KR101796111B1 - Skull deformity analyzing system using a 3d morphological descriptor and a method for analyzing skull deformity using the same - Google Patents

Abstract

In an automatic skull deformity determining system using a 3D morphological descriptor and an automatic skull deformity determining method using the same, the automatic skull deformity determining system includes an image capturing unit, a skull model generating unit, an operating unit, and a skull deformity determining unit. The image capturing unit captures a skull image of a patient. The skull model generating unit obtains actual skull information from the image captured by the image capturing unit and generates an average skull model from a database. The operating unit includes a skull model deforming unit which deforms the average skull model using a plurality of referent points allocated to each actual skull and the average skull model and a skull deformity operating unit which operates a determination index of skull deformity based on the deformed amount of the average skull model. The skull deformity determining unit determines the actual skull deformity based on the determination index of the skull deformity. Accordingly, it is possible to give early treatment by quickly and accurately diagnosing the skull deformity.

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a skull shape automatic determination system using a three-dimensional shape descriptor, and a skull shape automatic determination method using the same.

The present invention relates to a system for automatically determining a skull shape, and a method for automatically determining a skull shape using the 3D morphological descriptor. More particularly, the present invention relates to a system and method for automatically determining a skull shape from a patient's skull shape using a 3D morphological descriptor And a method for automatically determining a skull shape using the same.

Craniosynostosis is a congenital disease that causes fusion of the suture line of the skull and obstructs the growth of the skull and brain tissue, and it is known that proper correction or surgery is required especially between 3 months and 1 year after birth.

Therefore, the diagnosis and treatment of cranial aneurysm is better as soon as possible. However, diagnosis of cranial aneurysm is difficult to diagnose early, I have been.

On the other hand, in relation to the diagnosis of cranial union, U.S. Patent No. 9370318 discloses a technique for quantitatively evaluating the skull, and a technique for analyzing computation of cranial fusion from a CT image. However, A simple technique for diagnosing an abnormal object by calculating a difference between a measured model and a shape model is disclosed. Accordingly, there is a problem in that the accuracy of diagnosis of cranial fusion is not high.

Therefore, in particular, recent 3-D imaging techniques such as CT imaging have been developed, and based on the circumstances in which the necessary information can be acquired immediately, the self-diagnosis or automatic diagnosis The need for a technique that can more accurately diagnose cingulate atrophy is emerging.

One. US Patent No. 9370318

SUMMARY OF THE INVENTION Accordingly, the present invention has been made in view of the above problems, and it is an object of the present invention to provide an automatic skull discrimination system using a 3D shape descriptor capable of more accurate and effective diagnosis.

It is another object of the present invention to provide a method for automatically determining a skeletal deformity using the above automatic determination system.

According to an embodiment of the present invention, an automatic skull placement determining system includes an image capturing unit, a skull model generating unit, a calculating unit, and a skull shape determining unit. The image capturing unit captures a skull image of the patient. The skull model generating unit obtains actual skull information from the image photographed by the image photographing unit, and generates an average skull model from the database. Wherein the calculation unit includes a skull model modification unit that modifies an average skull model using the average skull model and a plurality of reference points allocated to each of the actual skulls and a determination index of a skull formation variant based on the modified amount of the average skull model, And a skull deforming operation unit for calculating a skull deforming operation unit. The skull deformity judging unit judges the actual skull deformity based on the judgment index of the skull deformity.

In another aspect of the present invention, there is provided a method for automatically determining a skull shape, comprising: acquiring actual skull information from a photographed image; Generate an average skull model. And assigns a plurality of reference points to each of the average skull model and the actual skull. The mean skull model is modified using the reference points to match the actual skull. To derive a modified amount of the mean skull model. Based on the modified amount of the average skull model, a determination index of the skull deformity is calculated. Based on the judgment index of the skull deformity, the actual skull deformity is judged.

In one embodiment, the actual skull information and the mean skull model may not include a suture or an opening.

In one embodiment, in generating the average skull model, the average skull model may be generated based on normal skull information at a corresponding age stored in the database.

In one embodiment, in the step of allocating the plurality of reference points, the plurality of reference points may be Nasion, Basion, Opisthion, and Porion on both sides.

