CN106991659B - A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance change - Google Patents
A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance change Download PDFInfo
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Abstract
The present invention is a kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change：Astronomical target image or spatial target images at different moments after adaptive optics corrects are gathered, establishes the linear equation of solving system point spread function；Establish the convex Optimized model of system point spread function solution；System point spread function is solved using the convex convex optimization method of Optimization Solution Algorithm for Solving of classics；The estimate of target to be observed is solved according to the system point spread function solved, so as to recover image.The method of the present invention takes full advantage of the dynamic characteristic of turbulent flow, and timeliness to IMAQ, the real-time to system do not require, and image recovery process belongs to linear solution, computation amount.Without carrying out constantly alternately solving to observed object and system point spread function.Image recovery method stability is strong, in the absence of divergence problem；Algorithm is succinct and directly perceived.
Description
Technical field
The invention belongs to optical image security field, and it is extensive to be related to a kind of optical imagery that can adapt to atmospheric turbulance change
Compound method, astronomical target or spatial target images processing method particularly after adaptive optics system image correction.
Background technology
After the real-time aberration correction of adaptive optics system, the most of low order picture for causing image fuzzy is compensated already for
Difference.But generally, limited by system cost, finite bandwidth and detection noise etc., adaptive optics method is to air
The compensation of turbulent flow is incomplete, and the high-frequency information of target is still suppressed and decayed.The loss of high-frequency information then make it that target is thin
It is unintelligible to save feature, then the requirement that space is accurately positioned with target identification hardly results in satisfaction.
After-treatment is carried out to the adaptive optical image after above-mentioned partial correction, will not compensated by adaptive optics system
Wavefront residual error be corrected, to obtain more preferable image quality.It is extensive that stigmatic image technology has been successfully applied to astronomical point target
It is multiple, such as Knox-Thompson cross spectrums or bispectrum treatment technology.But reference target is needed in this method implementation process, and is needed
Handle thousands of width short exposed images.When for extending target, reference information is generally difficult to obtain, and has used its limitation
Property；Lucky imaging technique can be used in expansion target, but the precision that image recovers is to a certain extent dependent on acquisition " good fortune
The probability of fortune picture ", suitable for image-forming condition of turbulent flow when smaller.According to system point spread function whether, it is known that can be by convolution class
Method is divided into deconvolution (known), blind deconvolution (unknown) and Myopic uncoilings (knowable to fraction but unreliable).Deconvolution Method
Generally to use wave front detector, the requirement to hardware is higher.In actual conditions, system point spread function is often difficult to really
It is fixed.Therefore, in the case where neither knowing that dreamboat does not know dot system point spread function again, it is not necessary to which any priori is known
Know, the blind deconvolution algorithm low to system requirements is used widely, but its constringency performance also needs to improve, in low photon water
Flat or in the case that noise is larger, algorithm is often not sufficiently stable, very sensitive to noise.
Above-mentioned several image post-processing methods require that imaging system synthesis point expands corresponding to several short exposed images mostly
It is identical to dissipate function (including atmospheric transfer function and optical system transfer function), i.e., within the time that turbulent flow is freezed, completes several
The collection and processing of short exposed images, thus it is high to the requirement of real-time of algorithm.As speckle is imaged, lucky imaging, extension target
Imaging, but all be present the problems such as computationally intensive, poor real in these methods, be only applicable to real-time without desired occasion.
The content of the invention
The technical problems to be solved by the invention are in view of the shortcomings of the prior art, there is provided one kind adapts to atmospheric turbulance dynamic
The multi-frame self-adaption optical image restoration methods of change.
