CN105005965B - Natural image ultra-resolution method based on EM algorithm - Google Patents
Natural image ultra-resolution method based on EM algorithm Download PDFInfo
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T3/00—Geometric image transformation in the plane of the image
- G06T3/40—Scaling the whole image or part thereof
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- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
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- G06T3/40—Scaling the whole image or part thereof
- G06T3/4053—Super resolution, i.e. output image resolution higher than sensor resolution
Abstract
The invention discloses a kind of natural image ultra-resolution method based on EM algorithm.Its step is：(1) low resolution image is inputted；(2) interpolation image；(3) hidden image is obtained；(4) it is cut into hidden image block；(5) similar matrix of hidden picture block is obtained；(6) dictionary of hidden image block is obtained；(7) average and covariance of estimation image block are obtained；(8) the MAP estimation value of estimation image block is obtained；(9) full resolution pricture is obtained；(10) relative error is calculated；(11) judge whether to meet end condition；(12) update the data；(13) optimal full resolution pricture is exported.EM algorithm is incorporated into natural image super-resolution field by the present invention, obtains abundant restoring image detail information, is adapted to Image Super-resolution in complex situations.
Description
Technical field
The invention belongs to technical field of image processing, further relate in natural image processing technology field based on most
The natural image ultra-resolution method of big Expectation Algorithm.The present invention is that the natural image of low resolution is carried out into super-resolution, to obtain one
Width clearly high-resolution natural image, so as to for figure, as follow-up interpretation, target identification, target detection provides it is more accurate, comprehensive
Information.
Background technology
Image Super-resolution technology refers to reconstruct a width clearly high-resolution from single width or the image of several low resolution
The process of the image of rate.Low resolution image spatial resolution is relatively low, have impact on and scenery more comprehensively, is clearly described.Image surpasses
The purpose of resolution is to strengthen and enrich the detail content of scenery in order to obtain full resolution pricture, at for follow-up image
Reason provides more accurate, comprehensive information with interpretation.High-resolution image is obtained, can be realized by following two approach：When
Using the sensor of renewal, higher-quality image can be so obtained, but the cost of novel sensor is higher；Second, pass through algorithm
Super-resolution is carried out to low resolution image, higher-quality image, and skill can be so obtained on the basis of existing sensor technology
Art cost is low.Image Super-resolution technology can be largely classified into based on interpolation, based on reconstruction and based on study three major types.At present, it is single
Width image is based primarily upon reconstructed error and the method for prior information constraint realizes Image Super-resolution.
Paper " the Image Super-Resolution Via Sparse that Yang, Wright et al. are delivered
Representation”(IEEE Trans.on Image Processing vol.19no.11pp.2861-2873.2010)
It is middle to propose a kind of image super-resolution method based on rarefaction representation.The thought of compressed sensing is introduced super-resolution reconstruct by this method
In, low resolution and high-resolution dictionary pair are obtained by the method for rarefaction representation.When the rarefaction representation of low resolution image passes through
When the method for compressed sensing obtains, then the rarefaction representation of full resolution pricture also obtains accordingly, so as to effectively to low point
Distinguish that image is rebuild.But weak point is existing for this method, this method is in the construction of dictionary pair, it is necessary to gather a large amount of
External trainer data, and this is unpractical, while represents error be present between this hypothesis height resolution image, this
The whole structure that sample to recover to obtain is not fine.
Paper " the Single Image Super-Resolution using Gaussian that He and the people of Siu bis- deliver
Process Regression”(IEEE Conference on Computer Vision and Pattern
Recognition, pp.449-456.2011) in disclose one kind and using Gaussian process priori solve the problems, such as Image Super-resolution
Method.This method learns to obtain the mapping relations between low resolution image and full resolution pricture using Gaussian process.Gauss
Process adaptively can be found to obtain the inherence between low resolution image block and full resolution pricture block by Gaussian distribution model
Contact.The advantage of this method is when mapping relations are learnt, and does not need full resolution pricture to be trained.But should
The weak point that method still has is, this method with only the local message of image itself as training data, simultaneously
This method is without the prior information for fully excavating topography's block, so that when the feelings for the information deficiency that Gaussian process can provide
Under condition, cause that the obtained image result of reconstruct is unstable, details and Edge restoration are not fine, regional area reconstruction qualities
Decline.
