CN102579045A - Sparse imaging method for magnetic resonance over-sampling and under-sampling K data - Google Patents

Abstract

A sparse imaging method for magnetic resonance over-sampling and under-sampling K data is used for reconstructing unsampled K data through over-sampling and under-sampling K data to acquire complete K data and obtain final imaging results. The sparse imaging method includes the steps: (a) selecting sparse operators and sparsifying zero-stuffing reconstructed images of the under-sampling K data to obtain sparse images; (b) extracting sparse parameters from the sparse images according to reduced-order functions of a K space distribution structure; and (c) reconstructing the complete K data by the aid of the sparse parameters according to sparse representation of the K data to acquire final images.

Description

Magnetic resonance the is ultra rarefaction formation method of K data of owing to sample

Technical field

The present invention relates to the medical imaging field, relate in particular to nuclear magnetic resonance.The present invention is specifically related to the rarefaction formation method of the ultra K data of owing to sample of a kind of magnetic resonance.

Background technology

Nuclear magnetic resonance (MRI) has the incomparable advantages of formation method such as X-ray, CT as a kind of undamaged diagnostic means, has obtained using widely.

Yet, nuclear magnetic resonance sometimes because of the limitation of conditions, few partial data that people must not only gather carries out image reconstruction.This few partial data that satisfies conventional medical imaging desired data far away is called the ultra sampled data of owing exactly.The ultra image reconstruction of owing sampled data is called the ultra sampled data imaging of owing, and its application is very extensive.

In addition; The permanent magnet type magnetic resonance imaging device has low cost of manufacture; Advantage such as easy to maintenance, its shortcoming system causes scanning speed not high enough because of main field strength is limited, makes the function of many superconducting magnetic resonance imaging devices on the permanent magnet type magnetic resonance imaging device, not realize.Ultra owe the sampling imaging technology and need not increase equipment cost; Can be under the precondition that hardware device is not upgraded and scan mode is constant; Improve scanning speed exponentially, can make the permanent magnet type magnetic resonance imaging reach the suitable image taking speed of superconducting magnetic resonance imaging device, the function of imaging increases substantially.

Thus, surpass and owe the research content and research focus that the sampled data nuclear magnetic resonance is the medical imaging field, have great scientific meaning and application prospect.Current industry main approaches and present situation are following:

(1) zero padding method is the straightforward procedure of owing sampled data imaging.The zero padding method is that the not sampled data of owing in the sampled data is filled up with zero, forms images as partial data then.Be characterized in that algorithm is simple, error stable, but image quality.(2) neighbour's interpolation iterative method is traditional sampled data formation method of owing.All interpolation methods all require by the data break of interpolation smaller, can tolerate the error of neighbour's point interpolation.But form images for the ultra sampled data of owing, bigger by the interval of interpolation, interpolation method can't use.(3) method of regularization is under certain constraints, is optimized imaging by some objective criterias such as image error is minimum, entropy is maximum, the minimum methods of total differential.Its advantage is that can obtain for objective criteria is better image.But picture quality changes according to the objective function optimization strategy, and algorithm complex is high, and the result lacks objectivity.(4) (Compressed sensing CS) is the theory that grew up in recent years to compressed sensing, is in the developmental research stage at present.For the ease of storage and transmission, adopt the use data pattern of data acquisition-compression-transmission-decompression usually.Since will reduce redundant data wherein after the image data anyway, not as good as directly gathering these nonredundant data.Here it is so-called " compressed sensing ".If signal satisfies sparse property (if for K sparse), so as long as M observation of sampling (wherein N＞＞M >=K), just can in the equation group of a non-full rank, come reconstruction signal X through the L1 regularization method.Shortcoming is that optimization searching length computation time, reconstructed image result are responsive to expressing function base and optimisation strategy.(5) (the Projection Onto Convex Sets of iteration convex set back projection; POCS) method is a kind of iterative backprojection method; This method requires the phase place of correct estimated magnetic flux resonance image; To better re-configurability being arranged about former point-symmetric K data, better effects is arranged owing the sampled data imaging on a small quantity, very poor in ultra picture quality of owing sampling imaging.(6) being used for magnetic resonance owes the common method of sampled data imaging and also has locking phase bearing calibration and singular spectrum analysis method.Based on the method for phasing require data in the Partial K data more than 9/16; And because the slow change condition of MR image phase usually is difficult to satisfy in the entire image space; Cause the phase estimation error big; Cause reconstructed image than mistake so that up to the present can't extensive use in each field of clinical medicine.The singular spectrum analysis method is to be similar to the sparse expression method, and it expresses image with singular function, obtains the parameter of unusual expression image through the data of sampling, and then comes reconstructed image by singular function.It hopes that the image of reconstruct has the characteristic of local platform according to image, and responsive to noise.

