CN101236663B - Heart capacity calculation method based on NURBS curve surface integral - Google Patents

Abstract

A kind of cardiac volume calculations method based on nurbs surface integral, comprising the following steps: the 1) acquisition and processing of data: giving a large amount of cardiologic medical image, and the three-dimensional point of heart surface is obtained from these images; 2) it takes the preset of the three-dimensional point cloud of above-mentioned heart as control point, carries out nurbs surface fitting; 3) nurbs surface is indicated with matrix; 4) above-mentioned nurbs surface is integrated to obtain the volume of heart, enabling A (z) is the cross-sectional area at height z, then volume are as follows:

. The present invention provide a kind of computational accuracy is high, arithmetic speed is fast, meet clinical diagnosis needed for requirement the cardiac volume calculations method based on nurbs surface integral.

Description

Cardiac volume calculations method based on the nurbs surface integration

Technical field

The present invention relates to a kind of cardiac volume calculations method.

Background technology

Heart is extremely complicated system ensembles such as current collection physiology, dynamics, hemodynamics and nerve, biochemical control.Modeling and simulating is the effective means of research complex biological knowledge topic.In the past few years, people have had deep understanding to the physiological significance of cardiac structure and function, and have set up many mathematical models, make great efforts to quantize viewed myocardium mechanical behavior, conductivity behavior and biological chemistry behavior.But because the complicacy of heart physiological pathology system, generally speaking these models are separate development, and still nobody can integrate research to the various mechanism of heart at present.

The virtual heart research of rising in recent years is incorporated into the thought of virtual reality the research field of the such complexity of cardiovascular system, it is to utilize computing machine powerful computing ability and graphics process display capabilities, sets up virtual cardiac module and provides possibility for further investigation cardiomotility mechanism.Model not only will be from emulation heart on the form, and the motion process that should be able to simulate true heart, mechanical characteristics, the characteristics of electrical conductivity of heart and the hydrodynamic characteristic of chambers of the heart inner blood of the cardiac muscle of emulation heart, valve and chambers of the heart motion, and can emulation heart pathological state, for clinical diagnosis disease is given information.

There are some scholars to propose some methods at present, are used to obtain the body of heart and the description of motion based on model.Kyoungju Park, scholars such as Dimitris Metaxas have proposed the new theory of a kind of cardiac function analysis.Set up a basic cardiac module with the image of MRI, the method that has proposed finite element analysis is calculated whole and local functional parameter.Experiment shows, the structure that draws based on such model can characterize the motion and the dynamic rule of heart wall.Taratorin and Sideman then are divided into a large amount of cube infinitesimal sheets to myocardium and carry out modeling and analysis, and it is more satisfactory to obtain effect.

Yet because method itself, some does not also reach the required requirement of clinical diagnosis on computational accuracy based on the heart volume computing method of model for these, and some arithmetic speed is slow.

Summary of the invention

Clinical diagnosis requires, the deficiency of poor practicability for the computational accuracy that overcomes existing heart movement analytical approach or speed do not reach, and the invention provides a kind of computational accuracy height, fast operation, meets the cardiac volume calculations method based on the nurbs surface integration of the required requirement of clinical diagnosis.

The technical solution adopted for the present invention to solve the technical problems is:

A kind of cardiac volume calculations method based on the nurbs surface integration, described cardiac volume calculations method may further comprise the steps:

1), the obtaining and handling of data: given a large amount of heart medical image, from these images, obtain the three-dimensional point of heart surface, comprising:

(1.1), carry out smoothing processing with image filtering method, the removal noise;

(1.2), by given index file sectioning image is adjusted to correct order;

(1.3), the definition area-of-interest, from image, be partitioned into the target area by the gray scale threshold value method;

(1.4), obtain the gray-scale value of image, calculate the changing value of gray scale, get the cardiac boundary that is of grey scale change maximum;

(1.5), extract the three-dimensional point cloud of heart;

2), the three-dimensional point cloud of getting above-mentioned heart is as the reference mark, carries out the nurbs surface match, its formula is (1):

$s(u,v)=\frac{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}\underset{j=0}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\ω}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\Σ}}}\underset{j=0}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\ω}}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}---\left(1\right)$

In the following formula, p _{I, j}(i=0,1 ..., n; J=0,1 ..., m) be the reference mark of curved surface, promptly take from the frontier point cloud of heart, be topological rectangular array, ω _{I, j}Be the weight factor that interrelates with the reference mark; N _{I, k1}(u) and N _{J, k2}(v) be respectively k1 and k2 the reasonable B spline base function of standard;

The de Boor-Cox recursion formula of reasonable B spline base function, it is defined as follows:

3), with the nurbs surface matrix representation:

Be defined in knot vector U, on the V, be limited to node region [u _i, u _I+1) * [v _j, v _J+1) on k ₁* k ₂Inferior (k ₁+ 1, k ₂+ 1 rank) the NURBS knee-piece is expressed as follows:

$s_{\mathrm{ij}}(u,v)=\frac{\underset{l=i-k_1}{\overset{i}{\mathrm{\Σ}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\Σ}}}{\mathrm{\ω}}_{\mathrm{lr}}{p}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}{\underset{l=i-k_1}{\overset{i}{\mathrm{\Σ}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\Σ}}}{\mathrm{\ω}}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}---\left(3\right)$

Homogeneous coordinates are expressed as follows:

$s_{\mathrm{ij}}^h(u,v)=\underset{l=i-k_1+1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+1}{\overset{j}{\mathrm{\&Sigma;}}}{V}_{\mathrm{lr}}^h{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)---\left(4\right)$

V _Lr ^hBe homogeneous coordinates (ω _Lr, p _Lr, ω _Lr), p _LrBe the reference mark, ω _LrBe weight factor, i=k, k+1 ... n, j=k, k+1 ... m;

The nurbs surface parameter standardized change t=(u-u _i)/(u _I+1-u _i), w=(u-u _i)/(u _I+1-u _i), t ∈ [0,1), w ∈ [0,1), obtain the matrix representation of nurbs surface sheet (4):

$s_{i,j}^h(t,w)=T_{k_1}{M}_{i,u}^{k_1+1}{V}_{i,j}^h{\left(M_{j,v}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+1}\right)}^T{W}_{k_2}^T---\left(5\right)$

Wherein, $T_{k_1}=\left[\begin{array}{cccc}1& t& \&CenterDot;\&CenterDot;\&CenterDot;& t^{k_1}\end{array}\right],$ $W_{k_2}^T={\left[\begin{array}{cccc}1& w& \&CenterDot;\&CenterDot;\&CenterDot;& w^{k_2}\end{array}\right]}^T,$

M _{I, u} ^K1+1And M _{J, v} ^K2+1Be respectively u to v to matrix of coefficients;

The power basis function of formula (5) is represented:

$s_{i,j}^h(t,w)=\underset{l=0}{\overset{k_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(6\right)$

Wherein, $B_{i,j}(r,l)=\underset{b=0}{\overset{k_2}{\mathrm{\&Sigma;}}}(\underset{c=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{M}_{i,u}^{k_1+1}(l,c)\&times;V^h(i-(k_1-c),j-(k_2-b)))\&times;M_{j,v}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+1}(b,r)$

Nurbs surface on whole zone [0,1] * [0,1] is expressed as equation (7):

$S^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{s}_{i,j}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(7\right)$

That is:

$X^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^x(r,l)t^r{w}^l$

$Y^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^y(r,l)t^r{w}^l---\left(8\right)$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^r{w}^l$

The recursion formula of k ordered coefficients matrix:

Wherein, ${\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_{0,j}=\frac{t_i-t_j}{t_{j+(k+1)-1}-t_j},$ $d_{1,j}=\frac{t_{i+1}-t_j}{t_{j+(k+1)-1}-t_j}$