The content of the invention
It is an object of the invention to the deficiency for above-mentioned prior art, proposes a kind of nature based on EM algorithm
Image super-resolution method, the prior information of image is adequately bonded, in Image Super-resolution Reconstruction, can preferably remove and shake
Bell, drastically increase the recovery effects of Image Super-resolution.
To achieve the above object, the present invention realizes natural image super-resolution on the basis of based on EM algorithm, its
Technical scheme is that Image Super-resolution process is splitted into two subprocess by EM algorithm, i.e., hidden image it is expected maximum process
With estimation image posterior probability maximum process.In image posterior probability maximum process is estimated, we use Gaussian process method
Study obtains estimating the average and variance of image block, and then training obtains the dictionary of hidden image block, finally by mean square error most
The method of smallization obtains estimating the maximum a posteriori value of image.By general to hidden image expectation maximization process and estimation image posteriority
Rate maximum process the two processes carry out loop iteration, when iteration meets end condition, then jump out circulation, finally give optimal
Full resolution pricture.
The present invention's comprises the following steps that：
(1) a low resolution image to be restored is inputted；
(2) interpolation image：
Using the imresize functions in matlab softwares, low resolution image to be restored is interpolated into be restored low point
Distinguish image 3 times, obtain the low resolution image after interpolation；
(3) according to the following formula, hidden image is obtained：
Z=L+ λ HT(Y-HL)
Wherein, Z represents hidden image, and L represents the low resolution image after interpolation, and λ represents iteration step length, and λ=0.8, H represent to see
Matrix is surveyed, T represents transposition operation, and Y represents low resolution image to be restored；
(4) hidden image is cut into W hidden image blocks：
Hidden image is subjected to slide window processing, wherein hidden image block is sized to 6 × 6 pixels, sliding window step-length is set to 1 picture
Element, obtain W hidden image block collection；
(5) similar matrix of each hidden image block is obtained：
(5a) concentrates one hidden image block of any extraction from hidden image block, concentrates searching hidden with being extracted from hidden image block
Minimum preceding 30 image blocks of image block Euclidean distance, 30 image blocks are carried out to draw row are perpendicular to stack, obtain one 36 ×
30 similar matrix；
(5b) repeats step (5a) described process, until obtaining the similar matrix of each hidden image block；
(6) dictionary of each hidden image block is obtained：
(6a) inputs the similar matrix of any one hidden image block, and corresponding hidden image block is constructed using this similar matrix
Dictionary；
(6b) repeats step (6a) described process, until obtaining the dictionary of each hidden image block；
(7) average and covariance of each estimation image block are obtained：
Low resolution image to be restored is carried out stripping and slicing by (7a), and wherein stripping and slicing is sized to 2 × 2 pixels, obtained to be restored
Low resolution image block collection；
(7b) obtains hidden low resolution image, hidden image and hidden low resolution image is distinguished to hidden image premultiplication observing matrix
Block is cut into, stripping and slicing size is respectively 6 × 6 pixels and 2 × 2 pixels, is pulled into row, respectively obtains hidden image block collection and hidden low point
Image block collection is distinguished, forms image block to set；
The input of (7c) using hidden low resolution image block collection as Gaussian process method, using hidden image block collection as Gaussian process
Method exports, and the covariance for calculating Gaussian process method calculates function；
(7d) according to the following formula, obtains the average and covariance of each estimation image block：
μj=K (vj,y)K(y,y)-1f
Σj=K (vj,vj)-K(vj,yj)K(y,y)-1K(y,vj)
Wherein, μjThe average of j-th of estimation image block is represented, K () represents that covariance calculates function, and -1 represents the behaviour that inverts
Make, vjJ-th of low resolution image block to be restored is represented, y represents hidden image block collection, and f represents hidden low resolution image block collection, ΣjTable
Show the covariance of j-th of estimation image block, j=1,2 ..., W, W represents the number of hidden image block；
