CN101685158B - Hidden Markov tree model based method for de-noising SAR image - Google Patents

Abstract

The invention discloses a hidden Markov tree model based method for de-noising an SAR image, which relates to the field of image processing, and mainly solves the problems that details and texture information of an image are smoothed and an equivalent noise level in a homogeneous region is low. The method comprises the following steps: (1) carrying out logarithmic transformation and Contourlet decomposition; (2) carrying out HMT modeling on Contourlet coefficients, and training the model; (3) correcting the Contourlet coefficients by using estimated parameters; (4) establishing a background hidden Markov training model, correcting the coefficients again by using the estimated parameters, and carrying out Contourlet inverse transformation and exponential transformation to acquire a primary de-noised image; (5) de-noising differential images to obtain a secondary de-noised image; and (6) combining the two de-noised images, and carrying out rotary translation on the combined image to acquire and output a final de-noised image. The method well maintains the details and texture information of the SAR image, reduces speckle noise in the homogeneous region of the SAR image, and can be used for de-noising the SAR image.

Description

SAR image de-noising method based on concealed Markov tree model

Technical field

The invention belongs to technical field of image processing, particularly a kind of method that relates to the SAR image denoising can be used for the denoising to SAR image, natural image and medical image.

Background technology

Synthetic-aperture radar SAR is as the active radar; Have the advantages that the illumination of not receiving, weather condition influence; Can round-the-clock, round-the-clock earth observation; Can also obtain information through the face of land and vegetation, obtain using widely in agricultural, forestry, geology, environment, the hydrology, ocean, disaster, mapping and military field.In the SAR image because the existence of the speckle noise that the relevant effect of imaging scatterer scatter echo causes is unfavorable for the automatic analysis of image scene and the understanding of SAR image; Make the decipher work of image become difficult, especially very obvious to the influence at point target in the SAR image and edge.Therefore the removal of speckle noise to the subsequent treatment of SAR image for example rim detection be very important.

In recent years, SAR image coherent speckle noise suppresses technological develop rapidly, looks smoothing technique and imaging back filtering technique two big classes before can being divided into imaging more, and airspace filter and transform domain method have obtained studying widely in the post-processing technology.Airspace filter such as Lee; Gamma-MAP etc. carry out the simple easy to understand of denoising implementation procedure to the SAR image, can effectively weaken The noise to a certain extent, but this type of wave filter image had been produced cross smoothing effect in various degree all; Make image blur, detailed information is lost seriously.In recent years; The method of transform domain also is developed; Wherein become the focus of research based on method of wavelet; Like people such as Crouse wavelet transformation and HMM are connected, proposed the wavelet domain concealed Markov model, opened up the multi-scale transform domain statistical signal and handled this new research field.But for having the unusual objective function of wire, the edge in the image for example, wavelet coefficient is no longer sparse, so small echo can not keep the detailed information in the image well in the SAR image denoising.

In order to solve the problem that the wavelet basis function isotropy is brought; People such as Donoho have proposed multi-scale geometric analysis; It can effectively represent and handle the higher-dimension singular function, and has obtained using widely in Flame Image Process, and Contourlet is wherein a kind of analysis tool.Along with the development of multi-scale geometric analysis, numerous scholars have proposed HMT model and the HMT model of various improvement under the multiple dimensioned geometric transformation territory.The Contourlet conversion except the advantage with wavelet transformation, also has multi-direction and anisotropic characteristics as new multiple dimensioned signal indication method.Statistical nature with the Contourlet coefficient is the basis; Po and Do have proposed the HMT model that the Contourlet territory is used for image denoising; The Contourlet domain HMT model is compared with the small echo domain HMT model; Not only can describe the correlativity between yardstick, and can describe the correlativity of coefficient between different directions.But this model has only been caught the dependence in the small yardstick, and when being used for the SAR image denoising, some details, texture information are by level and smooth, and the noise remove in the homogeneous region thoroughly causes the homogeneous region equivalent number low inadequately, and denoising effect is unsatisfactory.

Summary of the invention

The objective of the invention is to overcome details, the texture information that above-mentioned existing method exists and smoothly reached the low shortcoming of homogeneous region equivalent number, propose a kind of SAR image de-noising method based on the hidden Markov tree-model,

Reaching denoising effect more completely, and well keep the details and the texture information of image.

