CN102629374A - Image super resolution (SR) reconstruction method based on subspace projection and neighborhood embedding - Google Patents

Abstract

The invention discloses an image super resolution (SR) reconstruction method based on subspace projection and neighborhood embedding. The method is characterized by: using first and secondary subspace projection methods to project original high-dimensional data to a low-dimensional space, using dimension reduction feature vectors to show a feature of a low-resolution image block so that global structure information and local structure information of original data can be maintained; comparing a Euclidean distance between the dimension reduction feature vectors in the low-dimensional space, finding a neighborhood block which is most matched with the low-resolution image block to be reconstructed, using a similarity and a scale factor between the feature vectors to construct an accurate embedded weight coefficient so that a searching speed and matching precision can be increased; then constructing the similarity and the scale factor between the feature vectors, calculating the accurate weight coefficient and acquiring more high frequency information from a training database; finally, according to the weight coefficient and the neighborhood block, estimating the high-resolution image block with high precision, reconstructing the image which has the high similarity with a real object, which is good for later-stage real object identification processing.

Description

Image super-resolution rebuilding method based on subspace projection and neighborhood embedding

Technical field

The present invention relates to image process method, relate in particular to a kind of image super-resolution rebuilding method based on subspace projection and neighborhood embedding.

Background technology

Image super-resolution (Super Resolution; SR) reconstruction technique is meant and is utilized in one or more low resolution (the Low Resolution that captures under the different observation angles of Same Scene, different observation time or the different sensors situation; LR) mutual information in the image; Adopt the method for Digital Image Processing to reconstruct panel height resolution (High Resolution, HR) image.This reconstruction technique can (Charge Coupled Device CCD) waits under the condition of hardware device qualification, estimates the high-frequency information that it is lost through the one or more low-resolution image at charge-coupled image sensor.In fields such as remote sensing satellite, military surveillance, medical imaging, security monitoring and traffic administrations, the image super-resolution rebuilding technology not only has important researching value, also is with a wide range of applications.

The essence of image super-resolution rebuilding is how to solve the accurate estimation problem of losing high-frequency information; This is the one-to-many ill-conditioning problem of classics; This theory problem and applied research still are in the exploratory stage, and the aspects such as quality of the time efficiency of image super-resolution rebuilding method and super-resolution rebuilding image all remain further to be improved.Existing image super-resolution rebuilding method mainly is divided into interpolation, reconstruct and study three major types.Method of interpolation is a kind of method of utilizing known neighbour's point value to estimate the interpolation point value, and these class methods are difficult to estimate the high-frequency information of losing.The reconstruct method is to utilize the mutual information between several low-resolution images comprehensively to estimate high-frequency information, and its reconstruction effect is better than method of interpolation.But, the reconstruct method needs the low-resolution image behind several accurate registrations, and the precision of present most image registration algorithms is not very high, can influence the quality of super-resolution rebuilding image.Learning method is the corresponding relation between low-resolution image piece and the high-definition picture piece in the first learning training storehouse, and estimates and wait to rebuild the corresponding high-definition picture piece of low-resolution image piece according to this relation.Learning method can be realized the single image super-resolution rebuilding, does not need image is carried out registration, and can from the training storehouse, get access to more high-frequency information, thereby better rebuild effect.Therefore, learning method is present image super-resolution rebuilding Study on Technology focus.

The image super-resolution rebuilding that embeds based on neighborhood is an important branch in the learning method, obtains good achievement in research in recent years.(document 1:H.Chang such as Chang; D.-Y.Yeung; And Y.Xiong, " Super-resolution through neighbor embedding ", IEEE Conference on Computer Vision Pattern Recognition; Pp.275-282; 2004) at first propose to utilize the thought of neighborhood embedding to realize image super-resolution rebuilding, this method is to have under the prerequisite of similar local geometric features hypothesis at low-resolution image piece and corresponding high-definition picture piece, waits to rebuild the low-resolution image piece to each; Utilize local linear (the Locally Linear Embedding of embedding; LLE) algorithm calculates several and the embedding weight coefficient of waiting to rebuild the similar training low-resolution image piece of low-resolution image piece, and each similar training low-resolution image piece is as a neighborhood piece, and then embeds the high-definition picture piece after weight coefficients and corresponding neighborhood piece come comprehensively to estimate reconstruction with these.But, this method is not considered the influence to the super-resolution rebuilding effect of image block type and neighborhood piece number.Subsequently, (document 2:T.-M.Chan, J.Zhang such as Chan; J.Pu, and H.Huang, " Neighbor embedding based super-resolution algorithm through edge detection and feature selection "; Pattern Recognition Letters, vol.30, pp.494-502; 2009) propose a kind of neighborhood and embed algorithm based on rim detection and Feature Selection; This method can confirm to choose the number of neighborhood piece according to the type (edge and non-edge) of image block, yet, when there is serious blurring effect in input low-resolution image to be rebuild; This algorithm can be difficult to judge the type of image block, makes the super-resolution rebuilding poor effect.(document 3:K.Zhang, X.Gao, X.Li such as Zhang; And D.Tao, " Partially supervised neighbor embedding for example-based image super-resolution ", IEEE Journal of Selected Topics in Signal Processing; Vol.5; No.2, pp.230-239,2011) utilize class label information to construct a semi-supervised distance function; With the neighborhood piece of waiting to rebuild low-resolution image piece coupling, the super-resolution rebuilding effect of this method depends on the accuracy of identification of sorter to a certain extent more than from the training storehouse, selecting more according to this semi-supervised distance function.Algorithm described in the document 1～3 all is on original high-dimensional feature space, to seek several neighborhood pieces; Because the dimension of original feature vector is very big; Only rely on certain distance function to seek the neighborhood piece, neighborhood piece that searches out and the similarity of waiting to rebuild between the low-resolution image piece are not high.The characteristic dimensionality reduction is a kind of feasible method that addresses the above problem; Based on this thinking, (document 4:X.Gao, K.Zhang such as Gao; D.Tao; And X.Li, " Joint learning for single image super-resolution via coupled constraint ", IEEE Transactions on Image Processing; 2011) proposition is low with training with the combination learning method, high-resolution features is vectorial projects on the low dimensional feature subspace simultaneously, and then on this low dimensional feature subspace, seeks out several and the neighborhood piece of waiting to rebuild low-resolution image piece coupling.Though document 4 has been used the method for characteristic dimensionality reduction; But when original low, high-resolution features is vectorial when projecting on the unified low dimensional feature subspace simultaneously; Because the dimension of original high resolution proper vector is bigger than original low resolution proper vector; The useful information in the original high resolution proper vector can be damaged inevitably, also harmful effect can be produced the neighborhood piece of seeking coupling.In addition; The algorithm of document 1～4 all is to adopt the local linear algorithm that embeds when calculating the embedding weight coefficient; The embedding weight coefficient that calculates might be negative value; This can influence the neighbour and train the high-definition picture piece that the degree of high-frequency information can be provided, and the computation complexity of this algorithm is high, and arithmetic speed is slow.