(8) the MAP estimation value of each estimation image block is obtained：
(8a) inputs any one hidden image block, according to the following formula, obtains and the estimation figure corresponding to the hidden image block of the input
As the coefficient matrix of block：
Λ=(diag (DTμ μTD+DTΣD))-1diag(DTμ μTD+DTΣD)
Wherein, Λ represents the coefficient matrix of estimation image block, and diag () represents diagonalization operation, and D represents hidden image block
Dictionary, T represent transposition operation, μ represent estimation image block average, Σ represent estimation image block covariance, -1 represent ask
Inverse operation；
(8b) according to the following formula, obtains the MAP estimation value of estimation image block corresponding to hidden image block：
X=D Λ DTz
Wherein, x represents the MAP estimation value of estimation image block, and D represents the dictionary of hidden image block, and Λ represents estimation figure
As the coefficient matrix of block, T represents transposition operation, and z represents hidden image block；
(8c) repeats step (8a), (8b) described process, until obtaining the maximum a posteriori of each estimation image block
Estimate；
(9) the MAP estimation value of all estimation image blocks is spliced into a panel height resolution image；
(10) following formula is utilized, calculates the relative error of hidden image and full resolution pricture：
Wherein, γ represents the relative error of hidden image and full resolution pricture, and Z represents hidden image, and T represents full resolution pricture, |
|·||2Represent that 2 norms operate；
(11) judge whether the relative error of hidden image and full resolution pricture meets end condition, if it is, performing step
(13)；Otherwise, step (12) is performed；
(12) update the data：
The pixel for the low resolution image pixel value of full resolution pricture being assigned to after interpolation, perform step (3)；
(13) an optimal full resolution pricture is exported.
The present invention has advantages below compared with prior art：
First, because the present invention uses EM algorithm, Image Super-resolution process is splitted into two processes：Hidden image obtains
The process of obtaining and the MAP estimation process for estimating image block, then by carrying out loop iteration to the two processes, carry out image
Super-resolution, overcome prior art and cause to recover the defects of image is unstable using single computing so that the present invention can obtain
Abundant restoring image detail information, enhance the definition for recovering image.
Second, due to present invention introduces training data of the similar matrix of hidden image block as dictionary learning, overcoming existing
There is the defects of external data ability training dictionary is needed in technology so that the present invention can be in outside data sample number deficiency feelings
Under condition, Image Super-resolution still can be carried out.
3rd, due to present invention introduces the MAP estimation method of image block, fully by the dictionary sparse table of image
Show that priori and Gaussian process priori are combined together, overcome the priori for only obtaining image by Gaussian process in the prior art
The defects of information so that the present invention can further enhance the Quality of recovery of image.
Brief description of the drawings
Fig. 1 is the flow chart of the present invention；
Fig. 2 is the low resolution chart pictures of Butterfly that the present invention uses in emulation experiment；
Fig. 3 is the Butterfly High resolution reconstruction images obtained using the present invention in emulation experiment；
Fig. 4 is using the method based on rarefaction representation, the Butterfly obtained in its emulation experiment are high in the prior art
Resolved reconstruction image；
Fig. 5 is that solve the problems, such as the method for Image Super-resolution using Gaussian process priori of the prior art, in its emulation
The Butterfly High resolution reconstruction images obtained in experiment.
Embodiment
The present invention will be further described below in conjunction with the accompanying drawings.
Referring to the drawings 1, specific implementation step of the present invention is as follows.
Step 1, a low resolution image to be restored is inputted.
The low resolution image size to be restored inputted in the embodiment of the present invention is 86 × 86 pixels, referring to accompanying drawing 2.
Step 2, interpolation image.
Using the imresize functions in matlab softwares, low resolution image to be restored is interpolated into be restored low point
Distinguish image 3 times, obtain the low resolution image after interpolation.