Thus, industry is expected the method for reconstruct MRI fast, accurately to some extent.

Summary of the invention

The technical problem that the present invention will solve is that MRI is carried out fast accurate reconstruct.

In order to reach above-mentioned purpose; The invention provides the ultra K data rarefaction formation method of owing to sample of a kind of magnetic resonance; Said method obtains complete K data through the ultra unsampled K data of K data reconstruction of owing to sample; To obtain final imaging results, may further comprise the steps: (a) select the rarefaction operator, the zero padding reconstructed image of the said K data of owing to sample is carried out rarefaction handle and obtain the rarefaction image; (b) the depression of order function according to said K spatial distribution structure extracts the rarefaction parameter from said rarefaction image; And (c) according to the sparse expression formula of said K data, utilize the complete K data of said rarefaction parameter reconstruct, to get image to the end.

Ultra owe sampling and reach gratifying effect because method of the present invention is capable of using, therefore reconstruct MRI fast, accurately.

In conjunction with accompanying drawing, can know other aspects of the present invention and advantage based on the description of passing through example description purport of the present invention hereinafter.

Description of drawings

In conjunction with accompanying drawing, through the detailed description of stating of hereinafter, can more be expressly understood above-mentioned and other feature and advantage of the present invention, wherein:

Fig. 1 is the flow chart that the step of the inventive method is shown;

Fig. 2 is the sketch map of the image of the embodiment of the invention, and wherein: (a) being magnetic resonance real part image, (b) is that figure (a) is by rarefaction operator φ (i, j)=2 δ (i; J)-and δ (i-1, j)+(i, the j-1) image after the rarefaction (c) are the real part image that the zero padding method obtains to δ; (d) be figure (c) by rarefaction operator φ (i, j)=2 δ (i, j)-δ (i-1; J)+(i, the j-1) image after the rarefaction (e) are depression of order function real (δ to δ _z(i, j)) (f) is the indication function.

Fig. 3 owes the impact analysis figure of sampling degree to reconstruction accuracy.

Fig. 4 is the reconstruct histogram of error.

Fig. 5 is with reference to image and reconstructed image, wherein, is with reference to image (a), (b) is ZF (zero padding) method reconstructed image, is TV (total variation) method reconstructed image (c), (d) is present embodiment method reconstructed image.

Fig. 6 is the phase diagram with reference to image and reconstructed image, wherein, is with reference to image (a), (b) is ZF method reconstructed image, (c) is TV method reconstructed image, (d) is present embodiment method reconstructed image.

Fig. 7 is the K data with reference to image and reconstructed image;

Wherein, be (a) with reference to image, (b) be ZF method reconstructed image, (c) be TV method reconstructed image, (d) be present embodiment method reconstructed image.

Fig. 8 be ZF, TV and present embodiment method reconstructed image with reference to the mistake difference STD of image with the variation that adds noise STD.

Fig. 9 is with reference to image and ZF, TV and the contrast of present embodiment method reconstructed image;

Wherein, be (a) with reference to image, (b) be ZF method reconstructed image, (c) be TV method reconstructed image, (d) be present embodiment method reconstructed image, (e) be the scatterplot of TV method, (f) be the scatterplot of present embodiment method.