Technical scheme of the present invention is: be the basis all to have very big correlativity between Contourlet coefficient neighborhood and between yardstick; Utilize hidden Markov tree-model HMT model to come the dependence between the capture coefficient yardstick; Utilize background HMM CHMM to come the dependence in the capture coefficient neighborhood; Combine HMT and CHMM and set up the improved HMT statistical model in Contourlet territory, model structure is as shown in Figure 3, with this statistical model the SAR image is carried out denoising.Concrete performing step is following:

(1) input SAR image is carried out log-transformation and Contourlet conversion successively, obtain the Contourlet conversion coefficient of different directions;

(2) adopt folk prescription respectively the Contourlet coefficient of different directions to be carried out HMT modeling between yardstick to transmitting HMT model and multi-direction transmission HMT model;

(3) with of the HMT model training of EM algorithm to setting up; Obtain average, variance, state transition probability and state probability optimal estimation parameter; And utilize this optimal estimation parameter the Contourlet coefficient to be carried out atrophy according to Bayes's minimum mean square error criterion, initially there is not the spot coefficient;

(4) initial no spot coefficient is set up the background HMM CHMM in the yardstick; Utilize the EM algorithm that the CHMM in the yardstick is trained; Obtain the estimated parameter in the yardstick, utilize the estimated parameter in the yardstick that initial no spot coefficient is carried out atrophy, finally do not had the spot coefficient;

(5) final no spot coefficient is carried out Contourlet inverse transformation and exponential transform successively, obtain preliminary denoising image I 1;

(6) adopt the anisotropy method of diffusion that the error image of original SAR image and denoising image I 1 is carried out denoising, obtain second denoising image I 2;

(7) with modulus maximum method preliminary denoising image I 1 is merged with second denoising image I 2;

(8) image after merging is rotated translation with the CycleSpinning method and handles, obtain final denoising image, and output.

The present invention compared with prior art has following advantage:

1) the present invention has made full use of the correlativity between the Contourlet coefficient

Traditional HMT denoising method has only been considered correlativity in the small yardstick of Contourlet coefficient, does not make full use of the correlativity between the coefficient; Cause some details, the texture information quilt of denoising image level and smooth; It is not thorough that picture noise is removed, and equivalent number is lower, and the improvement HMT algorithm that the present invention proposes has overcome the shortcoming that the coefficient correlativity does not make full use of; The correlativity of utilized between Contourlet coefficient yardstick, yardstick is interior and direction is interior has obtained denoising result preferably.

2, the invention provides better coherent spot and suppress the result

Traditional airspace filter image had been produced in various degree crossed smoothing effect, and detailed information is lost seriously, traditional Contourlet territory Denoising Algorithm, and owing to the non-translation invariance of Contourlet conversion, cut can occur is pseudo-Gibbs phenomenon; The present invention adopts the anisotropy method of diffusion that error image is carried out denoising, has replenished the image detail information of losing in the denoising, makes the minutia of fused image also obtain keeping preferably; Introduced CycleSpinning, overcome the pseudo-Gibbs phenomenon that the non-translation invariance of Contourlet conversion brings, make noise remove in the homogeneous region thoroughly, be used to estimate after the denoising what equivalent number of noise remove in the homogeneous region and be improved.

The result of emulation experiment shows that the inventive method has better kept the details and the texture information of image, and noise remove is more thorough in the homogeneous region.

Description of drawings

Fig. 1 is a main operating process synoptic diagram of the present invention;

Fig. 2 is a HMT illustraton of model between Contourlet transform domain yardstick of the present invention;

Fig. 3 is the improved Contourlet transform domain of a present invention HMT illustraton of model;

Fig. 4 is to SAR1 image denoising effect comparison diagram with the present invention and existing method;

Fig. 5 is to SAR2 image denoising effect comparison diagram with the present invention and existing method.