Step 3, according to the following formula, hidden image is obtained：
Z=L+ λ HT(Y-HL)
Wherein, Z represents hidden image, and L represents the low resolution image after interpolation, and λ represents iteration step length, and λ=0.8, H represent to see
Matrix is surveyed, T represents the transposition operation of observing matrix, and Y represents low resolution image to be restored.
Observing matrix H is by the fspecial functions and downsampling in matlab softwares in the embodiment of the present invention
Function produces.
Step 4, hidden image is cut into W hidden image blocks.
Hidden image is subjected to slide window processing, wherein hidden image block is sized to 6 × 6 pixels, sliding window step-length is set to 1 picture
Element, obtain W hidden image block collection.
Hidden image block collection number W size depends on the size of hidden image, and the step of the size of hidden image block and sliding window
It is long.
The size of hidden image is taken into 258 × 258 pixels in the embodiment of the present invention, hidden image block is sized to 6 × 6 pictures
Element, sliding window step-length are set to 1 pixel, then hidden image block collection number W is 64009.
Step 5, the similar matrix of each hidden image block is obtained.
1st step, one hidden image block of any extraction is concentrated from hidden image block, concentrates what is found and extracted from hidden image block
Minimum preceding 30 image blocks of hidden image block Euclidean distance, this 30 image blocks are carried out to draw row are perpendicular to stack, obtain one
36 × 30 similar matrix.
2nd step, the process of the 1st step is repeated, until obtaining the similar matrix of each hidden image block.
Step 6, the dictionary of each hidden image block is obtained.
1st step, the similar matrix of any one hidden image block is inputted, corresponding hidden image block is constructed using this similar matrix
Dictionary.
According to the following formula, the dictionary of hidden image block is obtained：
subject to Di TDi=I
Wherein, min { } represents to minimize operation, DiThe dictionary of i-th of hidden image block to be solved is represented, T represents to turn
Put operation, PiThe similar matrix of i-th of hidden image block is represented, | | | |1,2The operation of 1,2 norms is represented, the row of matrix represents 1 model
Number, row represent 2 norms, and subject to represent Di TDi=I is solution | | Di TPi||1,2Restrictive condition, i=1,2 ..., W, W
The number of hidden image block is represented, I represents unit matrix.
2nd step, the process of the 1st step is repeated, until obtaining the dictionary of each hidden image block.
Step 7, the average and covariance of each estimation image block are obtained.
1st step, low resolution image to be restored is subjected to stripping and slicing, wherein stripping and slicing is sized to 2 × 2 pixels, obtains treating extensive
Multiple low resolution image block collection.
2nd step, to hidden image premultiplication observing matrix, hidden low resolution image is obtained, by hidden image and hidden low resolution image point
Block is not cut into, and stripping and slicing size is respectively 6 × 6 pixels and 2 × 2 pixels, is pulled into row, respectively obtains hidden image block collection and hidden low
Resolution image block collection, image block is formed to set.
3rd step, the input using hidden low resolution image block collection as Gaussian process method, using hidden image block collection as Gauss mistake
Cheng Fangfa is exported, and the covariance for calculating Gaussian process method calculates function.
It is as follows that the covariance of Gaussian process method calculates function：
G | X~N (m (X), K (X, X))
Wherein, g represents hidden low resolution image block collection, | represent to carry out conditional probability distribution operation, X represents hidden image block collection,
~represent to obey distribution symbol, N () represents normal distribution operation, and m () represents what is arrived by Gaussian process methodology acquistion
Mean value function, the covariance that K () represents to arrive by Gaussian process methodology acquistion calculate function.
It is by matlab kits " Gaussian that the covariance of Gaussian process, which calculates function, in the embodiment of the present invention
Processes for Machine Learning " are calculated.
4th step, according to the following formula, obtain the average and covariance of each estimation image block：
μj=K (vj,y)K(y,y)-1f
Σj=K (vj,vj)-K(vj,yj)K(y,y)-1K(y,vj)
Wherein, μjThe average of j-th of estimation image block is represented, K () represents that covariance calculates function, and -1 represents the behaviour that inverts
Make, vjJ-th of low resolution image block to be restored is represented, y represents hidden image block collection, and f represents hidden low resolution image block collection, ΣjTable
Show the covariance of j-th of estimation image block, j=1,2 ..., W, W represents the number of hidden image block.