The specific embodiment

Referring to the accompanying drawing that the embodiment of the invention is shown, hereinafter will be described the present invention in more detail.Yet the present invention can be with many multi-form realizations, and should not be construed as the restriction of the embodiment that receives in this proposition.On the contrary, it is abundant and complete open in order to reach proposing these embodiment, and makes the technical staff in present technique field understand scope of the present invention fully.In these accompanying drawings, for clarity sake, possibly amplify the size and the relative size in layer and zone.

According to formation method of the present invention, still can reconstruct gratifying imaging results through the ultra K data of owing to sample, reach quick, accurately image thus.

Refer now to Fig. 1 and describe rarefaction formation method according to the ultra K data of owing to sample of magnetic resonance of the present invention.Said method obtains complete K data through the ultra unsampled K data of K data reconstruction of owing to sample, to obtain final imaging results.

As shown in Figure 1, in step S101, select the rarefaction operator, the zero padding reconstructed image of the said ultra K data of owing to sample is carried out rarefaction handle and obtain the rarefaction image.

If most of discrete function values of a discrete function are zero, claim that then this discrete function is a sparse discrete function.Be converted into sparse discrete function to discrete function through rarefaction, sparse discrete function only needs its nonvanishing function value and coordinate thereof just can confirm, these nonvanishing function values and coordinate thereof can be used for recovering original discrete function at certain condition.

Discrete function g becomes the process for sparse discrete function through the convolution φ * g of another discrete function φ, is called the rarefaction of discrete function g.Wherein function phi is called the rarefaction operator of discrete function g, and " * " representes convolution.

In the present embodiment, at first the said K data of owing to sample are carried out zero padding reconstruct (hereinafter will detail).Then to carrying out Fuli's leaf inverse transformation, and represent the zero padding reconstructed image, further said two-dimensional discrete function is carried out rarefaction and handle with the form of two-dimensional discrete function through the K of zero padding reconstruct data.

Thus, utilize rarefaction operator φ (i, j), i, j=0,1 ..., (i j) carries out rarefaction and handles N-1 to the two-dimensional discrete function g of said K data.Sparse discrete function g after the rarefaction _φ(i j) is expressed as:

g _φ(i，j)＝φ(i，j)*g(i，j) (1)

The choice criteria of said rarefaction operator is that the zero point of its K spatial function is considerably less.

If g _φ(i, j) middle nonvanishing function value is respectively b={b ₁, b ₂..., b _q, corresponding coordinate is χ={ (i ₁, j ₁), (i ₂, j ₂) ..., (i _q, j _q), claim that then b and χ are respectively sparse value and sparse coordinate, are referred to as sparse discrete function g _φ(i, sparse parameter j), or claim discrete function g (i, rarefaction parameter j).Wherein q is the number of the nonvanishing function value of sparse discrete function.We with " || ₀" number of nonvanishing function value of expression statistical straggling function " ".Promptly

q＝|g _φ(i，j)| ₀。(2)

Introduce two-dimentional Dirac function

Then sparse discrete function g _φ(i, two-dimentional Dirac function sparse expression j) is:

$g_{\mathrm{\φ}}(i,j)=\underset{s=0}{\overset{q}{\mathrm{\Σ}}}{b}_s\mathrm{\δ}(i-i_s,j-j_s)---\left(4\right)$

G with leaf transformation in the fetching, is remembered in formula (4) both sides _φ(k _i, k _j)=F [g _φ(i, j)], δ _s(k _i, k _j)=F [δ (i-j _s, j-j _s)], wherein F [] representes fourier transform, then

$G_{\mathrm{\φ}}(k_i,k_j)=\underset{s=0}{\overset{q}{\mathrm{\Σ}}}{b}_s{\mathrm{\δ}}_s(k_i,k_j)---\left(5\right)$

Formula (5) has shown that the fourier transform (that is the K spatial function of sparse discrete function) of sparse discrete function can be { δ with function ₁(k _i, k _j), δ ₂(k _i, k _j) ..., δ _q(k _i, k _j) carry out linear rarefaction representation.Note Φ (k _i, k _j)=F [φ (i, j)], G (k _i, k _j)=F [g (i, j)], to g _φ(i, j)=(i, j) (i j) carries out fourier transform to * g to φ, and arrangement can obtain