The technical problems to be solved by the invention are realized by following technical scheme.The present invention is a kind of adaptation
The multi-frame self-adaption optical image restoration methods of atmospheric turbulance dynamic change, are characterized in, this method includes：
(1) multiple system point spread functions after adaptive optics correction at different moments, i.e. system under turbulent-flow conditions are gathered
Point target imaging, multiple images are fourier transformed and are transformed into frequency domain, analysis at different moments correct after system point spread function
Proximity and heterogeneite between number；Use the condition of the related Silvester Sylvester matrixes of different point spread functions
Number selects to provide reference come the degree of closeness between judging for regular terms；
(2) the astronomical target image or spatial target images after adaptive optics corrects at different moments are gathered, establishes and solves
The linear equation of system point spread function, image recovery problem is converted into linear solution；When the size of image is larger, matrix
Dimension is higher, is solved using the method based on Fast Fourier Transform (FFT) FFT；
(3) based on the heterogeneite degree and the hangover characteristic of point spread function between imaging noise, point spread function, addition
Regular terms is to obtain stable solution；Using least square as canonical bound term, point spread function after being corrected due to adaptive optics
Several spectrums generally has sparsity structure characteristic, uses l1Norm regularization model is handled between point spread function to a certain degree
Proximity and point spread function hangover；Establish the convex Optimized model of system point spread function solution；
(4) using the classical convex convex optimization method of Optimization Solution Algorithm for Solving, so as to solve system point spread function；
(5) objective function, adds regular terms, and deconvolution solves target to be observed；Pressed down very well using having to noise
The full variation regular terms of making, solution obtains the estimate of target to be observed, so as to recover image.
A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change of the present invention, it enters
The preferable technical scheme of one step is：Described convex optimized algorithm is using interior point method, projection subgradient algorithm or low-rank method.
A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change of the present invention, it enters
The preferable technical scheme of one step is：Gather astronomical target image or extraterrestrial target figure at different moments after adaptive optics corrects
Picture, the linear equation of solving system point spread function is established, such as formula (3)：
Image recovery problem is converted into linear solution；
Wherein subscript i and j represents that the i-th frame is imaged I respectivelyi(x, y) and jth frame are imaged IjThe sequence number of (x, y)；BiAnd BjPoint
It is not by being imaged Ii(x, y) and IjThe Teoplitz toeplitz matrixes that (x, y) is formed, the component of this toeplitz matrix are
Teoplitz block block toeplitz matrixes, T represent transposition；Hi(x, y) and Hj(x, y) is respectively that i-th and j two field pictures are corresponding
System point spread function.
A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change of the present invention, it enters
The preferable technical scheme of one step is：The linear equation of solving system point spread function is established, using least square (4) as just
Then bound term：
Wherein * is product calculation, | | | |2For l2Norm operator, l2Norm also known as not Luo Beini crow this
Frobenius norm operators.
A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change of the present invention, it enters
The preferable technical scheme of one step is：Use l1Norm regularization model is a certain degree of close between point spread function to handle
The hangover of property and point spread function；Establish the convex Optimized model of system point spread function solution：Such as following formula：
Formula (5) is a convex optimization problem, in formula | | | |1For l1Norm operator, | | | |FFor F norms；λ is adjustment
Coefficient, fft () are Fourier transform, and s.t. is subject to abbreviation, and implication is " constrained in ", constraintsBy the energy of point spread function and 1 is constrained to, to avoid the appearance of null solution.
A kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change of the present invention, it enters
The preferable technical scheme of one step is：Using the full variation regular terms to noise with fine inhibitory action, solution obtains to be observed
The estimate of target, so as to recover image；Objective function is as follows：
WhereinFor regular terms,It is symbol of differentiating, O (x, y) is observed object, solves above formula and obtains waiting to see
The estimate of target is surveyed, so as to recover image.