Summary of the invention

The object of the present invention is to provide a kind of image super-resolution rebuilding method based on subspace projection and neighborhood embedding; This method realizes more effective characteristic dimensionality reduction; Estimate more accurately and rebuild back high-definition picture piece; Similarity between reconstructed image and the real-world object is higher, the super-resolution rebuilding better effects if.

The present invention solves its technical matters, and the technical scheme that is adopted is:

A, training:

L width of cloth resolution is identical, that size is identical high-definition picture is as the training high-definition picture

L=1,2 ..., L, L=3～80; To every width of cloth training high-definition picture

Carry out obtaining N behind the overlapping piecemeal ₁Individual size is the training high-definition picture piece of z*z, N ₁=1000～7000, z=6,9,12,15, obtain N=L*N altogether ₁Individual training high-definition picture piece; Extract behind standardization brightness of each training high-definition picture piece as a training high resolving power standardization brightness image block, convert i training high resolving power standardization brightness image block into i by the order that be listed as and train high-resolution features vectorial

I=1,2 ..., N, each training high-resolution features vector

Dimension be d ₁=z ², all training high-resolution features vectors

(1≤i≤N) form to train the high-resolution features matrix

(1≤l≤L) width of cloth is trained high-definition picture to l

Do a times of down-sampling and handle, a=2,3,4,5, obtain corresponding l width of cloth training low-resolution image

Again to every width of cloth training low-resolution image

Do and obtain corresponding training interpolation image after a times of up-sampling handled

Extract every width of cloth training interpolation image The vertical and second order level of single order level, single order, second order VG (vertical gradient) characteristic, simultaneously this four width of cloth gradient characteristic image is carried out overlapping piecemeal, every width of cloth training interpolation image

Obtain 4*N behind the overlapping piecemeal ₁Individual size is the training low resolution gradient characteristic image piece of z*z; Obtain 4*N training low resolution gradient characteristic image piece altogether; Per four training low resolution gradient characteristic image pieces are as a training low resolution gradient characteristic image piece group, convert i training low resolution gradient characteristic image piece group into i by the order that is listed as and train the low resolution proper vector Each training low resolution proper vector

Dimension be d ₂=4*z ², all training low resolution proper vectors

(1≤i≤N) form to train the low resolution eigenmatrix $X^s=[x_1^s,x_2^s,\·\·\·,x_N^s];$

B, pre-service:

Input waits to rebuild low-resolution image R ^d, its resolution and all training low-resolution images

(resolution of 1≤l≤L) is identical, will wait to rebuild low-resolution image R ^dCarry out obtaining waiting to rebuild interpolation image R after a times of up-sampling handled ^c, extract and wait to rebuild interpolation image R ^cThe vertical and second order level of single order level, single order, second order VG (vertical gradient) characteristic, simultaneously this four width of cloth gradient characteristic image is carried out obtaining 4*N altogether behind the overlapping piecemeal ₂Individual size be z*z wait to rebuild low resolution gradient characteristic image piece, N ₂=1000～7000; Wait to rebuild low resolution gradient characteristic image piece and wait to rebuild low resolution gradient characteristic image piece group for per four, wait to rebuild low resolution gradient characteristic image piece group with j and convert by the order that be listed as that j is individual to wait to rebuild the low resolution proper vector into as one

J=1,2 ..., N ₂, each waits to rebuild the low resolution proper vector

Dimension and all training low resolution proper vector

(dimension of 1≤i≤N) is identical, is d ₂=4*z ², institute remains to be rebuild the low resolution proper vector (1≤j≤N ₂) form one and wait to rebuild the low resolution eigenmatrix $X^t=[x_1^t,x_2^t,\·\·\·,x_{N_2}^t];$

C, super-resolution rebuilding:

C1, subcharacter matrix generate: each that obtains among the step B waited to rebuild the low resolution proper vector (1≤j≤N ₂), the training low resolution eigenmatrix that obtains in steps A successively

In seek out with j and wait to rebuild the low resolution proper vector

Between the minimum n-1=100～300 training low resolution proper vector of Euclidean distance, wait to rebuild the low resolution proper vector with j

As the subcharacter vector x _1j, and with the n-1 that searches out a training low resolution proper vector by with j wait to rebuild the low resolution proper vector

Between Euclidean distance from small to large successively as the subcharacter vector x _2j, x _3j..., x _Nj, form j sub-eigenmatrix thus

C2, dimensionality reduction eigenmatrix generate: adopt I and II subspace projection method, with j sub-eigenmatrix

Carry out the characteristic dimensionality reduction, each subcharacter vector x _{I ' j}(1≤i '≤n) project on the low n-dimensional subspace n, obtain n dimensionality reduction proper vector altogether

Each dimensionality reduction proper vector

Dimension be m＜d ₂Each subcharacter vector x _{I ' j}With corresponding dimensionality reduction proper vector Form mapping relations one by one, behind the I and II subspace projection, dimension is d ₂The subcharacter vector x _{I ' j}Converting dimension to is the dimensionality reduction proper vector of m

Realize the characteristic dimensionality reduction, all dimensionality reduction proper vectors (1≤i '≤n) form j dimensionality reduction eigenmatrix $X_j^m={\left\{x_{i^{\′}j}^m\right\}}_{i^{\′}=1}^n;$

C3, weight coefficient calculate: the dimensionality reduction proper vector in that step C2 obtains is gathered

In seek out and the dimensionality reduction proper vector

Between k minimum dimensionality reduction proper vector of Euclidean distance, k=5～10 are according to step C1 neutron proper vector x _{I ' j}With dimensionality reduction proper vector among the step C2

Between mapping relations one by one, and the call number of the k that searches out a dimensionality reduction proper vector, low in the training that steps A obtains successively, high-resolution features matrix X ^sAnd Y ^sIn seek out that corresponding k is low to training, high-resolution features is vectorial, and, high-resolution features low to training by this k is vectorial forms j low, the high-resolution features matrix of neighbour respectively

