CN107993205A - A kind of MRI image reconstructing method based on study dictionary with the constraint of non-convex norm minimum - Google Patents

A kind of MRI image reconstructing method based on study dictionary with the constraint of non-convex norm minimum Download PDF

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CN107993205A
CN107993205A CN201711211074.XA CN201711211074A CN107993205A CN 107993205 A CN107993205 A CN 107993205A CN 201711211074 A CN201711211074 A CN 201711211074A CN 107993205 A CN107993205 A CN 107993205A
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munderover
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刘书君
曹建鑫
沈晓东
杨婷
李勇明
张奎
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Chongqing University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T5/00—Image enhancement or restoration
    • G06T5/90—Dynamic range modification of images or parts thereof
    • G06T5/92—Dynamic range modification of images or parts thereof based on global image properties
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06V—IMAGE OR VIDEO RECOGNITION OR UNDERSTANDING
    • G06V10/00—Arrangements for image or video recognition or understanding
    • G06V10/40—Extraction of image or video features
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06V—IMAGE OR VIDEO RECOGNITION OR UNDERSTANDING
    • G06V10/00—Arrangements for image or video recognition or understanding
    • G06V10/40—Extraction of image or video features
    • G06V10/50—Extraction of image or video features by performing operations within image blocks; by using histograms, e.g. histogram of oriented gradients [HoG]; by summing image-intensity values; Projection analysis
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/10—Image acquisition modality
    • G06T2207/10072—Tomographic images
    • G06T2207/10088—Magnetic resonance imaging [MRI]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/20—Special algorithmic details
    • G06T2207/20048—Transform domain processing
    • G06T2207/20064—Wavelet transform [DWT]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/30—Subject of image; Context of image processing
    • G06T2207/30004—Biomedical image processing
    • G06T2207/30016—Brain
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/30—Subject of image; Context of image processing
    • G06T2207/30168—Image quality inspection
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06V—IMAGE OR VIDEO RECOGNITION OR UNDERSTANDING
    • G06V10/00—Arrangements for image or video recognition or understanding
    • G06V10/40—Extraction of image or video features
    • G06V10/513—Sparse representations

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Abstract

The invention discloses a kind of MRI image reconstructing method based on study dictionary with the constraint of non-convex norm minimum.Belong to digital image processing techniques field.It is that one kind carries out structure group adaptive sparse expression using dictionary is learnt, and carries out non-convex norm minimum constraint to sparse coefficient in the method for reconstructed image.The similar image set of blocks i.e. structure group of target image block is found first, then the non-convex norm minimum restricted model of image is established, and orthogonal dictionary is gone out to improve sparse expression ability based on the model learning, finally solve sparse coefficient and reconstructed image in the model；The present invention is indicated structure group by learning dictionary, the degree of rarefication after rarefaction representation can be effectively improved, further the coefficient that estimates can be made closer to true coefficient using the constraint of non-convex norm minimum, the MRI image reconstructed by the present invention integrally becomes apparent from, and detailed information is more rich, the accuracy higher of reconstruct, therefore available for the reconstruct of medical image.

Description

It is a kind of to be reconstructed based on study dictionary and the MRI image of non-convex norm minimum constraint Method
Technical field
The invention belongs to digital image processing techniques field, it more particularly to carries out sparse table using learning dictionary to image Show and the method to reconstruct MRI image is constrained to the non-convex norm minimum of sparse coefficient progress, for the high-quality of medical image Amount is recovered.
Background technology
The medical imaging technology that magnetic resonance imaging (MRI) technology is protruded as extremely important at present and effect, extensive use In medical imaging field.For conventional medical imaging technology, magnetic resonance imaging is with the obvious advantage, it is to human body without ionization Radiation injury, and imaging parameters are more, the diagnostic message more horn of plenty that can be provided, contributes to soft tissue to differentiate；Blemish in an otherwise perfect thing It is that the speed of magnetic resonance imaging is slow, and artifact phenomenon is more serious, this constrains the application of MRI technique to a certain extent.