$G(k_i,k_j){|}_{\mathrm{\Φ}(k_i,k_j)\≠0}=\underset{s=0}{\overset{q}{\mathrm{\Σ}}}{b}_s\frac{{\mathrm{\δ}}_s(k_i,k_j)}{\mathrm{\Φ}(k_i,k_j)}{|}_{\mathrm{\Φ}(k_i,k_j)\≠0}---\left(6\right)$

(6) formula be discrete function g (i, K spatial function sparse expression j), show except

Outward, all K spatial data G (k _i, k _j) can calculate by following formula.If in whole K space,

Φ (k _i, k _jThen there is Fourier transform in) ≠ 0:

F wherein ^-1[] expression Fourier transform; Formula can be derived so:

Wherein

That is to say if in whole K space the K spatial function Φ (k of rarefaction operator _i, k _j) there is not zero point, then (i j) can be discrete function g with translation function

Rarefaction representation.It's a pity that all there is zero point in the K spatial function of the effective rarefaction operator of great majority.For example, can make image sparseization to simple function below most of two-dimensional medical images:

φ _i-(i，j)＝δ(i，j)-δ(i-1，j) (9)

φ _j-(i，j)＝δ(i，j)-δ(i，j-1) (10)

φ _i+(i，j)＝δ(i，j)-δ(i+1，j) (11)

φ _j+(i，j)＝δ(i，j)-δ(i，j+1) (12)

Its corresponding K function is respectively:

${\mathrm{\Φ}}_{i-}(k_i,k_j)=1-e^{-\frac{2\mathrm{\π}{k}_i}{N}\sqrt{-1}}---\left(13\right)$

${\mathrm{\Φ}}_{j-}(k_i,k_j)=1-e^{-\frac{{2\mathrm{\πk}}_j}{N}\sqrt{-1}}---\left(14\right)$

${\mathrm{\&Phi;}}_{i+}(k_i,k_j)=1-{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000140791680000032.png"\; state="\{\{state\}\}">e</figure-callout>}}^{\frac{{2\mathrm{\&pi;k}}_i}{N}\sqrt{-1}}---\left(15\right)$

${\mathrm{\&Phi;}}_{j+}(k_i,k_j)=1-{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA0000140791680000032.png"\; state="\{\{state\}\}">e</figure-callout>}}^{\frac{{2\mathrm{\&pi;k}}_j}{N}\sqrt{-1}}---\left(16\right)$

Find out that from above K function the K function of simple rarefaction operator all has N zero point.By formula (6) these zero points and near K data G (k thereof _i, k _j) be incalculable.Therefore, be suitable for effective rarefaction operator through generating of operator with mode.For example, and φ (i, j)=φ _I-(i, j)+φ _J-(i j) is:

φ(i，j)＝2δ(i，j)-δ(i-1，j)+δ(i，j-1) (17)

Its K function is:

$\mathrm{\&Phi;}(k_i,k_j)=2-e^{-\frac{2\mathrm{\&pi;}{k}_j}{N}\sqrt{-1}}-e^{-\frac{2\mathrm{\&pi;}{k}_i}{N}\sqrt{-1}}---\left(18\right)$

A Φ at zero point (0,0)=0 is only arranged.It is applicable to most of Partial K data imagings.

Be applicable to that rarefaction operator of the present invention is not so limited, as long as can make the zero point of the rarefaction of two-dimensional discrete function and its K spatial function considerably less just passable.

For ease of describing the magnetic resonance K space coordinates of being sampled, we define an indicator function δ _z(k _i, k _j) come labelling to sample and unsampled K space coordinates point (k _i, k _j), the coordinate points (k that is gathered _i, k _j) be defined as 1, otherwise be defined as 0.Promptly

δ _z(k _i, k _j) the contravariant of Fuli's leaf be changed to:

δ _z(i，j)＝F ^-1[δ _z(k _i，k _j)] (20)

When whole sampling δ then _z(i, j) become for Dirac function δ (i, j), when more approaching complete K space, sample space, δ _z(i, j) just approach more δ (i, j), δ _z(i is that (i, approximate or depression of order j) is so claim δ for δ j) _z(i j) is the depression of order function.As indicator function δ _z(k _i, k _j) be during about former point symmetry, δ _z(i j) is a real function (referring to Fig. 2).