With

According to j neighbour's low resolution eigenmatrix

Wait to rebuild the low resolution proper vector with j

Calculate j similarity group

With j scale factor group And associating

With

Calculate j normalization and embed the weight coefficient group

C4, reconstruction high-definition picture piece: the linear weighted function form that embeds weight coefficient group

and j neighbour's high-resolution features matrix

according to j normalization; Estimate j rebuild back high-resolution features vector

again with j brightness average

addition of waiting to rebuild the low-resolution image piece; And convert the image block form into, can draw j and rebuild back high-definition picture piece;

Repeat above step C1～step C4, obtain N altogether ₂Individual size is a high-definition picture piece after the reconstruction of z*z, obtains the preliminary image H of super-resolution rebuilding behind the lap splice ₀

D, aftertreatment:

Adopt the preliminary image H of iteration back-projection algorithms to super-resolution rebuilding ₀Carry out Q iterative computation and handle, Q=10～30 obtain the final image H of super-resolution rebuilding ^*

Compared with prior art, the invention has the beneficial effects as follows:

One, the present invention adopts I and II subspace projection method that the subcharacter matrix is projected on the lower dimensional space; Obtain corresponding dimensionality reduction eigenmatrix; Reduce the dimension of original low resolution proper vector to greatest extent, and more effectively represent the characteristic of low-resolution image piece.Therefore, the time and the space complexity of the inventive method reduce, and arithmetic speed improves.

Two, the direct Euclidean distance between the dimensionality reduction proper vector relatively at lower dimensional space but not on the original higher dimensional space of the present invention; Owing to comprised the overall situation and the local structural information of raw data in the dimensionality reduction eigenmatrix, can seek out and wait to rebuild the neighborhood piece that the low-resolution image piece matees the most more efficiently according to the dimensionality reduction proper vector.It is thus clear that the present invention has higher matching precision to seeking the neighborhood piece.

Three, the present invention utilizes the similarity between weight coefficient and two proper vectors that approximate proportional relation is arranged; Construct the similarity waiting to rebuild between low resolution proper vector and neighbour's low resolution proper vector and corresponding scale factor; Calculate more accurate weight coefficient, help estimating the high-definition picture piece.It is thus clear that the present invention has higher estimated accuracy and stronger adaptability.

In a word; The inventive method adopts I and II subspace projection method that original high dimensional data is projected on the lower dimensional space; Make the character representation of low-resolution image piece become more succinct, efficient, can search out the more neighborhood piece of coupling, and construct more accurate weight coefficient through similarity between proper vector and scale factor; Rebuild back high-definition picture piece thereby estimate more accurately; The super-resolution rebuilding better effects if, the similarity of reconstructed image and real-world object is higher, helps later stage real-world object identification and handles.

Adopt I and II subspace projection method among the above-mentioned step C2, the specific practice of j sub-eigenmatrix

being carried out obtaining behind the characteristic dimensionality reduction j dimensionality reduction eigenmatrix is:

The one-level subspace projection:

Calculate j j the radially basic kernel function matrix that sub-eigenmatrix

is corresponding $K_j={\left\{K_{i^{\′}{j}^{\′}j}\right\}}_{i^{\′}{,j}^{\′}=1}^n={\left\{\exp (-{\left|\right|x_{i^{\′}j}-x_{j^{\′}j}\left|\right|}^2/{\mathrm{\β}}_j)\right\}}_{i^{\′},j^{\′}=1}^n,$ Wherein the individual radially parameter of basic kernel function of j does

Then to j radially basic kernel function matrix K _jCarry out obtaining j centralization kernel function matrix after centralization is handled

G=I-1/n in the formula, I are unit matrixs that size is n*n, and 1 is that a size is n*n, and all values is 1 matrix; Find the solution j centralization kernel function matrix

Descending characteristic value decomposition equation

I '=1,2,3 ..., n, j=1,2,3 ..., N ₂, λ in the formula _{I ' j}For by the individual eigenwert of i ' after the descending sort, α _{I ' j}It is the individual eigenvalue of i ' _{I ' j}The corresponding individual proper vector of i '; Again to the individual proper vector α of i ' _{I ' j}Carry out obtaining the orthogonalization proper vector after the orthogonalization process

R eigenvalue of maximum λ before choosing _{I ' j}Corresponding orthogonalization proper vector

R=50～n, the r that selects orthogonalization proper vector

Form j one-level subspace projection matrix

R eigenvalue of maximum before described

Sum accounts for all eigenwerts

More than 99% of sum; By formula

Draw j one-level mapping matrix T is the matrix transpose computing, all one-level mapping vectors (1≤i '≤n) dimension is r;

The secondary subspace projection:

Calculate j corresponding nuclear distance function matrix of j one-level mapping matrix

$D_j^r={\left\{D_{i^{\′}{j}^{\′}j}^r\right\}}_{i^{\′},j^{\′}=1}^n={\left\{\sqrt{2-2*\mathrm{Exp}(-||x_{i^{\′}j}^r-x_{j^{\′}j}^r{\left|\right|}^2/{\mathrm{\γ}}_j)}\right\}}_{i^{\′},j^{\′}=1}^n,$ Wherein the parameter of j nuclear distance function does

Construct j adjacency matrix then

As i ' ∈ Λ _{J ' j}Or j ' ∈ Λ _{I ' j}The time,

Otherwise, S _{I ' j ' j}=0, Λ wherein _{J ' j}And Λ _{I ' j}Be expressed as respectively The neighborhood tally set, the element number of neighborhood tally set is b=5～10; Calculate j adjacency matrix S _jJ corresponding Laplce's matrix F _j=diag (S _j* 1 _n)-S _j, wherein 1 _nBeing expressed as a size is n*1, and all values is 1 column vector, and diag () is expressed as the diagonalization of matrix computing; Then calculate j symmetric matrix respectively $U_j=X_j^r*F_j*F_j*{\left(X_j^r\right)}^T$ With j diagonal matrix $V_j=X_j^r*\left\{\mathrm{Diag}(S_j*1_n)\right\}*{\left(X_j^r\right)}^T,$ J symmetric matrix U _jWith j diagonal matrix V _jAll be that size is the matrix of r*r, unite to solve U _jAnd V _jBroad sense ascending order characteristic value decomposition equation

In the formula

Be the individual eigenwert of i ' after arranging by ascending order, p _{I ' j}It is the individual eigenwert of i ' The corresponding individual proper vector of i '; M minimal eigenvalue before choosing

Characteristic of correspondence vector p _{I ' j}, m=10～r, the m that a selects proper vector

Form j secondary subspace projection matrix P _j=[p _1j, p _2j..., p _Mj], by formula

Can draw j dimensionality reduction eigenmatrix All dimensionality reduction proper vectors

(1≤i '≤n) dimension is m.

