CN102547073B - Self-adaptive threshold value video denoising method based on surfacelet conversion - Google Patents

Abstract

The invention discloses a self-adaptive threshold value video denoising method based on surfacelet conversion, which is mainly used for solving the problems of edge blur and insufficient denoising in a video image existing in the prior art. The implementation process of the method comprises the following steps of: (1) inputting a noise-containing video image, and performing surfacelet conversion on the video image; (2) calculating the initial threshold value of a sub-band coefficient in each layer along each direction after surfacelet conversion; (3) adjusting the initial threshold value by using the space neighborhood information of the coefficient to obtain a self-adaptive threshold value; (4) performing soft threshold value processing on the sub-band coefficient in each layer along each direction after surfacelet conversion by using the self-adaptive threshold value; and (5) performing surfacelet inverse conversion on the coefficient which is subjected to soft threshold value processing to obtain a denoised video image. Compared with the prior art, the method has the advantages that: the noise inhibiting capability on a video image is improved remarkably, detailed information in the video image and the smooth effect of a moving object are better kept, and the method can be applied to video image compression, video image texture detection, video image watermark extraction and target identification and tracking in the video.

Description

Adaptive threshold video denoising method based on surface wave conversion

Technical field

The invention belongs to image processing field, relate to image denoising, can be used for the denoising of video image and 3-D view.

Background technology

Video image usually can be subject to the impact of various noises in collection and transmission, makes video image quality variation, causes the important information in video image to be lost.When video image is processed or is applied, how to retain the useful information in video image, how the curved surface in captured video image is unusual, is that a focus is also a difficult point.

Minh N.Do and Martin Vetterli have proposed the method for a double-smoothing device group in calendar year 2001, thereby obtain having the rarefaction representation of the typical image of smooth contoured.The framework that it is comprised of profile fragment the expansion of image, so be called as Contourlet.Contourlet conversion is the image two-dimensional representation method of a kind of " really ", for image representation provide a kind of flexibly, that differentiate, part and expansion directivity more.But because down-sampling operation has been taked in Contourlet conversion, lack translation invariance, and there is 4/3 redundancy, can produce Gibbs phenomenon, make the image fault after denoising.The people such as Cunha, by conjunction with the tower decomposition of non-lower sampling and the DFB of non-lower sampling, have realized non-downsampling Contourlet conversion NSCT.NSCT has inherited the multiple dimensioned and multidirectional of Contourlet, also possesses translation invariant characteristic simultaneously, can effectively eliminate Gibbs phenomenon.But Contourlet conversion and non-downsampling Contourlet conversion all can only be processed 2D signal, can not be directly to three dimensional signal even multidimensional signal process.

Yue M.Lu and Mimh N.Do extend to higher-dimension by DFB, thereby form multi-dimensional direction bank of filters NDFB, combination by multi-Scale Pyramid and NDFB has proposed surface wave Surfacelet, the multiple dimensioned decomposition of Surfacelet conversion is defined in frequency domain, it can effectively catch and represent smooth surface Signal Singularity, there is the character such as multi-direction decomposition, high efficiency tree bank of filters, Perfect Reconstruction and end redundancy, be applicable to very much Video processing.

In above-mentioned transform domain, adopting threshold process is a kind of the most frequently used reasonable image de-noising method.Choosing of threshold value is usually the key in Threshold Filter Algorithms, and the too small meeting that threshold value is chosen causes removing fully noise; On the contrary, the conference of crossing that threshold value is chosen produced and strangled phenomenon signal, and the part important information of lossing signal, can cause the image after denoising to cross blooming.

In transform domain, conventional threshold method has the Visual shrinkage method that the people such as Donoho propose, the 3 σ methods that the people such as Q.Pan propose, the BayesShrink method based on minimum risk that M.Vetterli and B.Yu propose, the NormalShrink method that the people such as Lakhwinder Kaur propose.

Therefore above-mentioned threshold method is not considered direction structure information and spatial neighborhood information, can not obtain desirable denoising effect in details and the abundant video image denoising of background information.

Utilizing Surfacelet transfer pair video image to carry out in the process of denoising, traditional threshold method does not consider that the neighborhood correlation characteristic of coefficient in yardstick and direction structure information and noise are in the distribution characteristics of different scale, therefore noise can not be removed fully, the phenomenon such as cause that edge blurry and material particular information are strangled.