In the present embodiment, said the K data of owing to sample carried out zero padding reconstruct and the zero padding reconstructed image carried out rarefaction handle particular content such as following:

The zero padding reconstruct of the ultra K data of owing to sample is expressed as:

g _z(i，j)＝F ^-1[G(k _i，k _j)δ _z(k _i，k _j)] (21)

Press F ^-1Convolution character can be written as:

g _z(i，j)＝g(i，j)*δ _z(i，j) (22)

Formula (22) has been explained and the zero padding imaging of owing to sample has been equivalent to discrete function g (i j) has received depression of order function δ _z(i, convolution j) is polluted.To convolution rarefaction operator φ (i, j) (rarefaction), and general simultaneously of formula (22) both sides Substitution utilizes convolution character, and considers δ (i-i _h, j-j _h) * δ _z(i, j)=δ _z(i-i _h, j-j _h) put in order:

$\mathrm{\&phi;}(i,j)*g_z(i,j)=\underset{h=1}{\overset{q}{\mathrm{\&Sigma;}}}{b}_h{\mathrm{\&delta;}}_z(i-i_h,j-j_h)---\left(23\right)$

Wherein, δ _z(i-i _h, j-j _h) represent that the origin translation of depression of order function arrives (i _h, j _h).Formula (23) has provided the zero padding method discrete function of the K data of owing to sample, and ((i is j) about discrete function g (i, rarefaction parameter χ={ (i j) still can to use sparse operator φ for i, the j) discrete function after the convolution by φ ₁, j ₁), (i ₂, j ₂) ..., (i _q, j _q) and b={b ₁, b ₂..., b _qCarry out δ _z(just (i is j) by price reduction function δ for Dirac function δ for i, rarefaction representation j) _z(i, j) convolution is stained.

In step S102, the depression of order function of constructing according to said K spatial distribution extracts the rarefaction parameter from said K data.

Depression of order function δ _z(i-i _h, j-j _h) be δ (i-i _h, j-j _h) an approximate oscillatory extinction function.δ _z(i-i _h, j-j _h) always at (i _h, j _h) get maximum on the point, decay rapidly in neighborhood is only to the g in the vibration main lobe neighborhood _φ(i j) produces bigger pollution.| b _h| be worth greatly more, the pollution that the neighborhood functional value is produced is also big more.As long as χ={ (i ₁, j ₁), (i ₂, j ₂) ..., (i _q, j _q) coordinate is enough sparse, δ _z(i-i _h, j-j _h) the phase mutual interference just can ignore.Note g _{Z φ}(i, j)=φ (i, j) * g _z(i, j), then

Be that the rarefaction coordinate parameters is big probability event, can make so

Obtain one group of rarefaction parameter { b _h, (i _h, j _h).Structure depression of order function b _hδ _z(i-i _h, j-j _h), then from g _{Z φ}(i deducts b in j) _hδ _z(i-i _h, j-j _h), promptly carry out iteration

$g_{\mathrm{z\&phi;}}(i,j)\&DoubleLeftArrow;g_{\mathrm{z\&phi;}}(i,j)-b_h\&CenterDot;{\mathrm{\&delta;}}_z(i-i_h,j-j_h)---\left(24\right)$