Hidden image block collection number W size depends on the size of hidden image, and the step of the size of hidden image block and sliding window
It is long.
The size of hidden image is taken into 258 × 258 pixels in the embodiment of the present invention, hidden image block is sized to 6 × 6 pictures
Element, sliding window step-length are set to 1 pixel, then hidden image block collection number W is 64009.
Step 8, the MAP estimation value of each estimation image block is obtained.
1st step, any one hidden image block is inputted, according to the following formula, obtained and the estimation corresponding to the hidden image block of the input
The coefficient matrix of image block：
Λ=(diag (DTμ μTD+DTΣD))-1diag(DTμ μTD+DTΣD)
Wherein, Λ represents the coefficient matrix of estimation image block, and diag () represents diagonalization operation, and D represents hidden image block
Dictionary, T represent transposition operation, μ represent estimation image block average, Σ represent estimation image block covariance, -1 represent ask
Inverse operation.
2nd step, according to the following formula, obtain the MAP estimation value of estimation image block corresponding to hidden image block：
X=D Λ DTz
Wherein, x represents the MAP estimation value of estimation image block, and D represents the dictionary of hidden image block, and Λ represents estimation figure
As the coefficient matrix of block, T represents transposition operation, and z represents hidden image block；
3rd step, the 1st step is repeated, process described in the 2nd step, until obtaining maximum a posteriori of each estimation image block
Estimate.
Step 9, the MAP estimation value of all estimation image blocks is spliced into a panel height resolution image.
Step 10, using following formula, the relative error of hidden image and full resolution pricture is calculated：
Wherein, γ represents the relative error of hidden image and full resolution pricture, and Z represents hidden image, and T represents full resolution pricture, |
|·||2Represent that 2 norms operate.
Step 11, judge whether the relative error of hidden image and full resolution pricture meets end condition, if it is, performing step
Rapid 13；Otherwise, step 12 is performed.
End condition refers to：γ≤ε, wherein γ represent the relative error of hidden image and full resolution pricture, and ε represents tolerance
The limit, its value span are ε ∈ (10-4,10-3) positive number.
Step 12, update the data.
The pixel for the low resolution image pixel value of full resolution pricture being assigned to after interpolation, perform step 3.
Step 13, an optimal full resolution pricture is exported.
The effect of the present invention can be further illustrated by following emulation experiment.
1. emulation experiment condition：
The present invention experiment simulation environment be：
Software：MATLAB R2012a
Processor：Intel(R)Core(TM)i5-3470MCPU@3.20GHz
Internal memory：4GB RAM
Image used in the emulation experiment of the present invention：From standard picture storehouse.
2. emulation experiment content：
The emulation experiment of the present invention is specifically divided into three emulation experiments.
Emulation experiment one：Super-resolution Reconstruction is carried out to low resolution image using the present invention, as a result as shown in Figure 3.
Emulation experiment two：Super-resolution Reconstruction, knot are carried out to low resolution image based on the method for rarefaction representation using existing
Shown in fruit Fig. 4.
Emulation experiment three：Solves the problems, such as the method for Image Super-resolution to low point using Gaussian process priori using existing
Distinguish that image carries out Super-resolution Reconstruction, as a result as shown in Figure 5.
In emulation experiment, the quality of super-resolution result is evaluated using Y-PSNR PSNR evaluation indexes, its PSNR determines
Justice is：
Wherein, u represents picture rich in detail, and v represents the image after rebuilding, and A represents picture rich in detail u number of lines of pixels, and B represents clear
Clear image u pixel columns.
It is right respectively using the present invention with the method based on rarefaction representation of prior art, using Gaussian process transcendental method
Image Butterfly, Leaves carry out Super-resolution Reconstruction emulation.Commented using Y-PSNR PSNR rebuilding result figure
Valency, evaluation result is as shown in table 1, and the method that Alg1 represents the present invention in table 1, Alg2 represents the method based on rarefaction representation,
Alg3 represents to use Gaussian process transcendental method.