In one embodiment, the step of matching the mean skull model to the actual skull includes moving or rotating the mean skull model using the reference points, and moving the mean skull model to one of the reference points And reducing or enlarging the size of the image.

In one embodiment, in matching the average skull model to the actual skull,

식 (1)

Equation (1)

값이 최소가 되도록 정합을 수행할 수 있다. In the formula (1)

The matching can be performed so that the value becomes minimum.

In one embodiment, in matching the average skull model to the actual skull,

식 (2)

Equation (2)

의 평균값이 최소가 되도록 정합을 수행할 수 있다. In the formula (2)

Lt; RTI ID = 0.0 > minimum < / RTI >

In one embodiment, in deriving a modified amount of the mean skull model,

식 (3)

Equation (3)

Using Equation (3), the degree of local deformation between the average skull model and the actual skull can be derived.

In one embodiment, in deriving a modified amount of the mean skull model,

식 (4)

Equation (4)

Using equation (4) above, the directionality can be determined from the local deformation between the average skull model and the actual skull.

In one embodiment, in the step of calculating the judgment index of the skull deformity, the judgment index of the skull deformity includes an average value of positive distance values in each patch area, an average value of negative distance values, A ratio of the distance value and a ratio of the negative distance value.

In one embodiment, the mean value of the positive distance value is given by the following equation (5)

식 (5)

Equation (5)

The average value of the negative distance values is represented by the following equation (6)

식 (6)

Equation (6)

The ratio of the positive distance value is given by the following equation (7)

식 (7)

Equation (7)

The ratio of the negative distance value is expressed by the following equation (8)

식 (8)

Equation (8)

Lt; / RTI >

According to the embodiments of the present invention, it is possible to automatically determine whether or not the skull is deformed by extracting a judgment index capable of judging the skull deformity while matching the average skull model with the actual skull, Based on subjective judgment, it is possible to diagnose the diagnosis of skull deformity more quickly and accurately. Therefore, it is possible to treat early by early detection of skull deformation such as craniofibrosis.

In particular, by defining a skull with a skeleton model and defining an average skull model by removing sutures or openings that may be included in the actual skull information, it is possible to minimize errors in the operation when applying the above-mentioned judgment index, Quick judgment is possible.

In addition, in the step of matching the average skull model to the actual skull, since it includes the step of reducing or enlarging the scale with respect to any one reference point in addition to the movement and rotation using the reference points, more accurate skull matching is possible .

값 및

의 평균값을 최소가 되도록 하므로, 두개골 정합의 타당을 평가하는 기준을 통해 최적 정합을 수행할 수 있다. Further, as a means for judging whether the skull matching is valid or not

Value and

So that optimal matching can be performed based on criteria for evaluating the validity of skull matching.

Further, as an index for determining the amount of modification of the mean skull model, a mean value of positive distance values, a mean value of negative distance values, a ratio of positive distance values, and a negative distance value in each patch area By including the ratio, it is possible to determine whether or not the skull is automatically released based on the index value.

FIG. 1 is a block diagram illustrating an automatic determination system of a skull shape according to an exemplary embodiment of the present invention. Referring to FIG.
FIG. 2 is a flowchart illustrating a method of automatically determining a skull shape using the skull shape automatic determination system of FIG. 1;
3 is a flow chart illustrating the step of matching the mean skull model of FIG. 2 to the actual skull.
FIG. 4A is an image showing a skull suture of infancy, and FIG. 4B is an image showing an example of a skull divided by dividing the skull of FIG.
FIG. 5 is an image showing an example of an average skull model generated in the step of generating the average skull model of FIG. 2. FIG.
FIGS. 6A and 6B are images showing an example in which a plurality of reference points are allocated to the average skull model in the step of assigning the plurality of reference points in FIG. 2, and an example in which a plurality of reference points are allocated in the actual skull.
FIG. 7A is an image showing an example of moving or rotating the mean skull model of FIG. 3, and FIG. 7B is an image showing an example of reducing or enlarging the mean skull model of FIG.
FIGS. 8A and 8B are plan and side views illustrating an example of deriving the local deformation degree between the average skull model and the actual skull in the step of deriving the modified amount of the average skull model of FIG. 2. FIG.
FIG. 9A is a schematic diagram showing coordinates for determining a direction in a local deformation between an average skull model and an actual skull in a step of deriving a modified amount of the average skull model of FIG. 2, and FIG. Is an image showing an example in which the directionality is determined.
FIG. 10 is an image showing an example of a patch area for calculating a modified state of the mean skull model of FIG. 2 and a determination index of a skull deformity based thereon.
FIGS. 11A and 11B are graphs showing judgment indices of skull deformity. FIG.

