JP5564643B2 - Information processing method and information processing apparatus - Google Patents

Description

  The present invention relates to an information processing method and an information processing apparatus for processing higher-order energy that can be expressed by a polynomial of a fourth or higher order in which each variable is binary.

  As a method for efficiently solving problems in the image field such as region segmentation, stereo, and image restoration as a maximum posterior probability estimation problem of a stochastic model such as Markov Random Field (MRF), and as an energy minimization problem Graph cuts (for example, see Non-Patent Document 1), belief propagation methods (for example, see Non-Patent Document 2), tree weight reallocation message transmission methods (for example, see Non-Patent Document 3), and the like have been proposed. Although these approaches can minimize energy relatively efficiently, the form of energy that can be used is limited.

Y. Boykov, O. Veksler, R. Zabih. "Fast Apporoximate Energy Minimization via Graph Cuts." IEEE Transactions on Pattern Analysis and Machine Intelligence 23: 1222-1239: 2001. P. Felzenszwalb and D. Huttenlocher. "Efficient Belief Propagation for Early Vision." International Journal of Computer Vision 70: 41-54,2006. V. Kolmogorov. "Convergent Tree-Reweighted Mesage Passing for Energy Minimization." IEEE Transactions on Pattern Analysis and Machine Intelligence 28 (10): 1568-1583,2006.

  When energy can be expressed by the sum of functions depending on up to k points, it is called (k-1) -order energy in MRF terminology, but the number of energies that can be minimized by the above-described method is at most 2nd-order. In fact, only the first floor is used. For example, in the field of image processing, most energy on the first floor is represented by the sum of a term that depends only on each pixel and a term that depends only on two adjacent pixels. In recent years, in image processing, the importance of higher-order statistics has been pointed out in order to capture the characteristics of natural images with higher accuracy, and it is desired to construct a method for efficiently minimizing higher-order energy.

  An information processing method and an information processing apparatus according to the present invention are mainly intended to minimize data composed of higher-order energy represented by a polynomial having a binary fourth or higher order with each variable at a low calculation cost.

  The information processing method and information processing apparatus of the present invention employ the following means in order to achieve the main object described above.

The information processing method of the present invention includes:
An information processing method for processing higher-order energy in which each variable can be expressed by a binary fourth-order or higher-order polynomial,
Storing a data representation of the higher order energy in a storage means;
By changing the stored data representation by a calculation means so as to correspond to the addition of binary external variables of a number corresponding to the order using a symmetric formula for the higher order energy , the higher order energy is changed to the same first order energy. Converted into energy
Each variable that minimizes the higher-order energy is set by applying a predetermined energy minimizing method to the converted first-order energy.

  In this information processing method of the present invention, by adding a higher-order energy that can be expressed by a polynomial having a fourth or higher order of each variable to a binary, a binary external variable of a number corresponding to the order is added using a symmetric expression. Each variable that minimizes the higher-order energy is set by converting the energy to the equivalent first-order energy and applying a predetermined energy minimization method to the converted first-order energy. As a result, higher-order energy can be minimized with less calculation cost. Here, “equivalent” means that the setting of a variable that minimizes the energy of the first floor after conversion also minimizes the energy of the higher floor before conversion. The “binary external variable of the number corresponding to the order” is specifically the maximum integer that does not exceed (d−1) / 2 when the order is d for each term. Further, the “predetermined energy minimization method” includes a graph cut method, a belief propagation method, a tree weight redistribution message transmission method, and the like.

In such an information processing method according to the present invention, the primary variable is a primary basic symmetric expression where the order is d, the coefficient is a, the variable is x ₁ ,..., X _d, and the external variable taking either 0 or 1 is w. Is S ₁ and the second-order basic symmetry expression is S _{2. For} each term of the higher-order energy polynomial, when the coefficient d is an even number for a term with a positive coefficient a, the expression (1) To the first-order energy polynomial, and when the order d of the term is an odd number, it can be converted to the first-order energy polynomial using equation (2). This is based on the fact that any second-order symmetric expression can be expressed by the first-order and second-order basic symmetric expressions.

  In the information processing method of the present invention of this aspect, a term having a negative coefficient a for each term of the higher-order energy polynomial is converted to the first-order energy polynomial using Equation (3). You can also.

