linear map L _m is described by an equation as shown in Equation (6).

The edge vector generation linear mapping L _m, since also equal operation multiplying a sparse matrix of 2 rows and N columns shown in equation (7) from the left of the variable U, the matrix side vector generation linear mapping L _m and the formula (7) are the same Can be regarded as

The side vector angle evaluation function g evaluates the size of the angle formed by each side vector with respect to the side vector matrix P _m of 2 rows and D columns. As the side vector angle evaluation function g, a function exemplified by Expression (8) or Expression (9) can be used.

In Expression (8) and Expression (9), σ ₁ (P _m ) and σ ₂ (P _m ) represent two singular values of the edge vector matrix P _m . In the functions shown in Expression (8) and Expression (9), the function value decreases as the angle formed by the two side vectors decreases, and the function value increases as the angle increases.

  The multiplier 1114 outputs a function obtained by multiplying the linear alignment regularization function of Expression (5) by the trade-off parameter λ.

  The adder 1115 linearly combines the vertex position change loss function f and the function output from the multiplier 1114, and outputs an objective function on the right side of the minimum in Equation (1).

  The constraint condition expression unit 1116 receives the original embedded graph G = (V, E), and outputs a constraint condition expression on the right side of the subject to of the formula (1). The constraint condition expression is a constraint determined in advance by the user. The constraint condition limits some vertex information as in Expression (10), and the movable range of vertex information as in Expression (11) and Expression (12). You can impose restrictions that limit

  In Expression (11) and Expression (12), ε (n) is the maximum value of the movable range for the vertex whose vertex ID is n, which is predetermined by the user. It should be noted that some of these constraint formulas (10) to (12) can be combined into a constraint formula. It is also possible to define a constraint equation that does not impose any constraint.

  The optimization problem generation unit 1117 receives the objective function and the constraint condition expression, and generates an optimization problem for line alignment. That is, the optimization problem generating unit 1117 adds “minimize” representing minimization to the left side of the objective function output from the adder 1115, and sets a constraint on the left side of the constraint condition expression output from the constraint condition formulating unit 1116. “Subject to” expressed is added, and the optimization problem shown in the equation (1) is output.

  The embedded graph can be effectively linearly aligned by applying the kernel type norm function of Expression (8) as the above edge vector angle evaluation function g. Hereinafter, the optimization problem solving unit 112 when the kernel type norm function of Expression (8) is applied as the edge vector angle evaluation function g will be described. In this case, when the above-described PDS method is used, in order to obtain the solution of the optimization problem shown in Expression (1) in the optimization problem solving unit 112, a local linear alignment problem as shown in the following Expression (13) is used. The solving algorithm is applied.

  In Expression (13), the matrix X indicated by bold letters corresponds to the variable U in Expression (1) described above. In the following equation (14), which is the third arithmetic expression, the part of the equation (15) in parentheses is a matrix Y described later. That is, the equation (14) is obtained by dividing the matrix Y with a singular value decomposition of a matrix with 2 rows and D columns, threshold processing, a matrix with 2 rows and D columns × a matrix with D rows and D columns × a matrix with D rows and D columns. It is shown that the close mapping calculation of the karyotype norm function, which is a calculation including the above, is applied and the calculation is repeated.

Here, a more detailed configuration of the straight line alignment regularization function equation unit 1113 will be described.
FIG. 4 is a block diagram showing a detailed functional configuration of the straight line alignment regularization function equation unit 1113 shown in FIG. As shown in FIG. 4, the straight line alignment regularization function equation unit 1113 includes an edge vector generation linear map generation device 11131, an edge vector matrix generation device 11132, a singular value decomposition device 11133, and a sum of singular value indexes. A generating device 11134 and a sum generating device 11135 for the vertex index are provided.

The edge vector generation linear map generator 11131 generates and outputs an edge vector generation linear map L _m (m = 1,..., M) based on the edge information E.

The edge vector matrix generator 11132 generates and outputs an edge vector matrix P _m = L _m U based on the edge vector generation linear map L _m output from the edge vector generation linear map generator 11131. This edge vector matrix P _m is shown by the equation (6) in the above description.