The problem of for MRI image taking speeds, in addition to being improved from hardware, it is exactly profit to also have a kind of important method With the part K space data of image come reconstructed image, that is, make full use of fractional-sample signal under Fourier transform domain as far as possible True picture is recovered, due to need to only utilize fractional-sample signal, can largely reduce the time of signal acquisition. Traditional K space image reconstruction method has zero padding method, Phase Correction Method, signal estimation technique etc., but the figure that these methods reconstruct Picture and not up to ideal effect.Compressive sensing theory be applied to MRI technique after, can not only further reduce sampled signal when Between, moreover it is possible to using the sparse characteristic reconstructed image of image, the image effect after reconstruct is had larger carry compared to compared with conventional method Rise.
Conventional compressed sensing MRI image reconstruct carries out rarefaction representation using the conversion such as small echo, discrete cosine to image, though Good effect is so achieved, but this conversion lacks the adaptability to different images；What is then proposed is learnt based on redundancy The method that image is reconstructed in dictionary, compared to fixed dictionary, the advantage that it represents different images energy adaptive sparse is bright It is aobvious but this limited based on global redundant dictionary one side reconstructed velocity, while be also difficult to characterize various images well Partial structurtes.
The content of the invention
It is an object of the invention to for deficiency existing for existing MRI image reconstructing method, propose a kind of based on study word Allusion quotation and the MRI image reconstructing method of non-convex constraint.This method take into full account image transform domain it is openness between image block it is non- Local similarity, the structure group that similar image set of blocks is obtained carry out non-convex sparse constraint, while to needed for rarefaction representation Dictionary realizes renewal using singular value decomposition, can further improve the degree of rarefication of coefficient after rarefaction representation.Specifically include following step Suddenly：
(1) width MRI raw k-spaces observation data are inputted, it is discrete using total variation method or translation invariant to input data y Wavelet Transform carries out initial reconstitution, obtains the image x after initial reconstitution(0)；
(2) to each target image block x in reconstructed image xiEuclidean distance and other figures are utilized in its search range As block progress similarity-rough set, and the m-1 image block most like with target image block is formed together with target image block One structure groupWhereinMatrix is extracted for image block；
(3) the MRI reconstruction model of non-convex norm minimum constraint is established：
Wherein FuFor down-sampling Fourier transform matrix, D is orthogonal study dictionary, AiFor structure group XiIn orthogonal study word Sparse coefficient under allusion quotation D, i.e. Xi=DAi, λ and β are regularization parameter, and M is the number of structure group,αkTable Show coefficient matrices AiIn row k, the value range of p is (0,1), andRepresent αkThe quadratic sum of middle all elements opens radical sign P power computings are done again later；
(4) for the non-convex norm minimum restricted model in (3), can be analyzed on optimized variable AiWith the optimization of D Problem, and the optimization problem of optimized variable x are iterated solution：
(5a) in the case of given x, on sparse coefficient AiWith optimization problem, that is, sparse coding model of orthogonal dictionary D For：
Wherein I is unit battle array, and orthogonal dictionary D can learn to obtain by singular value decomposition and inequality characteristic, sparse coefficient AiIt can be shunk and be estimated by threshold value；
(5b) is by learning to update orthogonal dictionary D and obtaining structure group sparse coefficient estimate AiAfterwards, become on optimization Amount x optimization problem, that is, image reconstruction model be：
The model is least square model, can be solved to obtain x with conjugate gradient method；
(5) repeat step (2)~(5), until estimation image meets that condition or iterations reach preset upper limit.
The innovative point of the present invention is that structure group is carried out in transform domain using image local openness and non local similitude Rarefaction representation；To strengthen rarefaction representation performance, rarefaction representation is carried out to structure group using orthogonal study dictionary；Then to sparse system Number carries out non-convex minimum norm constraint, further improves the estimated accuracy of sparse coefficient；Threshold value is recycled to shrink to sparse system Number is estimated, and this method is applied to the reconstruct of magnetic resonance image (MRI).
