CN103198521A - Caput femoris surface three-dimensional model reconstruction method for individualized design - Google Patents

Abstract

The invention belongs to the medical image processing field, and particularly relates to a caput femoris surface three-dimensional model reconstruction method for individualized design. The method comprises the steps of carrying out contour extraction and interior filling on caput femoris and acetabulum in each two-dimensional CT image in input articulatio coxae CT sequential images, carrying out segmentation operation on acetabulum in each two-dimensional CT image by means of connectedness among pixels, carrying out acetabulum three-dimensional model reconstruction by means of segmented acetabulum sequential images, carrying out acetabulum concave surface data point sampling by means of the space radial line method, and achieving individualized caput femoris surface reconstruction by means of the ellipsoid fitting method based on general quadratic surface constraint. The caput femoris surface three-dimensional model reconstruction method can provide an accurate data model for the individualized design of a caput femoris prosthesis.

Description

A kind of femoral head surface three dimension model reconstructing method towards personalized design

Technical field

The invention belongs to field of medical image processing, particularly a kind of femoral head surface three dimension model reconstructing method towards personalized design.

Background technology

Avascular necrosis of femoral head is a kind of worldwide disease, the number that the using artificial femoral head replacement performs the operation to save hip joint function increases day by day, but because prosthetic loosening usually makes artificial thigh bone lose efficacy, its main cause is the instability coupling between prosthese and the acetabular bone.The long-time stability that effectively improve between femoral head prosthesis and the acetabular bone are to improve the key of artificial femoral head replacement surgery's long-term effect.

Femoral head prosthesis and acetabular bone matching effect are not good, are that the individual difference by human organ causes.Different patients' hip joint shape has nothing in common with each other, and specification, the kind of the artificial femoral head prosthesis of producing both at home and abroad at present are limited, therefore the artificial femoral head prosthesis of implanting for the patient mostly can not form with acetabular bone and dissect coupling, contact area is little, what form between prosthese and the acetabular bone is contacting of several points or regional area, rather than face contact on a large scale, cause the stress distribution of prosthese and acetabular bone surface of contact extremely unreasonable, thereby the appreciable impact prosthesis stability makes the activity of patient's postoperative femur not freely simultaneously.Only have 3～15 years the serviceable life of single femoral head prosthesis, brings huge life influence and economic pressures for the caput femoris necrosis patient.Therefore, individual organ parameter according to patient self, with the femoral head surface reconstruction of necrosis be with acetabular bone matched well state under three-dimensional model, for the femoral head prosthesis production of " individuation ", " formula customized " provides the geometric data parameter, become problem demanding prompt solution in the rehabilitation of current caput femoris necrosis.

At the personalized three-dimensional reconstruct of femoral head surface model, domestic scholars has proposed some solutions, and the document of delivering mainly comprises: " based on the research of the individuality couplingization bone manufacture method of reverse engineering " of " XI AN JIAOTONG UNIVERSITY Subject Index "; " human femoral head based on CT image reverse technology is repaired modeling " of " Dalian University of Technology's journal ".This type of technology is mainly utilized CT(or MRI) the volume data image carries out three-dimensionalreconstruction to the femoral head of necrosis, according to the biomechanical analysis result resulting model carried out the reconstruct of curved surface discretize secondary then.The precision of these class methods is compared traditional prosthetic designs method and is increased, but its design cycle is longer, and restructuring procedure is complicated.

The foreign scholar comprises for the method that the personalized three-dimensional reconstruct of femoral head surface proposes: at " the Articular surface remodeling of the hip after periacetabular osteotomy " and " An integrated approach for reconstructing a surface model of the proximal femur from sparse input data and a multi-resolution point distribution model:an in vitro study " of " International Journal of Computer Assisted Radiology and Surgery ".This type of technology is mainly utilized the sparse point data in the X-ray sheet of calibration back, adopts the some distributed model (MR-PDM) of more piece resolution to rebuild the near end of thighbone surface model of personalized patient.The surface reconstruction precision of these class methods further improves, but the stability of its reconstruction result is lower, and the data environment of restructuring procedure has been proposed harsh requirement.

Summary of the invention

Purpose of the present invention is just in order to overcome the deficiencies in the prior art, solve and how to utilize the biological geometric parameter of patient self fast, carry out the personalization of femoral head surface and stablize the problem of reconstruct, and propose a kind of femoral head surface three dimension model reconstructing method towards personalized design.The resulting femoral head surface model of this method can be realized good matching status with acetabular bone, has advantages such as robustness is strong, computation process is convenient, and model can offer precise data for the personalized design of femoral head prosthesis.