The singular value decomposition apparatus 11133 receives the edge vector matrix P _m = L _m U, obtains and outputs two singular values σ ₁ (P _m ) and σ ₂ (P _m ) of the edge vector matrix P _m .

The sum generation apparatus 11134 for the singular value index calculates and outputs {σ ₁ (P _m ) + σ ₂ (P _m )} which is the sum of the two singular values output from the singular value decomposition apparatus 11133.

The sum generation unit 11135 for the vertex index inputs the sum of singular values {σ ₁ (P _m ) + σ ₂ (P _m )} output from the sum generation unit 11134 for the singular value index, and the vertex index is obtained. Calculate the sum of.

  The output from the sum generation device 11135 for the vertex index is expressed by Expression (16) or Expression (17).

  That is, the sum generation device 11135 for the vertex index generates the sum of all the vertices of the sum or the square of the sum output from the sum generation device 11134 for the singular value index as a linearly aligned regularization function. . These equations (16) and (17) correspond to the linear alignment regularization function shown in equation (5). Expressions (16) and (17) correspond to cases where the side vector angle evaluation functions g shown in Expression (8) and Expression (9) are used, respectively.

  Next, simplification processing by the embedded graph simplification apparatus 1 according to the present embodiment will be described with reference to FIGS.

FIG. 5 is a flowchart of an embedded graph simplification process performed by the embedded graph simplification apparatus 1. The embedded graph linear alignment unit 11 inputs the original embedded graph G = (V, E) and the trade-off parameter λ, and outputs the linearly aligned embedded graph G ′ = (V ′, E) ( Step S11). The unnecessary vertex removal unit 12 inputs the straight line-aligned embedded graph G ′ = (V ′, E), and outputs a simplified embedded graph G ^* = (V ^* , E ^* ) (step S12). .

  FIG. 6 is a flowchart detailing the embedded graph straight line alignment processing by the embedded graph straight line aligning unit 11 performed in step S11 of FIG. The optimization problem formulation unit 111 for linear alignment inputs the original embedded graph G = (V, E) and the trade-off parameter λ, and outputs an optimization problem for linear alignment (step) S111). The optimization problem solving unit 112 receives the optimization problem for the linearity sequence as an input, and outputs the vertex information V ′ aligned in a straight line (step S112). The vertex information replacement unit 113 receives the line-aligned vertex information V ′ and the original embedded graph G = (V, E), and outputs the line-aligned embedded graph G ′ = (V ′, E). (Step S113).

  FIG. 7 is a flowchart detailing the optimization problem formulation process for linear alignment performed by the optimization problem formulation unit 111 for linear alignment performed in step S111 of FIG. The vertex side information decomposition unit 1111 receives the original embedded graph G = (V, E), decomposes it into vertex information V and edge information E, and outputs it (step S1111). The vertex position change loss function equation unit 1112 receives the vertex information V and outputs the vertex position change loss function f (step S1112). The straight line alignment regularization function equation unit 1113 receives the edge information E and outputs a straight line alignment regularization function (step S1113). The multiplier 1114 receives the linear alignment regularization function and the trade-off parameter λ, and outputs a function obtained by multiplying these two (step S1114). The adder 1115 linearly combines the vertex position change loss function f and the function output from the multiplier 1114, and outputs an objective function (step S1115). The constraint condition expression unit 1116 receives the original embedded graph G = (V, E) and outputs a constraint condition expression (step S1116). The optimization problem generation unit 1117 receives the objective function and the constraint condition expression, generates and outputs an optimization problem for line alignment (step S1117).

  Next, the embedding graph simplification apparatus 1 shown in FIG. 1, the embedding graph straight line alignment unit 11 shown in FIG. 2, and the optimization problem formulating unit 111 for straight line alignment shown in FIG. 5, the operation when the processing shown in FIGS. 6 and 7 is performed will be described.

  Here, the embedding graph simplification for encoding the image region is considered, the embedding graph is a graph embedded on a two-dimensional plane, the vertex position change loss function f is the cost of the equation (4), the edge vector angle evaluation function g is the cost of equation (8), and the constraint condition is that the vertex located at the image edge is constrained in the form of equation (10). It is assumed that the PDS method is used for the algorithm of the optimization problem solving unit.