Beneficial effects of the present invention：Rarefaction representation is carried out to structure group using dictionary is learnt, and using singular value decomposition more New dictionary, enhances the adaptability represented image sparse；Sparse coefficient is constrained using non-convex bound term, improves and is Several estimated accuracies, therefore not only overall visual effect is good for the image finally estimated, also retains a large amount of inside image Details, makes whole estimated result closer to actual value.
It is of the invention mainly to be verified that all steps, conclusion are all verified on MATLAB8.0 using the method for emulation experiment Correctly.
Brief description of the drawings
Fig. 1 is the workflow block diagram of the present invention；
Fig. 2 is the MRI people's brain image artwork used in present invention emulation；
Fig. 3 is the reconstruction result to sample rate for 20% MRI people's brain image with RecPF methods；
Fig. 4 is the reconstruction result to sample rate for 20% MRI people's brain image with PANO methods；
Fig. 5 is reconstruction result of the NLR methods to sample rate for 20% MRI people's brain image；
Fig. 6 is the reconstruction result to sample rate for 20% MRI people's brain image with the method for the present invention.
Embodiment
With reference to Fig. 1, the present invention is the MRI image reconstructing method based on study dictionary with the constraint of non-convex norm minimum, is had Body step includes as follows：
Step 1, initial reconstitution is carried out to image, and establishes the corresponding structure group of each target image block.
(1a) inputs width MRI raw k-spaces observation data y, utilizes total variation method or shift-invariant spaces Method carries out initial reconstitution to it, obtains initial reconstructed image x(0)；
Reconstructed image x is by (1b) according to sizeImage block extracted, and to each target image block xi Euclidean distance comparison is carried out with other image blocks in search range；
(1c) takes out and target image block xiM-1 image block of Euclidean distance minimum, and form and tie with target image block Structure groupWherein xi,0=xi。
Step 2, non-convex sparse constraint model is established, orthogonal study dictionary is updated and estimates sparse coefficient, and reconstructed image.
(2a) establishes the non-convex norm minimum restricted model of entire image：
Wherein FuFor fractional-sample Fourier transform matrix, D is orthogonal study dictionary, AiFor structure group XiIn orthogonal study Sparse coefficient under dictionary D, λ and β are regularization parameter, and M is the number of structure group,αkThe kth of expression OK, the value range of p is (0,1), which can be analyzed on optimized variable AiWith the optimization of the optimization problem of D, and x Problem is iterated solution；
(2b) can be broken down into sparse coefficient A for total model in (2a)iWith the optimization problem of orthogonal dictionary D：
With the subproblem on reconstructed image x：
Carry out alternating iteration solution；
(2c) is for AiWith the subproblem of D, it is necessary to be updated to orthogonal study dictionary D, then to sparse coefficient AiEstimated Meter：
(2c1) in the case of given x, on sparse coefficient AiSubproblem with orthogonal dictionary D is：
Wherein I is unit battle array；
(2c2) in (2c1) on AiSubproblem in Section 1According to von Neumann mark not Equation can transform it into：
WhereinAi=UiΔiVi H, expression pairAnd AiProgress singular value decomposition, and equal sign establishment Condition is DUi=FiWith Vi=Gi；
(2c3) is due to UiAnd ViWith the characteristic that unit is orthogonal, i.e.,ViVi H=I, then basis The Jensen ineguality of concave function, in (2c1) | | Ai||p,2Meet：
Wherein τjFor ΔiIn j-th of coefficient, the condition U that is that equal sign is set upiIn every a line and each show and only There is the nonzero element that an absolute value is 1, U may be selectedi=I is as the special case met under the conditions of this；
(2c4) is according in (2c2) and (2c3)With | | Ai||p,2The inequality of satisfaction, can be by (2c1) Subproblem be transformed to：
Equal sign establishment condition is DUi=Fi, Vi=GiAnd Ui=I, therefore D=Fi, therefore orthogonal study dictionary should beIt is unusual Left orthogonal transform matrix after value decomposition.
