# Effective feature extraction and representation for robust face recognition

Document Type:Thesis

Subject Area:Computer Science

Introduction ………………………………………………………………. Background information ………………………………………. Basics principles for face recognition …………………………. Development experience and research and current situation…. Framework………………………………………………………………………. what kind of work you do……………………………. Result and analysis…………………………………………………. Conclusion. Reference. 6 Abstract Face recognition requires a lot knowledge in computing, Factors that make face recognition a problem are identifiable like occultations, pose variation, and misalignment, some it require one to look at it in different dimension. Development experience and research and current situation In this research paper, i bring about the idea of SLF which stands for statistical local feature with NRK or robust kernel function representation model. Some rigorous teste is then done on benchmark face databases which include the likes of extended YALE B, LFW, FRGC, FERET, multi-pie and ar. These will then provide for the various variations of expression, occultations, pose, and lighting.

Following these is the promised performance than demonstrated in the merit of this proposed method. Framework 2. To top this, a reading kernel function primarily based really subspace changed into advised for aspect recognition. As an example, yang recommended a kernel function based certainly discriminant shape for recognition and extraction for verification. (yang et. Al. , 60) The leaning techniques preserve in mind the rounded character of the aspect photographs. Those close by sample primarily based definitely sincerely statistical talents have confirmed very promising outcomes in big scale databases, which includes FERET [23-23] and FRGC [36]. Other than hired talents, the active classifier is also essential to the overall everyday regular overall presentation of FR. NN or nearest neighbor, hidden MARKOV and SVM models are the considerably used distinguishers in FR[33][35-38][59][27].

Furthermore, that permits you to better make the most the preceding records that aspect pictures from similar undertaking acquire a closest, nearest subspace or ns distinguishers [29][37-39][52][59] were moreover superior, which can be commonly advanced to the well-known NN classifier. In recent times an interesting classifier, in particular, sparse instance, primarily based in truth certainly genuinely kind (sparse demonstration based classifier), changed into recommended with the beneficial useful resource of the manner of wright [20] for sturdy face recognition. Second, i suggest a sturdy kernel function instance framework, which uses kernel function samples to virtually take gain of the discriminated information embedded in the community abilities, but even further it adopts a sturdy regression study because of the truth the degree to correctly address the occlusion in facial pix.

Equated to the preceding techniques, for example NN with sparse demonstration based classifier with rounded skills and SLF abilities, the recommended SLF based genuinely RKR version suggests a whole lot stronger robustness to several aspect photo variations (for example – misalignment, occlusion, expression and illumination), as confirmed in our experiments completed on benchmark aspect catalogs. The rest of the research is ready as follows. Segment 2 in short evaluations some associated paintings. Section 3 offers the recommended SLF based in reality honestly actually robust kernel function instance set of suggestions. A collaborative instance or sparse example based totally on classifier In assessment to nearest subspace (NS) and nearest neighbor (NN) classifiers, which prohibits representing the query instance within the direction of really taken into consideration one in each of a kind schooling, the nowadays superior l2-regularized sparse example [20] or l2-regularized collaborative depiction [33] signifies the inquiry photo through the drill samples from all classifications, that would ineffectively overwhelm the little-pattern-duration or big problem of nearest subspace (ns) and nearest neighbor (NN) allow x I = [s , s , s ] ∈mani Denote the set of drill samples of the itch object I, 2 i, 2 i, n i 3 A differential, in which in sum, j =2, 2, in, is an m-length trajectory stretched via the jth tester of the itch class2.

Y ∈amis a questions sample to be classified. The representation model of collaborative representation based classifier (CRC) or sparse demonstration based classifier (sparse demonstration based classifier) or could be written as Αˆ=arg minα{ Y 0− xα 2 } (1) 2 + λ Α Lap Then allow/allow y ∈ℜm be a question pattern to be labeled. The instance model of sparse example primarily based completely definitely genuinely truly Classifier sparse demonstration based classifier collaborative instance primarily based definitely absolutely no truth in truth surely collaborative classifier representation based classifier is probably illustrated as Αˆ = arg minα y zero − xα 2 (2) 2 + λ α lp Wherein x=[x2, x2, xc] and c is the number of numerous commands; ⋅lp is the lap-nom, and p=2 for sparse demonstration based classifier in [20], on the identical time as P=2 for collaborative representation based classifier in [33].