The original embedded graph G = (V, E), which is the input of the embedded graph simplification apparatus 1, is assumed to be a graph embedded on the two-dimensional plane shown in FIG. The circle symbol in FIG. 8 represents the position of each vertex of the embedded graph, and the number on the lower right of the circle symbol represents the vertex ID. The vertex position is described in an orthogonal coordinate system in which the downward direction in FIG. 8 is the first coordinate axis direction and the right direction is the second coordinate axis direction. For example, the vertex at the top left in FIG. 8 has a vertex ID “2” and its coordinates are (1, 0). The vertex information V is an N-row / 2-column matrix, the row number indicates the vertex ID, and the column number indicates the coordinate axis number. For example, the information about the vertex whose vertex ID is “2” is stored in the second row of the matrix indicating the vertex information V, and is (v _2,1 = 1), and (v _2,2 = 0) is there.

  A solid line connecting the vertices represents each side of the embedded graph. For example, an edge connecting the two vertices exists between the vertex having the ID “ID = 2” and the vertex having the “ID = 3”. The edge information E is stored as an adjacent list. As the trade-off parameter λ, which is an input of the embedded graph simplification apparatus 1, any real value such as 1.5 can be selected.

  In processing step S11 performed by the embedded graph straight line alignment unit 11, a straight line aligned embedded graph G ′ = (V ′, E) is output. In this example, an embedded graph as shown in FIG. 9 is output as a linearly aligned embedded graph G ′ = (V ′, E). The circle symbol in FIG. 9 represents the position of each vertex of the embedded graph as in FIG. 8, and the number on the lower right of the circle symbol represents the vertex ID. As can be seen from a comparison between FIG. 8 and FIG. 9, the processing step S11 performed by the embedded graph linear alignment unit 11 maintains the edge information so that the edge information between the vertices is maintained and the embedded graph is linearly aligned. Is fluctuating. Further, for the vertices whose vertex IDs are (ID = 1, 2, 42,..., 141), the variation positions are restricted by the restriction conditions.

In processing step S12 performed by the unnecessary vertex removing unit 12, a vertex having a small angle formed by connected edges is selected from among the vertices of the embedded graph G ′ with a predetermined threshold value (for example, π / 100 radians). The simplified embedded graph G ^* = (V ^* , E ^* ) is generated by removing the vertices.

  The detailed operation of the processing step S11 performed by the embedded graph linear alignment unit 11 will be described. In the processing step S111 performed by the optimization problem formulation unit 111 for straight line alignment, for example, an optimization problem of Expression (13) is generated.

  Here, the vertex position change loss function f uses the cost of equation (4), and the edge vector angle evaluation function g uses the cost of equation (8). Further, the constraint condition is that the vertex located at the outer edge of the image can only be moved on the outer edge, and the vertex ID of the vertex (ID = 1, 2, 42,..., 141) in the form of Expression (10) is restricted. doing.

In processing step S112 performed by the optimization problem solving unit 112, the optimization problem of Expression (18) is solved by the PDS method to generate vertex information V ^* .

In the PDS method for Expression (18), the proximity map of the vertex position change loss function f in Expression (4), the proximity map of the edge vector angle evaluation function g in Expression (8), the edge vector generation linear map L _m and its conjugate map. L _m ^T (m = 1,..., 132), the projection of the constraint set C to the set representing the constraint condition needs to be calculated. If these can be calculated, a solution can be obtained by iterative calculation.

  The proximity map of the vertex position change loss function f in equation (4) can be calculated by equation (19) for a given arbitrary step size μ (> 0).

  Here, the function sign is a function that extracts the sign of each element of the input matrix, associates it with -1, 0, +1, stores it in a matrix of the same size as the input, and outputs it. An operator indicated by a circle is a Hadamard product, which is an operation of multiplying the elements of the left and right matrices, storing them in a matrix of the same size, and outputting them. The function max is a function that compares two input real numbers and outputs the larger one.