  In the information processing method of the present invention for processing image data, a pixel having a first value is assigned to a pixel that satisfies a predetermined condition among the pixels constituting the image data, and a pixel that does not satisfy the predetermined condition A higher-order polynomial including a first-order term that depends only on each pixel and a fourth-order or more term that depends on a set of four or more pixels, with a binary label that assigns a second-value label to the variable The binary label may be assigned by setting energy and setting the variable so that the set higher-order energy is minimized. In this way, it is possible to perform image data processing using a predetermined condition with less calculation cost. Here, the “predetermined condition” includes a condition for dividing a region of the image, a condition for removing noise included in the image, and the like.

  In the information processing method of the present invention that processes image data, the higher-order energy of the polynomial is set by determining the coefficient of each term so that the energy decreases as the probability of meeting the predetermined condition increases. You can also. In this aspect of the information processing method of the present invention, the coefficient of each term may be determined so that the energy is −logP, where P is the probability of meeting the predetermined condition. By doing so, it is possible to set higher-order energy that conforms to predetermined conditions.

  Further, in the information processing method of the present invention that processes image data, a proposed image that changes the current image is generated, and pixels of the current image are replaced with corresponding pixels of the generated proposed image. Sometimes assigning the label of the first value and repeating the process of generating a new image by assigning the second label when not replacing the pixel of the current image with the corresponding pixel of the generated proposed image It is also possible to restore the image. In the information processing method according to the aspect of the present invention, the proposed image may be generated by using a re-steep descent method or by applying a predetermined blur filter to the current image. It can also be.

The information processing apparatus of the present invention
An information processing apparatus for processing higher-order energy, each variable being represented by a binary fourth or higher order polynomial,
Storage means for storing a data representation of the higher order energy;
A calculating means for changing the data representation that is the stored so as to correspond to the addition of external variable binary number corresponding to the order by using a symmetric polynomial in the higher-order energy,
Energy conversion means for converting the higher-order energy into equivalent first- order energy by changing the data representation ;
Energy minimizing means for setting each variable that minimizes the higher order energy by applying a predetermined energy minimizing method to the converted first floor energy;
It is a summary to provide.

  In the information processing apparatus according to the present invention, by adding a higher-order energy that can be expressed by a polynomial having a binary fourth or higher order to each variable, a binary external variable having a number corresponding to the order is added using a symmetric expression. Each variable that minimizes the higher-order energy is set by converting the energy to the equivalent first-order energy and applying a predetermined energy minimization method to the converted first-order energy. As a result, higher-order energy can be minimized with less calculation cost.

It is a block diagram which shows the outline of a structure of the information processing apparatus as one Example of this invention. It is explanatory drawing which shows an example of the pattern of the label allocated to a pixel. 4 is a flowchart illustrating an example of higher-order energy setting processing executed by a computer 20. It is explanatory drawing which shows the relationship between the pixel data value D and energy E (D). 4 is a flowchart illustrating an example of energy conversion processing executed by a computer 20.

  Next, embodiments of the present invention will be described using examples.

  FIG. 1 is a configuration diagram showing an outline of the configuration of an information processing apparatus as an embodiment of the present invention. The information processing apparatus according to the embodiment is a general-purpose computer 20 including a CPU, a ROM, a RAM, an input / output processing circuit, a hard disk, a keyboard 40, a mouse 50, and the like, and includes a digital camera, CT (Computed Tomography), and MRI (Magnetic Resonance Imaging). ), PET (Positron Emission Tomography) and other imaging devices and storage devices for storing image data are connected via an input processing circuit and are connected to output devices such as a display 30 and a printer via an output processing circuit. . In the embodiment, the computer 20 is installed with an image processing program, and functions as an image processing apparatus.

  Data obtained from CT, MRI, PET, etc. give numerical values (data values) to three-dimensional lattice points. For example, in the case of CT, the data value represents how much each grid point in the space absorbs X-rays. In the case of a two-dimensional cross-sectional image, the entire image can be seen by changing the color according to the value of each point, but it is difficult to get a panoramic view of the three-dimensional data. Therefore, for example, if it is desired to see only the bone in the CT image, only the bone portion is left and the other portions are made transparent and displayed in three-dimensional graphics. For this purpose, it is necessary to divide the original image into a bone portion and other portions (image division problem). This can be considered as a label allocation problem in which a value of 0 or 1 is given to each grid point such that value 0 represents the bone portion and value 1 represents the rest. In this case, if a threshold value is determined and a value greater than the threshold value is assigned a value of 0 and a value less than or equal to the threshold value is assigned a value of 1, the surface shape becomes rough due to noise or the like. For this reason, generally, there is a tendency that the value 0 and the value 1 are not switched very vigorously between adjacent grid points, that is, the boundary becomes smooth. One of the methods for this is a method based on energy minimization. Hereinafter, in the embodiment, an image division problem in which a region is divided by assigning a label with a value of 0 to pixels corresponding to bones and assigning a label with a value of 1 to other pixels of a human CT image is minimized. An example of solving by the conversion method will be described.