While the present invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not limited to the disclosed embodiments. It is to be understood, however, that the invention is not intended to be limited to the particular forms disclosed, but on the contrary, is intended to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention. Like reference numerals are used for like elements in describing each drawing. The terms first, second, etc. may be used to describe various components, but the components should not be limited by the terms.

The terms are used only for the purpose of distinguishing one component from another. The terminology used in this application is used only to describe a specific embodiment and is not intended to limit the invention. The singular expressions include plural expressions unless the context clearly dictates otherwise.

In the present application, the term "comprises" or "comprising ", etc. is intended to specify that there is a stated feature, figure, step, operation, component, But do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, parts, or combinations thereof.

Unless defined otherwise, all terms used herein, including technical or scientific terms, have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Terms such as those defined in commonly used dictionaries are to be interpreted as having a meaning consistent with the contextual meaning of the related art and are to be interpreted as either ideal or overly formal in the sense of the present application Do not.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings.

FIG. 1 is a block diagram illustrating an automatic determination system of a skull shape according to an exemplary embodiment of the present invention. Referring to FIG. FIG. 2 is a flowchart illustrating a method of automatically determining a skull shape using the skull shape automatic determination system of FIG. 1; 3 is a flow chart illustrating the step of matching the mean skull model of FIG. 2 to the actual skull. FIG. 4A is an image showing a skull suture of infancy, and FIG. 4B is an image showing an example of a skull divided by dividing the skull of FIG. FIG. 5 is an image showing an example of an average skull model generated in the step of generating the average skull model of FIG. 2. FIG. FIGS. 6A and 6B are images showing an example in which a plurality of reference points are allocated to the average skull model in the step of assigning the plurality of reference points in FIG. 2, and an example in which a plurality of reference points are allocated in the actual skull. FIG. 7A is an image showing an example of moving or rotating the mean skull model of FIG. 3, and FIG. 7B is an image showing an example of reducing or enlarging the mean skull model of FIG. FIGS. 8A and 8B are plan and side views illustrating an example of deriving the local deformation degree between the average skull model and the actual skull in the step of deriving the modified amount of the average skull model of FIG. 2. FIG. FIG. 9A is a schematic diagram showing coordinates for determining a direction in a local deformation between an average skull model and an actual skull in a step of deriving a modified amount of the average skull model of FIG. 2, and FIG. Is an image showing an example in which the directionality is determined. FIG. 10 is an image showing an example of a patch area for calculating a modified state of the mean skull model of FIG. 2 and a determination index of a skull deformity based thereon.

Referring to FIGS. 1 and 2, the automatic diagnosis system for clavicular joint according to the present embodiment includes an image capturing unit 100, a database 200, a skull model generating unit 300, an operation unit 400 And a skull deformity determination unit 430. The operation unit 400 includes a skull model modification unit 410 and a skull shape operation unit 420 and automatically determines whether or not the skull is deformed.

More specifically, the image capturing unit 100 photographs the patient's skull using a computerized tomographic (CT) or MRI image, and the captured image is provided to the skull model generating unit to obtain actual skull information S10).

In this case, the CT image or the MRI image may be a two-dimensional or three-dimensional image, and the image data photographed in this manner is also provided in the database 200.

The database 200 stores information photographed by the image capturing unit 100. In particular, in a state in which information about a skull in a normal state is stored for each age group, Provide information about the skull in steady state.

Thereafter, the skull model generation unit 300 receives information on the skull in a steady state from the database 200, and generates an average skull model (step S20).

Referring to FIG. 4A, in general, the skull includes a suture, and in the case of infant cranium, in many cases, the suture is open, so that the actually taken skull may include a suture or an opening.

However, as shown in FIG. 4B, in this embodiment, the one-piece skull shape is obtained so that the suture or the opening is not included in the actual skull information.