The answer of y is finished through Identity (y) = arg mini y − xi δi (αˆ) 2 (2) In which δi (⋅): ℜ n →ℜni is the distinguishing study that picks from αˆ the coefficients related to The ith difference [20]. Five. Changing using l2-regular to standardize the records fidelity in the term in the encrypting model, yang signal instance as an MLE-like estimator: Minα ∑m= ρθ (i − ria) s. t. Α 2 ≤σ I 2 (6) In which in ri is the ith column vector of you and x is the ith component of y. This sturdy sparse encrypting is probably correctly resolved with the useful resource or manner of an iterative re-weighted sparse encrypting logarithms. On this phase, we endorse some clean but powerful pooling technique to this save you. The pooling techniques are quickly finished in object and photograph type to remove invariant talents.

In class, instructions of pooling strategies are provided, max pooling [39] [56-57] and sum pooling [50] [57]. Denote via manner of using fi the ith characteristic vector in a pool, and through fj the jth component of the characteristic vector = f. Inside sum pooling, the output encrypt of vector fs is computed through this way of fsj=f2j +f2j +…+fnj, at the identical time as within the case of max pooling the output function fm is fmj =maxf2, f2jfnj. 2, for example, the divided of the sample map (for example lbp) may be made as 2×2, three×3 and 3×3, correspondingly, with 28 blocks of the unique sizes in total. This form of divided need to flexibly set the number of blocks in each scale and is predicted to Capture greater spatial discrimination statistics than the spatial pyramid.

Within the projected mpm p primarily based completely honestly definitely clearly statistical close by having a study (sl f) extraction, we undertake s +2 degree block divided, in which s=0, 2, … , s. Mention, inside the sth diploma, the entire picture is divided into ps ×qs blocks, every of it clearly is in addition divided into ps ×sq. Sub-blocks. Take the featured era in a single sub-block as an instance. Denote with the useful beneficial or beneficial useful resource of fi the function vector (for example the histogram look at) extracted from the ith sliding trouble, and anticipate that there are n observe vectors, f2, f2, fn, which might be extracted from all viable sliding packing and then the very last output function vector, denoted via f, following max pooling is {f}j =max{|{f1}j|, |{f2}j|, … , |{fn}j}|} Coding for face recognition clearall; clc; class_style = 'linear'; addpath('.

/PCALDA'); addpath('. /libsvm-3. 17'); testpath = '. 17'); testpath = '. /2 pics'; trainpath = '. /ORL'; trainimagenames = dir(trainpath); testimagenames = dir(testpath); %extracting training image features for i=3:size(trainimagenames,1) temp_img = imread([trainpath, '/', trainimagenames(i). name] ); train_fea(i-2,:) = double(temp_img(:)); pos = strfind( trainimagenames(i). name, '_'); labelstr = trainimagenames(i). name] ); train_fea(i-2,:) = double(temp_img(:)); pos = strfind( trainimagenames(i). name, '_'); labelstr = trainimagenames(i). name(1:pos-1); train_label(i-2,1) = str2num(labelstr); end %extracting test image features for i=3:size(testimagenames,1) temp_img = imread([testpath, '/', testimagenames(i). name] ); test_fea(i-2,:) = double(temp_img(:)); pos = strfind( testimagenames(i). name, '_'); labelstr = testimagenames(i). name] ); train_fea(i-2,:) = double(temp_img(:)); pos = strfind( trainimagenames(i).

name, '_'); labelstr = trainimagenames(i). name(1:pos-1); train_label(i-2,1) = str2num(labelstr); end %extracting test image features for i=3:size(testimagenames,1) temp_img = imread([testpath, '/', testimagenames(i). name] ); test_fea(i-2,:) = double(temp_img(:)); pos = strfind( testimagenames(i). name, '_'); labelstr = testimagenames(i). The generally used classifiers, including the linear svm, nn and ns classifiers [19] [36-38] [51] [58], further to the sparse demonstration based totally classifier and CRC classifiers [10] [33], often adopt the l2-norm to a diploma space (i. e. , eucli-dean distance). Other than l2-norm primarily based sizes, kernel function techniques have grown to be increasingly more well-known for pattern category, especially face reputation [3] [60]. The kernel function trick ought to map the non-linearly separable skills right into an immoderate dimensional function region, wherein abilities of numerous instructions can be extra without troubles separated via linear classifiers.