After adopting above one-level subspace projection method, comprise the global structure information of original high dimensional data in the one-level mapping matrix, kept more partial structurtes information in the dimensionality reduction eigenmatrix that behind above secondary subspace projection, obtains.Adopt I and II subspace projection method; The dimensionality reduction eigenmatrix has not only kept the global structure information of raw data but also has kept partial structurtes information; Realize the characteristic dimensionality reduction, can more effectively express the characteristic of low-resolution image piece, thereby make super-resolution rebuilding become succinct and efficient.

Step C3 joint above the j-th similarity group

and j scale factor group calculate the j th normalized weighting coefficients embedded

the specific approach is:

Calculate j neighbour's low resolution eigenmatrix that step C3 obtains respectively

In k training low resolution proper vector

(1≤i "≤k) wait to rebuild the low resolution proper vector with j

Between similarity

With corresponding scale factor

Wherein

With Be expressed as training low resolution proper vector respectively

Wait to rebuild the low resolution proper vector with j

In u brightness value, ε is 1*10 ^-4～1*10 ^-2Positive number; The k that calculates is to similarity

With scale factor c _{I " j}Form j similarity group respectively With j scale factor group

Unite j similarity group

With j scale factor group

Calculate j and embed the weight coefficient group

And to j embedding weight coefficient group Do and obtain j normalization embedding weight coefficient group after normalization is handled ${\left\{{\hat{w}}_{i^{\′\′}j}\right\}}_{i^{\′\′}=1}^k={\{w_{i^{\′\′}j}/{\mathrm{\Σ}}_{i^{\′\′}j}^k{w}_{i^{\′\′}j}\}}_{i^{\′\′}=1}^k.$

Construct through above method and to wait to rebuild low resolution proper vector and similarity and the scale factor between the training low resolution proper vector of coupling the most; Calculate more accurate weight coefficient with them; Satisfy the requirement that weight coefficient and similarity become to be similar to direct ratio; Can from the training storehouse, get access to more high-frequency information, help improving the estimated accuracy of rebuilding back high-definition picture piece, thereby obtain better super-resolution rebuilding effect.

Come the present invention is done further detailed description below in conjunction with accompanying drawing and embodiment.

Description of drawings

Fig. 1 (a)～Fig. 1 (f) is employed training image and a test pattern in the embodiment of the invention.

Fig. 2 (A)～Fig. 2 (E) realizes the The simulation experiment result of image super-resolution rebuilding for adopting algorithms of different.

In Fig. 2 (A)～Fig. 2 (E), the intermediate rectangular block diagram is the regional area on the reconstructed image, and the corner block diagram amplifies 2 times design sketch for this regional area, and Fig. 2 (A) is for adopting the figure as a result of NESR algorithm (document 1); Fig. 2 (B) is for adopting the figure as a result of NeedFS algorithm (document 2); Fig. 2 (C) is for adopting the figure as a result of CSNE algorithm (document 3); Fig. 2 (D) is for adopting the figure as a result of JLNE algorithm (document 4); Fig. 2 (E) is the figure as a result of the inventive method.

Fig. 3 carries out image block root-mean-square error value E behind the super-resolution rebuilding to (a)～(f) among Fig. 1 six width of cloth images respectively for existing four kinds of methods and the inventive method _pHistogram.

Fig. 4 carries out image block structural similarity value S behind the super-resolution rebuilding to (a)～(f) among Fig. 1 six width of cloth images respectively for existing four kinds of methods and the inventive method _pHistogram.

Embodiment

Embodiment

A kind of image super-resolution rebuilding method based on subspace projection and neighborhood embedding may further comprise the steps:

A, training:

L width of cloth resolution is identical, that size is identical high-definition picture is as the training high-definition picture

L=1,2 ..., L, L=3～80; To every width of cloth training high-definition picture

Carry out obtaining N behind the overlapping piecemeal ₁Individual size is the training high-definition picture piece of z*z, N ₁=1000～7000, z=6,9,12,15, obtain N=L*N altogether ₁Individual training high-definition picture piece; Extract behind standardization brightness of each training high-definition picture piece as a training high resolving power standardization brightness image block, convert i training high resolving power standardization brightness image block into i by the order that be listed as and train high-resolution features vectorial

I=1,2 ..., N, each training high-resolution features vector

Dimension be d ₁=z ², all training high-resolution features vectors

(1≤i≤n) form to train the high-resolution features matrix

(1≤l≤L) width of cloth is trained high-definition picture to l

Do a times of down-sampling and handle, a=2,3,4,5, obtain corresponding l width of cloth training low-resolution image

Again to every width of cloth training low-resolution image Do and obtain corresponding training interpolation image after a times of up-sampling handled

Extract every width of cloth training interpolation image

The vertical and second order level of single order level, single order, second order VG (vertical gradient) characteristic, simultaneously this four width of cloth gradient characteristic image is carried out overlapping piecemeal, every width of cloth training interpolation image

Obtain 4*N behind the overlapping piecemeal ₁Individual size is the training low resolution gradient characteristic image piece of z*z; Obtain 4*N training low resolution gradient characteristic image piece altogether; Per four training low resolution gradient characteristic image pieces are as a training low resolution gradient characteristic image piece group, convert i training low resolution gradient characteristic image piece group into i by the order that is listed as and train the low resolution proper vector Each training low resolution proper vector

Dimension be d ₂=4*z ², all training low resolution proper vectors

(1≤i≤N) form to train the low resolution eigenmatrix $X^s=[x_1^s,x_2^s,\·\·\·,x_N^s];$

B, pre-service:

Input waits to rebuild low-resolution image R ^d, its resolution and all training low-resolution images

(resolution of 1≤l≤L) is identical, will wait to rebuild low-resolution image R ^dCarry out obtaining waiting to rebuild interpolation image R after a times of up-sampling handled ^c, extract and wait to rebuild interpolation image R ^cThe vertical and second order level of single order level, single order, second order VG (vertical gradient) characteristic, simultaneously this four width of cloth gradient characteristic image is carried out obtaining 4*N altogether behind the overlapping piecemeal ₂Individual size be z*z wait to rebuild low resolution gradient characteristic image piece, N ₂=1000～7000; Wait to rebuild low resolution gradient characteristic image piece and wait to rebuild low resolution gradient characteristic image piece group for per four, wait to rebuild low resolution gradient characteristic image piece group with j and convert by the order that be listed as that j is individual to wait to rebuild the low resolution proper vector into as one

J=1,2 ..., N ₂, each waits to rebuild the low resolution proper vector

Dimension and all training low resolution proper vector

(dimension of 1≤i≤N) is identical, is d ₂=4*z ², institute remains to be rebuild the low resolution proper vector