The objective of the invention is to be achieved through the following technical solutions.

A kind of femoral head surface three dimension model reconstructing method towards personalized design said method comprising the steps of:

1. import the hip joint CT sequence image of being gathered by conventional Medical CT machine;

2. the femoral head in each width of cloth two-dimensional ct image in the sequence image of having imported and acetabular bone are carried out profile extraction and inner filling; Utilize the profile of the interior femoral head of Sobel operator extraction two-dimensional ct image and acetabular bone earlier, the recycling scanning Beam Method realizes the inside filling of femoral head and acetabular bone profile;

3. in the femoral head of having filled and acetabular bone, utilize connectedness between pixel to realize the cutting operation of acetabular bone in each width of cloth two-dimensional ct image, concrete steps are as follows:

1) in femoral head and acetabular bone closed region, appoints and get two pixels, if these two pixels have the path of connection, then these two pixels are classified as same class, if these two pixels not have the path of connection, then these two pixels are classified as two classes;

2) all pixels in traversal femoral head and the acetabular bone closed region are classified as two classes with all pixels;

3) give up the pixel set that belongs in the femoral head closed region, keep the pixel set that belongs in the acetabular bone closed region, realize the cutting operation of acetabular bone;

4. utilize and cut apart the acetabular bone sequence image that finishes, carry out the acetabular bone three-dimensional model reconfiguration;

5. on the acetabular bone three-dimensional model that has obtained, utilize the space radiation collimation method, carry out the sampling of acetabular bone concave surface data point, comprise the steps:

1) in the concave surface of acetabular bone three-dimensional model, selects a volume coordinate point, centered by this coordinate points, to complete all spatial emission radiation straight lines;

Obtain an intersection point when 2) every radiation straight line and acetabular bone three-dimensional model intersect, the set of all intersection points is acetabular bone concave surface data sampling point;

6. utilize the acetabular bone concave surface data point that has obtained, adopt the ellipsoid approximating method based on the general quadric surface constraint, realize personalized femoral head surface reconstruction, concrete steps are as follows:

1) general quadric surface is constrained to Ellipsoidal Surface, the equation of establishing general quadric surface is:

ax ²+by ²+cz ²+2fyx+2gxz+2hxy+2px+2qy+2rz+d=0 （1）

If:

I=a+b+c （2）

J=ab+bc+ac-f ²-g ²-h （3）

$K=\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|---\left(4\right)$

For any one ellipsoid α 〉=4 are arranged, make α J-I ²0, select for use α=4 that quadric surface is constrained to Ellipsoidal Surface;

2) utilize acetabular bone concave surface data sampling point, adopt and carry out the Ellipsoidal Surface match based on the least square method of algebraically distance, obtain all unknowm coefficients in the formula (1);

3) establish X=(x, y, z) ^T, ellipsoid general quadric surface equationization is classified as matrix form:

d+(2p,2q,2r)X+X ^TKX=0 （5）

For matrix K, use Jacobi orthogonal transformation method to ask its characteristic root (λ ₁, λ ₂, λ ₃), and 3 * 3 proper vector R, order

(a ₁,a ₂,a ₃)=(2p,2q,2r)R （6）

Have

$a_0=d-\frac{1}{4}(\frac{{a_1}^2}{{\mathrm{\λ}}_1}+\frac{{a_2}^2}{{\mathrm{\λ}}_2}+\frac{{a_3}^2}{{\mathrm{\λ}}_3})---\left(7\right)$

Obtain the length of three main shafts of ellipsoid

$a=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_1}}$ $b=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_2}}$ $c=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_3}}---\left(8\right)$

If the rotation angle of three main shafts of ellipsoid is (α, beta, gamma), obtain the transformation relation of ellipsoid centre coordinate system and master coordinate system

X′=X ₀+R ₁(α)R ₁(β)R ₁(γ)X （9）

$X_0={(\frac{a_1}{{2\mathrm{\λ}}_1},\frac{a_2}{{2\mathrm{\λ}}_2},\frac{a_3}{{2\mathrm{\λ}}_3})}^T---\left(10\right)$

R ₁(α)R ₁(β)R ₁(γ)=R ^T （11）

Description of drawings

Fig. 1 is femoral head surface three dimension model reconstructing method process flow diagram of the present invention.

Fig. 2 is hip joint CT image related in the embodiment of the invention.