If we placed into impact that αi = αj for exquisite blocks i, i. e. , we expect that the most effective of a type blocks yi extracted from the same test pattern has the same instance over their related matrix ai, then kernel function illustration of the question image via combining all of the block functions is probably written as Min φ y; φ y; φ y − φ a; φ a; φ a α 2 s. t. Α ≤σ (11) Α ( 1) ( 2) ( b) ( 1) ( 2) ( b) 2 lap In which α is the encrypting coefficient vector of the query pattern. e. , ei = φ (yi) − φ (ai) α 2. We expect that 2 Ea. Is impartial from ej if i≠j for the cause that they represent the instance residuals of various blocks. The proposed sturdy kernel function example can then be formulated as Min ρ (e) s.

We can also set ρ (ei) = ei (i. e. Ρ (e) = e 1). As can be seen in discern three, ρ (ea. )= ea. 5 desired charge characteristic 3 2 Charge ρ (ea. ) = (ea. 5 Feature 2 1. 5 Price 1 0. five zero -1 0 1 2 -2 ea. It needs to be stated that the weight values of every attempting out pattern are anticipated online, and there isn't always a schooling section of them. The corresponding price feature ρ to the weight function in eq. (13) maybe differentiable and bounded, due to the truth the blue curve examined in determine 3. With the above improvement, the ideal strong kernel function instance in eq. (12) may be approximated Via: Min w 1 2e 2 set α ≤σ (15) Α 2 lp Following a few derivations, eq. (17) we are capable of seeing that the weighted-sum kernel function terms, collectively with ∑bit=1ωi okay (yen, yen), ∑bit=1ωi adequate aiai, And ∑ib=1ωi kai yi, need to make the maximum the discrimination data in the mapped better dimensional characteristic Space; at the identical time, the load ωi'm capable of effectively eliminating the outlier’s impact on computing the encrypting Vector.

The encrypting vector α is regularized with the useful resource of lap-norm. In this paper, we talk vital times: p=1 for sparse regularization and p=2 for non-sparse regularization. At the equal time as p=1, l1-norm minimization strategies which incorporate the green characteristic-sign are searching for a set of regulations [30] might be used to remedy the sparse encrypting trouble of eq. At the same time as p=2, a closed-form solution of eq. The coefficient is computed through eq. (17) with diagnosed weight cost. As soon as you have were given got the solution αˆ following a few iterations, the magnificence of the question pattern is carried out via identification(y) = min b ωε i, j (18) j, ∑i=1 i In which ε i, j = φ ( yen )−φ(ai, j )αˆ j 2 is the ith-block kernel function example residual associated with the jth splendor, 2 ai,j being the sub-matrix of ai associated with the j th α i, 1 , α i, 2 , with elegance, and Α i=,i, c Αˆ = [αˆ1; αˆ 2; αˆc] with αˆ j being the instance coefficient vector associated with the jth beauty.

From Eq. (18) it is able to be seen that the category requirements are based on a weighted sum of kernel function instance residuals, which uses every the discrimination strength of kernel function illustration in the excessive dimensional characteristic area and the insensitiveness of sturdy example to outliers. We denote with the aid of slf-rkr_l1 and slf-rkr_l2 the implementations of slf-rkr model with l1-norm regularization and l2-norm regularization, respectively. The time complexity of slf-rkr mainly lies in mpmp primarily based slf extraction and fixing the robust kernel function instance. Consistent with the dispositions of histogram feature, we are able to adopt the critical photo technique [53] to hurry up mpmp primarily based slf extraction. For every pixel in a sub-block, handiest 2 additions are Thirteen Desired in computing vital image and three additions are preferred in computing histogram bin cost.

So the computing of each histogram bin for this sub-block goals 3hw (1-ratio) 2 additions and 1 max operation, in which hand we are the peak and width of the sub-block, and the ratio is the parameter of the sliding subject. For slf-rkr_l1 with up to date weight, the step a) (i. e. , weighted kernel function instance with p=1) is an iterative technique itself, and the steps b), c) and d) can be operated in every new release of step a). Fashionable, the time complexity of slf-rkr_l1 with updated weight is a form of just like that of slf-rkr_l1 with ωi=1, because of the truth the preceding has almost the equal fixing technique due to the reality the latter with first-ratean in addition step to update weight in each era.