The principle of the present invention is as follows：
It is right when the transmission function of different passages meets relatively prime property in multichannel it was found from multichannel blind recognition knowwhy
Multiple fuzzy signals, can be to be attributed to a relatively easy blind identification problem, directly by simply handling
Connect and estimate the point spread function of system, then carry out deconvolution computing again and obtain high-resolution echo signal.It is astronomical
In target or extraterrestrial target imaging, due to the dynamic of atmospheric turbulance, imaging system corresponding to the image gathered at different moments is comprehensive
Chalaza spread function is different, and different systems synthesis point spread function can be regarded as different passages, just meet that multichannel is blind
The requirement of identification signal restoration methods, this point be exactly with existing astronomic graph as or spatial target images love your processing method
Fundamental difference.During noiseless, by taking two width figures as an example, observed object O (x, y), system point spread function H (x, y) and imaging I (x,
Y) following relation is met
WhereinFor convolution algorithm, subscript i and j represent that the i-th frame is imaged I respectivelyi(x, y) and jth frame are imaged Ij(x's, y)
Sequence number, Hi(x, y) and Hj(x, y) is respectively system point spread function corresponding to i-th and j two field pictures.Then
Equation does not contain original object information in (2).If multiple sampled datas of equation (2), one can be write out
On Hi(x, y) and HjThe linear equation of (x, y).The equation left side will be moved on on the right of equation (2), convolution algorithm is melted into matrix
The form of multiplication, a linear equation can be obtained：
Wherein BiAnd BjIt is by being imaged I respectivelyi(x, y) and IjTeoplitz (toeplitz) matrix that (x, y) is formed, this
The component of individual toeplitz matrix is Teoplitz block (block toeplitz) matrix, and T represents transposition.Can by mathematical theory
Know, matrix [B during noiselessi-Bj] zero singular value corresponding to singular vector be exactly Hi(x, y) and HjThe solution of (x, y), therefore can
To realize Hi(x, y) and HjThe complete recovery of (x, y).More generally, it is unusual corresponding to minimum singular value in the presence of noise
It is then the solution of the two to be worth vector.
Compared with prior art, the advantage of the invention is that：
1st, require that system point spread function is consistent, needs into corresponding to multiple blurred pictures of collection relative to existing method
For the collection for completing image in turbulent flow freeze-off time constant as system, method of the invention then takes full advantage of turbulent flow
Dynamic characteristic, the timeliness of IMAQ is not required.
2nd, it is computationally intensive, poor real to the requirement of real-time height and image recovery process of system relative to existing method
For problem, the present invention is not required the real-time of system, and image recovery process belongs to linear solution, and amount of calculation subtracts significantly
It is few.
3rd, the inventive method constantly alternately solves without being carried out to observed object and system point spread function.
4th, the image recovery method stability of the inventive method is strong, in the absence of divergence problem；Algorithm is succinct and directly perceived.
Brief description of the drawings
The flow chart of Fig. 1 the inventive method；
The instance graph of Fig. 2-6 the inventive method image restoration results；Wherein：Fig. 2 and Fig. 3 is fuzzy graph at different moments
Picture, the I in formula (1) is corresponded to respectivelyi(x, y) and Ij(x, y), directly being gathered by imaging system to fall, and Fig. 4 and Fig. 5 are to use this
The system point spread function figure that inventive method is recovered, solves to obtain, corresponds to respectively in formula (1) to the convex optimization method established
Hi(x, y) and Hj(x, y)；Fig. 6 is the target imaging figure O (x, y) recovered using the inventive method, based on convex Optimization Solution
Point spread function, obtained using deconvolution method.
Embodiment
Referring to the drawings, the concrete technical scheme of the present invention is further described, in order to which those skilled in the art enters
Understand the present invention to one step.
Embodiment 1, reference picture 1, a kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change,
This method includes：
(1) multiple system point spread functions after adaptive optics correction at different moments, i.e. system under turbulent-flow conditions are gathered
Point target imaging, multiple images are fourier transformed and are transformed into frequency domain, analysis at different moments correct after system point spread function
Proximity and heterogeneite between number；Judge that using the conditional number of the related Sylvester matrixes of different point spread functions
Degree of closeness between this, reference is provided for regular terms selection；
(2) the astronomical target image or spatial target images after adaptive optics corrects at different moments are gathered, establishes and solves
The linear equation of system point spread function, image recovery problem is converted into linear solution；When the size of image is larger, matrix
Dimension is higher, is solved using the method based on Fast Fourier Transform (FFT) FFT；
(3) based on the heterogeneite degree and the hangover characteristic of point spread function between imaging noise, point spread function, addition
Regular terms is to obtain stable solution；Using least square as canonical bound term, point spread function after being corrected due to adaptive optics
Several spectrums generally has sparsity structure characteristic, uses l1Norm regularization model is handled between point spread function to a certain degree
Proximity and point spread function hangover；Establish the convex Optimized model of system point spread function solution；
(4) using the classical convex convex optimization method of Optimization Solution Algorithm for Solving, so as to solve system point spread function；
(5) objective function, adds regular terms, and deconvolution solves target to be observed；Pressed down very well using having to noise
The full variation regular terms of making, solution obtains the estimate of target to be observed, so as to recover image.