Fig. 3 is for utilizing the synoptic diagram of the interior femoral head of Sobel operator extraction two-dimensional ct image and acetabular bone profile in the embodiment of the invention.

The synoptic diagram that Fig. 4 fills for the inside that utilizes scanning Beam Method to carry out femoral head and acetabular bone profile in the embodiment of the invention.

Fig. 5 is the synoptic diagram after fill the inside of realizing femoral head and acetabular bone profile in the embodiment of the invention.

Fig. 6 carries out the synoptic diagram of pixel assorting process for utilize connectedness between pixel in the embodiment of the invention in femoral head and acetabular bone zone.

Fig. 7 is all pixels in traversal femoral head and the acetabular bone closed region in the embodiment of the invention, and all pixels are classified as synoptic diagram after two classes.

Fig. 8 is for giving up the pixel set that belongs in the femoral head closed region, keep the pixel set that belongs in the acetabular bone closed region, the synoptic diagram behind the cutting operation of realization acetabular bone in the embodiment of the invention.

Fig. 9 has been cut apart the acetabular bone sequence image that finishes for utilizing in the embodiment of the invention, carries out the synoptic diagram of acetabular bone three-dimensional model reconfiguration.

Figure 10 in the embodiment of the invention on the acetabular bone three-dimensional model that has obtained, utilize the space radiation collimation method, carry out the synoptic diagram of acetabular bone concave surface data point sampling.

Figure 11 adopts the ellipsoid approximating method based on the general quadric surface constraint for utilizing the acetabular bone concave surface data point that has obtained in the embodiment of the invention, realizes the synoptic diagram behind the personalized femoral head surface reconstruction.

Figure 12 is the synoptic diagram of personalized femoral head and acetabular bone matched well in the embodiment of the invention.

Embodiment

Below in conjunction with accompanying drawing preferred embodiment of the present invention is described in detail, thereby so that our bright advantage and feature can be easier to be it will be appreciated by those skilled in the art that protection scope of the present invention is made more explicit defining.

Please refer to Fig. 1, the invention provides a kind of femoral head surface three dimension model reconstructing method towards personalized design, this method comprises the steps: to import hip joint CT sequence image; Femoral head and acetabular bone rim detection are filled with inner; Acetabular bone based on connectedness is cut apart; The acetabular bone three-dimensional model reconfiguration; The sampling of acetabular bone concave surface data point; The match of personalized femoral head ellipsoid.To introduce this a kind of femoral head surface three dimension model reconstructing method towards personalized design below in detail.

Please refer to Fig. 2～12, will specifically introduce this a kind of femoral head surface three dimension model reconstructing method towards personalized design below.

The ill hip joint sequence C T image that input is collected by conventional medical multilayer spiral CT machine.

At first, the femoral head in each width of cloth two-dimensional ct image in the sequence image of having imported and acetabular bone are carried out profile extraction and inner filling, concrete steps are:

Utilize the profile of the interior femoral head of Sobel operator extraction two-dimensional ct image and acetabular bone.Definition Sobel operator is:

s(i,j)=|Δ _xf|+|Δ _yf|=|(f(i-1,j-1)+2f(i-1,j)+f(i-1,j+1))-(f(i+1,j-1)+2f(i+1,j)+f(i+1,j+1))|+|(f(i-1,j-1)+2f(i,j-1)+f(i+1,j-1))-(f(i-1,j+1)+2f(i,j+1)+f(i+1,j+1))|

Its convolution operator is:

${\mathrm{\Δ}}_{x}f=\left[\begin{array}{ccc}-1& 0& 1\\ -2& 0& 2\\ -1& 0& 1\end{array}\right],$ ${\mathrm{\Δ}}_{y}f=\left[\begin{array}{ccc}-1& -2& -1\\ 0& 0& 0\\ 1& 2& 1\end{array}\right]$

Suitable T=160 in this example of selected threshold T(), do as judge: s (i, j)〉during T, (i j) is the step-like marginal point, and s (i, j) } be edge image.

In the image at femoral head and acetabular bone outline line place, from top to bottom, generate horizontal scan line line by line, every surface sweeping line can produce several intersection points when intersecting with femoral head and acetabular bone profile, N intersection point of every sweep trace and N+1(N=1,3,5,7,9,) can be judged to be femoral head and acetabular bone profile inside between the individual intersection point, all the other are outside.Accordingly, utilize scanning Beam Method to realize the inside filling of femoral head and acetabular bone profile.