(1≤j≤N ₂) form one and wait to rebuild the low resolution eigenmatrix $X^t=[x_1^t,x_2^t,\·\·\·,x_{N_2}^t];$

C, super-resolution rebuilding:

C1, subcharacter matrix generate: each that obtains among the step B waited to rebuild the low resolution proper vector

(1≤j≤N ₂), the training low resolution eigenmatrix that obtains in steps A successively

In seek out with j and wait to rebuild the low resolution proper vector

Between the minimum n-1=100～300 training low resolution proper vector of Euclidean distance, wait to rebuild the low resolution proper vector with j

As the subcharacter vector x _1j, and with the n-1 that searches out a training low resolution proper vector by with j wait to rebuild the low resolution proper vector

Between Euclidean distance from small to large successively as the subcharacter vector x _2j, x _3j..., x _Nj, form j sub-eigenmatrix thus

C2, dimensionality reduction eigenmatrix generate: adopt I and II subspace projection method, with j sub-eigenmatrix

Carry out the characteristic dimensionality reduction, each subcharacter vector x _{I ' j}(1≤i '≤n) project on the low n-dimensional subspace n, obtain n dimensionality reduction proper vector altogether

Each dimensionality reduction proper vector

Dimension be m＜d ₂Each subcharacter vector x _{I ' j}With corresponding dimensionality reduction proper vector Form mapping relations one by one, behind the I and II subspace projection, dimension is d ₂The subcharacter vector x _{I ' j}Converting dimension to is the dimensionality reduction proper vector of m

Realize the characteristic dimensionality reduction, all dimensionality reduction proper vectors

(1≤i '≤n) form j dimensionality reduction eigenmatrix $X_j^m={\left\{x_{i^{\′}j}^m\right\}}_{i^{\′}=1}^n;$

More than adopt I and II subspace projection method, the specific practice of j sub-eigenmatrix

being carried out obtaining behind the characteristic dimensionality reduction j dimensionality reduction eigenmatrix is:

The one-level subspace projection:

Calculate j j the radially basic kernel function matrix that sub-eigenmatrix

is corresponding $K_j={\left\{K_{i^{\′}{j}^{\′}j}\right\}}_{i^{\′},j^{\′}=1}^n={\left\{\exp ({-\left|\right|x_{i^{\′}j}-x_{j^{\′}j}\left|\right|}^2/{\mathrm{\β}}_j)\right\}}_{i^{\′},j^{\′}=1}^n,$ Wherein the individual radially parameter of basic kernel function of j does

Then to j radially basic kernel function matrix K _jCarry out obtaining j centralization kernel function matrix after centralization is handled

G=I-1/n in the formula, I are unit matrixs that size is n*n, and 1 is that a size is n*n, and all values is 1 matrix; Find the solution j centralization kernel function matrix Descending characteristic value decomposition equation

I '=1,2,3 ..., n, j=1,2,3 ..., N ₂, λ in the formula _{I ' j}For by the individual eigenwert of i ' after the descending sort, α _{I ' j}It is the individual eigenvalue of i ' _{I ' j}The corresponding individual proper vector of i '; Again to the individual proper vector α of i ' _{I ' j}Carry out obtaining the orthogonalization proper vector after the orthogonalization process

R eigenvalue of maximum λ before choosing _{I ' j}Corresponding orthogonalization proper vector

R=50～n, the r that selects orthogonalization proper vector

Form j one-level subspace projection matrix

R eigenvalue of maximum before described

Sum accounts for all eigenwerts

More than 99% of sum; By formula

Draw j one-level mapping matrix

T is the matrix transpose computing, all one-level mapping vectors

(1≤i '≤n) dimension is r;

The secondary subspace projection:

Calculate j corresponding nuclear distance function matrix of j one-level mapping matrix

$D_j^r={\left\{D_{i^{\′}{j}^{\′}j}^r\right\}}_{i^{\′},j^{\′}=1}^n={\left\{\sqrt{2-2*\mathrm{Exp}({-\left|\right|x_{i^{\′}j}^r-x_{j^{\′}j}^r\left|\right|}^2/{\mathrm{\γ}}_j)}\right\}}_{i^{\′},j^{\′}=1}^n,$ Wherein the parameter of j nuclear distance function does

Construct j adjacency matrix then

As i ' ∈ Λ _{J ' j}Or j ' ∈ Λ _{I ' j}The time, Otherwise, S _{I ' j ' j}=0, Λ wherein _{J ' j}And Λ _{I ' j}Be expressed as respectively

The neighborhood tally set, the element number of neighborhood tally set is b=5～10; Calculate j adjacency matrix S _jJ corresponding Laplce's matrix F _j=diag (S _j* 1 _n)-S _j, wherein 1 _nBeing expressed as a size is n*1, and all values is 1 column vector, and diag () is expressed as the diagonalization of matrix computing; Then calculate j symmetric matrix respectively $U_j=X_j^r*F_j*{\left(X_j^r\right)}^T$ With j diagonal matrix $V_j=X_j^r*\left\{\mathrm{Diag}(S_j*1_n)\right\}*{\left(X_j^r\right)}^T,$ J symmetric matrix U _jWith j diagonal matrix V _jAll be that size is the matrix of r*r, unite to solve U _jAnd V _jBroad sense ascending order characteristic value decomposition equation

In the formula

Be the individual eigenwert of i ' after arranging by ascending order, p _{I ' j}It is the individual eigenwert of i '

The corresponding individual proper vector of i '; M minimal eigenvalue before choosing

Characteristic of correspondence vector p _{I ' j}, m=10～r, the m that a selects proper vector

Form j secondary subspace projection matrix P _j=[p _1j, p _2j..., p _Mj], by formula Can draw j dimensionality reduction eigenmatrix

All dimensionality reduction proper vectors

(1≤i '≤n) dimension is m;

C3, weight coefficient calculate: the dimensionality reduction proper vector in that step C2 obtains is gathered

In seek out and the dimensionality reduction proper vector

Between k minimum dimensionality reduction proper vector of Euclidean distance, k=5～10 are according to step C1 neutron proper vector x _{I ' j}With dimensionality reduction proper vector among the step C2

Between mapping relations one by one, and the call number of the k that searches out a dimensionality reduction proper vector, low in the training that steps A obtains successively, high-resolution features matrix X ^sAnd Y ^sIn seek out that corresponding k is low to training, high-resolution features is vectorial, and, high-resolution features low to training by this k is vectorial forms j low, the high-resolution features matrix of neighbour respectively

With

According to j neighbour's low resolution eigenmatrix Wait to rebuild the low resolution proper vector with j