Utilize formula (9), obtain following matrix of coefficients:

$M_i^1=\left[1\right],$ $M_i^2=\left[\begin{array}{cc}1& 0\\ -1& 0\end{array}\right],$

$M_i^3=\left[\begin{array}{ccc}\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& \frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}}& 0\\ \frac{-2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& \frac{2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& 0\\ \frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& -(u_{i+1}-u_i)(\frac{1}{u_{i+1}-u_{i-1}}+\frac{1}{u_{i+2}-u_i})& \frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

$M_i^4=\left[\begin{array}{cccc}\frac{{(u_{i+1}-u_i)}^2}{(u_{i+1}-u_{i-1})(u_{i+1}-u_{i-2})},& 1-a_{00}-a_{02}& \frac{u_i-u_{i-1}}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ -3a_{00},& 3a_{00}-a_{12},& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ 3a_{00},& -3a_{00}-a_{22}& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}},& 0\\ -a_{00},& a_{31},& a_{32}& \frac{u_{i+1}-u_i}{u_{i+3}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

Wherein, $a_{31}=a_{00}+\frac{1}{3}{a}_{12}+\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ $a_{32}=-\frac{1}{3}{a}_{12}-a_{33}-\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ a _{I, j}It is matrix M _i ⁴The capable j of an i row element;

4), to above-mentioned nurbs surface, promptly formula (8) carries out the volume that integration obtains heart,

Make A (z) be the cross-sectional area at height z place, then volume is:

$V={\&Integral;}_{z_1}^{z_2}A\left(z\right)\mathrm{dz}---\left(10\right).$

As preferred a kind of scheme: in described step 4), in conjunction with the power basis function expression formula of nurbs surface, order

With (u is v) separately to the partial derivative of t and w for curved surface S;

With

By difference formula (8) t and w are asked local derviation:

$\frac{\&PartialD;X(t,w)}{\&PartialD;t}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{r=1}{\overset{k_1}{\mathrm{\&Sigma;}}}r\&times;B_{i,j}^x(r-1,l)t^{r-1}{w}^l$

$\frac{\&PartialD;Z(t,w)}{\&PartialD;w}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}l\&times;B_{i,j}^z(r,l-1)t^r{w}^{l-1}$

In the cross-sectional area A (z (w)) that height z (w) locates, as follows:

$A\left(z\left(w\right)\right)=\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^{1}Y(t,w)\&times;\frac{\&PartialD;X(t,w)}{\&PartialD;t}\mathrm{dt}$

Top formula substitution (10) is obtained:

$V=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}{\&Integral;}_0^{1}A\left(z\left(w\right)\right)z^{\&prime;}\left(w\right)\mathrm{dw}$

$=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^{1}Y(t,w)\&times;\frac{\&PartialD;X(t,w)}{\&PartialD;t}\&times;\frac{\&PartialD;Z(t,w)}{\&PartialD;w}\mathrm{dtdw}$

$=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^1\underset{l_1=0}{\overset{(3\&times;k_2-1)}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2=0}{\overset{(3\&times;k_1-1)}{\mathrm{\&Sigma;}}}{C}_{i,j}(l_1,l_2)\&times;{\mathrm{<figure-callout\; id="2"\; label="t\; l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{l_{}}\&times;w^{l_1}\mathrm{dtdw}$

k ₁* k ₂The cubature formula of the heart that inferior nurbs surface is represented is as follows:

$V=\underset{i=k1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k2}{\overset{n}{\mathrm{\&Sigma;}}}(\underset{l_1=1}{\overset{(3\&times;k_2-1)}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2=1}{\overset{(3\&times;k_1-1)}{\mathrm{\&Sigma;}}}{C}_{i,j}(l_1,l_2)\&times;\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+1}\&times;\frac{1}{l_1+1})---\left(11\right)$

Wherein, C _{I, j}Be 3k ₁* 3k ₂The rank matrix, it is a polynomial expression

With After multiplying each other, about the coefficient before t and the w variable, that is, and l ₁Row l ₂Column element C _{I, j}(l ₁, l ₂) expression t ^L1w ^L2Preceding coefficient; C _{I, j}(l ₁, l ₂) calculate acquisition by expression (12):

$C_{i,j}(l_1,l_2)=\underset{0\&le;d\&le;{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\overset{2k_2-1}{\underset{d=l_1-\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}{\underset{f=0}{\mathrm{\&Sigma;}}}}}\underset{0\&le;{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_2\&le;k_1-1}{\overset{2k_1}{\underset{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_2=l_2-{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_2}{\underset{{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_2=0}{\mathrm{\&Sigma;}}}}}(\underset{0\&le;s\&le;k_2-1}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\underset{s=f-\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}{\underset{r=0}{\mathrm{\&Sigma;}}}}}\underset{0\&le;{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">s</figure-callout>}}_2\&le;k_1}{\overset{k_1}{\underset{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">s</figure-callout>}}_2=f_2-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}{\underset{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=0}{\mathrm{\&Sigma;}}}}}{B}_{i,j}^y(r,r_2)\&times;s\&times;B_{i,j}^x(s,s_2))\&times;{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_2\&times;B_{i,j}^z(d,d_2)---\left(12\right).$

As preferred another kind of scheme: in described step 2) in, with (n+1) * (m+1) reference mark array (x _Ij, y _Ij, z _Ij) (i=0,1...n, j=0,1...m) translation causes to such an extent that curved surface is the center with the z axle, and Cartesian coordinates is represented a little to convert to point (r under the cylindrical coordinates _Ij, θ _Ij, z _Ij) (i=0,1...n, j=0,1...m), its conversion formula following (13):

$r_{\mathrm{ij}}=\sqrt{x_{\mathrm{ij}}^2+y_{\mathrm{ij}}^2},$

${\mathrm{\&theta;}}_{\mathrm{ij}}=\left\{\begin{array}{cc}\mathrm{\&pi;}/2& \mathrm{if}{x}_{\mathrm{ij}}=0\mathrm{and}{y}_{\mathrm{ij}}>0\\ 3\mathrm{\&pi;}/2& \mathrm{if}{x}_{\mathrm{ij}}=0\mathrm{and}{y}_{\mathrm{ij}}<0\\ a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}{x}_{\mathrm{ij}}>0\mathrm{and}{y}_{\mathrm{ij}}\&GreaterEqual;0\\ \mathrm{\&pi;}+a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}{x}_{\mathrm{ij}}<0\\ 2\mathrm{\&pi;}+a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}{x}_{\mathrm{ij}}>0\mathrm{and}{y}_{\mathrm{ij}}<0\end{array}\right.---\left(13\right)$

z _ij＝z _ij

The formula of cylindrical coordinates nurbs surface match is (14):

$[r(u,v),\mathrm{\&theta;}(u,v),z(u,v)]=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}---\left(14\right)$

Wherein, p _{I, j}=[r _{I, j}, θ _{I, j}, z _{I, j}].