Embodiment

With reference to Fig. 1, concrete performing step of the present invention is following:

Step 1: the SAR image is made log-transformation, multiplicative noise is converted into additive white Gaussian noise handles: logy=log z+log x, wherein y representes to import the SAR image, and z representes noise image; X representes the not image of noisy, adopts direction two prescriptions inequality to wave filter, and this two prescription is respectively 4 to the direction of wave filter; 4,4 and 4,8; 8, the data after log-transformation are carried out Contourlet decompose, the Contourlet conversion coefficient that obtains is respectively y ₁And y ₂

Step 2: to conversion coefficient y ₁Set up folk prescription to transmitting the HMT model, shown in Fig. 2 (a), black square is a father node, and four short side shapes are its child nodes, and the parameter set of this model is Θ ₁To conversion coefficient y ₂Set up multi-direction transmission HMT model, shown in Fig. 2 (b), black square is a father node, and four short side shapes are its child nodes, and four node that each father node is corresponding are distributed in two different direction subbands, and the parameter set of this model is Θ ₂

Step 3: to two parameter set Θ of the original state probability that comprises average, variance, state transition probability and root node ₁And Θ ₂Carry out initialization, again with the EM algorithm to folk prescription to transmitting HMT model and the training of multi-direction transmission HMT model, obtain Θ ₁And Θ ₂The optimal estimation parameter; Utilize this optimal estimation parameter according to Bayes's minimum mean square error criterion the Contourlet coefficient to be carried out atrophy, initially do not had the spot coefficient, the atrophy formula is following:

${\hat{c}}_{j,k,i}=\underset{m=\left\{\mathrm{1,2}\right\}}{\mathrm{\Σ}}p(S_{j,k,i}=m)\×\frac{{\mathrm{\σ}}_{j,k,i,m}^2}{{\mathrm{\σ}}_{j,\mathrm{speckle}}^2+{\mathrm{\σ}}_{j,k,i,m}^2}{c}_{s_{j,k,i}}$

Wherein m representes the state of node, and value is 1 or 2, p (S _{J, k, i}=m) be the j yardstick, k direction, the node state probability of i position, σ _{J, k, i, m} ²Be that state is m, j yardstick, k direction, the variance of the node of i position, σ _{J, speckle} ²Be the noise variance of yardstick j,

Be the j yardstick, the k direction, the Contourlet coefficient of i position,

It is the initial no spot coefficient after the atrophy.

Step 4: initial no spot coefficient is set up the background HMM CHMM in the yardstick, and as shown in Figure 3, the node of black is the Contourlet conversion coefficient; The circular node of white is the latent state variable of the coefficient of Contourlet, and status number is 2, and the rhombus node is the background variable of the coefficient of Contourlet; The state of corresponding 4 node of the state of a father node; The j yardstick, k direction, the background v of the node of i position _{J, k, i}Be by the decision of the local energy mean value of eight nodes in the neighborhood, as shown in the formula:

$v_{j,k,i}=\left\{\begin{array}{c}1,{\mathrm{\λ}}_{j,k,i}^2>{\mathrm{\δ}}_j^2\\ 0,\mathrm{others}\end{array}\right.$

Wherein, λ _{J, k, i} ²Be coefficient c _{J, k, i}Eight fields in the local energy mean value of Contourlet coefficient, δ _j ²It is the average energy of coefficient in the yardstick j; Utilize the EM algorithm that the CHMM in the yardstick is trained; The initial parameter that average, variance and the state probability optimal estimation parameter that earlier step (3) is obtained trained as CHMM; Train then; Obtain the estimated parameter in the yardstick, utilize the estimated parameter in the yardstick that initial no spot coefficient is carried out atrophy, finally do not had the spot coefficient.

Step 5: the final no spot coefficient that (4) step was obtained carries out Contourlet inverse transformation and exponential transform successively, obtains preliminary denoising image I 1.

Step 6: with the anisotropy method of diffusion error image of original SAR image and denoising image I 1 is carried out denoising, obtain second denoising image I 2.Utilize gradient operator to distinguish edge and noise to error image, then error image is spread denoising, diffusion denoising equation is following:

$v_{i,j}^{k+1}=v_{i,j}^k+\mathrm{\τ}\underset{\underset{(r,s)\≠\left(\mathrm{0,0}\right)}{r,s=-1}}{\overset{1}{\mathrm{\Σ}}}\frac{g\left({\sqrt{2}}^{1-\left|r\right|-\left|s\right|}\left(\right|v_{i+r,j+s}-v_{i,j}\left|\right)\right)(v_{i+r,j+s}-v_{i,j})}{r^2+s^2}$

V wherein _{I, j} ^kBe to carry out anisotropy diffusion error image pixel value before, v _{I, j} ^K+1Be to carry out the error image pixel value after the anisotropy diffusion one time; τ is a scale parameter; Span is: 0＜τ＜1/6, and r, s are the position of impact point neighborhood coefficient coordinate with respect to impact point; G is the diffusivity function, and the diffusivity function that in following formula, uses is Perona-Malik diffusivity: g (x)=1/ (1+x ²/ rr ²), rr is the reduced parameter in the diffusivity.