Calculate j similarity group

With j scale factor group

And associating

With

Calculate j normalization and embed the weight coefficient group

Joint above the j-th similarity group

and j scale factor group calculate the j th normalized weighting coefficients embedded

The specific approach is:

Calculate j neighbour's low resolution eigenmatrix that step C3 obtains respectively

In k training low resolution proper vector (1≤i " waits to rebuild the low resolution proper vector for≤k and j

Between similarity

With corresponding scale factor

Wherein With

Be expressed as training low resolution proper vector respectively

Wait to rebuild the low resolution proper vector with j

In u brightness value, ε is 1*10 ^-4～1*10 ^-2Positive number; The k that calculates is to similarity

With scale factor c _{I " j}Form j similarity group respectively

With j scale factor group

Unite j similarity group

With j scale factor group

Calculate j and embed the weight coefficient group And to j embedding weight coefficient group

Do and obtain j normalization embedding weight coefficient group after normalization is handled ${\left\{{\hat{w}}_{i^{\′\′}j}\right\}}_{i^{\′\′}=1}^k={\{w_{i^{\′\′}j}/{\mathrm{\Σ}}_{i^{\′\′}=1}^k{w}_{i^{\′\′}j}\}}_{i^{\′\′}=1}^k;$

C4, reconstruction high-definition picture piece: the linear weighted function form that embeds weight coefficient group

and j neighbour's high-resolution features matrix

according to j normalization; Estimate j rebuild back high-resolution features vector

again with j brightness average

addition of waiting to rebuild the low-resolution image piece; And convert the image block form into, can draw j and rebuild back high-definition picture piece;

Repeat above step C1～step C4, obtain N altogether ₂Individual size is a high-definition picture piece after the reconstruction of z*z, obtains the preliminary image H of super-resolution rebuilding behind the lap splice ₀

D, aftertreatment:

Adopt the preliminary image H of iteration back-projection algorithms to super-resolution rebuilding ₀Carry out Q iterative computation and handle, Q=10～30 obtain the final image H* of super-resolution rebuilding.

Emulation experiment:

The condition of emulation experiment with concrete parameter is:

The natural image of in Fig. 1, choosing the big or small 384*510 of being of the L=5 width of cloth is as the training high-definition picture, and a remaining width of cloth has been done 6 times emulation experiment successively by turns altogether as the test reference image.

The sampling multiple a=3 of training high-definition picture, the image block size is z*z=9*9, obtains N=L*N behind the doubling of the image piecemeal altogether ₁=5*5440=27200 image block, the dimension of training high-resolution features vector is d ₁=z ²=9 ²=81, the dimension of training low resolution proper vector is d ₂=4*z ²=4*9 ²=324.

As low-resolution image to be rebuild, size is 128 * 170 with the image of test reference image behind 3 times of down-samplings.The interpolation image that low-resolution image to be rebuild is carried out obtaining behind 3 times of up-samplings carries out obtaining N behind the overlapping piecemeal ₂Wait to rebuild the low-resolution image piece for=5440, the dimension of waiting to rebuild the low resolution proper vector is d ₂=324.All subcharacter matrix X _jVectorial number in (1≤j≤5440) is n=101, and the dimension of one-level mapping vector is r=100, in secondary subspace projection process; The element number of neighborhood tally set is b=5; The dimension of dimensionality reduction proper vector is m=22, and the number of the dimensionality reduction proper vector of searching is k=9, ε=1*10 ^-3, the employed iterations of aftertreatment is Q=20.

Adopt existing four kinds of methods simultaneously; Be respectively NESR algorithm (document 1), NeedFS algorithm (document 2), CSNE algorithm (document 3) and JLNE algorithm (document 4) and the inventive method (a)～(f) among Fig. 1 six width of cloth images are similarly being carried out super-resolution emulation reconstruction under the emulation experiment condition.

The result is as shown in Figure 2 for the subjective vision effect comparison, Fig. 2 only provide (b) among Fig. 1, (c) and (d) three width of cloth images carry out the result of emulation experiment.Fig. 2 (A)～Fig. 2 (D) is respectively the The simulation experiment result of document 1～4, and Fig. 2 (E) is the The simulation experiment result of the inventive method.In Fig. 2 (A)～Fig. 2 (E), the intermediate rectangular block diagram is the regional area on the reconstructed image, and the corner block diagram amplifies 2 times design sketch for this regional area.Relatively partial enlarged drawing can be found out: fuzzy and texture aliasing can appear in the reconstructed image of NESR, NeedFS and CSNE algorithm; The texture edge is level and smooth inadequately in the reconstructed image of JLNE algorithm; Detailed information in the reconstructed image of the inventive method is more clear, and the edge is more level and smooth.It is thus clear that the inventive method is better than existing four kinds of methods on the subjective vision effect.

In order more accurately the whole bag of tricks to be carried out objective evaluation, below with image block root-mean-square error value E _pWith image block structural similarity value S _pAs estimating the good and bad objective indicator of the whole bag of tricks, wherein E _pAnd S _pComputing formula following:

$E_p=\frac{1}{N_2}\underset{j=1}{\overset{N_2}{\mathrm{\Σ}}}\sqrt{\frac{1}{d_1}\underset{v=1}{\overset{d_1}{\mathrm{\Σ}}}{(y_{\mathrm{jv}}-y_{\mathrm{jv}}^t)}^2}$

$S_p=\frac{1}{N_2}\underset{j=1}{\overset{N_2}{\mathrm{\Σ}}}\frac{({2\mathrm{\μ}}_{1j}{\mathrm{\μ}}_{2j}+{\mathrm{\ϵ}}_1)({2\mathrm{\σ}}_{12j}+{\mathrm{\ϵ}}_2)}{({\mathrm{\μ}}_{1j}^2+{\mathrm{\μ}}_{2j}^2+{\mathrm{\ϵ}}_1)({\mathrm{\σ}}_{1j}^2+{\mathrm{\σ}}_{2j}^2+{\mathrm{\ϵ}}_2)}$

In the same form, y _JvBe v brightness value of j image block on the original high resolution image,

Be v brightness value of j image block on the super-resolution rebuilding image, d ₁Be the number of the contained brightness value of each image block, identical with the dimension of training high-resolution features vector, be d ₁=81, N ₂For rebuilding back high-definition picture piece number.

In two formulas, μ _1jAnd μ _2jBe respectively the brightness average of j image block on original high resolution image and the super-resolution rebuilding image, σ _1jAnd σ _2jBe respectively the standard deviation of j image block on original high resolution image and the super-resolution rebuilding image, σ _12jBe the standard covariance of j image block on j image block on the original high resolution image and the super-resolution rebuilding image, ε ₁=6.5, ε ₂=58.5.