  The proximity map of the edge vector angle evaluation function g in Expression (8) can be calculated by Expression (20) for a certain arbitrary step size μ.

  Here, the function diag is a function that outputs a diagonal matrix in which input vectors are arranged in diagonal elements. S (P), T (P), and σ (P) are singular value decompositions of the matrix P, and Equation (21) is established.

The edge vector generation linear map L _m and its conjugate map L _m ^T (m = 1,..., 132) can be calculated by multiplying the matrix of Expression (7) and its transposed matrix from the left.

  A set representing a constraint condition of the constraint set C is Expression (22).

  Therefore, the projection onto C can be calculated by equation (23).

  In processing step S113 performed by the vertex information replacement unit 113, the vertex information V ′ aligned in a straight line and the original embedded graph G = (V, E) are input, and the embedded graph G ′ = (V ′, E) aligned in a straight line. ) Is output. That is, the processing step S113 performed by the vertex information replacement unit 113 replaces the vertex information V of the original embedded graph G with the linearly aligned vertex information V ′ to generate the linearly aligned embedded graph G ′.

  The detailed operation of the processing step S111 performed by the optimization problem formulation unit 111 for straight line alignment will be described. In processing step S1111 performed by the vertex side information decomposing unit 1111, the original embedded graph G = (V, E) is input, decomposed into vertex information V and edge information E, and output.

  In processing step S1112 performed by the vertex position change loss function equation unit 1112, the original vertex information V is input and the vertex position change loss function f is output. As the vertex position change loss function f, for example, the cost of equation (4) is used.

In processing step S1113 performed by the straight line alignment regularization function equation unit 1113, the original edge information E is input, and the straight line alignment regularization function of Expression (5) is output. The edge vector generation linear map L _m (m is the vertex index, m = 1,..., M) is generated as a matrix of Expression (7) by obtaining the adjacent vertex ID of each degree 2 vertex from the edge information E. can do. Further, as the side vector angle evaluation function g, for example, the cost of Expression (8) is used.

  In processing step S1114 performed by the multiplier 1114, a function obtained by multiplying the linearity sequence regularization function of Expression (5) by the trade-off parameter λ is output.

  In processing step S1115 performed by the adder 1115, the vertex position change loss function f and the function output from the multiplier 1114 are linearly combined to output an objective function on the right side of the minimize in the equation (18).

  In processing step S1116 performed by the constraint condition expression unit 1116, the original embedded graph G = (V, E) is input, and the constraint condition expression on the right side of the subject to of the equation (18) is output.

  In processing step S1117 performed by the optimization problem generation unit 1117, an objective function and a constraint condition expression are input, and an optimization problem for line alignment is generated. That is, the processing step S1117 performed by the optimization problem generating unit 1117 outputs “minimize” representing minimization and “subject to” representing constraints to the left of each of the objective function and the constraint condition expression.

  In this way, an optimization problem for line alignment is generated according to the processing procedure shown in FIG. 7, the embedded graph is aligned according to the processing procedure shown in FIG. 6, and the embedded graph is converted according to the processing procedure shown in FIG. It can be simplified.

[Second Embodiment]
Next, a second embodiment will be described. In addition, description is abbreviate | omitted about the matter similar to the above-mentioned embodiment, and the matter peculiar to this embodiment is demonstrated below.

  When the local linear regularization method according to the first embodiment is used, there may be a problem that a steep curve composed of a large number of vertices is short-circuited to greatly deteriorate the graph shape. This is due to the singular value sum in the straight line regularization function equation part. Of the two singular values of the input matrix, the first singular value contributes to the size of the column vector of the input matrix, and the second singular value contributes to the angle formed by the column vector. Here, since the column vector of the input matrix is two adjacent side vectors, by making the kernel norm regular, the angle formed by these side vectors approaches 0 degree or 180 degrees and at the same time, the magnitude of the vector The effect of reducing the value will work. As described above, the effect of reducing the size of each side vector works, so that a steep curve may be short-circuited and the graph shape may be greatly deteriorated.

  The present embodiment avoids the problem of greatly degrading the graph shape due to short-circuiting as a result of the simplification process even when the graph is composed of a large number of vertices and the shape has a steep curve.