Summary of the invention

The object of the invention is to overcome above-mentioned existing methods shortcoming, proposed a kind of adaptive threshold video denoising method based on surface wave conversion, to improve the phenomenons such as soft edge and noise remove be inadequate, improve the denoising effect of video.

For achieving the above object, the present invention includes following steps:

(1) input noisy video image, and it is carried out to Surfacelet conversion, decomposing the number of plies is 4 layers, and every layer of corresponding direction number is respectively 192,192,48,12;

(2) coefficient in Surfacelet transform domain is calculated as follows to the initial threshold of each layer of all directions sub-band coefficients:

$T_{l,k}=\sqrt{\log (L_{l,k}/J)}{\left({\mathrm{\δ}}_{l,k}^n\right)}^2/{\mathrm{\δ}}_{l,k},$

Wherein: T _{l, k}be initial threshold corresponding to l layer k directional subband coefficient, l ∈ 1,2,3,4}, in layers 1 and 2 subband: k ∈ 1,2,3 ..., 192}, in the 3rd straton band: k ∈ 1,2,3 ..., 4 8}, in the 4th straton band: k ∈ 1,2,3 ..., 12};

L _{l, k}be the total number of coefficient in l layer k directional subband, J=4 is for decomposing total number of plies;

${\mathrm{\δ}}_{l,k}=\sqrt{\frac{1}{\mathrm{MNP}-1}\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\underset{p=1}{\overset{P}{\mathrm{\Σ}}}{(w_{l,k}(m,n,p)-\stackrel{\‾}{w_{l,k}})}^2}$ The signal standards that is l layer k directional subband is poor, wherein w _{l, k}(m, n, p) is the locational coefficient of l layer k directional subband mid point (m, n, p), m ∈ 1,2,3 ..., M}, n ∈ 1,2,3 ..., N}, p ∈ 1,2,3 ..., P},

be the average of l layer k directional subband coefficient, M, N, P are respectively subband length, width and height;

the noise criteria that is l layer k directional subband is poor, wherein

the noise criteria that is the 1st layer of k directional subband is poor, and the available intermediate value estimation technique is calculated;

(3) to initial threshold T _{l, k}adjust as follows, calculate adaptive threshold corresponding to each coefficient in each subband of each layer:

$T_{\mathrm{adapt}}(m,n,p)=T_{l,k}\frac{(\max \left(z\right)-z(m,n,p))(\mathrm{mean}\left(z\right)-\min \left(z\right))}{(z(m,n,p)-\min \left(z\right))(\max \left(z\right)-\mathrm{mean}\left(z\right))},$

Wherein: T _adapt(m, n, p) represents adaptive threshold corresponding to coefficient on l layer k directional subband mid point (m, n, p) position;

Mean (z) represents the average of z (m, n, p) in l layer k directional subband, min (z) represents the minimum value of z (m, n, p) in l layer k directional subband, max (z) represents the maximum of z (m, n, p) in l layer k directional subband;

for neighboring mean value, R is spatial neighborhood, and S is the coefficient number in spatial neighborhood R;

(4) utilize the adaptive threshold T obtaining in step (3) _adapt(m, n, p), carries out soft-threshold processing to each layer of all directions sub-band coefficients after Surfacelet conversion respectively;

(5) coefficient after soft-threshold processing is carried out to Surfacelet inverse transformation, obtain the video image after denoising.

The present invention compared with prior art has the following advantages:

1. the present invention is D S urfacelet conversion due to what adopt, compared with using the two-dimensional transforms such as small echo, Contourlet, NSCT more direct and convenient aspect video denoising, need not consider the motion compensation relation between each two field picture in video, the present invention simultaneously can effectively catch and represent smooth surface Signal Singularity, so can retain marginal information in video image and the smooth effect of moving object well;

2. the present invention is due to by selectively adding spatial information and the direction structure information of different neighborhoods, rather than all directions sub-band coefficients in transform domain is calculated to corresponding threshold value simply, therefore the present invention can calculate corresponding threshold value exactly to each coefficient of all directions subband, and therefore the noise for video has stronger inhibition ability.