The cylindrical coordinates nurbs surface equation that can obtain matrix form by step 3) is:

$r^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^r(r,l)t^f{w}^l$

${\mathrm{\&theta;}}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^{\mathrm{\&theta;}}(r,l)t^f{w}^l---\left(15\right)$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^f{w}^l$

Further, in described step 4), in conjunction with the power basis function expression formula of nurbs surface, order

With

(u is v) separately to the partial derivative of t and w for curved surface S;

With

By difference formula (15) t and w are asked local derviation:

$A\left(z\left(w\right)\right)=\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1\frac{1}{2}{r}^2(t,w)\&times;\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}\mathrm{dt},$

$\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{f=1}{\overset{k_1}{\mathrm{\&Sigma;}}}f\&times;B_{i,j}^{\mathrm{\&theta;}}(f-1,l)t^{f-1}{w}^l$

Obtain cubature formula:

$V=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^1\frac{1}{2}{r}^2(t,w)\&times;\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}\&times;\frac{\&PartialD;Z(t,w)}{\&PartialD;w}\mathrm{dtdw}---\left(16\right)$

With r (t, w),

With

Cubature formula above the substitution (16), the volume expression formula that obtains under the cylindrical coordinates is (17):

$V=\frac{1}{2}\underset{i=k1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k2}{\overset{n}{\mathrm{\&Sigma;}}}(\underset{l_1=1}{\overset{4\&times;k_2-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2=1}{\overset{4\&times;k_1-1}{\mathrm{\&Sigma;}}}{C}_{i,j}^{\mathrm{\&theta;}}(l_1,l_2)\&times;\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+1}\&times;\frac{1}{l_1+1})---\left(17\right).$

In described step 1), medical image adopts SPECT medical image, nuclear magnetic resonance image, CT image, spiral CT image, ultrasonoscopy or PET image.

Described heart is left ventricle, right ventricle, atrium sinistrum, atrium dextrum, the surfaces externally and internally of heart partly or completely.

Technical conceive of the present invention is: NURBS claims non-uniform rational B-spline again, and it also has lot of advantages except the characteristics that possess the B batten: 1) both resolve shape for standard and also provide a public mathematical form for the accurate expression of free type curved surface with design; 2) not only can utilize weight factor again, therefore have bigger dirigibility by adjusting the shape that the reference mark changes curve and surface; 3) the suitable popularization of right and wrong reasonable B batten form and your form of reasonable and non-reasonable Betsy etc.

Nurbs curve: a k nurbs curve can be expressed as one section rational polynominal vector function:

$c_i\left(t\right)=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{p}_i{N}_{i,k}\left(u\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{N}_{i,k}\left(u\right)}---\left(1\right)$

Wherein, ω _i(i=0,1 ..., n) be called weight factor, respectively with control vertex p _i(i=0,1 ..., n) interrelate.First and last weight factor ω ₀, ω _n＞0, all the other ω _i〉=0, and k weight factor of order be not zero simultaneously, with prevent that denominator from being zero, reservation convex closure character and curve not the reason weight factor deteriorate to a bit.N _{I, k}(u) be k standard B spline base function.

Nurbs curve also has the local property adjusted, convex closure, character such as geometric invariance.In addition, owing to introduced weight factor, make that the adjustment of curve is more flexible, this will say at the 4th chapter.

Nurbs surface a: k ₁* k ₂The rational fraction of inferior nurbs surface is represented:

$s(u,v)=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}---\left(2\right)$

P wherein _{I, j}(i=0,1 ..., n; J=0,1 ..., m) be topological rectangular array, form a Control Network.ω _{I, j}Be the weight factor that interrelates with the reference mark.N _{I, k1}(u) and N _{J, k2}(v) be respectively k ₁And k ₂The reasonable B spline base function of inferior standard.

Reasonable B-spline surface has and the similar geometric properties of non-reasonable B-spline surface.And, being similar to nurbs curve, its weight factor can be used to adjust the shape of curved surface.

Another advantage that nurbs surface is different from B-spline surface is exactly that its accurately expression standard is resolved body (as cylinder, circular cone, ball, surface of revolution etc.).

The NURBS technology has been introduced weight factor, thereby solve the problem that B-spline surface can not accurately be represented elementary analytic surface, yet, for the free type curved surface, the weight factor of nurbs surface is not brought into play very big effect yet, and weight factor adjust unreasonable, will cause very bad parametrization, even destroy curved-surface structure subsequently.So we are difficult to accurately represent by weight factor the body of heart, left ventricle, owing to heart, left ventricle likeness in form column body, therefore, introduce the nurbs surface of cylindrical coordinate in addition.The nurbs surface of cylindrical coordinate is fit to expression heart, left ventricle more than the nurbs surface of cartesian coordinate system, especially, is providing under the situation of a small amount of marginal point, and the advantage of the nurbs surface of cylindrical coordinate is more obvious.

The matrix representation of nurbs surface: be defined in knot vector U, on the V, be limited to node region [u _i, u _I+1) * [v _j, v _J+1) on k ₁* k ₂Inferior (k ₁+ 1, k ₂+ 1 rank) the NURBS knee-piece is expressed as follows:

$s_{\mathrm{ij}}(u,v)=\frac{\underset{l=i-k_1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{lr}}{p}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}{\underset{l=i-k_1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}---\left(3\right)$

Homogeneous coordinates are expressed as follows:

$s_{\mathrm{ij}}^h(u,v)=\underset{l=i-k_1+1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+1}{\overset{j}{\mathrm{\&Sigma;}}}{V}_{\mathrm{lr}}^h{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)---\left(4\right)$

V _Lr ^hBe homogeneous coordinates (ω _Lrp _Lr, ω _Lr), p _LrBe the reference mark, ω _LrBe weight factor, i=k, k+1 ... n, j=k, k+1 ... m

The nurbs surface parameter standardized change t=(u-u _i)/(u _I+1-u _i), w=(u-u _i)/(u _I+1-u _i), t ∈ [0,1), w ∈ [0,1) obtain the matrix representation of nurbs surface sheet at last:

$s_{i,j}^h(t,w)=T_{k_1}{M}_{i,u}^{k_1+1}{V}_{i,j}^h{\left(M_{j,\mathrm{<figure-callout\; id="2"\; label="v\; k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">v</figure-callout>}}^{k_{}}\right)}^T{W}_{k_2}^T---\left(5\right)$

Wherein, $T_{k_1}=\left[\begin{array}{cccc}1& t& \&CenterDot;\&CenterDot;\&CenterDot;& t^{k_1}\end{array}\right],$ $W_{k_2}^T={\left[\begin{array}{cccc}1& w& \&CenterDot;\&CenterDot;\&CenterDot;& w^{k_2}\end{array}\right]}^T,$

M _{I, u} ^K1+1And M _{J, v} ^K2+1Be respectively u to v to matrix of coefficients.

The power basis function of formula (5) is represented:

$s_{i,j}^h(t,w)=\underset{l=0}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(6\right)$

Wherein, $B_{i,j}(r,l)=\underset{b=0}{\overset{k_2}{\mathrm{\&Sigma;}}}(\underset{c=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{M}_{i,u}^{k_1+1}(l,c)\&times;V^h(i-(k_1-c),j-(k_2-b)))\&times;M_{j,\mathrm{<figure-callout\; id="2"\; label="v\; k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">v</figure-callout>}}^{k_{}}(b,r)$

Nurbs surface on whole zone [0,1] * [0,1] is expressed as equation (7):

$S^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{s}_{i,j}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(7\right)$

That is:

$X^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^x(r,l)t^r{w}^l$

$Y^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^y(r,l)t^r{w}^l---\left(8\right)$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^r{w}^l$

The expression of matrix of coefficients: the following recursion formula that provides k ordered coefficients matrix:

Wherein, ${\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_{0,j}=\frac{t_i-t_j}{t_{j+(k+1)-1}-t_j},$ $d_{1,j}=\frac{t_{i+1}-t_j}{t_{j+(k+1)-1}-t_j}$

Utilize formula (9), we can cross and obtain following matrix of coefficients:

$M_i^1=\left[1\right],$ ${\mathrm{<figure-callout\; id="2"\; label="M\; i"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">M</figure-callout>}}_i^{}=\left[\begin{array}{cc}1& 0\\ -1& 0\end{array}\right],$