(2c5) after orthogonal study dictionary D is obtained, on sparse coefficient AiSubproblem can be equivalent to：
Wherein
(2c6) is to solve in (2c5) on optimized variable ΔiOptimization problem, using ΔiIn each coefficient it is relatively only Vertical characteristic, will be to ΔiOptimal Decomposition be to ΔiIn each coefficient optimization：
Wherein σkAnd δkΠ is represented respectivelyiAnd ΔiIn k-th of coefficient；
(2c7) can estimate each coefficient in (2c6), therefore sparse coefficient A using threshold value shrinkage methodiEstimated result For：
WhereinFor ΠiThreshold value shrink as a result, ΠiIn k-th of factor sigmakThreshold value shrink expression formula For：
δ(l)For according to δ(l)=| σk|-p(δ(l-1))p-1/ β is by l iteration as a result, τthFor threshold value, its expression formula is：
(2d) tries to achieve sparse coefficientAfterwards, the subproblem on x is：
The model is least square model, can be solved to obtain with conjugate gradient method：
Step 3, the non-convex sparse constraint model foundation of multiimage Block- matching and the process of solution and reconstructed image, until As a result stopping criterion for iteration is met.
The effect of the present invention can be further illustrated by following emulation experiment：
First, experiment condition and content
Experiment condition：Experiment uses pseudo- radial direction sampling matrix；Experimental image uses true Brain MRI image, such as Fig. 2 institutes Show；Experimental result evaluation index using Y-PSNR PSNR and relative norm error RLNE come objective evaluation reconstruction result, its Middle RLNE is defined as：
WhereinFor reconstruct as a result, x is original image, PSNR values are higher and RNLE values are smaller represents that reconstruction result is more preferable, Closer to true picture.
Experiment content：Under these experimental conditions, reconstruction result uses representative in MRI image reconstruction field at present RecPF methods, PANO methods and NLR methods and the method for the present invention contrasted.
Experiment 1：Image after being sampled respectively to Fig. 2 with the method for the present invention and RecPF methods, PANO methods and NLR methods It is reconstructed.Wherein RecPF methods traditional carry out l using wavelet transformation and total variation method to be a kind of to whole image1 The method of norm constraint sparse constraint, its reconstruction result are Fig. 3；PANO methods are a kind of typically three-dimensional to the progress of structure group small Wave conversion simultaneously uses l1The reconstructing method of norm constraint sparse coefficient, its reconstruction result are Fig. 4；And NLR methods utilize structure group Low-rank characteristic, and using non-convex bound terms of the logdet () as structure group, its reconstruction result is Fig. 5.Present invention side in experiment Method sets tile sizeImage block number m=32 in structure group, maximum iteration T=100, iteration Terminate coefficient η=5 × 10-8；Final reconstruction result is Fig. 6.
Contrast RecPF is can be seen that from the reconstruction result and regional area enlarged drawing of Fig. 3, Fig. 4, Fig. 5 and Fig. 6 each method Method, PANO methods, NLR methods can be seen that the method for the present invention with the method for the present invention and be higher than in the detail section of reconstruction result Other control methods.
The PSNR indexs of the different reconstructing methods of table 1
Image RecPF methods PANO methods NLR methods The method of the present invention
MRI people's mind map 31.40 33.64 35.07 37.27
Table 1 gives the PSNR index situations of each method reconstruction result, and wherein PSNR values are higher represents that quality reconstruction is better； It can be seen that the method for the present invention contrast other methods PSNR values improve a lot, illustrate this method reconstruction result closer to truly Image, this result match with quality reconstruction figure.