Also, as shown in FIG. 5, the average skull model generated by the skull model generation unit 300 is also formed as one integral skull shape so that the suture or the opening is not included.

Thus, when the average skull model is fitted to the actual skull through the matching step described later and the degree of deformation is determined, it is possible to minimize the error of the degree of deformation by the suture or the opening.

Referring to FIGS. 1, 2, 6A and 6B, the skull model modification unit 410 allocates a plurality of reference points from the average skull model and the actual skull, respectively (step S30).

In this case, the plurality of reference points may be Nasion, Basion, Opisthion, and Porion on both sides. 6a and 6b, Nasion (P1), Basion (P2), Opisthion (P3), Left Porion (P4) and Right Porion (P5) in the average skull model and the actual skull, Five reference points can be assigned respectively.

Referring to FIGS. 1, 2, 7A and 7B, the skull model modification unit 410 modifies the average skull model using the reference points to match the actual skull (step S40).

More specifically, referring to FIGS. 3 and 7A, the skull model modification unit 410 moves or rotates the average skull model using the reference points (step S41).

That is, in order to match the five reference points of the average skull model to the five reference points of the actual skull, the average skull model is moved in parallel with respect to each of the XYZ axes in the three-dimensional space, As shown in Fig.

Since the four reference points P2, P3, P4 and P5 among the five reference points are located at the back of the skull relative to each other as shown in the figure, the average skull model is moved using the reference points There is a limit in matching the average skull model to the actual skull.

Accordingly, referring to FIGS. 3 and 7B, the skull model modification unit 410 reduces or enlarges the average skull model with respect to any one of the reference points (step S42).

In this case, a scale change, such as enlargement or reduction, may be performed on any one of the five reference points, thereby modifying the average skull model more closely to the actual skull, can do.

Meanwhile, in the step of matching the average skull model to the actual skull as described above, it is possible to determine a criterion to what extent the movement / rotation / scaling should be progressed to a certain level, that is, Standards are needed.

값, 즉 기준 정합 오차(fiducial registration error: FRE)가 최소가 되도록 정합을 수행한다. For this purpose, in the present embodiment, when the average skull model is matched to the actual skull, the following equation (1)

Value, that is, the fiducial registration error (FRE), is minimized.

식 (1)

Equation (1)

In this case, N is the number of the base point, s is the scale quantity, T is the XYZ movement amount of each axis, R is XYZ a rotation amount based on each axis, x _i is the reference point of the average skull model, y _i is the actual skull Reference points.

값, 즉 가우시안 가중치 거리(Gaussian-weighted distance)의 평균값이 최소가 되도록 정합을 수행한다. In addition, in the present embodiment, when the average skull model is matched to the actual skull, in addition to Equation (1), Equation (2)

Value, that is, the average value of the Gaussian-weighted distance is minimized.

식 (2)

Equation (2)

In this case, GW means the number of reference points of the mean skull model, MNM (i) is the reference points of the mean skull model, and T _s means the same scale value.

As described above, in this embodiment, it is possible to judge whether or not the optimal matching is performed by matching the average skull model to the actual skull in consideration of the above equations (1) and (2).

The results of the above matching are shown in FIGS. 8A and 8B. As shown in FIG. 8B, it can be seen that the average skull model has been modified and optimized for the actual skull.

Referring to FIGS. 1 and 2, the clavicular deforming part 420 derives a deformed amount of the average cranial model (step S50).

In the previous step, the average skull model is deformed so as to be matched to the actual skull, and thus, if the modified amount of the average skull model is derived, the actual skull formation state can be determined.

In this case, since the average skull model is a virtual model, it is possible to easily carry out the related calculation. Therefore, it is possible to more easily obtain the actual skull through the modified amount of the average skull model without directly analyzing the image information about the skull. It is possible to judge whether or not it is released.

Specifically, in order to derive the modified amount of the average skull model, the degree of local deformation between the average skull model and the actual skull is derived using the following equation (3).

식 (3)

Equation (3)

In this case, B ₁ , B _m and B _n are respectively defined as the third basis functions of the B-spline as follows, where φ is the control point, u, v and w are the X, .