  The computer 20 includes, as its functional blocks, a higher-order energy setting unit 22 that sets a polynomial (higher-order energy) for processing region division of image data using an energy minimization method, and sets the higher-order energy to a binary value. An image conversion area 24 for adding external variables to convert the energy to the first floor and assigning a label of value 0 or value 1 to each pixel so that the energy of the first floor after conversion is minimized. An energy minimizing unit 26 that performs division is provided, and these functions such as hardware such as a CPU, a ROM, a RAM, and a hard disk and an image processing program function together.

  The higher-order energy setting unit 22 is a processing unit that sets a higher-order polynomial for solving the above-described region division problem using a value 0 or value 1 label assigned to each pixel as a binary variable. In the embodiment, A primary term that sets energy based on only one point of each pixel of image data, a secondary term that sets energy based on a pattern of two adjacent pixels (see FIG. 2A), and adjacent Based on the pattern of 4 points of 2 × 2 pixels to be set (see FIG. 2B), it is set by the sum with the fourth-order term that sets energy.

  The energy conversion unit 24 is configured as a processing unit that converts a third-order or higher-order (second-order or higher) energy polynomial composed of binary variables into second-order (first-order) energy, and this conversion is performed for each term of the polynomial. This is performed by adding binary external variables of a number corresponding to the maximum integer not exceeding (d-1) / 2 when the order is d using the basic symmetry formula.

Here, when a term of degree 3 having binary variables x, y, z is converted to a quadratic polynomial, according to the cited reference 1, both sides when the coefficient a is negative based on the following equation (4) “Max” is calculated by the following equation (6) calculated by replacing x, y, and z with 1-x, 1-y, and 1-z, respectively, when the coefficient a is positive. "Can be replaced with" min ". Here, “w” in the equations (4) to (6) is an external variable that takes only the value B = {0, 1}, that is, takes only the value 0 or the value 1. In the equations (5) and (6), when the cubic term axyz appears in the minimization problem, the minimum value and the variables x, y, and z for taking the same do not change even if the left side is replaced with the right side. Means that. That is, the variables x, y, and z that give the minimum value in the converted function also give the minimum value in the original function. However, considering a fourth-order term with binary variables x, y, z, and t as shown in Equation (7), in the above method, the coefficient a is positive and the even-order term is x , Y, z, and t can be replaced with 1-x, 1-y, 1-z, and 1-t, respectively, so that “max” cannot be changed to “min” and cannot be used. Even if an odd order of 5 or more is considered, an even-order term such as the fourth order appears in the expression after expansion, and thus cannot be used in the same manner.
Citation 1: D. Freedman and P. Drineas. “Energy minimization via graph cuts: Settling what is possible.” Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR2005) 2: 939-946.

  In contrast, in the embodiment, when the coefficient a is positive, the number of binary external variables corresponding to the largest integer not exceeding (d-1) / 2 is added to the equation (8). The following equation (9) is converted when the order d is an even number and the following equation (10) is converted when the order d is an odd number using the first-order basic symmetry equation S1 and the second-order basic symmetry equation S2 It is determined as a formula. For example, a term having a degree of 4 where x, y, z, and t are binary variables can be converted by the following equation (11) by adding one binary external variable w. A term of order 5, which is x, y, z, t, u, can be converted by the following equation (12) by adding two binary external variables v, w. These equations make use of the fact that an arbitrary second-order symmetric expression can be expressed by a first-order and a second-order basic symmetric expression. Thereby, even if the coefficient a is positive, the energy polynomial of an arbitrary floor can be converted into the energy polynomial of the first floor. When the coefficient a is negative, the following equation (13) obtained by generalizing the above equations (4) and (7) is defined as the conversion equation.