  FIG. 10 is a block diagram illustrating a functional configuration of the optimization problem formulation unit 111 for linear alignment in the present embodiment. As shown in FIG. 10, the optimization problem formula unit 111 for straight line alignment in the present embodiment has a configuration similar to the optimization problem formula unit 111 for straight line alignment in the first embodiment. However, the optimization problem formulating unit 111 for straight line alignment in the present embodiment includes a straight line alignment weighted regularization function formulating unit 1118 instead of the straight line alignment regularizing function formulating unit 1113. That is, the optimization problem equation unit 111 for straight line alignment in FIG. 10 is the same as the optimization problem equation unit 111 for straight line alignment in FIG. 3 except for the straight line alignment weighted regularization function equation unit 1118. The configuration is the same process. For this reason, description of each part other than the straight line alignment weight regularization function formula part 1118 is omitted. The straight line weighted regularization function equation unit 1118 performs addition with weights added to each of two kinds of singular values when performing straight line alignment processing on the vertices of the graph.

  Here, the straight line alignment weight regularization function equation unit 1118 will be described in detail. FIG. 11 is a block diagram showing a detailed functional configuration of the straight line alignment weighted regularization function equation unit 1118. As shown in FIG. 11, the straight line alignment weight regularization function equation unit 1118 includes an edge vector generation linear map generation device 11131, an edge vector matrix generation device 11132, a singular value decomposition device 11133, and weights for singular value indexes. A sum generation device 11184 and a sum generation device 11135 for the vertex index are provided. That is, the straight line alignment weighted regularization function equation unit 1118 replaces the sum generation device 11134 for the singular value index in the straight line alignment regularization function equation unit 1113 with a weighted sum generation device 11184 for the singular value index. Prepare. The functions of the respective units other than the weighted sum generating apparatus 11184 for the singular value index shown in FIG. 11 are the same as the functions of the respective units having the same reference numerals in FIG.

  The straight line alignment weighted regularization function equation unit 1118 obtains an edge vector representing the direction and size of the edge connected to each vertex included in the embedded graph, and the size of the two edges connected to the vertex and the two edges are calculated. The straight line alignment regularization function is generated by weighting and synthesizing the formed angle so that the effect of the two sides is relatively small and the effect of the two sides is relatively large.

The weighted sum generation device 11184 for the singular value index inputs the singular value, which is the output of the singular value decomposition device 11133, and the weights w ₁ (first weight value) and w ₂ (second weight value), Generate and output a weighted sum of singular values. The weighted sum of the singular values obtained by the weighted sum generation device 11184 for the singular value index is as shown in Expression (24).

That is, the weight w ₁ is a weight applied to the first singular value (σ ₁ ) corresponding to the size of the two sides connected to the vertex. The weight w ₂ is a weight applied to the second singular value (σ ₂ ) corresponding to the angle formed by these two sides.

As the weights w ₁ and w ₂ in the equation (24), values satisfying the relationship (w ₂ ≧ w ₁ ≧ 0) are given. Note that the relationship between w ₂ and w ₁ may be (w ₂ > w ₁ ). Further, the relationship between w ₁ and 0 may be (w ₁ > 0). In the method of the first embodiment, a graph that is regularized in line alignment may converge to a shape in which an exponential function is graphed. On the other hand, the method according to the present embodiment avoids such a case because the vector angle weight w _{2 is set} to be equal to or greater than the vector magnitude weight w ₁ (the weights w ₁ and w ₂ are both non-negative values). be able to. That is, in this embodiment, since (w ₂ ≧ w ₁ ≧ 0), when the graph data is simplified, it is avoided that the exponential function is simplified to a graph shape regardless of the original data. That is, the weight of the vector angle is set larger than the weight of the vector size.

  Since the weighted sum generation device 11184 for the singular value index performs the weighted addition as described above, in this embodiment, an example of the total value output by the sum generation device 11135 for the vertex index is expressed by Expression (25). ) And formula (26).