That is, Equation (3) is a B-spline-based level deformation matching that deforms an object by moving the grids of the respective control points constituting the B-spline. The degree of local deformation of the average skull model can be derived by comparing the modified mean skull model with the initial mean skull model.

On the other hand, FIG. 9A shows the coordinates for determining the direction in the local deformation of the average skull model in the step of deriving the modified amount of the average skull model, and based on the coordinates, Can be used to determine the directionality in the local deformation between the average skull model and the actual skull.

식 (4)

Equation (4)

In this case, p is the point of the skull in the matched average skull model, R _d is the inner area of the actual skull, and V _p is the size of the deformation vector when the mean skull model is deformed.

That is, it is possible to determine whether the local deformity of the average skull model has been enlarged (positive value) or reduced (negative value) through the calculation of the equation (4). As shown in FIG. 9B, 4), the deformation value of the average skull model can be schematized according to the degree of enlargement and reduction.

1, 2, and 10, the skull deformer calculator 420 calculates a determination index of the skull deformity based on the modified amount of the average skull model (step S60).

In general, the actual skull can be divided into five patch areas as shown in FIG. 4A, and the average skull model is also divided into five patch areas A, B, C, D, and E And calculates a judgment index of the skull formation in each of the patch regions.

At this time, the determination index of the skull deformity may be an average value of positive distance values, a mean value of negative distance values, a ratio of positive distance values, and a ratio of negative distance values.

That is, the four determination indices can be derived from each of the five patch regions, and as a result, a total of 20 determination indices of the skull deformity in the present embodiment can be derived.

Meanwhile, the four judgment indices are defined as follows.

That is, the positive local distance (PLD) of the positive distance value means an average value of the positive distance value in each patch region, and is defined by the following equation (5).

식 (5)

Equation (5)

Where N _p is the number of positive distance values and D _p is the deformation value with a positive distance value.

Further, the negative local distance (NLD) of the negative distance value means an average value of negative distance values in each patch region, and is defined by the following equation (6).

식 (6)

Equation (6)

In this case, N _n is the number of negative distance values, and D _n is a deformation value having a negative distance value.

On the other hand, the positive local area ratio (PLAR) of the positive distance ratio means a ratio of a region having a positive distance value in each patch region, and is defined by the following equation (7).

식 (7)

Equation (7)

In addition, the negative local area ratio (NLAR) of the negative distance value is a ratio of a region having a positive distance value in each patch region, and is defined by the following equation (8).

식 (8)

Equation (8)

Referring to FIGS. 1 and 2, the skull variant determining unit 430 determines the actual skull deformity based on the determination index of the skull deformity (step S70).

That is, as described above, it is possible to automatically determine the actual skull release based on the four determination indices derived from each of the five patch regions.

11A, 11B, 11C, and 11D are graphs showing determination indices of skull deformity.

As described above, with respect to the average skull model divided into five patch regions (A to E), the average skull model is matched to the actual skull, and as a result, To determine the deformation of the actual skull.

For example, FIGS. 11A and 11B show examples of calculation of a positive local distance (PLD) of positive distance values for A region and B region among five patch regions of the above-described average skull model It is a graph.

As shown in FIGS. 11A and 11B, when the skull shape is bicoronal or unicoronal, the mean values of the positive distances are both large in both positive and negative values Can be confirmed.

11C and 11D are graphs showing examples of calculation of a positive local distance (PLD) of positive distance values for the C region and the D region among the five patch regions of the average skull model described above to be.

As shown in FIGS. 11C and 11D, when the skull shape is sagittal, it can be seen that the mean values of the positive distance are relatively large as a negative value.

As described above, based on the result of calculating the above-mentioned determination indices in each of the five patch regions for the average skull model modified by the matching after the average skull model is matched with the actually photographed actual skull, The branch can automatically determine whether or not the release occurs.

According to the embodiments of the present invention, it is possible to automatically determine whether or not the skull is deformed by extracting a judgment index capable of judging a skull deformity while matching an average skull model with an actual skull model, The diagnosis of skull deformity can be made more quickly and accurately. Therefore, it is possible to treat early by early detection of skull deformation such as craniofibrosis.