Fig. 3 carries out image block root-mean-square error value E behind the super-resolution rebuilding to (a)～(f) among Fig. 1 six width of cloth images respectively for existing four kinds of methods and the inventive method _pHistogram, can find out the image block root-mean-square error value E of 6 width of cloth reconstructed images of the inventive method from this figure _pAll be lower than existing four kinds of methods, the difference minimum of the reconstructed image and the original high resolution image of the inventive method is described, the reconstruction effect is best.

Fig. 4 carries out image block structural similarity value S behind the super-resolution rebuilding to (a)～(f) among Fig. 1 six width of cloth images respectively for existing four kinds of methods and the inventive method _pHistogram, can find out the image block structural similarity value S of 6 width of cloth reconstructed images of the inventive method from this figure _pAll be higher than existing four kinds of methods, explained that also the inventive method can reconstruct more high-frequency information, more approaches the original high resolution image.

Above The simulation experiment result shows, subjective vision effect and objective evaluation index this aspect two on, the inventive method all is better than existing four kinds of methods, in the application of image super-resolution rebuilding, has feasibility and applicability.

Claims (3)

1. image super-resolution rebuilding method that embeds based on subspace projection and neighborhood may further comprise the steps:

A, training:

L width of cloth resolution is identical, that size is identical high-definition picture is as the training high-definition picture

L=1,2 ..., L, L=3～80; To every width of cloth training high-definition picture

Carry out obtaining N behind the overlapping piecemeal ₁Individual size is the training high-definition picture piece of z*z, N ₁=1000～7000, z=6,9,12,15, obtain N=L*N altogether ₁Individual training high-definition picture piece; The standardization brightness that extracts each training high-definition picture piece is as a training high resolving power standardization brightness image block, converts i training high resolving power standardization brightness image block into i by the order that be listed as and trains high-resolution features vectorial

I=1,2 ..., N, each training high-resolution features vector Dimension be d ₁=z ², all training high-resolution features vectors (1≤i≤N) form to train the high-resolution features matrix

(1≤l≤L) width of cloth is trained high-definition picture to l Do a times of down-sampling and handle, a=2,3,4,5, obtain corresponding l width of cloth training low-resolution image

Again to every width of cloth training low-resolution image

Do and obtain corresponding training interpolation image after a times of up-sampling handled

Extract every width of cloth training interpolation image

The vertical and second order level of single order level, single order, second order VG (vertical gradient) characteristic, simultaneously this four width of cloth gradient characteristic image is carried out overlapping piecemeal, every width of cloth training interpolation image

Obtain 4*N behind the overlapping piecemeal ₁Individual size is the training low resolution gradient characteristic image piece of z*z; Obtain 4*N training low resolution gradient characteristic image piece altogether; Per four training low resolution gradient characteristic image pieces are as a training low resolution gradient characteristic image piece group, convert i training low resolution gradient characteristic image piece group into i by the order that is listed as and train the low resolution proper vector

Each training low resolution proper vector

Dimension be d ₂=4*z ², all training low resolution proper vectors

(1≤i≤N) form to train the low resolution eigenmatrix $X^s=[x_1^s,x_2^s,\·\·\·,x_N^s];$

B, pre-service:

Input waits to rebuild low-resolution image R ^d, its resolution and all training low-resolution images (resolution of 1≤l≤L) is identical, will wait to rebuild low-resolution image R ^dCarry out obtaining waiting to rebuild interpolation image R after a times of up-sampling handled ^c, extract and wait to rebuild interpolation image R ^cThe vertical and second order level of single order level, single order, second order VG (vertical gradient) characteristic, simultaneously this four width of cloth gradient characteristic image is carried out obtaining 4*N altogether behind the overlapping piecemeal ₂Individual size be z*z wait to rebuild low resolution gradient characteristic image piece, N ₂=1000～7000; Wait to rebuild low resolution gradient characteristic image piece and wait to rebuild low resolution gradient characteristic image piece group for per four, wait to rebuild low resolution gradient characteristic image piece group with j and convert by the order that be listed as that j is individual to wait to rebuild the low resolution proper vector into as one

J=1,2 ..., N ₂, each waits to rebuild the low resolution proper vector

Dimension and all training low resolution proper vector

(dimension of 1≤i≤N) is identical, is d ₂=4*z ², institute remains to be rebuild the low resolution proper vector

(1≤j≤N ₂) form one and wait to rebuild the low resolution eigenmatrix $X^t=[x_1^t,x_2^t,\·\·\·,x_{N_2}^t];$

C, super-resolution rebuilding:

C1, subcharacter matrix generate: each that obtains among the step B waited to rebuild the low resolution proper vector

(1≤j≤N ₂), the training low resolution eigenmatrix that obtains in steps A successively

In seek out with j and wait to rebuild the low resolution proper vector

Between the minimum n-1=100～300 training low resolution proper vector of Euclidean distance, wait to rebuild the low resolution proper vector with j As the subcharacter vector x _1j, and with the n-1 that searches out a training low resolution proper vector by with j wait to rebuild the low resolution proper vector

Between Euclidean distance from small to large successively as the subcharacter vector x _2j, x _3j..., x _Nj, form j sub-eigenmatrix thus

C2, dimensionality reduction eigenmatrix generate: adopt I and II subspace projection method, with j sub-eigenmatrix Carry out the characteristic dimensionality reduction, each subcharacter vector x _{I ' j}(1≤i '≤n) project on the low n-dimensional subspace n, obtain n dimensionality reduction proper vector altogether

Each dimensionality reduction proper vector

Dimension be m＜d ₂Each subcharacter vector x _{I ' j}With corresponding dimensionality reduction proper vector

Form mapping relations one by one, behind the I and II subspace projection, dimension is d ₂The subcharacter vector x _{I ' j}Converting dimension to is the dimensionality reduction proper vector of m

Realize the characteristic dimensionality reduction, all dimensionality reduction proper vectors

(1≤i '≤n) form j dimensionality reduction eigenmatrix $X_j^m={\left\{x_{i^{\′}j}^m\right\}}_{i^{\′}=1}^n;$

C3, weight coefficient calculate: the dimensionality reduction proper vector in that step C2 obtains is gathered

In seek out and the dimensionality reduction proper vector

Between k minimum dimensionality reduction proper vector of Euclidean distance, k=5～10 are according to step C1 neutron proper vector x _{I ' j}With dimensionality reduction proper vector among the step C2