${\mathrm{<figure-callout\; id="3"\; label="M\; i"\; filenames="S2007103072478E00011.png,S2007103072478E00041.png"\; state="\{\{state\}\}">M</figure-callout>}}_i^{}=\left[\begin{array}{ccc}\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& \frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}}& 0\\ \frac{-2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& \frac{2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& 0\\ \frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& -(u_{i+1}-u_i)(\frac{1}{u_{i+1}-u_{i-1}}+\frac{1}{u_{i+2}-u_i})& \frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

${\mathrm{<figure-callout\; id="4"\; label="M\; i"\; filenames="S2007103072478E00021.png"\; state="\{\{state\}\}">M</figure-callout>}}_i^{}=\left[\begin{array}{cccc}\frac{{(u_{i+1}-u_i)}^2}{(u_{i+1}-u_{i-1})(u_{i+1}-u_{i-2})},& 1-a_{00}-a_{02}& \frac{u_i-u_{i-1}}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ -3a_{00},& 3a_{00}-a_{12},& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ 3a_{00},& -3a_{00}-a_{22}& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}},& 0\\ -a_{00},& a_{31},& a_{32}& \frac{u_{i+1}-u_i}{u_{i+3}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

Wherein, $a_{31}=a_{00}+\frac{1}{3}{a}_{12}+\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ $a_{32}=-\frac{1}{3}{a}_{12}-a_{33}-\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ a _{I, j}It is matrix M _i ⁴The capable j of an i row element.

And the matrix of above-mentioned nurbs surface carried out the volume that integration obtains heart.

Beneficial effect of the present invention mainly shows: 1) nurbs surface is represented convenience very, given reference mark is used than low order NURBS just can obtain a desirable curved surface, and this cardiac module is than truer based on simple geometric body or the cardiac module that utilizes simple mathematical to represent;

2) heart represented of NURBS is smooth, a continuous model, for the static state that is subsequently applied to and the analysis of dynamic function parameter provide the foundation;

3) local modification of nurbs surface can be done local modification to the heart body under the situation that does not change global shape;

4) nurbs surface has very strong dirigibility, changes the heart body by changing reference mark or weight factor, for the research of heart deformation provides possibility;

5) can utilize the Horner method to calculate point and derivative thereof on the nurbs curve curved surface; In the time will calculating a large amount of points, this method is more efficient than traditional de Boor formula.

Description of drawings

Fig. 1 is the area-of-interest synoptic diagram that obtains from the CT section.

Fig. 2 is the synoptic diagram of the point of the heart surface that extracts.

Fig. 3 is the synoptic diagram of nurbs surface match plastics heart.

Fig. 4 is a synoptic diagram of playing up nurbs surface match plastics heart.

Fig. 5 is the left ventricle inside and outside wall nurbs surface match synoptic diagram of seven states in the cardiac cycle.

Fig. 6 is the round platform design sketch of cylindrical coordinates NURBS match.

Fig. 7 is the cylindrical coordinates NURBS fitting result chart of heart.

Fig. 8 is the left ventricle unfaithful intention pilaster coordinate nurbs surface match synoptic diagram of seven states in the cardiac cycle.

Fig. 9 is the left ventricle heart pilaster coordinate nurbs surface match synoptic diagram of seven states in the cardiac cycle.

Figure 10 is the change curve synoptic diagram of left ventricular mass in cardiac cycle.

Embodiment

Below in conjunction with accompanying drawing the present invention is further described.

Embodiment 1

With reference to Fig. 1～Fig. 5, a kind of cardiac volume calculations method based on the nurbs surface integration, described cardiac volume calculations method may further comprise the steps:

1), the obtaining and handling of data: given a large amount of heart medical image, from these images, obtain the three-dimensional point of heart surface, comprising:

(1.1), carry out smoothing processing with image filtering method, the removal noise;

(1.2), by given index file sectioning image is adjusted to correct order;

(1.3), the definition area-of-interest, from image, be partitioned into the target area by the gray scale threshold value method;

(1.4), obtain the gray-scale value of image, calculate the changing value of gray scale, get the cardiac boundary that is of grey scale change maximum;

(1.5), extract the three-dimensional point cloud of heart;

2), the three-dimensional point cloud of getting above-mentioned heart is as the reference mark, carries out the nurbs surface match, its formula is (1):

$s(u,v)=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}---\left(1\right)$

In the following formula, p _{I, j}(i=0,1 ..., n; J=0,1 ..., m) be the reference mark of curved surface, promptly take from the frontier point cloud of heart, be topological rectangular array, ω _{I, j}Be the weight factor that interrelates with the reference mark; N _{I, k1}(u) and N _{J, k2}(v) be respectively k ₁And k ₂The reasonable B spline base function of inferior standard;

The de Boor-Cox recursion formula of reasonable B spline base function, it is defined as follows:

3), with the nurbs surface matrix representation:

Be defined in knot vector U, on the V, be limited to node region [u _i, u _I+1) * [v _j, v _J+1) on k ₁* k ₂Inferior (k ₁+ 1, k ₂+ 1 rank) the NURBS knee-piece is expressed as follows:

$s_{\mathrm{ij}}(u,v)=\frac{\underset{l=i-k_1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{lr}}{p}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}{\underset{l=i-k_1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}---\left(3\right)$

Homogeneous coordinates are expressed as follows:

$s_{\mathrm{ij}}^h(u,v)=\underset{l=i-k_1+1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+1}{\overset{j}{\mathrm{\&Sigma;}}}{V}_{\mathrm{lr}}^h{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)---\left(4\right)$

V _Lr ^hBe homogeneous coordinates (ω _Lrp _Lr, ω _Lr), p _LrBe the reference mark, ω _LrBe weight factor, i=k, k+1 ... n, j=k, k+1 ... m

The nurbs surface parameter standardized change t=(u-u _i)/(u _I+1-u _i), w=(u-u _i)/(u _I+1-u _i), t ∈ [0,1), w ∈ [0,1), obtain the matrix representation of nurbs surface sheet:

$s_{i,j}^h(t,w)=T_{k_1}{M}_{i,u}^{k_1+1}{V}_{i,j}^h{\left(M_{j,\mathrm{<figure-callout\; id="2"\; label="v\; k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">v</figure-callout>}}^{k_{}}\right)}^T{W}_{k_2}^T---\left(5\right)$

Wherein, $T_{k_1}=\left[\begin{array}{cccc}1& t& \&CenterDot;\&CenterDot;\&CenterDot;& t^{k_1}\end{array}\right],$ $W_{k_2}^T={\left[\begin{array}{cccc}1& w& \&CenterDot;\&CenterDot;\&CenterDot;& w^{k_2}\end{array}\right]}^T,$

M _{I, u} ^K1+1And M _{J, v} ^K2+1Be respectively u to v to matrix of coefficients;

The power basis function of formula (5) is represented:

$s_{i,j}^h(t,w)=\underset{l=0}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(6\right)$

Wherein, $B_{i,j}(r,l)=\underset{b=0}{\overset{k_2}{\mathrm{\&Sigma;}}}(\underset{c=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{M}_{i,u}^{k_1+1}(l,c)\&times;V^h(i-(k_1-c),j-(k_2-b)))\&times;M_{j,\mathrm{<figure-callout\; id="2"\; label="v\; k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">v</figure-callout>}}^{k_{}}(b,r)$

Nurbs surface on whole zone [0,1] * [0,1] is expressed as equation (7):

$S^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{s}_{i,j}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(7\right)$

That is:

$X^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^x(r,l)t^r{w}^l$

$Y^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^y(r,l)t^r{w}^l---\left(8\right)$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^r{w}^l$

The recursion formula of k ordered coefficients matrix:

Wherein, ${\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_{0,j}=\frac{t_i-t_j}{t_{j+(k+1)-1}-t_j},$ $d_{1,j}=\frac{t_{i+1}-t_j}{t_{j+(k+1)-1}-t_j}$

Utilize formula (9), obtain following matrix of coefficients:

$M_i^1=\left[1\right],$ ${\mathrm{<figure-callout\; id="2"\; label="M\; i"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">M</figure-callout>}}_i^{}=\left[\begin{array}{cc}1& 0\\ -1& 0\end{array}\right],$

${\mathrm{<figure-callout\; id="3"\; label="M\; i"\; filenames="S2007103072478E00011.png,S2007103072478E00041.png"\; state="\{\{state\}\}">M</figure-callout>}}_i^{}=\left[\begin{array}{ccc}\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& \frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}}& 0\\ \frac{-2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& \frac{2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& 0\\ \frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& -(u_{i+1}-u_i)(\frac{1}{u_{i+1}-u_{i-1}}+\frac{1}{u_{i+2}-u_i})& \frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

${\mathrm{<figure-callout\; id="4"\; label="M\; i"\; filenames="S2007103072478E00021.png"\; state="\{\{state\}\}">M</figure-callout>}}_i^{}=\left[\begin{array}{cccc}\frac{{(u_{i+1}-u_i)}^2}{(u_{i+1}-u_{i-1})(u_{i+1}-u_{i-2})},& 1-a_{00}-a_{02}& \frac{u_i-u_{i-1}}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ -3a_{00},& 3a_{00}-a_{12},& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ 3a_{00},& -3a_{00}-a_{22}& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}},& 0\\ -a_{00},& a_{31},& a_{32}& \frac{u_{i+1}-u_i}{u_{i+3}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

Wherein, $a_{31}=a_{00}+\frac{1}{3}{a}_{12}+\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ $a_{32}=-\frac{1}{3}{a}_{12}-a_{33}-\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ a _{I, j}It is matrix M _i ⁴The capable j of an i row element;

4), the matrix of above-mentioned nurbs surface is carried out the volume that integration obtains heart,

Make A (z) be the cross-sectional area at height z place, then volume is:

$V={\&Integral;}_{z_1}^{z_2}A\left(z\right)\mathrm{dz}---\left(10\right)$

In described step 4), in conjunction with the power basis function expression formula of nurbs surface, order

With

(u is v) separately to the partial derivative of t and w for curved surface S;

With

By difference formula (8) t and w are asked local derviation:

$\frac{\&PartialD;X(t,w)}{\&PartialD;t}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{r=1}{\overset{k_1}{\mathrm{\&Sigma;}}}r\&times;B_{i,j}^x(r-1,l)t^{r-1}{w}^l$

$\frac{\&PartialD;Z(t,w)}{\&PartialD;w}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}l\&times;B_{i,j}^z(r,l-1)t^r{w}^{l-1}$

In the cross-sectional area A (z (w)) that height z (w) locates, as follows:

$A\left(z\left(w\right)\right)=\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^{1}Y(t,w)\&times;\frac{\&PartialD;X(t,w)}{\&PartialD;t}\mathrm{dt}$

Top formula substitution (10) is obtained:

$V=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}{\&Integral;}_0^{1}A\left(z\left(w\right)\right)z^{\&prime;}\left(w\right)\mathrm{dw}$

$=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^{1}Y(t,w)\&times;\frac{\&PartialD;X(t,w)}{\&PartialD;t}\&times;\frac{\&PartialD;Z(t,w)}{\&PartialD;w}\mathrm{dtdw}$

$=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^1\underset{l_1=0}{\overset{(3\&times;k_2-1)}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2=0}{\overset{(3\&times;k_1-1)}{\mathrm{\&Sigma;}}}{C}_{i,j}(l_1,l_2)\&times;{\mathrm{<figure-callout\; id="2"\; label="t\; l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{l_{}}\&times;w^{l_1}\mathrm{dtdw}$

k ₁* k ₂The cubature formula of the heart that inferior nurbs surface is represented is as follows:

$V=\underset{i=k1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k2}{\overset{n}{\mathrm{\&Sigma;}}}(\underset{l_1=1}{\overset{(3\&times;k_2-1)}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2=1}{\overset{(3\&times;k_1-1)}{\mathrm{\&Sigma;}}}{C}_{i,j}(l_1,l_2)\&times;\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+1}\&times;\frac{1}{l_1+1})---\left(11\right)$

Wherein, C _uBe 3k ₁* 3k ₂The rank matrix, it is a polynomial expression

With

After multiplying each other, about the coefficient before t and the w variable, that is, and l ₁Row l ₂Column element C _u(l ₁, l ₂) expression t ^L1w ^L2Preceding coefficient; C _{I, j}(l ₁, l ₂) calculate acquisition by expression (12):

$C_{i,j}(l_1,l_2)=\underset{0\&le;d\&le;{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\overset{2k_2-1}{\underset{d=l_1-\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}{\underset{f=0}{\mathrm{\&Sigma;}}}}}\underset{0\&le;{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_2\&le;k_1-1}{\overset{2k_1}{\underset{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_2=l_2-{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_2}{\underset{{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_2=0}{\mathrm{\&Sigma;}}}}}(\underset{0\&le;s\&le;k_2-1}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\underset{s=f-\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="S2007103072478E00021.png,S2007103072478E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}{\underset{r=0}{\mathrm{\&Sigma;}}}}}\underset{0\&le;s_2\&le;k_1}{\overset{k_1}{\underset{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">s</figure-callout>}}_2=f_2-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}{\underset{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=0}{\mathrm{\&Sigma;}}}}}{B}_{i,j}^y(r,r_2)\&times;s\&times;B_{i,j}^x(s,s_2))\&times;{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">d</figure-callout>}}_2\&times;B_{i,j}^z(d,d_2)---\left(12\right)$

In described step 1), medical image adopts SPECT medical image, nuclear magnetic resonance image, CT image, spiral CT image, ultrasonoscopy or PET image.

Described heart is left ventricle, right ventricle, atrium sinistrum, atrium dextrum, the surfaces externally and internally of heart partly or completely.

In the present embodiment, at first, the CT slice map of given a large amount of plastic cement heart, this plastic cement heart is placed on the wooden support, and the syringe that oil is housed on the support is used for heart deformation.The plastic cement heart of this experiment under the nurbs surface match state.

Secondly, from the CT slice map, obtain the three-dimensional point of plastic cement heart surface.This process is divided into 6 steps:

1) with filtering picture is carried out smoothing processing, remove some noises;

2) by given index file the CT section is adjusted into correct order;

3) area-of-interest of definition, purpose is to be partitioned into this process of target area (Fig. 1) to be partitioned into heart surface based on the brightness thresholding;

4) gray-scale value of acquisition image, the changing value of calculating gray scale is got the cardiac boundary that is of grey scale change maximum;

5) extract three-dimensional point cloud, manually delete some obvious noise points;

6) show these points (Fig. 2).

Get 30 * 35 points among Fig. 2 as reference mark (wherein 30 for counting on every layer, and 35 for obtaining the number of plies), heart such as Fig. 3 and 4 after 3 * 3 rank nurbs surface matches.

In order to obtain some functional parameters of heart, heart wall inside and outside the left ventricle of 7 states in the cardiac cycle is carried out nurbs surface match (Fig. 5).Being divided into 7 time intervals a cardiac cycle, each is 100ms at interval, so the left ventricle of each at interval corresponding state.

Utilize the nurbs surface of matrix form then, calculate the volume of heart by the curve surface integral formula.