Then, in the femoral head of having filled and acetabular bone closed region, appoint and get two pixels, with one of them pixel as the seed point, carry out iterative expansion growth to 8 neighborhood directions, if comprise another pixel in the zone after the final growth, judge that then these two pixels are communicated with, it is the path that these two pixels have connection, then these two pixels are classified as same class, if these two pixels do not have the path of connection, then these two pixels are classified as two classes.

All pixels in traversal femoral head and the acetabular bone closed region finally can be classified as all pixels two classes.

Give up the pixel set that belongs in the femoral head closed region, keep the pixel set that belongs in the acetabular bone closed region, can realize the cutting operation of acetabular bone.Accordingly, utilize connectedness between pixel to realize the cutting operation of acetabular bone tissue in each width of cloth two-dimensional ct image.

Utilize and cut apart the acetabular bone sequence image that finishes, adopt Marching Cubes algorithm (establishing threshold value is 60), carry out the acetabular bone three-dimensional model reconfiguration, obtain the three-dimensional model of acetabular bone tissue.

On the acetabular bone three-dimensional model that has obtained, in the acetabular fossa concave surface, choose a volume coordinate point (initial sampled point central point), be launching centre with this central point, to complete all spatial dimension emitted radiation straight lines (two angle is 20 ° between radiant rays), every radiation straight line and acetabular bone three-dimensional model can obtain an intersection point when intersecting, and the set of all intersection points is acetabular bone concave surface data sampling point set.

At last, utilize the acetabular bone concave surface data point that has obtained, adopt the ellipsoid approximating method based on the general quadric surface constraint, realize personalized femoral head surface reconstruction, concrete steps are:

Under right angle three-dimensional system of coordinate X-Y-Z, the general quadric surface equation

ax ²+by ²+cz ²+2fyx+2gxz+2hxy+2px+2qy+2rz+d=0

If:

I=a+b+c

J=ab+bc+ac-f ²-g ²-h

$K=\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$

By the quadric surface theory as can be known, I, J, K are the invariant of quadric surface under coordinate transform, work as J〉0, I * K〉0 o'clock, this quadric surface one is decided to be ellipsoid.When minor axis is at least a half of major axis, 4J-I must be arranged ²0, so 4J-I ²The 0th, quadric surface is the necessary condition of ellipsoid.For the femoral head curved surface, its general shape is similar with spheroid, that is to say that in general the minor axis of femoral head ellipsoid is at least the general of major axis, therefore selects α=4, i.e. 4J-I for use ²0 quadric surface is constrained to Ellipsoidal Surface is feasible for this example.

If the acetabular bone concave surface data sampling point set of pending match is

$\{p_i(x_i,y_i,z_i){\}}_{i=1}^n$

According to the coordinate of concentrated each point of point, make up matrix

$x_i={(x_i^2,y_i^2,x_i^2,{2y}_i{z}_i,{2x}_i{y}_i,{2x}_i,{2y}_i,{2z}_i,1)}^T$

D=(X ₁,X ₂,X ₃,...,X _n)

By aforementioned general quadric surface equation, establish

v=(a,b,c,f,g,h,p,q,r,d) ^T

Can be described as with formula apart from least square method based on algebraically under the ellipsoid constraint

minDv ²subject tokJ-I ²=1

KJ-I wherein ²=1 can be with the matrix equivalent representation

v ^TCv=1

$C=\left(\begin{array}{cc}{C}_1& 0_{6\×4}\\ 0_{4\×6}& 0_{4\×4}\end{array}\right)$

$C_1=\left(\begin{array}{cccccc}-1& \frac{k}{2}-1& \frac{k}{2}-1& 0& 0& 0\\ \frac{k}{2}-1& -1& \frac{k}{2}-1& 0& 0& 0\\ \frac{k}{2}-1& \frac{k}{2}-1& -1& 0& 0& 0\\ 0& 0& 0& -k& 0& 0\\ 0& 0& 0& 0& -k& 0\\ 0& 0& 0& 0& 0& -k\end{array}\right)$

Use method of Lagrange multipliers, the group of solving an equation

$\left\{\begin{array}{c}{\mathrm{DD}}^{T}v=\mathrm{\λCv}\\ v^T\mathrm{Cv}=1\end{array}\right.$

At k〉3 o'clock, first equation in this system of equations has only a solution, is C ^-1DD ^TIn the proper vector of a unique positve term eigenwert.Noticing that simultaneously C is a sparse matrix, all is 0 at ranks number greater than the value in 6 o'clock matrixes, therefore with DD ^TBe decomposed into