Accompanying drawing explanation

Fig. 1 is realization flow figure of the present invention;

Fig. 2 is for calculating five kinds of selected neighbour structure figure of neighboring mean value in the present invention;

Fig. 3 is the comparison diagram after using the present invention and existing Lung biopsy to video Mobile denoising;

Fig. 4 is the video image PSNR value curve chart obtaining after to video Mobile denoising with the present invention and existing Lung biopsy.

Fig. 5 is the comparison diagram after using the present invention and existing Lung biopsy to video Coastguard denoising;

Fig. 6 is the video image PSNR value curve chart obtaining after to video Coastguard denoising with the present invention and existing Lung biopsy.

Embodiment

With reference to Fig. 1, specific embodiment of the invention process is as follows:

The noisy image of step 1. input, carries out Surfacelet conversion to it.

1a) input noisy video image, by the non-decimation filter group of triple channel, its frequency spectrum is divided into the subband of three hourglass shape;

1b) hourglass shape subband is decomposed by twice two-dimensional directional bank of filters travel direction respectively, decomposing the number of plies is 4 layers, and every layer of corresponding direction number is respectively 192,192,48,12.

Step 2, the initial threshold of all directions sub-band coefficients after calculating Surfacelet conversion.

2a) calculate the poor δ of signal standards of l layer k directional subband coefficient _{l, k}:

${\mathrm{\δ}}_{l,k}=\sqrt{\frac{1}{\mathrm{MNP}-1}\underset{m=1}{\overset{M}{\mathrm{\Σ}}}\underset{n=1}{\overset{N}{\mathrm{\Σ}}}\underset{p=1}{\overset{P}{\mathrm{\Σ}}}{(w_{l,k}(m,n,p)-\stackrel{\‾}{w_{l,k}})}^2},$

Wherein, l represents l layer, l ∈ 1,2,3,4}, k represents k directional subband, in layers 1 and 2 subband: k ∈ 1,2,3 ..., 192}, in the 3rd straton band: k ∈ 1,2,3 ..., 4 8}, in the 4th straton band: k ∈ 1,2,3 ..., 12}, w _{l, k}(m, n, p) is (m, n, p) locational coefficient in l layer k directional subband, m ∈ 1,2,3 ..., M}, n ∈ 1,2,3 ..., N}, p ∈ 1,2,3 ..., P},

be the average of l layer k directional subband coefficient, M, N, P are respectively subband length, width and height;

2b) noise criteria of the 1st layer of k directional subband coefficient of calculating is poor

${\mathrm{\&delta;}}_{1,k}^n\text{=median}\left(\right|{\mathrm{<figure-callout\; id="1"\; label="w"\; filenames="HDA0000128454550000031.png,HDA0000128454550000061.png"\; state="\{\{state\}\}">w</figure-callout>}}_{1,k}(m,n,p)\left|\right)/0.6745,$

Wherein, w _{1, k}(m, n, p) is the 1st layer of k directional subband coefficient, and median represents to ask coefficient intermediate value;

2c) noise criteria of calculating l layer k directional subband coefficient is poor

2d) calculate the initial threshold T of l layer k directional subband coefficient _{l, k}:

$T_{l,k}=\sqrt{\log (L_{l,k}/J)}{\left({\mathrm{\&delta;}}_{l,k}^n\right)}^2/{\mathrm{\&delta;}}_{l,k},$

Wherein: T _{l, k}be initial threshold corresponding to l layer k directional subband coefficient, L _{l, k}be the total number of coefficient in l layer k directional subband, J=4 is for decomposing total number of plies.

Step 3. is calculated the adaptive threshold of coefficient in each subband of each layer.

3a) according to the multi-direction characteristic of Surfacelet conversion, choose five kinds of neighborhood Q _i, { 1,2,3,4,5}, as shown in Figure 2, wherein, Fig. 2 (a) is neighborhood Q to five kinds of neighbour structures to i ∈ ₁structure chart, Fig. 2 (b) is neighborhood Q ₂structure chart, Fig. 2 (c) is neighborhood Q ₃structure chart, Fig. 2 (d) is neighborhood Q ₄structure chart, Fig. 2 (e) is neighborhood Q ₅structure chart, center blockage is selected coefficient, grey blockage is the neighbour coefficient that this coefficient is corresponding, calculates respectively the average u of these five kinds of neighborhoods _iwith standard deviation δ _i, and according to this average u _iwith standard deviation δ _icalculate neighborhood Selecting All Parameters ρ _i:

${\mathrm{\&rho;}}_i=\frac{{[u_i/{\mathrm{\&delta;}}_i]}^2}{1+{[u_i/{\mathrm{\&delta;}}_i]}^2},$

3b) calculate neighborhood and determine parameter:

from these five kinds of neighborhoods, choose neighborhood and determine that the corresponding neighborhood of parameter η is as satisfactory spatial neighborhood R;

3c) calculate neighboring mean value z (m, n, p):

$z(m,n,p)=\frac{1}{S}\underset{(m,n,p)\&Element;R}{\mathrm{\&Sigma;}}\left|w(m,n,p)\right|,$

Wherein, S is the coefficient number in spatial neighborhood R, and w (m, n, p) is (m, n, p) locational coefficient in subband;

3d) calculate adaptive threshold T _adapt(m, n, p):

$T_{\mathrm{adapt}}(m,n,p)=T_{l,k}\frac{(\max \left(z\right)-z(m,n,p))(\mathrm{mean}\left(z\right)-\min \left(z\right))}{(z(m,n,p)-\min \left(z\right))(\max \left(z\right)-\mathrm{mean}\left(z\right))},$

Wherein, mean (z) represents the average of z (m, n, p) in l layer k directional subband, min (z) represents the minimum value of z (m, n, p) in l layer k directional subband, max (z) represents the maximum of z (m, n, p) in l layer k directional subband.

Step 4. is utilized the adaptive threshold obtaining in step 3, and each layer of all directions sub-band coefficients after Surfacelet conversion carried out to soft-threshold processing by following formula, obtains the coefficient w ' after soft-threshold is processed:

$w^{\&prime;}=\left\{\begin{array}{cc}\mathrm{sign}\left(w\right)\&CenterDot;\left(\right|w\left|\right)-T_{\mathrm{adapt}}),& \left|w\right|\&GreaterEqual;T_{\mathrm{adapt}}\\ 0,& \left|w\right|<T_{\mathrm{adapt}}\end{array}\right.,$

Wherein, w is the coefficient of not processing through soft-threshold, | the amplitude that w| is w, sign (w) represents to ask the symbol of w, T _adaptrepresent adaptive threshold corresponding to w.

Coefficient w ' after step 5. pair soft-threshold is processed carries out Surfacelet inverse transformation, obtains the video image after denoising.

Effect of the present invention can further illustrate by following experiment:

1. experiment condition

Experiment simulation environment is: MATLAB R2008b, CPU AMD Athlon * 23.00GHz, internal memory 4G, Window XP Professional.

2. experiment content

NormalShrink method (NSCT-NS) in Visual shrinkage method (Visual-SH), BayesShrink threshold method (Bayes-SH), 3 σ hard-threshold denoising methods (ST-3 σ), non-downsampling Contourlet, 3D-CMST method based on hard-threshold function are applied to respectively in Surfacelet conversion video image is carried out to denoising, by the present invention and above-mentioned five kinds of denoising methods, compare.

Experiment 1: in the former video Mobile that is 192 * 192 * 192 in size as shown in Fig. 3 (a), adding standard deviation is 30 noise, adds video image after making an uproar as shown in Fig. 3 (b) is.

With the present invention and above-mentioned Lung biopsy, the noisy video image shown in Fig. 3 (b) is carried out to denoising, comparison diagram after denoising is as shown in Fig. 3 (c)-(h), wherein Fig. 3 (c) is the video image after the denoising of Visual-SH method, Fig. 3 (d) is the video image after the denoising of ST-3 σ method, Fig. 3 (e) is the video image after the denoising of Bayes-SH method, Fig. 3 (f) is the video image after the denoising of NSCT-NS method, Fig. 3 (g) is the video image after the denoising of 3D-CMST method, and Fig. 3 (h) is the video image with after denoising of the present invention.

With the present invention and above-mentioned Lung biopsy, the noisy video image shown in video Fig. 3 (b) is carried out to denoising, the video image PSNR value curve after denoising, as shown in Figure 4.

Experiment 2: in the former video Coastguard that is 192 * 192 * 192 in size as shown in Fig. 5 (a), adding standard deviation is 30 noise, adds video image after making an uproar as shown in Fig. 5 (b) is.