  According to the present embodiment described above, even when the graph is composed of a large number of vertices and the shape of the graph has a steep curve, as a result of the graph simplification processing, the arrangement of the vertices is short-circuited to change the graph shape. The problem of significant deterioration can be avoided.

The reason is as follows. That is, of the two singular values of the edge vector matrix P _m =, the first singular value σ ₁ (P _m ) contributes to the size of the column vector of the input matrix, and the second singular value σ ₂ (P _m ) is This contributes to the angle formed by the column vectors of the input matrix. Since the column vector of the input matrix is two adjacent side vectors, regularizing the nuclear norm brings the effect of reducing the size of the vector at the same time that the angle formed by the side vectors approaches 0 or 180 degrees. End up. Then, the effect of reducing the size of each side vector works, so that a steep curve is short-circuited and the graph shape may be greatly deteriorated. On the other hand, in the method of the present embodiment, the influence of the first singular value is reduced and the influence of the second singular value is increased by weighting the sum calculation of the singular values. Thereby, the effect of reducing the size of each side vector is suppressed, and a short-circuit of a steep curve is suppressed. That is, in the method according to the present embodiment, even when the graph to be simplified includes a steep curve, there are few short circuits and deterioration is unlikely to occur.

  FIG. 12 is a schematic diagram for explaining a problem that the embedded graph simplification apparatus 1 according to the present embodiment avoids. In the same figure, the broken line shown with a dashed-dotted line shows the shape of the example of the graph used as the input to the embedded graph simplification apparatus 1. FIG. In addition, a small circle (vertex) and a straight line (side) by a solid line that sequentially connects these circles indicate the possibility of one processing result when the graph simplification process according to the first embodiment is performed. Show. As shown in the figure, the portion indicated by the arrow in this result is particularly short-circuited in the original graph, and it can be considered that the shape has deteriorated. In the present embodiment, as described above, the influence of the first singular value is reduced and the influence of the second singular value is increased, so that the shape deterioration of the graph can be avoided. Although an example in which the simplification process is applied to a graph on a two-dimensional plane is shown here, it goes without saying that the simplification process may be applied to a graph in a multi-dimensional (three-dimensional or more) space. Yes.

  The embedded graph simplification apparatus 1 having the configuration of any one of the above embodiments uses a graph indicating a shape by a plurality of vertices arranged at different positions and a connection between the plurality of vertices as input data, and the embedded graph linear alignment In the unit 11, when the vertex position change loss function equation unit 1112 moves a plurality of vertices and aligns them on a straight line as indicated by a combination of straight lines with less shape of the two-dimensional data, A vertex position change loss function for calculating a cost indicating the degree of position deviation is generated. In addition, when the straight line alignment regularization function equation unit 1113 or the straight line alignment weight regularization function equation unit 1118 aligns a plurality of vertices on a straight line, the cost indicating the smoothness of the linear shape formed by the plurality of vertices. Generate a line alignment regularization function that computes. The optimization problem generation unit 1117 generates an optimization problem for line alignment shown in Expression (1) based on the vertex position change loss function, the line alignment regularization function, and the constraint condition expression. The optimization problem solving unit 112 converts the matrix Y included in the algorithm used in the process of solving the optimization problem for line alignment into the matrix Z by the proximity map fast calculation unit 1122 of the karyotype norm function, Under the constraint condition, the vertex position where the sum of the cost calculated by the vertex position change loss function and the cost calculated by the line alignment regularization function is minimized is calculated. The vertex information replacement unit 113 moves the vertex to the calculated position. The unnecessary vertex removing unit 12 selects and removes unnecessary points from the points based on the angle formed by the sides formed by the connection between the moved vertices. Thereby, vertices and edges can be reduced even in a complicated embedded graph, and the vertex position can be prevented from being limited to the vertex position of the original embedded graph. In addition, the proximity map high-speed calculation unit 1122 of the karyotypic norm function converts the matrix Y into the matrix Z, so that the calculation time can be reduced as described below.