${\mathrm{DD}}^T=\left(\begin{array}{cc}{S}_{11}& S_{12}\\ {S_{12}}^T& S_{22}\end{array}\right)$

$v=\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)$

S ₁₁, S ₁₂, S ₂₂Be respectively 6 * 6,6 * 4,4 * 4 matrix, v ₁, v ₂Be respectively the column vector of 6 row and 4 row, the system of equations that will find the solution can further be reduced to so

$\left\{\begin{array}{c}(S_{11}-{\mathrm{\λC}}_1)v_1+S_{12}{v}_2=0\\ S_{12}^T{v}_1+S_{22}{v}_2=0\end{array}\right.$

Because treat that the acetabular bone concave surface data sampling point set of match is not coplanar, so S ₂₂Be reversible, the group of therefore solving an equation can obtain

$v_2=-S_{22}^{-1}{S}_{12}^T{v}_1$

The substitution system of equations, thus obtain

$(S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^T)v_1={\mathrm{\λC}}_1{v}_1$

In actual solution procedure, get k=4, can guarantee like this

Be reversible.And work as C ₁When being reversible, v ₁Be A unique positve term eigenwert characteristic of correspondence vector.Simultaneously with v ₁Solution bring formula into

Can find the solution all unknowm coefficients in the aforementioned general quadric surface equation.

General quadric surface equationization with ellipsoid is classified as canonical form below, establish X=(x, y, z) ^T, the general quadric surface The Representation Equation is that matrix form is so

d+(2p,2q,2r)X+X ^TKX=0

For matrix K, use Jacobi orthogonal transformation method to ask its characteristic root (λ ₁, λ ₂, λ ₃), and 3 * 3 proper vector R, so have

$R^T\mathrm{KR}=\left(\begin{array}{ccc}{\mathrm{\λ}}_1& & \\ & {\mathrm{\λ}}_2& \\ & & {\mathrm{\λ}}_3\end{array}\right)$

Establish Y=R simultaneously ^TX, together with following formula, substitution d+ (2p, 2q, 2r) X+X ^TKX=0 can get

$d+(a_1,a_2,a_3)Y+Y^T\left(\begin{array}{ccc}{\mathrm{\λ}}_1& & \\ & {\mathrm{\λ}}_2& \\ & & {\mathrm{\λ}}_3\end{array}\right)Y=0$

(a ₁,a ₂,a ₃)=(2p,2q,2r)R

Because R is proper vector, coordinate X, the pass between the Y is

$X=Y+{(\frac{a_1}{{2\mathrm{\λ}}_1},\frac{a_2}{{2\mathrm{\λ}}_2},\frac{a_3}{{2\mathrm{\λ}}_3})}^T$

Substitution $d+(a_1,a_2,a_3)Y+Y^T\left(\begin{array}{ccc}{\mathrm{\λ}}_1& & \\ & {\mathrm{\λ}}_2& \\ & & {\mathrm{\λ}}_3\end{array}\right)Y=0,$ Can get

$a_0+X^T\left(\begin{array}{ccc}{\mathrm{\λ}}_1& & \\ & {\mathrm{\λ}}_2& \\ & & {\mathrm{\λ}}_3\end{array}\right)X=0$

$a_0=d-\frac{1}{4}(\frac{{a_1}^2}{{\mathrm{\λ}}_1}+\frac{{a_2}^2}{{\mathrm{\λ}}_2}+\frac{{a_3}^2}{{\mathrm{\λ}}_3})$

Compare with the quadric matrix equation of ellipsoid standard, three main axis length that obtain ellipsoid are

$a=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_1}},$ $b=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_2}},$ $c=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_3}}$

If the rotation angle on three axles of ellipsoid is (α, beta, gamma), the transformation relation that obtains ellipsoid centre coordinate system and master coordinate system is

X′=X ₀+R ₁(α)R ₁(β)R ₁(γ)X

$X_0={(\frac{a_1}{{2\mathrm{\λ}}_1},\frac{a_2}{{2\mathrm{\λ}}_2},\frac{a_3}{{2\mathrm{\λ}}_3})}^T$

R ₁(α)R ₁(β)R ₁(γ)=R ^T

Accordingly, three axial lengths, centre coordinate and position angles of target ellipsoid have been calculated.