In FRwith occlusion/disguise, sparse demonstration based totally definitely classifier desires an in addition occlusion matrix to encrypt the occlusion, and consequently, its time complexity may be very high. 3 for the particular experimental placing), the not unusual on foot time of slf-rkr_l2 and slf-rkr_l1 is zero. 8073 2nd and zero. 8339 2d, respectively, which can be hundreds an entire lot an awful lot less than that of sparse demonstration primarily based truly classifier (1. 8800 seconds). 13 Desk 1: a set of regulations of statistical close by feature-based sturdy kernel function instance (slf-rkr). Checking convergence situation: ∑ i ( ωi( t ) −ωi( t −1) )2 ∑i (ωi( t −1) )2 < γ , Where γ is a small superscalar and ωi (t) is the burden fee of block I in technology t. Surrender at the equal time as Three.

Do class: Identification = mind ∑ ib=1 ωi ok (Yi, you) + αˆ tj ∑ ib=1 ωi good enough ai, j ai, j αˆ j − 2αˆtj ∑ib=1ωi kai, j you Wherein an imp is the sub-matrix of a related to the jth beauty and αˆ j is the example coefficient vector i Associated with the jet beauty. Desk 2: commonplace jogging time (2nd) of slf-rkr and sparse demonstration based totally definitely classifier. Method ar database prolonged yale b with 50% occlusion Slf+sparse demonstration based classifier zero. Then in segment four. Four, we 15 Take a look at FRin opposition to dam occlusion and real conceal. In the long run, the complete evaluations on big-scale face databases, which embody feret [22-23], frgc [35] and lfw [32], are supplied in segment 3. five. Four.

eight for FRwithout occlusion. The lagrange multiplier λ of slf-rkr_l1 (communicate to eq. (17)) Is ready as 0. zero. five, on the same time because of the truth the lagrange multiplier λ of slf-rkr_l2 is typically set as a bigger charge (as an instance zero. 1) prolonged yale b database: the prolonged yale b database consists of 2,332 frontal-face pics of 38 people (each situation has sixty-four samples), captured under various laboratory-managed lights conditions [58] [20]. For every project, nor samples are randomly determined on as education samples and 32 of the final photos are randomly decided on due to the reality the attempting out information. Proper right here the photographs are normalized to 96×80 four and the test for every ntr runs 10 times. The FR result, which includes advocate reputation accuracy and preferred variance, of all of the competing strategies are listed in desk four.

The proposed slf-rkr achieves the extremely good traditional commonplace normal overall performance, with greater than 2% development over all of the others on the identical time as ntr is small (as an example 5, and 10). forty nine slf+nn 59. 70 seventy six. eight±1. 7±zero. 87 slf+lrc fifty nine. 90 ninety 5. 5±zero. 88 ninety-nine. 32 slf+sparse demonstration based absolutely completely classifier 80. eighty 90 five. five±0. 18 2) ar database: the ar database consists of over four, 000 frontal photographs from 126 human beings [21]. For everyone, 26 snapshots have been taken in separate commands. As in [10], inside the take, a look at us decided on a subset of the dataset consisting of fifty male topics and 50 woman subjects. For each trouble, the seven snapshots with illumination exchange and expressions from session 1 have been used for training, and the opportunity seven photos with handiest illumination exchange and expression from consultation 2 have been used for finding out.

3 90 seven. zero 98. 3 slf+lrc eighty 3. three eighty two. 7 eighty 5. 8 slf-rkr_l1 90. zero 90. four 90 seven. 3 ninety nine. Four slf-rkr_l2 ninety. 3% on are through the use of lbp as its slf. This definitely suggests that using sturdy kernel function illustration drastically will grow the popularity costs. In addition, improvement might be completed for slf-rkr if photograph gradient orientation is used to layout the statistical close by characteristic. Further, on this check the l1-norm regularization and l2-norm regularization in slf-rkr purpose little difference inside the reputation expenses, however, the later has a whole lot lots tons less time complexity. Robustness to misalignment and pose In this section, we test the robustness of the proposed technique to shut through deformation, which consists of picture misalignment delivered through the manner of face detector and poses the model.