For choosing of rr, according to repeatedly experiment, through finding that relatively it is best that rr gets 50 o'clock effects.

Step 7: with modulus maximum method preliminary denoising image I 1 is merged with second denoising image I 2, concrete steps are following:

(7a) preliminary denoising image I 1 and second denoising image I 2 are carried out Contourlet respectively and decompose, the filter direction of selection is 4,4,4, obtains the Contourlet conversion coefficient of preliminary denoising image I 1 and second denoising image I 2;

(7b) the details item to the Contourlet conversion coefficient of two denoising images carries out the details item that modulus maximum merges the Contourlet coefficient that obtains fused images, fuzzy of the Contourlet conversion coefficient of two denoising images directly addition obtain fuzzy of fused images Contourlet coefficient;

(7c) the Contourlet coefficient to fused images carries out the image after the Contourlet inverse transformation obtains merging.

Step 8: the image after merging is rotated translation with the CycleSpinning method handles, concrete steps are following:

(8a) obtain a different denoising result s through every group of translational movement on the row and column direction _{I, j}:

s _i，j＝S _-i，-j(T ^-1(Λ[T(S _i，j(x))]))

Wherein, S is the circulation translation operator, and T is the Contourlet transformation operator, T ^-1Be Contourlet inverse transformation operator, Λ is a Contourlet territory HMT denoising operator, subscript-i, and-j, i and j are respectively the translational movement on the row and column direction;

(8b) all denoising results are carried out linear averaging, the denoising result of the pseudo-Gibbs phenomenon that is inhibited:

$s_t=\frac{1}{K_1{K}_2}\underset{i=0,j=0}{\overset{K1,K2}{\mathrm{\Σ}}}{s}_{i,j}$

Wherein, K ₁, K ₂Represent the maximal translation amount on the row and column direction respectively, be 4.

Step 9: the denoising result s that obtains in the step (8) _tAs final denoising result, and output.

Below provide the The simulation experiment result analysis, to further specify effect of the present invention:

1, simulated conditions

Two width of cloth testing SA R images among the present invention all are interceptings on the Horse track SAR image that obtains from U.S. Sandia Labs website, and size is 256 * 256, respectively called after SAR1 and SAR2.Adopt Gamma_MAP filtering method, Lee filtering method, wavelet field HMT denoising method, Contourlet territory HMT denoising method and the inventive method 1 to improve folk prescription and SAR1 and SAR2 image are carried out denoising to transmission HMT model and the multi-direction transmission HMT model of the inventive method 2 improvement.

2, analysis of simulation result

The result of emulation such as Fig. 4 and Fig. 5, wherein:

Fig. 4 (a) is the SAR1 original image,

Fig. 4 (b) be the SAR1 image through the filtered denoising image of Gamma_MAP,

Fig. 4 (c) be the SAR1 image through the filtered denoising image of Lee,

Fig. 4 (d) is the denoising image of SAR1 image after wavelet field HMT denoising,

Fig. 4 (e) is the denoising image of SAR1 image after the HMT denoising of Contourlet territory,

Fig. 4 (f) improves the denoising image of folk prescription after transmitting the denoising of HMT model method for the SAR1 image through the present invention,

Fig. 4 (g) is the denoising image of SAR1 image after the present invention improves the denoising of multi-direction transmission HMT model method.

Fig. 5 (a) is the SAR2 original image,

Fig. 5 (b) be the SAR2 image through the filtered denoising image of Gamma MAP,

Fig. 5 (c) be the SAR2 image through the filtered denoising image of Lee,

Fig. 5 (d) is the denoising image of SAR2 image after wavelet field HMT denoising,

Fig. 5 (e) is the denoising image of SAR2 image after the HMT denoising of Contourlet territory,

Fig. 5 (f) improves the denoising image of folk prescription after transmitting the denoising of HMT model method for the SAR2 image through the present invention,

Fig. 5 (g) is the denoising image of SAR2 image after the present invention improves the denoising of multi-direction transmission HMT model method.