  The energy minimizing unit 26 is configured as a processing unit that finds a variable that minimizes the first-order energy expressed as a quadratic polynomial having a binary variable having a value 0 or a value 1. As a processing algorithm, for example, Graph cut method, belief propagation method, tree weight redistribution message transmission method, etc. can be used.

  Next, the operation of the information processing apparatus according to the embodiment configured as described above, particularly, the operation of the higher-order energy setting unit 22 and the operation of the energy conversion unit 24 will be described. First, the operation of the higher-order energy setting unit 22 will be described. FIG. 3 is a flowchart showing an example of higher-order energy setting processing executed by the computer 20.

In the higher-order energy setting process, first, a term that depends on only one point of each pixel of the image data is set by the following equation (14) (step S100). Here, “X _u ” in Expression (14) is a binary variable as a label to be assigned to the pixel u, and “a ₁ ” and “a ₂ ” are real coefficients. The real coefficient a ₁ is X _u = 0, that is, the energy value E ₀ (D) when the pixel u is a bone, and the probability indicating the possibility that the data value D of the pixel u is a bone is P ₀ ( D) is set so that a _{1, that} is, E ₀ (D) is −logP ₀ (D). Similarly, the real coefficient a ₂ is a probability that the data value D of the pixel u may not be a bone. the P ₁ a ₂ i.e. E ₁ when the (D) (D) is set -log P ₁ (D) and so as. That is, the higher the possibility that the pixel u is a bone, the smaller the energy when X _u = 0, and the lower the possibility that the pixel u is not a bone, the smaller the energy when X _u = 1. The real number coefficients a ₁ and a ₂ are set as follows. Of course, if the possibility that the pixel u is a bone is high, the energy when X _u = 0 is small, and if the possibility that the pixel u is not a bone is high, the energy when X _u = 1 is small. Therefore, the method is not limited to the above-described method. For example, when the relationship between the data value D (or probability P) and the energy E is obtained in advance and stored as a map in the ROM, the data value D is given. The corresponding energy E may be derived from the map and set based on the derived energy E. An example of the relationship between the data value D and the energy E is shown in FIG.

Subsequently, a term depending on two pixels adjacent to the upper and lower sides or the left and right sides of the image data is set by the following equation (15) (step S110). Here, “X _u ” and “X _v ” in Equation (15) are binary variables as labels assigned to the adjacent pixels u and v, respectively, and “b _{1 to} b ₄ ” are real coefficients. It is. The real number coefficient b ₁ is an energy value E ₁₁ (D _u , D _v ) when X _u = 1, X _v = 1, that is, the pixels u and v are not bones, and the data value D of the pixels u and v. _When P ₁₁ (D _u , D _v ) is the probability that _u and D _v are not bones, the pixel u and v are both b _{1, that} is, E ₁₁ (D _u , D _v ) is −logP ₁₁ (D _u , D _v ). The real number coefficient b ₂ is an energy value E ₀₁ (D _u , D _v ) when X _u = 0, X _v = 1, that is, the pixel u is a bone and the pixel v is not a bone. When the data values D _u and D _v are P ₀₁ (D _u , D _v ), the probability that the pixel u is a bone and the pixel v is not a bone is b _{2, that} is, E ₀₁ (D _u , D _v ). Is set to be −logP ₀₁ (D _u , D _v ). The real coefficient b ₃ is an energy value E ₁₀ (D _u , D _v ) when X _u = 1, X _v = 0, ie, the pixel u is not a bone and the pixel v is a bone, and the pixels u, v B _{3, that} is, E ₁₀ (D _u , D _v ), where P ₁₀ (D _u , D _v ) is a probability that the data value D _u , D _v of the pixel v is a bone and the pixel v is a bone. _v ) is set to be -logP ₁₀ (D _u , D _v ). The real coefficient b ₄ is an energy value E ₀₀ (D _u , D _v ) when X _u = 0, X _v = 0, that is, the pixels u and v are both bones, and the data value of the pixels u and v When P ₀₀ (D _u , D _v ) is a probability indicating that both D _u and D _v are bones, b _{4, that} is, E ₀₀ (D _u , D _v ) is −logP ₀₀ (D _u , D _v ) was set to be Also in this case, for example, when the relationship between the data values D _u and D _v (or probability) and the energy E is obtained in advance and stored in the ROM as a map, the data values D _u and D _v are given. The corresponding energy E may be derived from the map and set based on the derived energy E.