The PSNR values (unit dB) that the present invention of table 1. and two kinds of control methods obtain in emulation experiment
3. the simulation experiment result is analyzed
From figure 3, it can be seen that the reconstructed results for the Butterfly that the present invention obtains, not only effectively supplement high frequency
Detailed information, and the sawtooth effect of image border is successfully inhibited, make image edge clear.
From fig. 4, it can be seen that the detail of the high frequency that the existing method based on rarefaction representation obtains loses serious, sawtooth
It is serious to change phenomenon, has severely impacted image Quality of recovery.
From fig. 5, it can be seen that the existing recovery imaging surface that is obtained using Gaussian process transcendental method is smoother, mould
Paste, image recovery effects unobvious.
Claims (5)
1. a kind of natural image ultra-resolution method based on EM algorithm, comprises the following steps：
(1) a low resolution image to be restored is inputted；
(2) interpolation image：
Using the imresize functions in matlab softwares, low resolution image to be restored is interpolated into low resolution figure to be restored
3 times of picture, obtain the low resolution image after interpolation；
(3) according to the following formula, hidden image is obtained：
Z=L+ λ HT(Y-HL)
Wherein, Z represents hidden image, and L represents the low resolution image after interpolation, and λ represents iteration step length, and λ=0.8, H represent observation square
Battle array, T represent transposition operation, and Y represents low resolution image to be restored；
(4) hidden image is cut into W hidden image blocks：
Hidden image is subjected to slide window processing, wherein hidden image block is sized to 6 × 6 pixels, sliding window step-length is set to 1 pixel,
Obtain W hidden image block collection；
(5) similar matrix of each hidden image block is obtained：
(5a) concentrates one hidden image block of any extraction from hidden image block, and the hidden image found with being extracted is concentrated from hidden image block
Minimum preceding 30 image blocks of block Euclidean distance, 30 image blocks are carried out to draw row are perpendicular to stack, obtain one 36 × 30
Similar matrix；
(5b) repeats step (5a) described process, until obtaining the similar matrix of each hidden image block；
(6) dictionary of each hidden image block is obtained：
(6a) inputs the similar matrix of any one hidden image block, and the word of corresponding hidden image block is constructed using this similar matrix
Allusion quotation；
(6b) repeats step (6a) described process, until obtaining the dictionary of each hidden image block；
(7) average and covariance of each estimation image block are obtained：
Low resolution image to be restored is carried out stripping and slicing by (7a), and wherein stripping and slicing is sized to 2 × 2 pixels, is obtained to be restored low
Resolution image block collection；
(7b) obtains hidden low resolution image, hidden image and hidden low resolution image is cut into respectively to hidden image premultiplication observing matrix
Block, stripping and slicing size are respectively 6 × 6 pixels and 2 × 2 pixels, are pulled into row, respectively obtain hidden image block collection and hidden low resolution figure
As block collection, image block is formed to set；
The input of (7c) using hidden low resolution image block collection as Gaussian process method, using hidden image block collection as Gaussian process method
Output, the covariance for calculating Gaussian process method calculate function；
(7d) according to the following formula, obtains the average and covariance of each estimation image block：
μj=K (vj,y)K(y,y)-1f
Σj=K (vj,vj)-K(vj,yj)K(y,y)-1K(y,vj)
Wherein, μjThe average of j-th of estimation image block is represented, K () represents that covariance calculates function, and -1 represents inversion operation, vj
J-th of low resolution image block to be restored is represented, y represents hidden image block collection, and f represents hidden low resolution image block collection, ΣjRepresent the
The covariance of j estimation image block, j=1,2 ..., W, W represent the number of hidden image block；
(8) the MAP estimation value of each estimation image block is obtained：
(8a) inputs any one hidden image block, according to the following formula, obtains and the estimation image block corresponding to the hidden image block of the input
Coefficient matrix：
Λ=(diag (DTμ μTD+DTΣD))-1diag(DTμ μTD+DTΣD)