Embodiment 2, a kind of multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change, by following step
It is rapid to carry out：
1st, the point target imaging of multiple point spread functions after adaptive optics correction under turbulent-flow conditions, i.e. system is gathered,
Multiple images are fourier transformed and are transformed into frequency domain, the proximity after analysis corrects at different moments between system point spread function
And heterogeneite.Between being judged using the conditional number of the related Sylvester matrixes of different point spread functions close to journey
Degree, reference is provided for regular terms selection.
2nd, the astronomical target image or spatial target images after adaptive optics corrects at different moments are gathered, establishes and solves system
The linear equation of system point spread function, such as formula (3).
Image recovery problem is converted into linear solution.When the size of image is larger, matrix dimension is higher, can use base
Solved in the method for Fast Fourier Transform (FFT) (FFT).
3rd, in view of the heterogeneite degree and the hangover characteristic of point spread function between imaging noise, point spread function, add
Add regular terms to obtain stable solution.The present invention is used as canonical bound term using least square (4) first.
The spectrum of point spread function generally has sparsity structure characteristic after being corrected due to adaptive optics, uses l1Norm canonical
Change model to handle the hangover of a certain degree of proximity and point spread function between point spread function.Establish system point expansion
Dissipate the convex Optimized model that function solves, such as following formula.
Formula (5) is a convex optimization problem, and the bound term in formula avoids the appearance of null solution.
4th, using the above-mentioned convex optimization method of convex Optimization Solution Algorithm for Solving of classics, so as to solve system point spread function.
Classical convex optimized algorithm can be interior point method (interior point methods), projection subgradient algorithm (projected
Sub-gradient method), low-rank method (low-rank parameterization) etc..
5th, obtained using full variation (total variation) regular terms to noise with fine inhibitory action, solution
The estimate of target to be observed, so as to recover image.Objective function is as follows：
Solution obtains the estimate of target to be observed, so as to recover image.
Fig. 2-6 is the instance graph that the inventive method recovers image.Fig. 2 and Fig. 3 is the fault image gathered at different moments, right
Answer different system point spread functions；Fig. 4 and Fig. 5 is the system point spread function solved；Fig. 6 is to be expanded using the point solved
Dissipate function and carry out the target imaging figure that deconvolution computing obtains.
Claims (5)
- A kind of 1. multi-frame self-adaption optical image restoration methods for adapting to atmospheric turbulance dynamic change, it is characterised in that this method Including：(1) point of the multiple system point spread functions, i.e. system under turbulent-flow conditions after adaptive optics correction at different moments is gathered Target imagings, multiple images are fourier transformed and are transformed into frequency domain, analysis at different moments correct after system point spread function it Between proximity and heterogeneite；Use the condition of the related Silvester Sylvester matrixes of different system point spread function Number selects to provide reference come the degree of closeness between judging for regular terms；(2) the astronomical target image or spatial target images after adaptive optics corrects at different moments are gathered, establishes solving system The linear equation of point spread function, image recovery problem is converted into linear solution；When the size of image is larger, matrix dimension It is higher, solved using the method based on Fast Fourier Transform (FFT) FFT；(3) based on the heterogeneite degree and the hangover characteristic of system point spread function between imaging noise, system point spread function, Regular terms is added to obtain stable solution；Using least square as canonical bound term, system after being corrected due to adaptive optics The spectrum of point spread function generally has sparsity structure characteristic, uses l1Norm regularization model come processing system point spread function it Between a certain degree of proximity and system point spread function hangover；Establish the convex optimization mould of system point spread function solution Type；(4) using the classical convex convex optimization method of Optimization Solution Algorithm for Solving, so as to solve system point spread function；Described warp The convex Optimization Solution algorithm of allusion quotation is using interior point method, projection subgradient algorithm or low-rank method；(5) objective function, adds regular terms, and deconvolution solves target to be observed；Suppress to make very well using having noise Full variation regular terms, solution obtains the estimate of target to be observed, so as to recover image.