[Modification]
Here, the modification applicable to each embodiment mentioned above is demonstrated.
In each of the embodiments described above, the embedded graph simplification device 1 includes the embedded graph straight line alignment unit 11 and the unnecessary vertex removal unit 12. As an example of this modification, the embedded graph simplification apparatus 1 may not have the unnecessary vertex removal unit 12 but may perform only the straight line alignment processing by the embedded graph straight line alignment unit 11. Also in this case, an effect that the embedded graph can be simplified can be obtained.

  In each of the embodiments described above, the optimization problem formulating unit 111 for straight line alignment includes the vertex position change loss function formulating unit 1112 and the straight line alignment regularizing function formulating unit 1113 (or the straight line alignment weighted regularizing function). Both vertical sections 1118) were provided. Then, the objective function is generated by weighting and adding the functions formulated in both of them (the weight depends on the trade-off parameter λ). As a modified example, the optimization problem formula unit 111 for straight line alignment does not have the vertex position change loss function formula unit 1112, but uses a function that is formulated only in the straight line regularization function formula unit 1113. Thus, an objective function may be generated. In such a modification, the change in the vertex position due to the simplification process may be larger, but a predetermined effect of linear alignment of the vertices can be obtained.

A program (computer program) for realizing all or part of the functions of the embedded graph simplification apparatus 1 described in each embodiment and modification is recorded on a computer-readable recording medium, and the recording medium is recorded on the recording medium. Processing of each unit may be performed by reading a recorded program into a computer system and executing the program. Here, the “computer system” includes an OS and hardware such as peripheral devices.
Further, the “computer system” includes a homepage providing environment (or display environment) if a WWW system is used.
The “computer-readable recording medium” refers to a storage device such as a flexible medium, a magneto-optical disk, a portable medium such as a ROM and a CD-ROM, and a hard disk incorporated in a computer system. Furthermore, the “computer-readable recording medium” dynamically holds a program for a short time like a communication line when transmitting a program via a network such as the Internet or a communication line such as a telephone line. In this case, a volatile memory in a computer system serving as a server or a client in that case, and a program that holds a program for a certain period of time are also included. The program may be a program for realizing a part of the functions described above, and may be a program capable of realizing the functions described above in combination with a program already recorded in a computer system.

  The embodiment of the present invention has been described in detail with reference to the drawings. However, the specific configuration is not limited to this embodiment, and includes design changes and the like without departing from the gist of the present invention.

DESCRIPTION OF SYMBOLS 1 ... Embedded graph simplification apparatus, 11 ... Embedded graph straight line alignment part (embedded graph alignment part), 12 ... Unnecessary vertex removal part, 111 ... Optimization problem equation part for straight line alignment (optimization problem equation part) 112 ... Optimization problem solving unit 113 ... Vertex information replacement unit 1111 ... Vertex edge information decomposition unit 1112 ... Vertex position change loss function equation unit 1113 ... Linear alignment regularization function equation unit (alignment regularization function) 1114... Multiplier, 1115... Adder, 1116... Constraint condition generator, 1117... Optimization problem generator, 1118 ... Linear alignment weighted regularization function equation (alignment regularization function equation part) ), 11131... Edge vector generation linear mapping generator (edge vector generation linear mapping generator), 11132... Edge vector matrix generator (edge vector matrix generator), 11133. 11134... Sum generating device for singular value index, 11135... Sum generating device for vertex index (sum generating unit), 11184... Weighted sum generating device for singular value index (weighted sum generating unit) )

Claims (6)