Look for the next one again , repeat with this, up to norm

Value is no longer because of extracting b _hδ _z(i-i _h, j-j _h) and minimizing is arranged, promptly | g _{Z φ}(i, j) || ₂≈ || g _{Z φ}(i, j)-b _hδ _z(i-i _h, j-j _h) || ₂, just obtain all rarefaction parameter χ={ (x ₁, y ₁), (x ₂, y ₂) ..., (x _q, y _q) and b={b ₁, b ₂..., b _q.This method is called depression of order function extraction method.Can find out, if rarefaction coordinate parameters χ={ (i ₁, j ₁), (i ₂, j ₂) ..., (i _q, j _q) overstepping the bounds of propriety each other diffusing, sparse more.Their extraction function { b _hδ _z(i-i _h, j-j _h), h=1,2 ..., the mutual interferential influence of q} is more little, and the error of introducing is just more little.Therefore to select appropriate rarefaction operator to make rarefaction coordinate parameters χ={ (i ₁, j ₁), (i ₂, j ₂) ..., (i _q, j _q) disperse each other as far as possible, sparse as far as possible.

In step S103, according to the sparse expression formula of said K data with the view data of rarefaction parameter reconstruct, to carry out the reconstruct of said image through ultra shortcoming sampling.

Will ${\mathrm{\&delta;}}_s(k_i,k_j)=F\left[\mathrm{\&delta;}(i-i_s,j-j_s)\right]=e^{-\frac{2\mathrm{\&pi;}(i_s{k}_i+j_s{k}_j)}{N}\sqrt{-1}},$ S=1,2 ..., the substitution formula of q

$G(k_i,k_j){|}_{\mathrm{\&Phi;}(k_i,k_j)\&NotEqual;0}=\underset{s=0}{\overset{q}{\mathrm{\&Sigma;}}}{b}_s\frac{{\mathrm{\&delta;}}_s(k_i,k_j)}{\mathrm{\&Phi;}(k_i,k_j)}{|}_{\mathrm{\&Phi;}(k_i,,k_j)\&NotEqual;0},$

Put in order:

$G(k_i,k_j){|}_{\mathrm{\&Phi;}(k_i,k_j)\&NotEqual;0}=b_1\frac{e^{-\frac{2\mathrm{\&pi;}(i_1{k}_i+j_1{k}_j)}{N}\sqrt{-1}}}{\mathrm{\&Phi;}(k_i,k_j)}+b_2\frac{e^{-\frac{2\mathrm{\&pi;}(i_2{k}_i+j_2{k}_j)}{N}\sqrt{-1}}}{\mathrm{\&Phi;}(k_i,k_j)}+...+b_q\frac{e^{-\frac{2\mathrm{\&pi;}(i_q{k}_i+j_q{k}_j)}{N}\sqrt{-1}}}{\mathrm{\&Phi;}(k_i,k_j)}{|}_{\mathrm{\&Phi;}(k_i,k_j)\&NotEqual;0}---\left(25\right)$

Formula (25) has shown corresponding G (k _x, k _y) can by

I=1,2 ..., the q sparse expression.In case acquisition g (i, j) rarefaction parameter, except

Outward, all G (k _i, k _j) can provide by formula (25).This crosses range request | Φ (k _i, k _j) |=0 G (k _i, k _j) data must gather.Same up-to-date style (25) shows, G (k _i, k _j) and Φ (k _i, k _j) be inversely proportional to, if χ={ (i ₁, j ₁), (i ₂, j ₂) ..., (i _q, j _q) and b={b ₁, b ₂..., b _qWhen containing error, | Φ (k _i, k _j) | be worth more little, at G (k _i, k _j) in to be introduced into error also just big more, therefore for those | Φ (k _i, k _j) | little data will be gathered as far as possible, for | Φ (k _i, k _j) |=0 G (k _i, k _j) data must gather.

The existing instance that carries out image reconstruction according to the inventive method of describing.

This instance adopts the ROSE track scanning, and it is the developed recently rapid scanning technology of getting up, and can save sweep time through sparse scan mode.Sparse ROSE track scanning data, through obtaining the sampled data of owing of right angle grid after INNG (inferior contiguous iteration gridding) the method gridding, this sampled data of owing all owes sampled data except that near zero point; Available rarefaction operator φ (i; J)=2 δ (i, j)-δ (i-1, j)+δ (i; J-1), because its K function is zero at initial point only.Adopt the above-mentioned sampled data of owing in the present embodiment, choose the rarefaction operator K data imaging of owing to sample.