The RNLE indexs of the different reconstructing methods of table 2
Image RecPF methods PANO methods NLR methods The method of the present invention
MRI people's mind map 0.1797 0.1373 0.1157 0.0913
Table 2 gives the RNLE index situations of each method reconstruction result, and wherein RNLE values are lower represents reconstruction result details Retain more preferable；It can be seen that the method for the present invention compares other methods, the reconstruction result of the method for the present invention and the error smaller of artwork, This result matches with quality reconstruction figure.
Above-mentioned experiment shows that not only reduction effect is obvious for reconstructing method of the present invention, but also reconstructed image is abundant in content, together When visual effect and objective evaluation index it is all preferable, it can be seen that the present invention to medical image reconstruct be effective.

Claims (2)

1. a kind of MRI image reconstructing method based on study dictionary with the constraint of non-convex norm minimum, comprises the following steps：
(1) width MRI raw k-spaces observation data are inputted, total variation method or translation invariant discrete wavelet are utilized to input data y Converter technique carries out initial reconstitution, obtains the image x after initial reconstitution(0)；
(2) to each target image block x in reconstructed image xiIn its search range using Euclidean distance and other image blocks into Row similarity-rough set, and with m-1 most like image block of target image block a knot will be formed together with target image block Structure groupWhereinMatrix is extracted for image block；
(3) the MRI reconstruction model of non-convex norm minimum constraint is established：
<mrow> <mtable> <mtr> <mtd> <mrow> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mrow> <mi>x</mi> <mo>,</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>D</mi> </mrow> </munder> <mfrac> <mi>&amp;lambda;</mi> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <mi>y</mi> <mo>-</mo> <msub> <mi>F</mi> <mi>u</mi> </msub> <mi>x</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <mi>&amp;beta;</mi> <mn>2</mn> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mi>x</mi> <mo>-</mo> <msub> <mi>DA</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mo>|</mo> <mo>|</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>p</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <msup> <mi>D</mi> <mi>H</mi> </msup> <mi>D</mi> <mo>=</mo> <mi>I</mi> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
Wherein FuFor down-sampling Fourier transform matrix, D is orthogonal study dictionary, AiFor structure group XiUnder orthogonal study dictionary D Sparse coefficient, i.e. Xi=DAi, λ and β are regularization parameter, and M is the number of structure group,αkRepresent system Matrix number AiIn row k, the value range of p is (0,1), andRepresent αkThe quadratic sum of middle all elements is opened after radical sign P power computings are done again；
(4) for the non-convex norm minimum restricted model in (3), can be analyzed on optimized variable AiWith the optimization problem of D, And the optimization problem of optimized variable x is iterated solution：
(4a) in the case of given x, on sparse coefficient AiIt is with optimization problem, that is, sparse coding model of orthogonal dictionary D：
<mrow> <mtable> <mtr> <mtd> <mrow> <munder> <mi>min</mi> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>D</mi> </mrow> </munder> <mfrac> <mi>&amp;beta;</mi> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mi>x</mi> <mo>-</mo> <msub> <mi>DA</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>p</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <msup> <mi>D</mi> <mi>H</mi> </msup> <mi>D</mi> <mo>=</mo> <mi>I</mi> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
Wherein I is unit battle array, and orthogonal dictionary D can learn to obtain by singular value decomposition and inequality characteristic, sparse coefficient AiIt can lead to Cross threshold value and shrink and estimated；
(4b) is by learning to update orthogonal dictionary D and obtaining structure group sparse coefficient estimate AiAfterwards, on optimized variable x's Optimization problem, that is, image reconstruction model is：
<mrow> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>x</mi> </munder> <mfrac> <mi>&amp;lambda;</mi> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <mi>y</mi> <mo>-</mo> <msub> <mi>F</mi> <mi>u</mi> </msub> <mi>x</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <mi>&amp;beta;</mi> <mn>2</mn> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mi>x</mi> <mo>-</mo> <msub> <mi>DA</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>,</mo> </mrow>
The model is least square model, can be solved to obtain x with conjugate gradient method；
(5) repeat step (2)~(5), until estimation image meets that condition or iterations reach preset upper limit.