Can find out from Fig. 4 (b) and Fig. 4 (c), Gamma MAP filtering and Lee filtering to the speckle noise of SAR1 image remove better, but the SAR1 edge of image is by fuzzy, image detail information is lost more serious; By Fig. 4 (d) and Fig. 4 (e), the removal of noise can find out that Contourlet territory HMT method is better than wavelet field HMT method in the homogeneous region, but noise remove all thorough inadequately in the homogeneous region of these two kinds of methods after to the SAR1 image denoising.From Fig. 4 (f) and Fig. 4 (g), can find out after the SAR1 denoising the very thorough of noise remove in the homogeneous region, detailed information, marginal information and texture information have obtained good maintenance.

As can beappreciated from fig. 5, the present invention has more kept SAR2 edge of image and grain details when thoroughly removing noise; It is thus clear that the present invention is in details, there is sizable advantage texture maintenance aspect.

Evaluation index to SAR image denoising performance comprises equivalent number ENL, average and average ratio, and the definition of these evaluation criterions is following:

(1) equivalent number (ENL): $\mathrm{ENL}=A\frac{u^2}{{\mathrm{\σ}}^2}$

Wherein u and σ are respectively average and the standard deviation that coherent spot suppresses certain homogeneous region in the image of back, for intensity image A=1, for magnitude image A=4/ π-1.It is a kind of index of weighing coherent spot relative intensity in the image, and it is dark more that coherent spot suppresses degree, and equivalent number is big more.Select to be of a size of 40 * 40,50 * 50 for the SAR1 image, 40 * 40 homogeneous region indicates the zone as the test data of calculating ENL like the square box among Fig. 4 (a); Select to be of a size of 30 * 30,30 * 30 for the SAR2 image, 30 * 30 homogeneous region indicates the zone as the test data of calculating ENL like the square box among Fig. 5 (a).

(2) average of image: the mean flow rate of average reflection image, the image average before and after coherent spot suppresses will be consistent basically.

(3) average ratio: the ratio of image D average after original image I suppresses with coherent spot,

$\mathrm{\γ}=\frac{\mathrm{mean}\left(D\right)}{\mathrm{mean}\left(I\right)}$

Ratio γ is the statistics amplitude 1 of pure speckle noise under the ideal situation.Actual γ value and 1 differs big more, shows that the radiancy distortion is more severe.

Evaluation index behind SAR1 and the SAR2 image denoising is shown in table 1 and table 2:

The evaluation index that table 1 pair SAR1 image carries out denoising image after the denoising relatively

The SAR1 image	Zone	1 ENL	Zone 2 ENL	Zone 3 ENL	Average	The average ratio
							Gamma_MAP	217.1888	122.2114	107.9559	83.1247	0.9913
Lee filtering	217.1888	122.2114	107.9559	83.6654	0.9978
						Small echo HMT	78.8185	58.1684	36.5429	80.3143	0.9577
Contourlet?HMT	120.2160	80.0639	52.2560	79.9457	0.9533
						The inventive method 1	327.6217	158.3174	140.7742	83.7811	0.9992
The inventive method 2	338.5326	153.0471	143.3387	83.7741	0.9991

[0084]The evaluation index that table 2 pair SAR2 image carries out denoising image after the denoising relatively

The SAR2 image	Zone	1 ENL	Zone 2 ENL	Zone 3 ENL	Average	The average ratio
							Gamma_MAP	66.5108	222.8523	69.6148	72.7787	0.9805
Lee filtering	66.5108	222.8523	69.6148	73.7958	0.9942
						Small echo HMT	35.2699	71.6358	26.7679	71.1371	0.9584
Contourlet?HMT	39.7765	89.1000	30.8420	70.7065	0.9526
						The inventive method 1	83.4655	360.5115	71.6091	74.1435	0.9989
The inventive method 2	82.3402	334.9756	73.9066	74.1274	0.9987

Can find out from table 1 and table 2:

(a) all the equivalent number value than other several kinds of classical ways is big for equivalent number value of the present invention, and it is darker to show that the present invention suppresses degree to SAR image coherent speckle noise, and it is more thorough that speckle noise is removed, and denoising effect is better;

(b) average of average of the present invention and corresponding original image is basic identical, shows that the mean flow rate of image remains unchanged basically after the denoising of the present invention;

(c) average of the present invention shows that than little than other classical way with 1 gap the radiancy distortion is very little in the denoising process of the present invention, and detailed information is better with the texture information maintenance.