- A kind of 2. multi-frame self-adaption optical image recovery side for adapting to atmospheric turbulance dynamic change according to claim 1 Method, it is characterised in that：Astronomical target image or spatial target images at different moments, foundation after adaptive optics corrects is gathered to ask The linear equation of system point spread function is solved, such as formula (3)：<mrow> <mo>&lsqb;</mo> <msub> <mi>B</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>B</mi> <mi>j</mi> </msub> <mo>&rsqb;</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>H</mi> <mi>j</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>Image recovery problem is converted into linear solution；Wherein (x, y) is plane of delineation coordinate, and subscript i and j represent that the i-th frame is imaged I respectivelyi(x, y) and jth frame are imaged Ij(x, y) Sequence number；BiAnd BjIt is by being imaged I respectivelyi(x, y) and IjThe Teoplitz toeplitz matrixes that (x, y) is formed, this Top profit Hereby the component of matrix is Teoplitz block block toeplitz matrixes, and T represents transposition；Hi(x, y) and Hj(x, y) is respectively i-th With j two field pictures corresponding to system point spread function.
- A kind of 3. multi-frame self-adaption optical image recovery side for adapting to atmospheric turbulance dynamic change according to claim 2 Method, it is characterised in that：The linear equation of solving system point spread function is established, is constrained using least square (4) as canonical ：<mrow> <msub> <mi>min</mi> <mrow> <msub> <mi>H</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>H</mi> <mi>j</mi> </msub> </mrow> </msub> <mo>|</mo> <mo>|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>I</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>H</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>I</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>Wherein * is product calculation, | | | |2For l2Norm operator, l2This black Frobenius model of norm also known as not Luo Beini Number.
- A kind of 4. multi-frame self-adaption optical image recovery side for adapting to atmospheric turbulance dynamic change according to claim 3 Method, it is characterised in that：Use l1A certain degree of proximity that norm regularization model comes between processing system point spread function With the hangover of system point spread function；Establish the convex Optimized model of system point spread function solution：Such as following formula：<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>min</mi> <mrow> <msub> <mi>H</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>H</mi> <mi>j</mi> </msub> </mrow> </msub> <mo>|</mo> <mo>|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>I</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>H</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>I</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>F</mi> </msub> <mo>+</mo> <mi>&lambda;</mi> <mo>|</mo> <mo>|</mo> <mi>f</mi> <mi>f</mi> <mi>t</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mi>j</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>|</mo> <msub> <mo>|</mo> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <munder> <mi>&Sigma;</mi> <mi>i</mi> </munder> <munder> <mi>&Sigma;</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </munder> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>Formula (5) is a convex optimization problem, in formula | | | |1For l1Norm operator, fft () are Fourier transform, | | | |FFor F norms；λ is regulation coefficient, and s.t. is subject to abbreviation, and implication is " constrained in ", constraintsBy the energy of system point spread function and 1 is constrained to, to avoid the appearance of null solution.
- A kind of 5. multi-frame self-adaption optical image recovery side for adapting to atmospheric turbulance dynamic change according to claim 4 Method, it is characterised in that：Using the full variation regular terms to noise with fine inhibitory action, solution obtains estimating for target to be observed Evaluation, so as to recover image；Objective function is as follows：<mrow> <mi>min</mi> <mo>|</mo> <mo>|</mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi>I</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>I</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>*</mo> <mi>O</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>|</mo> <msup> <msub> <mo>|</mo> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mo>|</mo> <mo>&dtri;</mo> <mi>O</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>WhereinFor regular terms,It is symbol of differentiating, O (x, y) is observed object, solves above formula and obtains mesh to be observed Target estimate, so as to recover image.
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