Concrete steps are:

The first step is obtained indicator function δ from the ultra K data of owing to sample _z(k _i, k _j), and structure zero padding K data, G _z(k _i, k _j)=G (k _i, k _j) δ _z(k _i, k _j), calculate depression of order function δ _z(i, j)=F ^-1[δ _z(k _i, k _j)].

In second step, calculate g _{Z φ}(i, j)=F ^-1[G _z(k _i, k _j) Φ (k _i, k _j)], obtain

$g_{\mathrm{z\&phi;}}(i,j)=\underset{h=1}{\overset{q}{\mathrm{\&Sigma;}}}{b}_h{\mathrm{\&delta;}}_z(i-i_h,j-j_h),$

Obtain rarefaction parameter χ={ (x with depression of order function extraction method ₁, y ₁), (x ₂, y ₂) ..., (x _q, y _q) and b={b ₁, b ₂..., b _q,

In the 3rd step, calculate:

$G_{\mathrm{\&phi;}}(k_i,k_j)=\underset{s=0}{\overset{q}{\mathrm{\&Sigma;}}}{b}_s{\mathrm{\&delta;}}_s(k_i,k_j)$

Wherein ${\mathrm{\&delta;}}_s(k_i,k_j)=e^{\frac{-2\mathrm{\&pi;}\sqrt{-1}(i_s{k}_i+j_s{k}_s)}{N}}$

The 4th step, the complete K data of reconstruct

The 5th step, through inverse fourier transform reconstructed image g (i, j)=F ^-1[G (k _i, k _j)].

With the actual magnetic FANTAM complex image image as a reference that resonates, its Pixel Dimensions is 512 * 512, and pixel mould value scope is 0～1.How much represent to owe the sampling degree with K spacescan track number, the scan track number is more little, and the sparse degree of owing to sample is high more.With the sampling track number is 50 to 300 the K data of sampling of owing, and carries out reconstruct with ZF, TV and present embodiment method respectively, as with reference to image, carries out the STD calculating of mould error image with complete K data respective image respectively.Fig. 3 is the situation of change that the STD value increases with the sampling track number.As can be seen from Figure 3 no matter under how many bar scan track lines, the present embodiment method all has minimum STD, explains that the present embodiment method is more effective than TV, and the TV method is much better than the ZF method.Use the present embodiment method, reconstructed error STD is suitable under the STD of the reconstructed error under 120 scan tracks and TV method 300 scan tracks.Under 120 scan tracks, the STD of the reconstructed error of present embodiment method is merely 1/3rd of TV method.

Fig. 4 is under 120 tracks, and the rectangular histogram of the reconstructed error of ZF, TV and present embodiment method can be found out, the present embodiment method with reference to image minimum error is arranged, the TV method is taken second place.

Fig. 5 is with reference to the reconstructed image of image and ZF, TV and present embodiment method, can find out except that the ZF method, and TV method and present embodiment method are all very approaching with reference to image, but the image artifacts of TV method is faintly visible.

Fig. 6 is the phase diagram with reference to image and ZF, TV and present embodiment method reconstructed image; Can find out the ZF method almost level and smooth with reference to figure in original grid; The TV method has been blured reference figure discal patch grid a lot, and present embodiment method and very approaching with reference to image.

Fig. 7 is the K data with reference to image and ZF, TV and present embodiment method reconstructed image; Can find out that the ZF method does not have to recover to owe sampled data; TV has recovered most of K data, and also having the Partial K data is that bigger reconfiguring false is arranged, and the present embodiment method has then recovered to owe sampled data preferably.

As shown in Figure 8, for considering picture noise to algorithm affects, with the actual magnetic FANTAM complex image image as a reference that resonates, its Pixel Dimensions is 512 * 512, and pixel mould value scope is 0～1.Real part and imaginary part at reference picture are respectively the noise that adds the POISSON distribution, and its standard deviation is respectively 0 to 15.Be sampled as 120 roses and distinguish track.Fig. 7 be ZF, TV and present embodiment method reconstructed image with reference to the mistake difference STD of image with the variation that adds noise STD.Can find out that the present embodiment method is superior to the TV method, the excellent ZF method of TV method.But the present embodiment method is at the STD of big noise situations near the TV method.