An embedding graph simplification device for simplifying an embedding graph having vertex position information,
An embedded graph alignment unit that aligns vertex positions so that the angle formed by the edges connected to each vertex is small;
With
The embedded graph alignment unit includes:
An optimization problem formulation section that forms and outputs an optimization problem for alignment;
An optimization problem solving unit for solving the optimization problem for alignment output in the optimization problem equation unit to obtain the aligned vertex positions;
A vertex information replacement unit that replaces the position information of the vertices of the embedded graph with the vertex positions determined by the optimization problem solving unit;
With
The optimization problem formula part is
An alignment regularization function expression that generates an alignment regularization function that evaluates the degree to which the aligned embedded graph is close to a straight line using the vertex position of the aligned embedded graph as a variable;
A constraint expression unit that generates a constraint on the range of the vertex positions after being aligned;
An optimization problem generation unit that generates an optimization problem by combining the objective function based on the alignment regularization function generated by the alignment regularization function expression unit and the constraint condition;
With
The alignment regularization function equation unit obtains an edge vector representing the direction and size of an edge connected to each vertex included in the embedded graph, and the size of the two edges connected to the vertex and the two edges Are generated by weighting so that the influence of the angle between the two sides is relatively small and the influence of the angle between the two sides is relatively large, thereby generating the alignment regularization function. ,
Embedded graph simplification device. The optimization problem formula part is
A vertex position change loss function equation that generates a vertex position change loss function that evaluates how much the variable deviates from the original vertex position when the vertex position of the aligned embedded graph is a variable;
A multiplier that multiplies the alignment regularization function output by the alignment regularization function formulation part by a trade-off parameter;
An adder that linearly combines the vertex position change loss function and the alignment regularization function multiplied by the trade-off parameter to generate an objective function;
With
The optimization problem generation unit generates an optimization problem by combining the objective function output from the adder and the constraint condition.
The embedded graph simplification device according to claim 1. An unnecessary vertex removing unit that selects vertices having a small angle formed by connecting edges from the embedded graph arranged by the embedded graph aligning unit, and removes vertices having a small angle formed by the connecting edges;
The embedded graph simplification device according to claim 1, further comprising: The alignment regularization function equation part is:
An edge vector generation linear map L _m (m is an index of a vertex) representing the size and direction of each edge connected to the vertex of the vertex included in the embedded graph having two connected edges. , M = 1,..., M), an edge vector generation linear map generation unit;
The edge vector generation linear map L _m output from the edge vector generation linear map generation unit and the variable U indicating the vertex information of the embedded graph are input, and the edge vector matrix P _m (P _m = L _m U) is obtained. An edge vector matrix generation unit to generate;
A singular value decomposition unit for obtaining a first singular value corresponding to the size of the side and a second singular value corresponding to the angle formed by the two sides of the side vector matrix P _m output from the side vector matrix generation unit. When,
A weight for generating a weighted sum of the first singular value and the second singular value by using the first weight value w ₁ and the second weight value w ₂ respectively corresponding to the first singular value and the second singular value. A sum generation unit;
A sum generation unit that generates the sum of all vertices of the weighted sum output by the weighted sum generation unit or the square of the weighted sum as an aligned regularization function;
With
The weighted sum generation unit uses the first weight value w ₁ and the second weight value w ₂ satisfying a relationship of (w ₂ ≧ w ₁ ≧ 0),
The embedded graph simplification device according to any one of claims 1 to 3. An embedded graph simplification method for simplifying an embedded graph having vertex position information,
Embedded graph alignment processing step for aligning the vertex positions so that the angle formed by the edges connected to each vertex is small;
Have
The embedded graph alignment processing step includes:
Optimization problem formulation processing step that formulates and outputs an optimization problem for alignment, and
An optimization problem finding processing step for solving the optimization problem for alignment output in the optimization problem formula processing step to obtain aligned vertex positions;
Vertex information replacement processing step of replacing the position information of the vertices of the embedded graph with the vertex positions determined in the optimization problem solving processing step;
Have
The optimization problem formula processing step includes:
Using a vertex position of the aligned embedded graph as a variable, generating an aligned regularization function that evaluates the degree to which the aligned embedded graph is close to a straight line;
A constraint expression processing step for generating a constraint on the range of the vertex positions after being aligned; and
An optimization problem generation processing step for generating an optimization problem by combining the objective function based on the alignment regularization function generated in the alignment regularization function formulation processing step and the constraint condition;
Have
In the alignment regularization function equation processing step, an edge vector representing the direction and size of an edge connected to each vertex included in the embedded graph is obtained, and the size of the two edges connected to the vertex and the two The alignment regularization function is generated by weighting and synthesizing the angle formed by the sides so that the influence of the size of the two sides is relatively small and the influence of the angle formed by the two sides is relatively large. To
Embedded graph simplification method.   A computer program for causing a computer to function as the embedded graph simplification device according to any one of claims 1 to 4.

Images