Wherein, Λ represents the coefficient matrix of estimation image block, and diag () represents diagonalization operation, and D represents the word of hidden image block
Allusion quotation, T represent transposition operation, and μ represents the average of estimation image block, and Σ represents the covariance of estimation image block, and -1 represents the behaviour that inverts
Make；
(8b) according to the following formula, obtains the MAP estimation value of estimation image block corresponding to hidden image block：
X=D Λ DTz
Wherein, x represents the MAP estimation value of estimation image block, and D represents the dictionary of hidden image block, and Λ represents estimation image block
Coefficient matrix, T represents transposition operation, and z represents hidden image block；
(8c) repeats step (8a), (8b) described process, until obtaining the MAP estimation of each estimation image block
Value；
(9) the MAP estimation value of all estimation image blocks is spliced into a panel height resolution image；
(10) following formula is utilized, calculates the relative error of hidden image and full resolution pricture：
<mrow>
<mi>&gamma;</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>|</mo>
<mo>|</mo>
<mi>Z</mi>
<mo>-</mo>
<mi>T</mi>
<mo>|</mo>
<msub>
<mo>|</mo>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mo>|</mo>
<mi>Z</mi>
<mo>|</mo>
<msub>
<mo>|</mo>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mrow>
Wherein, γ represents the relative error of hidden image and full resolution pricture, and Z represents hidden image, and T represents full resolution pricture, | | |
|2Represent that 2 norms operate；
(11) judge whether the relative error of hidden image and full resolution pricture meets end condition, if it is, performing step (13)；
Otherwise, step (12) is performed；
(12) update the data：
The pixel for the low resolution image pixel value of full resolution pricture being assigned to after interpolation, perform step (3)；
(13) an optimal full resolution pricture is exported.
2. the natural image ultra-resolution method according to claim 1 based on EM algorithm, it is characterised in that：Step
(4), the size of the hidden image block collection number W described in step (7d) depends on the size of hidden image, and the size of hidden image block
With the step-length of sliding window.
3. the natural image ultra-resolution method according to claim 1 based on EM algorithm, it is characterised in that：Step
The dictionary that corresponding hidden image block is constructed using inputted similar matrix described in (6a) is to carry out according to the following formula：
<mrow>
<munder>
<mi>min</mi>
<msub>
<mi>D</mi>
<mi>i</mi>
</msub>
</munder>
<mo>{</mo>
<mo>|</mo>
<mo>|</mo>
<msup>
<msub>
<mi>D</mi>
<mi>i</mi>
</msub>
<mi>T</mi>
</msup>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
<mo>|</mo>
<msub>
<mo>|</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>}</mo>
</mrow>
subject to Di TDi=I
Wherein, min { } represents to minimize operation, DiThe dictionary of i-th of hidden image block to be solved is represented, T represents transposition behaviour
Make, PiThe similar matrix of i-th of hidden image block is represented, | | | |1,2The operation of 1,2 norms is represented, the row of matrix represents 1 norm, row
2 norms are represented, subject to represent Di TDi=I is solution | | Di TPi||1,2Restrictive condition, i=1,2 ..., W, W represent it is hidden
The number of image block, I represent unit matrix.
4. the natural image ultra-resolution method according to claim 1 based on EM algorithm, it is characterised in that：Step
It is as follows that the covariance of Gaussian process method described in (7c) calculates function：
G | X~N (m (X), K (X, X))
Wherein, g represents hidden low resolution image block collection, | represent to carry out conditional probability distribution operation, X represents hidden image block collection ,~table
Show obedience distribution symbol, N () represents normal distribution operation, and m () represents the average arrived by Gaussian process methodology acquistion
Function, the covariance that K () represents to arrive by Gaussian process methodology acquistion calculate function.
5. the natural image ultra-resolution method according to claim 1 based on EM algorithm, it is characterised in that：Step
(11) end condition described in refers to：γ≤ε, wherein, γ represents the relative error of hidden image and full resolution pricture, and ε represents to hold
Bear the limit, its value span is ε ∈ (10-4,10-3) positive number.
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