In particular, by defining a skull with a skeleton model and defining an average skull model by removing sutures or openings that may be included in the actual skull information, the error of the operation can be minimized in applying the decision index, Quick judgment is possible.

In addition, in the step of matching the average skull model to the actual skull, since it includes the step of reducing or enlarging the scale with respect to any one reference point in addition to the movement and rotation using the reference points, more accurate skull matching is possible .

값 및

의 평균값을 최소가 되도록 하므로, 두개골 정합의 타당을 평가하는 기준을 통해 최적 정합을 수행할 수 있다. Further, as a means for judging whether the skull matching is valid or not

Value and

So that optimal matching can be performed based on criteria for evaluating the validity of skull matching.

Further, as an index for determining the amount of modification of the mean skull model, a mean value of positive distance values, a mean value of negative distance values, a ratio of positive distance values, and a negative distance value in each patch area By including the ratio, it is possible to determine whether or not the skull is automatically released based on the index value.

It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the present invention as defined by the following claims. It can be understood that it is possible.

INDUSTRIAL APPLICABILITY According to the present invention, an automatic judgment system of a skull deformity type and an automatic judgment method of a skull type using this system have industrial applicability that can be used for judging cortical union.

10: Automatic determination system of skull shape
100: image capturing unit 200: database
300: skull model generation unit 400:
410: Skull model deforming unit 420: Skull deforming operation unit
430: skull deformity judgment unit

Claims (12)

An image capturing unit for capturing an image of a skull of a patient;
A skull model generating unit for acquiring actual skull information from the image photographed by the image photographing unit and generating an average skull model from a database;
A skull model modifying unit that modifies an average skull model using the average skull model and a plurality of reference points allocated to each of the actual skulls and a determination index of a skull formation variant based on the modified amount of the average skull model An arithmetic unit including a skull shape computing unit; And
And a skull shape determining unit for determining an actual skull deformity based on the judgment index of the skull deformity. Obtaining actual skull information from the photographed image;
Generating an average skull model;
Allocating a plurality of reference points to the average skull model and the actual skull;
Modifying the mean skull model using the reference points and matching the actual skull with the actual skull;
Deriving a modified amount of the mean skull model;
Calculating a determination index of the skull deformity based on the modified amount of the average skull model; And
And determining an actual skull deformity based on the judgment index of the skull deformity. 3. The method of claim 2,
Wherein the actual skull information and the average skull model do not include a suture or an opening. 3. The method of claim 2, wherein in the step of generating the average skull model,
Wherein the average skull model is generated based on normal skull information at a corresponding age stored in a database. 3. The method according to claim 2, wherein, in the step of allocating the plurality of reference points,
Wherein the plurality of reference points are Nasion, Basion, Opisthion, and Porion on both sides. 3. The method of claim 2, wherein matching the average skull model to an actual skull comprises:
Moving or rotating the average skull model using the reference points; And
And reducing or enlarging the average skull model with respect to any one of the reference points. 7. The method of claim 6, wherein, in matching the average skull model to an actual skull,

Equation (1)
In the formula (1)

Wherein the matching is performed so that the value is minimized. 7. The method of claim 6, wherein, in matching the average skull model to an actual skull,

Equation (2)
In the formula (2)

Wherein the matching is performed so that the average value of the skull portions is minimized. 3. The method of claim 2, wherein in deriving a modified amount of the mean skull model,

Equation (3)
Wherein the degree of local deformation between the average skull model and the actual skull is derived using the equation (3). 10. The method of claim 9, wherein in deriving a modified amount of the mean skull model,

Equation (4)
Wherein the directionality is determined in the local deformation between the average skull model and the actual skull using the equation (4). 11. The method according to claim 10, wherein, in calculating the determination index of the skull deformity,
The determination index of the skull shape variance includes an average value of positive distance values, a mean value of negative distance values, a ratio of positive distance values, and a ratio of negative distance values in each patch area A method of automatic determination of skull deformity. 12. The method of claim 11,
The average value of the positive distance values is expressed by the following equation (5)

Equation (5)
The average value of the negative distance values is represented by the following equation (6)

Equation (6)
The ratio of the positive distance value is given by the following equation (7)

Equation (7)
The ratio of the negative distance value is expressed by the following equation (8)

Equation (8)
And calculating the skull shape automatically.

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