Between mapping relations one by one, and the call number of the k that searches out a dimensionality reduction proper vector, low in the training that steps A obtains successively, high-resolution features matrix X ^sAnd Y ^sIn seek out that corresponding k is low to training, high-resolution features is vectorial, and, high-resolution features low to training by this k is vectorial forms j low, the high-resolution features matrix of neighbour respectively With

According to j neighbour's low resolution eigenmatrix

Wait to rebuild the low resolution proper vector with j Calculate j similarity group

With j scale factor group

And associating

With

Calculate j normalization and embed the weight coefficient group

C4, reconstruction high-definition picture piece: the linear weighted function form that embeds weight coefficient group and j neighbour's high-resolution features matrix

according to j normalization; Estimate j rebuild back high-resolution features vector

again with j brightness average

addition of waiting to rebuild the low-resolution image piece; And convert the image block form into, can draw j and rebuild back high-definition picture piece;

Repeat above step C1～step C4, obtain N altogether ₂Individual size is a high-definition picture piece after the reconstruction of z*z, obtains the preliminary image H of super-resolution rebuilding behind the lap splice ₀

D, aftertreatment:

Adopt the preliminary image H of iteration back-projection algorithms to super-resolution rebuilding ₀Carry out Q iterative computation and handle, Q=10～30 obtain the final image H of super-resolution rebuilding ^*

2. image super-resolution rebuilding method according to claim 1; It is characterized in that; Adopt I and II subspace projection method among the described step C2, the specific practice of j sub-eigenmatrix

being carried out obtaining behind the characteristic dimensionality reduction j dimensionality reduction eigenmatrix is:

The one-level subspace projection:

Calculate j j the radially basic kernel function matrix that sub-eigenmatrix

is corresponding $D_j^r={\left\{D_{i^{\′}{j}^{\′}j}^r\right\}}_{i^{\′},j^{\′}=1}^n={\left\{\sqrt{2-2*\exp ({-\left|\right|x_{i^{\′}j}^r-x_{j^{\′}j}^r\left|\right|}^2/{\mathrm{\γ}}_j)}\right\}}_{i^{\′},j^{\′}=1}^n,$ Wherein the individual radially parameter of basic kernel function of j does

Then to j radially basic kernel function matrix K _jCarry out obtaining j centralization kernel function matrix after centralization is handled

G=I-1/n in the formula, I are unit matrixs that size is n*n, and 1 is that a size is n*n, and all values is 1 matrix; Find the solution j centralization kernel function matrix

Descending characteristic value decomposition equation

I '=1,2,3 ..., n, j=1,2,3 ..., N ₂, λ in the formula _{I ' j}For by the individual eigenwert of i ' after the descending sort, α _{I ' j}It is the individual eigenvalue of i ' _{I ' j}The corresponding individual proper vector of i '; Again to the individual proper vector α of i ' _{I ' j}Carry out obtaining the orthogonalization proper vector after the orthogonalization process

R eigenvalue of maximum λ before choosing _{I ' j}Corresponding orthogonalization proper vector

R=50～n, the r that selects orthogonalization proper vector

Form j one-level subspace projection matrix

R eigenvalue of maximum before described

Sum accounts for all eigenwerts

More than 99% of sum; By formula

Draw j one-level mapping matrix

T is the matrix transpose computing, all one-level mapping vectors

(1≤i '≤n) dimension is r;

The secondary subspace projection:

Calculate j corresponding nuclear distance function matrix of j one-level mapping matrix

$D_j^r={\left\{D_{i^{\′}{j}^{\′}j}^r\right\}}_{i^{\′},j^{\′}=1}^n={\left\{\sqrt{2-2*\mathrm{Exp}({-\left|\right|x_{i^{\′}j}^r-x_{j^{\′}j}^r\left|\right|}^2/{\mathrm{\γ}}_j)}\right\}}_{i^{\′},j^{\′}=1}^n,$ Wherein the parameter of j nuclear distance function does

Construct j adjacency matrix then

As i ' ∈ Λ _{J ' j}Or j ' ∈ Λ _{I ' j}The time,

Otherwise, S _{I ' j ' j}=0, Λ wherein _{J ' j}And Λ _{I ' j}Be expressed as respectively

The neighborhood tally set, the element number of neighborhood tally set is b=5～10; Calculate j adjacency matrix S _jJ corresponding Laplce's matrix F _j=diag (S _j* 1 _n)-S _j, wherein 1 _nBeing expressed as a size is n*1, and all values is 1 column vector, and diag () is expressed as the diagonalization of matrix computing; Then calculate j symmetric matrix respectively $U_j=X_j^r*F_j*{\left(X_j^r\right)}^T$ With j diagonal matrix $V_j=X_j^r*\left\{\mathrm{Diag}(S_j*1_n)\right\}*{\left(X_j^r\right)}^T,$ J symmetric matrix U _jWith j diagonal matrix V _jAll be that size is the matrix of r*r, unite to solve U _jAnd V _jBroad sense ascending order characteristic value decomposition equation

In the formula

Be the individual eigenwert of i ' after arranging by ascending order, p _{I ' j}It is the individual eigenwert of i '

The corresponding individual proper vector of i '; M minimal eigenvalue before choosing Characteristic of correspondence vector p _{I ' j}, m=10～r, the m that a selects proper vector

Form j secondary subspace projection matrix P _j=[p _1j, p _2j..., p _Mj], by formula

Can draw j dimensionality reduction eigenmatrix

All dimensionality reduction proper vectors (1≤i '≤n) dimension is m.

3. image super-resolution rebuilding method according to claim 1; It is characterized in that associating j similarity group

and j scale factor group calculates the specific practice of j normalization embedding weight coefficient group and be among the described step C3:

Calculate j neighbour's low resolution eigenmatrix that step C3 obtains respectively

In k training low resolution proper vector

(1≤i "≤k) wait to rebuild the low resolution proper vector with j

Between similarity

With corresponding scale factor

Wherein

With

Be expressed as training low resolution proper vector respectively

Wait to rebuild the low resolution proper vector with j

In u brightness value, ε is 1*10 ^-4～1*10 ^-2Positive number; The k that calculates is to similarity

With scale factor c _{I " j}Form j similarity group respectively

With j scale factor group

Unite j similarity group

With j scale factor group

Calculate j and embed the weight coefficient group

And to j embedding weight coefficient group

Do and obtain j normalization embedding weight coefficient group after normalization is handled ${\left\{{\hat{w}}_{i^{\′\′}j}\right\}}_{i^{\′\′}=1}^k={\{w_{i^{\′\′}j}/{\mathrm{\Σ}}_{i^{\′\′}=1}^k{w}_{i^{\′\′}j}\}}_{i^{\′\′}=1}^k.$

Images