Embodiment 2

With reference to Fig. 1-Fig. 7, in the present embodiment, in described step 2) in, change by coordinate system: with (n+1) * (m+1) reference mark array (x _Ij, y _Ij, z _Ij) (i=0,1...n, j=0,1...m) translation causes to such an extent that curved surface is the center with the z axle, and Cartesian coordinates is represented a little to convert to point (r under the cylindrical coordinates _Ij, θ _Ij, z _Ij) (i=0,1...n, j=0,1...m), its conversion formula following (13):

$r_{\mathrm{ij}}=\sqrt{x_{\mathrm{ij}}^2+y_{\mathrm{ij}}^2},$

${\mathrm{\&theta;}}_{\mathrm{ij}}=\left\{\begin{array}{cc}\mathrm{\&pi;}/2& \mathrm{if}{x}_{\mathrm{ij}}=0\mathrm{and}{y}_{\mathrm{ij}}>0\\ 3\mathrm{\&pi;}/2& \mathrm{if}{x}_{\mathrm{ij}}=0\mathrm{and}{y}_{\mathrm{ij}}<0\\ a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}{x}_{\mathrm{ij}}>0\mathrm{and}{y}_{\mathrm{ij}}\&GreaterEqual;0\\ \mathrm{\&pi;}+a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}{x}_{\mathrm{ij}}<0\\ 2\mathrm{\&pi;}+a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}{x}_{\mathrm{ij}}>0\mathrm{and}{y}_{\mathrm{ij}}<0\end{array}\right.---\left(13\right)$

z _ij＝z _ij

The formula of cylindrical coordinates nurbs surface match is (14):

$[r(u,v),\mathrm{\&theta;}(u,v),z(u,v)]=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}---\left(14\right)$

Wherein, p _{I, j}=[r _{I, j}, θ _{I, j}, z _{I, j}].

The cylindrical coordinates nurbs surface equation that can obtain matrix form is:

$r^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^r(r,l)t^f{w}^l$

${\mathrm{\&theta;}}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^{\mathrm{\&theta;}}(r,l)t^f{w}^l---\left(15\right)$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^f{w}^l$

In described step 4), in conjunction with the power basis function expression formula of nurbs surface, order

With

(u is v) separately to the partial derivative of t and w for curved surface S;

With By difference formula (15) t and w are asked local derviation:

$A\left(z\left(w\right)\right)=\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1\frac{1}{2}{r}^2(t,w)\&times;\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}\mathrm{dt},$

$\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}{\mathrm{\&Sigma;}}}\underset{f=1}{\overset{k_1}{\mathrm{\&Sigma;}}}f\&times;B_{i,j}^{\mathrm{\&theta;}}(f-1,l)t^{f-1}{w}^l$

Obtain cubature formula:

$V=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^1\frac{1}{2}{r}^2(t,w)\&times;\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}\&times;\frac{\&PartialD;Z(t,w)}{\&PartialD;w}\mathrm{dtdw}---\left(16\right)$

With r (t, w),

With

Cubature formula above the substitution (16), the volume expression formula that obtains under the cylindrical coordinates is (17):

$V=\frac{1}{2}\underset{i=k1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k2}{\overset{n}{\mathrm{\&Sigma;}}}(\underset{l_1=1}{\overset{4\&times;k_2-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2=1}{\overset{4\&times;k_1-1}{\mathrm{\&Sigma;}}}{C}_{i,j}^{\mathrm{\&theta;}}(l_1,l_2)\&times;\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="S2007103072478E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_2+1}\&times;\frac{1}{l_1+1})---\left(17\right).$

Other steps of present embodiment are identical with embodiment 1.

Because the change more complicated of weight factor, and if the incorrect of adjustment can go wrong on the contrary.Therefore, under identical weight factor condition, the cylindrical coordinates nurbs surface is fit to the heart surface match more.Therefore, cylindrical coordinates nurbs surface volume calculated is more suitable.

Verify our calculation method of physical volume, we adopt the experimental subjects of round platform as us in this trifle.Adopt two purposes that mainly contain of round platform, the first, the volume energy of round platform is by formula π h (R ²+ Rr+r ²)/3 are calculated and are obtained; The second, round platform can be come out by the accurate match of nurbs surface.The bottom surface radius of getting round platform is respectively r=1cm, R=2cm, and height is h=4cm, its fitting result chart such as Fig. 6.The volume that calculates round platform with formula is 29.3215cm ³(getting π=3.1416).Form 1 is the comparison diagram of the round platform volume that calculates of nurbs surface integration method and Simpson method.

Table 1

In table 1, n and m are the minizone quantity that in the Simpson formula interval u ∈ [0,1] and interval v ∈ [0,1] is divided into, and n and m are big more, and the result who obtains is just accurate more, also need long more computing time simultaneously.We have known owing to the accurate match of nurbs surface round platform from table 1, just the same by the volume that the volume that obtains behind the nurbs surface integration and actual formula calculate, that is to say that the nurbs surface integration can calculate the volume of NURBS model accurately and (that is: can accurately be represented by nurbs surface owing to quadric surface, so, this nurbs surface integration method can be calculated quadric volume accurately, and for the free type curved surface, calculate the precision that the volume accuracy of trying to achieve depends on the NURBS match with this volume method, therefore, the nurbs surface integral and calculating heart of present case employing cylindrical coordinate and the volume of left ventricle).Although when we use n=200, the volume that the Simpson formula of m=200 obtains is very approaching with real volume, still has 0.0022 error and is cost to spend the bigger time.Need under the situation at more reference mark when object is complicated, the Simpson method need be spent longer computing time (seeing Table 2) than the method for present embodiment.

In addition, be not that the NURBS of high-order more is good more, form 1 shows that the nurbs surface on 3 * 3 to 4 * 4 rank is more effective.

Be applied to cardiac image: thus the volume algorithm of present embodiment is mainly used to ask the volume of left ventricle to improve the precision and the efficient of a lot of heart function parameters.Here, we adopt and the plastic cement cardiac data, and Fig. 7 is its fitted figure, and table 2 is comparisons of two kinds of method volumes.

Table 2

Associative list 1, the algorithm that we obtain this paper obtains better result under the situation of a small amount of time of cost.

The left ventricular function calculation of parameter: the volume of the left ventricle that the volume algorithm by present case obtains, thus obtain some functional parameters of left ventricle.Here, present case adopts left ventricle SPECT image, cuts apart a statistical model that obtains left ventricle by Three-Dimensional Dynamic body model (3-D ASM).The data that we just provide statistical model are as the data of this experiment, and being divided into 7 intervals cardiac cycle, each is 100ms at interval here, and the left ventricle of a constantly corresponding state just comes one cardiac cycle of approximate representation with these 7 states so.Fig. 8 and Fig. 9 are respectively outer, the heart pilaster coordinate nurbs surface fitted figure of left ventricle of seven states in the cardiac cycle.Table 3 is for obtaining the left ventricular mass of seven states in the cardiac cycle.Figure 10 is the change curve of left ventricular mass in cardiac cycle, and this change curve is that the left ventricular mass by seven phase places obtains by spline interpolation

, among Figure 10, the horizontal ordinate express time, each phase intervals 100ms, ordinate is represented left ventricular mass, unit/ml.

By table 3, left ventricular ejection fraction EF can obtain by calculating:

Normal value that it is generally acknowledged EF is 50%-70%.

Table 3.