Fig. 9 is with reference to the contrast of image and ZF, TV and present embodiment method reconstructed image, can find out, TV method and present embodiment method are all very approaching with reference to image, but the image artifacts of TV method is faintly visible.Contrast TV method reconstructed image and present embodiment method reconstructed image are seen (Fig. 9 (e) and Fig. 9 (f)) about the scatterplot with reference to image; The scatterplot that can find out TV has more multiple spot explain that away from diagonal the present embodiment method is superior to the TV method than the scatterplot about the present embodiment method.

Because of should be understood that the present invention, the technical staff in present technique field can realize not breaking away from the spirit or scope of the present invention with many other concrete forms.Although described embodiments of the invention already, the present invention should be understood and these embodiment should be restricted to, the technical staff in present technique field can like enclosed to make within the spirit and scope of the invention that claims define and change and revise.

Claims (10)

1. a magnetic resonance surpasses the K data rarefaction formation method of owing to sample, and said method obtains complete K data through the ultra unsampled K data of K data reconstruction of owing to sample, and to obtain final imaging results, it is characterized in that, may further comprise the steps:

(a) select the rarefaction operator, the zero padding reconstructed image of the said K data of owing to sample is carried out rarefaction handle and obtain the rarefaction image;

(b) the depression of order function according to said K spatial distribution structure extracts the rarefaction parameter from said rarefaction image; And

(c), utilize the complete K data of said rarefaction parameter reconstruct, to get image to the end according to the sparse expression formula of said rarefaction K data.

2. formation method according to claim 1; It is characterized in that; Step (a) comprises that the said K data of owing to sample are carried out zero padding reconstruct carries out Fuli's leaf inverse transformation then, and shows with the form of two-dimensional discrete function, handles further said two-dimensional discrete function is carried out rarefaction.

3. formation method according to claim 2 is characterized in that, the choice criteria of said rarefaction operator is that the zero point of its K spatial function is considerably less.

4. formation method according to claim 3 is characterized in that, said rarefaction operator can for:

φ(i，j)＝2δ(i，j)-δ(i-1，j)+δ(i，j-1)，

Wherein, δ (i j) is two-dimentional Dirac function, and its K spatial function is:

$\mathrm{\&Phi;}(k_i,k_j)=2-e^{-\frac{2\mathrm{\&pi;}{k}_j}{N}\sqrt{-1}}-e^{-\frac{2\mathrm{\&pi;}{k}_i}{N}\sqrt{-1}}.$

5. formation method according to claim 4 is characterized in that, the rarefaction in the step (a) is meant carries out convolution with two-dimensional discrete function and selected rarefaction operator, the sparse discrete function g after the rarefaction _φ(i j) is expressed as:

g _φ(i，j)＝φ(i，j)*g(i，j)

Wherein, " * " representes convolution, and (i is that (i j) is said rarefaction operator for two-dimensional discrete function and the φ of said K data j) to g.

6. formation method according to claim 1 is characterized in that, the described rarefaction parameter of step (b) comprises said sparse discrete function g _φ(i, sparse value j) and sparse coordinate.

7. formation method according to claim 6 is characterized in that, the method for the described extraction rarefaction of step (b) parameter is a depression of order function extraction method.

8. formation method according to claim 7; It is characterized in that; Described depression of order function obtains through the spatial coordinate points sampling of said K indicator function is carried out Fuli's leaf inverse transformation, and wherein said indicator function is in order to labelling sampling and unsampled said K space coordinates point.

9. formation method according to claim 1 is characterized in that, according to the sparse expression formula of said K data, utilizes the unsampled K data of said rarefaction parameter reconstruct, and then obtains final imaging results in the step (c).

10. formation method according to claim 9 is characterized in that, according to said rarefaction expression formula, the K spatial function that must gather the rarefaction operator belongs to the K data of coordinate zero point.

Images