With the present invention and above-mentioned Lung biopsy, the noisy video image shown in Fig. 5 (b) is carried out to denoising, comparison diagram after denoising is as shown in Fig. 5 (c)-(h), wherein Fig. 5 (c) is the video image after the denoising of Visual-SH method, Fig. 5 (d) is the video image after the denoising of ST-3 σ method, Fig. 5 (e) is the video image after the denoising of Bayes-SH method, Fig. 5 (f) is the video image after the denoising of NSCT-NS method, Fig. 5 (g) is the video image after the denoising of 3D-CMST method, and Fig. 5 (h) is the video image with after denoising of the present invention.

With the present invention and above-mentioned Lung biopsy, the noisy video image shown in video Fig. 5 (b) is carried out to denoising, the video image PSNR value curve after denoising, as shown in Figure 6.

3. experimental result and analysis

From Fig. 3 (d) and Fig. 3 (g), can find out, the video image after the denoising obtaining by ST-3 σ method and 3D-CMST method there will be ringing; From Fig. 3 (c), Fig. 3 (e) and Fig. 3 (f), can find out, the video image after the denoising obtaining by Visual-SH, Bayes-SH and tri-kinds of methods of NSCT-NS has excessively smoothly caused soft edge; From Fig. 3 (h), can find out, with the present invention, can effectively suppress to obtain noise in video image, the video image after the denoising obtaining has retained details and marginal information better.

The PSNR value of each two field picture obtaining after to video Mobile denoising with the present invention as can be seen from Figure 4, will be all higher than other five kinds of denoising methods.

From Fig. 5 (a)-(h) can find out, all can there is motion blur phenomenon in the video image after the denoising obtaining by the present invention and above-mentioned five kinds of denoising methods, but the denoising rear video image obtaining with the present invention is subject to motion blur phenomenon to affect minimum, the effect obtaining in details and marginal information reservation is best, inhibition ability to noise is also the strongest, and other five kinds of denoising methods are to the noise remove in video image insufficient.Because Figure of description of the present invention only can provide static image, in actual video display process, with the present invention, can eliminate video image in the effect of smear and scintillation more outstanding.

The PSNR value of each frame video image obtaining after to noisy video Coastguard denoising with the present invention as can be seen from Figure 6, will be all higher than other five kinds of denoising methods.

With the average Average PSNR of each frame video image PSNR after denoising as objective evaluation index, to video image Coastguard in the video image Mobile in Fig. 3 (a) and Fig. 5 (a), adding respectively standard deviation is 20,30,40,50 noise, and the Average PSNR value of the denoising rear video image obtaining with the present invention and above-mentioned Lung biopsy is as shown in table 1.

The video image Average PSNR value that table 1. obtains after to video image denoising with the present invention and above-mentioned Lung biopsy

As can be known from Table 1, compare with above-mentioned five kinds of denoising methods, the Average PSNR value of the denoising rear video image obtaining with the present invention has improved 0.4dB to 1dB, for the smaller video Mobile of motion amplitude, particularly outstanding to the denoising ability of video image with the present invention, Average PSNR value has at least improved 0.6dB.

To sum up, the adaptive threshold video denoising method based on surface wave conversion that the present invention proposes, because by Surfacelet coefficient, the neighborhood correlation characteristic in yardstick and noise combine in the distribution characteristics of different scale effectively, accurately calculate the threshold value of each coefficient of each layer of all directions subband, can to coefficient, carry out soft-threshold processing more exactly, so the present invention has very strong inhibition ability for the noise of video, also can be good at retaining the detailed information of video image and the smooth effect of moving object, video image after denoising is significantly improved in visual effect.