Claims (6)

1. cardiac volume calculations method based on the nurbs surface integration, it is characterized in that: described cardiac volume calculations method may further comprise the steps:

1), the obtaining and handling of data: given a large amount of heart medical image, from these images, obtain the three-dimensional point of heart surface, comprising:

(1.1), carry out smoothing processing with image filtering method, the removal noise;

(1.2), by given index file sectioning image is adjusted to correct order;

(1.3), the definition area-of-interest, from image, be partitioned into the target area by the gray scale threshold value method;

(1.4), obtain the gray-scale value of image, the variation of calculating gray scale, the position of getting the grey scale change maximum is a cardiac boundary;

(1.5), extract the three-dimensional point cloud of heart;

2), the three-dimensional point cloud of getting above-mentioned heart is as the reference mark, carries out the nurbs surface match, its formula is (1):

$s(u,v)=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{N}_{i,k_1}\left(u\right)N_{{j,k}_2}\left(v\right)}---\left(1\right)$

In the following formula, p _{I, j}Be the reference mark of curved surface, wherein, i=0,1 ..., n; J=0,1 ..., m promptly takes from the frontier point cloud of heart, is topological rectangular array, ω _{I, j}Be the weight factor that interrelates with the reference mark;

With

Be respectively k ₁And k ₂The reasonable B spline base function of inferior standard;

The de Boor-Cox recursion formula of reasonable B spline base function, it is defined as follows:

In the following formula, B batten N _{I, k}(u) be defined on the entire parameter u axle B batten N _{I, k}(u) by interior nodes u between its supporting area _i, u _I+1... u _I+k+1Decision;

3), with the nurbs surface matrix representation:

Be defined in knot vector U, on the V, be limited to node region [u _i, u _I+1) * [v _j, v _J+1) on k ₁* k ₂Inferior NURBS knee-piece is expressed as follows:

$s_{\mathrm{ij}}(u,v)=\frac{\underset{l=i-k_1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{lr}}{p}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}{\underset{l=i-k_1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2}{\overset{j}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{\mathrm{lr}}{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)}---\left(3\right)$

Homogeneous coordinates are expressed as follows:

$s_{\mathrm{ij}}^h(u,v)=\underset{l=i-k_1+1}{\overset{i}{\mathrm{\&Sigma;}}}\underset{r=j-k_2+1}{\overset{j}{\mathrm{\&Sigma;}}}{V}_{\mathrm{lr}}^h{N}_{l,k_1}\left(u\right)N_{r,k_2}\left(v\right)---\left(4\right)$

V _Lr ^hBe homogeneous coordinates (ω _Lrp _Lr, ω _Lr), p _LrBe the reference mark, ω _LrBe weight factor, i=k, k+1 ... n, j=k, k+1 ... m;

The nurbs surface parameter standardized change t=(u-u _i)/(u _I+1-u _i), w=(u-u _i)/(u _I+1-u _i), t ∈ [0,1), w ∈ [0,1), obtain the matrix representation of nurbs surface sheet (4):

$s_{i,j}^h(t,w)=T_{k_1}{M}_{i,u}^{k_1+1}{V}_{i,j}^h{\left(M_{j,v}^{k_2+1}\right)}^T{W}_{k_2}^T---\left(5\right)$

Wherein, $T_{k_1}=\left[\begin{array}{cccc}1& t& ...& t^{k_1}\end{array}\right],$ $W_{k_2}^T={\left[\begin{array}{cccc}1& w& ...& w^{k_2}\end{array}\right]}^T,$

With

Be respectively u to v to matrix of coefficients;

The power basis function of formula (5) is represented:

$s_{i,j}^h(t,w)=\underset{l=0}{\overset{k_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(6\right)$

Wherein, $B_{i,j}(r,l)=\underset{b=0}{\overset{k_2}{\mathrm{\&Sigma;}}}(\underset{c=0}{\overset{k_1}{\mathrm{\&Sigma;}}}{M}_{i,u}^{k_1+1}(l,c)\&times;V^h(i-(k_1-c),j-(k_2-b)))\&times;M_{j,v}^{k_2+1}(b,r)$

Nurbs surface on whole zone [0,1] * [0,1] is expressed as equation (7):

$S^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{s}_{i,j}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}(r,l)t^r{w}^l---\left(7\right)$

That is:

$X^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^x(r,l)t^r{w}^l$

$Y^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^y(r,l)t^r{w}^l---\left(8\right)$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{S}_{i,j}(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^r{w}^l$

The recursion formula of k ordered coefficients matrix:

Wherein, $d_{0,j}=\frac{t_i-t_j}{t_{j+(k+1)-1}-t_j},$ $d_{1,j}=\frac{t_{i+1}-t_i}{t_{j+(k+1)-1}{t}_j};$

Utilize formula (9), obtain following matrix of coefficients:

$M_i^1=\left[1\right],$ $M_i^2=\left[\begin{array}{cc}1& 0\\ -1& 0\end{array}\right],$

$M_i^3=\left[\begin{array}{ccc}\frac{u_{i+1}{-u}_i}{u_{i+1}-u_{i-1}}& \frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}}& 0\\ \frac{-2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& \frac{2(u_{i+1}-u_i)}{u_{i+1}-u_{i-1}}& 0\\ \frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}}& -(u_{i+1}-u_i)(\frac{1}{u_{i+1}-u_{i-1}}+\frac{1}{u_{i+2}-u_i})& \frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

$M_i^4=\left[\begin{array}{cccc}\frac{{(u_{i+1}-u_i)}^2}{(u_{i+1}-u_{i-1})(u_{i+1}-u_{i-2})},& 1-a_{00}-a_{02}& \frac{u_i-u_{i-1}}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ -3a_{00},& 3a_{00}-a_{12},& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_i-u_{i-1}}{u_{i+1}-u_{i-1}},& 0\\ 3a_{00},& -3a_{00}-a_{22}& 3\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+1}-u_{i-1}},& 0\\ -a_{00},& a_{31},& a_{32}& \frac{u_{i+1}-u_i}{u_{i+3}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\end{array}\right]$

Wherein, $a_{31}=a_{00}+\frac{1}{3}{a}_{12}+\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ $a_{32}=-\frac{1}{3}{a}_{12}-a_{33}-\frac{u_{i+1}-u_i}{u_{i+2}-u_i}\&CenterDot;\frac{u_{i+1}-u_i}{u_{i+2}-u_{i-1}},$ a _{I, j}It is matrix M _i ⁴The capable j of an i row element;

4), to above-mentioned nurbs surface, promptly formula (8) carries out the volume that integration obtains heart,

Make A (z) be the cross-sectional area at height z place, then volume is:

$V={\&Integral;}_{z_1}^{z_2}A\left(z\right)\mathrm{dz}---\left(10\right).$

2. a kind of cardiac volume calculations method based on the nurbs surface integration as claimed in claim 1 is characterized in that: in described step 4), and in conjunction with the power basis function expression formula of nurbs surface, order

With

(u is v) separately to the partial derivative of t and w for curved surface S;

With

By formula (8) t and w are asked local derviation respectively:

$\frac{\&PartialD;X(t,w)}{\&PartialD;t}=\underset{{i=k}_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{{j=k}_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}r\&times;B_{i,j}^x(r-1,l)t^{r-1}{w}^l$

$\frac{\&PartialD;Z(t,w)}{\&PartialD;w}=\underset{{i=k}_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{{j=k}_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=1}{\overset{k_2}{\mathrm{\&Sigma;}}}\underset{r=0}{\overset{k_1}{\mathrm{\&Sigma;}}}l\&times;B_{i,j}^z(r,l-1)t^r{w}^{l-1}$

In the cross-sectional area A (z (w)) that height z (w) locates, as follows:

$A\left(z\left(w\right)\right)=\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^{1}Y(t,w)\&times;\frac{\&PartialD;X(t,w)}{\&PartialD;t}\mathrm{dt}$

Top formula substitution (10) is obtained:

$V=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}{\&Integral;}_0^{1}A\left(z\left(w\right)\right)z^{\&prime;}\left(w\right)\mathrm{dw}$