A kind of femoral head surface three dimension model reconstructing method towards personalized design of the present invention can be implemented on ordinary individual's computer, need not the support of special setting, and this makes that the present invention is easy to promote.The above embodiment has only expressed embodiments of the present invention, and it describes comparatively concrete and detailed, but can not therefore be interpreted as the restriction to claim of the present invention.Should be pointed out that for the person of ordinary skill of the art without departing from the inventive concept of the premise, can also make some distortion and improvement, these all belong to protection scope of the present invention.Therefore, the protection domain of patent of the present invention should be as the criterion with claims.

Claims (1)

1. femoral head surface three dimension model reconstructing method towards personalized design, its feature may further comprise the steps:

1. import the hip joint CT sequence image of being gathered by the Medical CT machine;

2. the femoral head in each width of cloth two-dimensional ct image in the sequence image of having imported and acetabular bone are carried out profile extraction and inner filling; Utilize the profile of the interior femoral head of Sobel operator extraction two-dimensional ct image and acetabular bone earlier, the recycling scanning Beam Method realizes the inside filling of femoral head and acetabular bone profile;

3. in the femoral head of having filled and acetabular bone, utilize connectedness between pixel to realize the cutting operation of acetabular bone in each width of cloth two-dimensional ct image, concrete steps are as follows:

1) in femoral head and acetabular bone closed region, appoints and get two pixels, if these two pixels have the path of connection, then these two pixels are classified as same class, if these two pixels not have the path of connection, then these two pixels are classified as two classes;

2) all pixels in traversal femoral head and the acetabular bone closed region are classified as two classes with all pixels;

3) give up the pixel set that belongs in the femoral head closed region, keep the pixel set that belongs in the acetabular bone closed region, realize the cutting operation of acetabular bone;

4. utilize and cut apart the acetabular bone sequence image that finishes, carry out the acetabular bone three-dimensional model reconfiguration;

5. on the acetabular bone three-dimensional model that has obtained, utilize the space radiation collimation method, carry out the sampling of acetabular bone concave surface data point, comprise the steps:

1) in the concave surface of acetabular bone three-dimensional model, selects a volume coordinate point, centered by this coordinate points, to complete all spatial emission radiation straight lines;

Obtain an intersection point when 2) every radiation straight line and acetabular bone three-dimensional model intersect, the set of all intersection points is acetabular bone concave surface data sampling point;

6. utilize the acetabular bone concave surface data point that has obtained, adopt the ellipsoid approximating method based on the general quadric surface constraint, realize personalized femoral head surface reconstruction, concrete steps are as follows:

1) general quadric surface is constrained to Ellipsoidal Surface, the equation of establishing general quadric surface is:

ax ²+by ²+cz ²+2fyx+2gxz+2hxy+2px+2qy+2rz+d=0 （1）

If:

I=a+b+c （2）

J=ab+bc+ac-f ²-g ²-h （3）

$K=\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|---\left(4\right)$

For any one ellipsoid α 〉=4 are arranged, make α J-I ²0, select for use α=4 that quadric surface is constrained to Ellipsoidal Surface;

2) utilize acetabular bone concave surface data sampling point, adopt and carry out the Ellipsoidal Surface match based on the least square method of algebraically distance, obtain all unknowm coefficients in the formula (1);

3) establish X=(x, y, z) ^T, ellipsoid general quadric surface equationization is classified as matrix form:

d+(2p,2q,2r)X+X ^TKX=0 （5）

For matrix K, use Jacobi orthogonal transformation method to ask its characteristic root (λ ₁, λ ₂, λ ₃), and 3 * 3 proper vector R, order

(a ₁,a ₂,a ₃)=(2p,2q,2r)R （6）

Have

$a_0=d-\frac{1}{4}(\frac{{a_1}^2}{{\mathrm{\λ}}_1}+\frac{{a_2}^2}{{\mathrm{\λ}}_2}+\frac{{a_3}^2}{{\mathrm{\λ}}_3})---\left(7\right)$

Obtain the length of three main shafts of ellipsoid

$a=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_1}}$ $b=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_2}}$ $c=\sqrt{-\frac{a_0}{{\mathrm{\λ}}_3}}---\left(8\right)$

If the rotation angle of three main shafts of ellipsoid is (α, beta, gamma), obtain the transformation relation of ellipsoid centre coordinate system and master coordinate system

X′=X ₀+R ₁(α)R ₁(β)R ₁(γ)X （9）

$X_0={(\frac{a_1}{{2\mathrm{\λ}}_1},\frac{a_2}{{2\mathrm{\λ}}_2},\frac{a_3}{{2\mathrm{\λ}}_3})}^T---\left(10\right)$

R ₁(α)R ₁(β)R ₁(γ)=R ^T （11）。

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