Claims (2)

1. the adaptive threshold video denoising method converting based on surface wave, comprises following steps:

(1) input noisy video image, and it is carried out to Surfacelet conversion, decomposing the number of plies is 4 layers, and every layer of corresponding direction number is respectively 192,192,48,12;

(2) coefficient in Surfacelet transform domain is calculated as follows to the initial threshold of each layer of all directions sub-band coefficients:

$T_{l,k}=\sqrt{\log (L_{l,k}/J)}{\left({\mathrm{\&delta;}}_{l,k}^n\right)}^2/{\mathrm{\&delta;}}_{l,k},$

Wherein: T _l,kbe initial threshold corresponding to l layer k directional subband coefficient, l ∈ 1,2,3,4}, in layers 1 and 2 subband: k ∈ 1,2,3 ..., 192}, in the 3rd straton band: k ∈ 1,2,3 ..., 48}, in the 4th straton band: k ∈ 1,2,3 ..., 12};

L _l,kbe the total number of coefficient in l layer k directional subband, J=4 is for decomposing total number of plies;

${\mathrm{\&delta;}}_{l,k}=\sqrt{\frac{1}{\mathrm{MNP}-1}\underset{m=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{p=1}{\overset{P}{\mathrm{\&Sigma;}}}{(w_{l,k}(m,n,p)-\stackrel{\&OverBar;}{w_{l,k}})}^2}$ The signal standards that is l layer k directional subband is poor, wherein w _l,k(m, n, p) is the locational coefficient of l layer k directional subband mid point (m, n, p),

M ∈ 1,2,3 ..., M}, n ∈ 1,2,3 ..., N}, p ∈ 1,2,3 ..., P},

be the average of l layer k directional subband coefficient, M, N, P are respectively subband length, width and height;

the noise criteria that is l layer k directional subband is poor, wherein

the noise criteria that is the 1st layer of k directional subband is poor, adopts the intermediate value estimation technique to calculate:

${\mathrm{\&delta;}}_{l,k}^n=\mathrm{median}\left(\right|w_{l,k}(m,n,p)\left|\right)/0.6745,$

Wherein, w _{1, k}(m, n, p) is the 1st layer of k directional subband coefficient, and median represents to ask coefficient intermediate value;

(3) to initial threshold T _l,kadjust as follows, calculate adaptive threshold corresponding to each coefficient in each subband of each layer:

$T_{\mathrm{adapt}}(m,n,p)=T_{l,k}\frac{(\max \left(z\right)-z(m,n,p))(\mathrm{mean}\left(z\right)-\min \left(z\right))}{(z(m,n,p)-\min \left(z\right))(\max \left(z\right)-\mathrm{mean}\left(z\right))},$

Wherein: T _adapt(m, n, p) represents adaptive threshold corresponding to coefficient on l layer k directional subband mid point (m, n, p) position;

Mean (z) represents the average of z (m, n, p) in l layer k directional subband, min (z) represents the minimum value of z (m, n, p) in l layer k directional subband, max (z) represents the maximum of z (m, n, p) in l layer k directional subband;

for neighboring mean value, R is spatial neighborhood, and S is the coefficient number in spatial neighborhood R;

(4) utilize the adaptive threshold T obtaining in step (3) _adapt(m, n, p), respectively each layer of all directions sub-band coefficients after Surfacelet conversion carried out to soft-threshold processing:

$w^{\&prime;}=\left\{\begin{array}{cc}\mathrm{sign}\left(w\right)\&CenterDot;\left(\right|w\left|\right)-T_{\mathrm{adapt}}),& \left|w\right|\&GreaterEqual;T_{\mathrm{adapt}}\\ 0,& \left|w\right|<T_{\mathrm{adapt}}\end{array}\right.,$

Wherein, w is the coefficient of not processing through soft-threshold, | the amplitude that w| is w, sign (w) represents to ask the symbol of w, T _adaptrepresent adaptive threshold corresponding to w.

(5) coefficient after soft-threshold processing is carried out to Surfacelet inverse transformation, obtain the video image after denoising.

2. according to the method described in claims 1, the described spatial neighborhood R of step (3) wherein, chooses as follows:

First, according to the multi-direction characteristic of Surfacelet conversion, choose five kinds of neighborhood Q _i, { 1,2,3,4,5} calculates respectively the average u of these five kinds of neighborhoods to i ∈ _iwith standard deviation δ _i, then calculate as follows neighborhood Selecting All Parameters ρ _i:

${\mathrm{\&rho;}}_i=\frac{{[u_i/{\mathrm{\&delta;}}_i]}^2}{1+{[u_i/{\mathrm{\&delta;}}_i]}^2};$

Finally, calculate neighborhood and determine parameter

from these five kinds of neighborhoods, choose the corresponding neighborhood of this η as satisfactory spatial neighborhood R.

Images