Then, a term depending on four points adjacent to the upper, lower, left, and right sides of the image data is set by the following equation (16) (step S120). Here, “X _u ”, “X _v ”, “X _s ”, and “X _t ” in Equation (16) are binary variables assigned to the adjacent pixels u, v, s, and t, respectively. “C _{1 to} c ₁₆ ” are real coefficients. As in the above-described step S110, the real coefficients c _{1 to} c ₁₆ are determined by whether each pixel determined for each pattern in which X _u , X _v , X _s , and X _t are given values 0 or 1 is a bone. The energy becomes smaller as the probability that the data values D _u , D _v , D _s , and D _{t of} the pixels u, v, s, and t indicate the possibility of the combination is higher. It is determined as follows.

  When each term is set in this way, a fourth-order polynomial is set by taking the sum of the terms and expanding them (step S130). This routine ends. Thereby, the energy of the 3rd floor is set. In the embodiment, by finding a binary variable corresponding to each pixel that minimizes this energy, the image data is divided into a bone portion and other portions.

  Next, the energy conversion process will be described. FIG. 5 is a flowchart illustrating an example of the energy conversion process. In this energy conversion process, first, a third or higher order conversion target term is set (step S200), whether the coefficient a of the set conversion target term is positive (step S210), and whether the order d is an odd number. (Step S220) is determined. When the coefficient a of the conversion target term is not positive, that is, negative, it is converted into a second-order polynomial, that is, first-order energy using the above-described equation (13) (step S230). When the order d is an even number, it is converted to the first-order energy using the above-described formula (9) (step S240). When the coefficient a of the conversion target term is positive and the order d is an odd number, the above-described formula (10) is used to convert the energy to the first floor (step S250). When there is an unconverted term having an order d of 3 or more, the process returns to step S200 to newly set a term to be converted, and the processing of steps S210 to 250 is repeated, and an unconverted term having an order d of 3 or more is found. This is the end of the process when there is no more.

When converted to the first-order energy polynomial in this way, a binary variable corresponding to each pixel is set using an algorithm such as an existing graph cut method, and the variable X _u has a value 0, that is, a color is set for the pixel u which is a bone. At the same time, the variable X _u has a value of 1, that is, a non-bone pixel u is subjected to a transparent process and displayed on the display 30 to display only the bone portion of the input image. Although there is a possibility that an error in which the variable is unknown may occur even by applying the graph cut method, etc., the energy minimization algorithm is applied by making a predetermined allocation or generating energy again only with the variable in which the error has occurred. It can respond by doing.

  According to the information processing apparatus of the embodiment described above, when a region is divided by assigning a binary label to each pixel of image data, a higher-order polynomial that is a fourth or higher order polynomial using a binary variable corresponding to the label. By setting the energy and adding binary external variables by the maximum integer number that does not exceed (d-1) / 2 for each term in which the order d is 3 or more, the primary and secondary symmetric equations are obtained. Since higher-order energy is converted into first-order energy with an order of 2 or less, energy minimization algorithms such as a graph cut method that can minimize the first-order energy can be applied appropriately with less computational cost. Image data can be divided into regions.

  In the information processing apparatus of the embodiment, a fourth-order (third floor) energy polynomial that depends on four adjacent pixels is set. However, a fifth-order (fourth floor) or more that depends on five or more neighboring pixels. It is also possible to set energy, for example, eighth-order (seventh floor) energy that depends on eight adjacent pixels in a three-dimensional grid for pixels (voxels) arranged in a three-dimensional space.

  In the information processing apparatus according to the embodiment, the present invention is applied to the image division problem that divides an area of image data. However, the present invention is not limited to this, and for example, for image restoration, stereo, video analysis, The present invention can be applied, and can also be applied to data processing other than the image field. Here, when the present invention is applied to image restoration, for example, a proposed image with a slight change is generated from the current image, and the current image and the generated proposed image are merged to generate a new image. A fusion move algorithm that repeats the process can be used. Here, “fusion” is to set the data value of the smaller energy of the data value of the pixel in the current image and the data value of the corresponding pixel in the proposed image to the corresponding pixel of the new image. By this, a new image is generated. Therefore, the energy minimization method can be used because it can be considered as a label allocation problem in which the value 0 is assigned to the pixel when maintaining the data value of the current image and the value 1 is assigned when replacing the data value of the proposed image. Can be solved. Here, the real number coefficient of the energy (polynomial) can be set so that the energy becomes smaller as the pattern is less likely to contain noise in the group of pixels constituting the image data. In addition, the proposed image can be set by blurring the current image using a smoothing filter (Gaussian kernel), or generate a gradient vector with the same dimensions as the number of variables by partial differentiation of energy with each variable. And a re-steep descent method in which the value is shifted from the data value of the current image in the direction of the generated vector. Of course, the present invention is not limited to these methods, and any method such as setting a random number image can be adopted.