$=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^{1}Y(t,w)\&times;\frac{\&PartialD;X(t,w)}{\&PartialD;t}\&times;\frac{\&PartialD;Z(t,w)}{\&PartialD;w}\mathrm{dtdw}$

$=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^1\underset{l_1=0}{\overset{(3\&times;k_2-1)}{\mathrm{\&Sigma;}}}\underset{l_2=0}{\overset{(3\&times;k_1-1)}{\mathrm{\&Sigma;}}}{C}_{i,j}(l_1,l_2)\&times;t^{l_2}\&times;w^{l_1}\mathrm{dtdw}$

k ₁* k ₂The cubature formula of the heart that inferior nurbs surface is represented is as follows:

$V=\underset{i=k1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k2}{\overset{n}{\mathrm{\&Sigma;}}}(\underset{l_1=1}{\overset{(3\&times;k_2-1)}{\mathrm{\&Sigma;}}}\underset{l_2=1}{\overset{(3\&times;k_1-1)}{\mathrm{\&Sigma;}}}{C}_{i,j}(l_1,l_2)\&times;\frac{1}{l_2+1}\&times;\frac{1}{l_1+1})---\left(11\right)$

Wherein, C _{I, j}Be 3k ₁* 3k ₂The rank matrix, it be polynomial expression Y (t, w), With

After multiplying each other, the coefficient before t and the w variable, that is, and l ₁Row l ₂Column element C _{I, j}(l ₁, l ₂) expression

Preceding coefficient; C _{I, j}(l ₁, l ₂) calculate acquisition by expression (12):

$C_{i,j}(l_1,l_2)=\underset{\underset{\underset{0\&le;d\&le;k_2}{d=l_1-f}}{f=0}}{\overset{2k_2-1}{\mathrm{\&Sigma;}}}\underset{\underset{\underset{0\&le;d_2\&le;k_1-1}{d_2=l_2-f_2}}{f_2=0}}{\overset{{2k}_1}{\mathrm{\&Sigma;}}}(\underset{\underset{\underset{0\&le;s\&le;k_2-1}{s=f-r}}{r=0}}{\overset{k_2}{\mathrm{\&Sigma;}}}\underset{\underset{\underset{0\&le;s_2\&le;k_1}{s_2=f_2-r_2}}{r_2=0}}{\overset{k_1}{\mathrm{\&Sigma;}}}{B}_{i,j}^y(r,r_2)\&times;s\&times;B_{i,j}^x(s,s_2))\&times;d_2\&times;B_{i,j}^z(d,d_2)---\left(12\right).$

3. a kind of cardiac volume calculations method based on the nurbs surface integration as claimed in claim 2 is characterized in that: in described step 2) in, with (n+1) * (m+1) reference mark array (x _Ij, y _Ij, z _Ij), i=0 wherein, 1...n, j=0, the 1...m translation causes to such an extent that curved surface is the center with the z axle, and Cartesian coordinates is represented a little to convert to point (r under the cylindrical coordinates _Ij, θ _Ij, z _Ij), i=0 wherein, 1...n, j=0,1...m, its conversion formula following (13):

$r_{\mathrm{ij}}=\sqrt{x_{\mathrm{ij}}^2+y_{\mathrm{ij}}^2},$

${\mathrm{\&theta;}}_{\mathrm{ij}}=\left\{\begin{array}{ccccc}\mathrm{\&pi;}/2& \mathrm{if}& x_{\mathrm{ij}}=0& \mathrm{and}& y_{\mathrm{ij}}>0\\ 3\mathrm{\&pi;}/2& \mathrm{if}& x_{\mathrm{ij}}=0& \mathrm{and}& y_{\mathrm{ij}}<0\\ a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}& x_{\mathrm{ij}}>0& \mathrm{and}& y_{\mathrm{ij}}\&GreaterEqual;0\\ \mathrm{\&pi;}+a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& & & \mathrm{if}& x_{\mathrm{ij}}<0\\ 2\mathrm{\&pi;}+a\tan (y_{\mathrm{ij}}/x_{\mathrm{ij}})& \mathrm{if}& x_{\mathrm{ij}}>0& \mathrm{and}& y_{\mathrm{ij}}<0\end{array}\right.---\left(13\right)$

z _ij＝z _ij

The formula of the nurbs surface match of cylindrical coordinate is (14):

$[r(u,v),\mathrm{\&theta;}(u,v),z(u,v)]=\frac{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{p}_{i,j}{N}_{i,k_1}\left(u\right)N_{j,k_2}\left(v\right)}{\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_{i,j}{N}_{i,k_1}\left(u\right)N_{{j,k}_2}\left(v\right)}---\left(14\right)$

Wherein, p _{I, j}=[r _{I, j}, θ _{I, j}, z _{I, j}].

The cylindrical coordinates nurbs surface equation that obtains matrix form is:

$r^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^r(r,l)t^f{w}^l$

${\mathrm{\&theta;}}^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^{\mathrm{\&theta;}}(r,l)t^f{w}^l---\left(15\right).$

$Z^h(t,w)=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k2}{\mathrm{\&Sigma;}}}\underset{f=0}{\overset{k1}{\mathrm{\&Sigma;}}}{B}_{i,j}^z(r,l)t^f{w}^l$

4. a kind of cardiac volume calculations method based on the nurbs surface integration as claimed in claim 3 is characterized in that: in described step 4), and in conjunction with the power basis function expression formula of nurbs surface, order

With

(u is v) separately to the partial derivative of t and w for curved surface S;

With

By difference formula (15) t and w are asked local derviation:

$A\left(z\left(w\right)\right)=\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1\frac{1}{2}{r}^2(t,w)\&times;\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}\mathrm{dt},$

$\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_2}{\overset{n}{\mathrm{\&Sigma;}}}\underset{l=0}{\overset{k_2}{\mathrm{\&Sigma;}}}\underset{f=1}{\overset{k_1}{\mathrm{\&Sigma;}}}f\&times;B_{i,j}^{\mathrm{\&theta;}}(f-1,l)t^{f-1}{w}^l$

Obtain cubature formula:

$V=\underset{i=k_1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k_{\text{2}}}{\overset{n}{\mathrm{\&Sigma;}}}{\&Integral;}_0^1{\&Integral;}_0^1\frac{1}{2}{r}^2(t,w)\&times;\frac{\&PartialD;\mathrm{\&theta;}(t,w)}{\&PartialD;t}\&times;\frac{\&PartialD;Z(t,w)}{\&PartialD;w}\mathrm{dtdw}---\left(16\right)$

With r (t, w),

With

Cubature formula above the substitution (16), the volume expression formula that obtains under the cylindrical coordinates is (17):

$V=\frac{1}{2}\underset{i=k1}{\overset{m}{\mathrm{\&Sigma;}}}\underset{j=k2}{\overset{n}{\mathrm{\&Sigma;}}}(\underset{l_1=1}{\overset{4\&times;k_2-1}{\mathrm{\&Sigma;}}}\underset{l_2=1}{\overset{4\&times;k_1-1}{\mathrm{\&Sigma;}}}{C}_{i,j}^{\mathrm{\&theta;}}(l_1,l_2)\&times;\frac{1}{l_2+1}\&times;\frac{1}{l_1+1})---\left(17\right)$

5. a kind of cardiac volume calculations method as claimed in claim 4 based on the nurbs surface integration, it is characterized in that: in described step 1), medical image adopts SPECT medical image, nuclear magnetic resonance image, CT image, spiral CT image, ultrasonoscopy or PET image.

6. a kind of cardiac volume calculations method based on the nurbs surface integration as claimed in claim 4 is characterized in that: described heart is left ventricle, right ventricle, atrium sinistrum, atrium dextrum, the surfaces externally and internally of heart partly or completely.

Images