To sum up, the inventive method had both kept minutia and texture informations such as point target and the edge thereof in the SAR image preferably, had also significantly reduced the speckle noise in the SAR image homogeneous region.

Claims (6)

1. the SAR image de-noising method based on concealed Markov tree model HMT comprises the steps:

(1) input SAR image is carried out log-transformation and Contourlet conversion successively, obtain the Contourlet conversion coefficient of different directions;

(2) adopt folk prescription respectively the Contourlet coefficient of different directions to be carried out HMT modeling between yardstick to TRANSFER MODEL and multi-direction TRANSFER MODEL;

(3) with of the HMT model training of EM algorithm to setting up; Obtain average, variance, state transition probability and state probability optimal estimation parameter; And utilize this optimal estimation parameter the Contourlet coefficient to be carried out atrophy according to Bayes's minimum mean square error criterion, initially there is not the spot coefficient;

(4) initial no spot coefficient is set up the background HMM CHMM in the yardstick; Utilize the EM algorithm that the CHMM in the yardstick is trained; Obtain the estimated parameter in the yardstick, utilize the estimated parameter in the yardstick that initial no spot coefficient is carried out atrophy, finally do not had the spot coefficient;

(5) final no spot coefficient is carried out Contourlet inverse transformation and exponential transform successively, obtain preliminary denoising image I 1;

(6) adopt the anisotropy method of diffusion that the error image of original SAR image and denoising image I 1 is carried out denoising, obtain second denoising image I 2;

(7) with modulus maximum method preliminary denoising image I 1 is merged with second denoising image I 2;

(8) image after merging is rotated translation with the CycleSpinning method and handles, obtain final denoising image, and output.

2. SAR image de-noising method according to claim 1, the wherein Contourlet conversion described in the step (1) adopts direction two prescriptions inequality to wave filter; This two prescription is respectively 4,4 to the direction of wave filter, and 4 and 4; 8,8, the Contourlet conversion coefficient that obtains is respectively y ₁And y ₂

3. SAR image de-noising method according to claim 1, wherein the described employing folk prescription of step (2) carries out HMT modeling between yardstick to the Contourlet coefficient of different directions respectively to TRANSFER MODEL and multi-direction TRANSFER MODEL, is to conversion coefficient y ₁Set up folk prescription to TRANSFER MODEL, the parameter set of this model is Θ ₁To conversion coefficient y ₂Set up multi-direction TRANSFER MODEL, the parameter set of this model is Θ ₂, these two kinds of parameter set Θ ₁And Θ ₂Interior parameter comprises the original state probability of average, variance, state transition probability and root node.

4. according to claim 1 or 3 described SAR image de-noising methods, wherein described in the step (3) with of the HMT model training of EM algorithm to setting up, be earlier to comprising average and variance, state transition probability, two parameter set Θ of the original state probability of root node ₁And Θ ₂Carry out initialization, again with the EM algorithm to the training of HMT model, obtain Θ ₁And Θ ₂The optimal estimation parameter.

5. SAR image de-noising method according to claim 1; Wherein the EM algorithm that utilizes described in the step (4) is trained the CHMM in the yardstick, is the initial parameter that average, variance, state transition probability and state probability optimal estimation parameter that step (3) obtains are trained as CHMM.

6. SAR image de-noising method according to claim 1 wherein is rotated the translation processing to the image after merging with the CycleSpinning method described in the step (8), carries out as follows:

(6a) obtain a different denoising result s through every group of translational movement on the row and column direction _{I, j}:

s _i，j＝S _-i，-j(T ^-1(Λ[T(S _i，j(x))]))

Wherein, S is the circulation translation operator, and T is the Contourlet transformation operator, T ^-1Be Contourlet inverse transformation operator, Λ is a Contourlet territory HMT denoising operator, subscript-i, and-j, i and j are respectively the translational movement on the row and column direction;

(6b) all denoising results are carried out linear averaging, the denoising result S of the pseudo-Gibbs phenomenon that is inhibited _t:

$s_t=\frac{1}{K_1{K}_2}\underset{i=0,j=0}{\overset{K1,K2}{\mathrm{\Σ}}}{s}_{i,j}$

Wherein, K ₁, K ₂Represent the maximal translation amount on the row and column direction respectively.

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