  The embodiments of the present invention have been described using the embodiments. However, the present invention is not limited to these embodiments, and can be implemented in various forms without departing from the gist of the present invention. Of course you get.

  The present invention can be used in the manufacturing industry of information processing apparatuses and image processing apparatuses.

  20 computers, 22 higher energy setting units, 24 energy conversion units, 26 energy minimizing units, 30 displays, 40 keyboards, 50 mice.

Claims (11)

An information processing method for processing higher-order energy in which each variable can be expressed by a binary fourth-order or higher-order polynomial,
Storing a data representation of the higher order energy in a storage means;
By changing the stored data representation by a calculation means so as to correspond to the addition of binary external variables of a number corresponding to the order using a symmetric formula for the higher order energy , the higher order energy is changed to the same first order energy. Converted into energy
An information processing method comprising: setting each variable that minimizes the higher-order energy by applying a predetermined energy minimizing method to the converted first-floor energy. An information processing method according to claim 1,
The degree is d, the coefficient is a, the variable is x ₁ ,..., X _d , the external variable taking either 0 or 1 is w, the primary basic symmetry formula is S _1, and the secondary basic symmetry is Assuming that the equation is S ₂ , for each term of the higher-order energy polynomial, when the coefficient d is an even number for a term with a positive coefficient a, the first-order energy polynomial is expressed by using equation (1). And when the order d of the term is an odd number, the information is converted into the first-order energy polynomial using equation (2).

An information processing method according to claim 2,
An information processing method, wherein a term having a negative coefficient a for each term of the higher-order energy polynomial is converted into the first-order energy polynomial using Equation (3).

  The information processing method according to any one of claims 1 to 3, wherein the predetermined energy minimization method is any one of a graph cut method, a belief propagation method, and a tree weight redistribution message transmission method. The information processing method according to any one of claims 1 to 4, wherein the image data is processed.
A binary label that assigns a label of a first value to a pixel that matches a predetermined condition among pixels constituting the image data and assigns a label of a second value to a pixel that does not meet the predetermined condition A higher order energy of a polynomial including a first order term that depends only on each pixel and a fourth order or more term that depends on a set of four or more pixels is set as a variable so that the set higher order energy is minimized. An information processing method, wherein the binary label is assigned by setting the variable.   6. The information processing method according to claim 5, wherein the higher-order energy of the polynomial is set by determining the coefficient of each term so that the energy becomes smaller as the probability of meeting the predetermined condition is higher.   The information processing method according to claim 6, wherein the coefficient of each term is determined so that the energy becomes −logP when P is a probability of meeting the predetermined condition. An information processing method according to any one of claims 5 to 7,
When generating a proposed image with a change to the current image and replacing the pixel of the current image with the corresponding pixel of the generated proposed image, a label of the first value is assigned and the pixel of the current image An information processing method characterized by reconstructing an image by repeating a process of generating a new image by assigning the second label when not replacing a corresponding pixel of the generated proposed image.   The information processing method according to claim 8, wherein the proposal image is generated using a re-steep descent method.   The information processing method according to claim 8, wherein the proposed image is generated by applying a predetermined blur filter to the current image. An information processing apparatus for processing higher-order energy, each variable being represented by a binary fourth or higher order polynomial,
Storage means for storing a data representation of the higher order energy;
A calculating means for changing the data representation that is the stored so as to correspond to the addition of external variable binary number corresponding to the order by using a symmetric polynomial in the higher-order energy,
Energy conversion means for converting the higher-order energy into equivalent first- order energy by changing the data representation ;
Energy minimizing means for setting each variable that minimizes the higher order energy by applying a predetermined energy minimizing method to the converted first floor energy;
An information processing apparatus comprising:

Images