A kind of 2. MRI image reconstruct side based on study dictionary with the constraint of non-convex norm minimum according to claim 1 Method, it is characterised in that the step of learning with inequality characteristic by singular value decomposition in step (4a) and obtain orthogonal dictionary D be：
(4a1) is understood in (4a) using singular value decomposition and von Neumann Trace inequalitiesMeet
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mi>x</mi> <mo>-</mo> <msub> <mi>DA</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mi>x</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>DA</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <msubsup> <mi>xA</mi> <mi>i</mi> <mi>H</mi> </msubsup> <msup> <mi>D</mi> <mi>H</mi> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mo>&amp;Pi;</mo> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mo>&amp;Pi;</mo> <mi>i</mi> </msub> <msubsup> <mi>G</mi> <mi>i</mi> <mi>H</mi> </msubsup> <msub> <mi>V</mi> <mi>i</mi> </msub> <msubsup> <mi>&amp;Delta;</mi> <mi>i</mi> <mi>H</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>DU</mi> <mi>i</mi> </msub> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <msub> <mi>F</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;GreaterEqual;</mo> <mo>|</mo> <mo>|</mo> <msub> <mo>&amp;Pi;</mo> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mo>&amp;Pi;</mo> <mi>i</mi> </msub> <msubsup> <mi>&amp;Delta;</mi> <mi>i</mi> <mi>H</mi> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mo>&amp;Pi;</mo> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
WhereinAi=UiΔiVi H, the condition that equal sign is set up is DUi=FiAnd Vi=Gi；
(4a2) on the basis of (4a1), due to matrix UiAnd matrix ViWith unit orthogonal property, i.e.,ViVi H=I, using the Jensen ineguality of concave function, | | Ai||p,2Meet：
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>p</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>U</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <msubsup> <mi>V</mi> <mi>i</mi> <mi>H</mi> </msubsup> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>p</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>U</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>p</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo>(</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mo>|</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msup> <mo>|</mo> <mn>2</mn> </msup> <msubsup> <mi>&amp;tau;</mi> <mi>j</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mrow> <mi>p</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;GreaterEqual;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mo>|</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msup> <mo>|</mo> <mn>2</mn> </msup> <msubsup> <mi>&amp;tau;</mi> <mi>j</mi> <mi>p</mi> </msubsup> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mi>&amp;tau;</mi> <mi>j</mi> <mi>p</mi> </msubsup> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mo>|</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mi>&amp;tau;</mi> <mi>j</mi> <mi>p</mi> </msubsup> <mo>=</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>p</mi> <mi>p</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> </mrow>
Wherein τjFor ΔiIn j-th of coefficient, equal sign set up condition be UiIn often row and each column one and only one is absolute It is worth the nonzero element for 1, and Ui=I can be as the special case met under the conditions of this；
(4a3) is according in (4a1) and (4a2)With | | Ai||p,2The inequality of satisfaction, it is known that the son in (4a) is asked Topic meets：
<mrow> <mfrac> <mi>&amp;beta;</mi> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mi>x</mi> <mo>-</mo> <msub> <mi>DA</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>|</mo> <msub> <mo>|</mo> <mrow> <mi>p</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>&amp;GreaterEqual;</mo> <mfrac> <mi>&amp;beta;</mi> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <msub> <mo>&amp;Pi;</mo> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;Delta;</mi> <mi>i</mi> </msub> <mo>|</mo> <msubsup> <mo>|</mo> <mi>p</mi> <mi>p</mi> </msubsup> <mo>,</mo> </mrow>
The condition that wherein equal sign is set up is DUi=Fi, Vi=GiAnd Ui=I, therefore D=Fi, i.e., orthogonal study dictionary D should be Left orthogonal transform matrix after singular value decomposition.
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