four fifty 8. 1 slf+hisvm sixty seven. 6 sixty four. Three slf+crc 77. 1 seventy four. 1(seventy-eight. Four) 80. 8(seventy-five. Zero) 1)massive-scale multi-pie database: the cmu multi-pie database [31] includes images of 337 topics captured in 3 instructions with simultaneous versions in the pose, expression, and illumination. Within the experiments, all the 239 topics in session 1 have been used. You can see that even without mpmp, slf-rkr_l1 even though outperforms slf+ sparse demonstration primarily based definitely classifier through 1. nine% in common, while slf-rkr_l2 outperforms slf+ crc via 2. three%. It is able to moreover be decided that the development added through the way of the usage of mpmpis over 3% in every session, which in fact show the effectiveness of the proposed mpmp in dealing with misalignment. 2) ferret pose database: in this test we use the feret pose dataset [22-23], which includes 1, four hundred pix from 198 topics (about 7 every).

The right sparse demonstration based totally classifier and slf+hisv m have the worst substantial standard performance considering that eigenface function is sensitive to pose version and hisvm cannot have a take a look at pose version from frontal schooling set. We also deliver in table 7 the effects of slf-rkr without mpmp on all poses. Similar save you to that during multi-pie is probably made, i. e. , significant improvements (for instance over 1 three% improvement at the same time some levels are ±25o) could be achieved with the useful resource of the use of mpmp. zero 3 6. 5 Slf+lrc forty five. Zero ninety-four. Zero seventy nine. five Slf+hisv m 12. zero (30. 0) 100(ninety nine. 5) ninety six. zero) 5 7. zero (39. On this segment, we check the overall standard performance of slf-rkr to several occlusions, which incorporates block occlusion and real cover.

In slf-rkr, the robustness to occlusion mainly comes from its iterative reweighed kernel function strong instance. In this section, the weight win every block is automatically updated. The cutting-edge-day strategies to deal with face occlusion, together with the robust model of sparse demonstration primarily based definitely classifier [10] (i. e. As an example, at the same time as occlusion is 50%, slf-rkr want to advantage at least ninety-four % recognition accuracy, in assessment to at most 87. four% for specific strategies. For slf-rkr_l1, at the identical time as there is 60% block occlusion, it can even though collect a reputation price of over 80 3%. This in truth demonstrates the effectiveness of the proposed slf-rkr technique to deal with face occlusion. Further, each KCRC and sparse demonstration based totally sincerely classifier may additionally need to get higher normal common overall, overall performance than CRC, but worse, not unusual normal well known average performance than sparse demonstration based completely classifier and slf-rkr.

zero eighty 5. 9 fifty seven. Four slf+lrc one hundred 100 ninety eight. 9 ninety six. 8 sixty nine. nine 88. 5 sixty six. Four 33. 7 slf+sdc a hundred a hundred 99. eight 98. zero seventy seven. 9 2) FR with hiding: a subset of 50 guys and 50 women are decided on from the are database [21]. For every hassle, seven samples without occlusion from session one are used for schooling, with all of the final samples with disguises for attempting out. The ones trying out samples (which encompass three samples with sunglass in session1, three samples with sunglass in consultation 2, three samples with a headband in consultation 1 and three samples with a headscarf in consultation two every day with trouble) not amazing have disguises, however furthermore have variations of time and illumination.

Proper here the photo length is normalized to 80 3×60. 22 Desk 9: face recognition charges (%) at the hard datasets with the real cowl. Sunglass-s1 scarf-s1 sunglass-s2 scarf-s2 sturdy sparse demonstration primarily based certainly classifier [10] 83. three 38. 7 forty nine. Zero 29. 7 slf+crc ninety nine. three 86. 7 slf+kcrc one hundred ninety eight. zero slf+sparse demonstration primarily based classifier a hundred 99. zero eighty 5. Considering that slf-rkr_l2 has comparable recognition accuracy to slf-rkr_l1, however, has thousands decrease time complexity, in this phase we simplest report the effects of slf-rkr_l2. We update the load of slf-rkr_l2 and set the shape of histogram bin in every sub-block as 30. 1) feret database: the feret database [22-23] is frequently used to validate an algorithms effectiveness because it consists of many varieties of photo variations.

Via taking fa subset as a gallery, the probe subsets facebook and fc had been captured with expression and illumination variations (the pix in fc had been captured with the beneficial aid of a brilliant virtual digital). Especially, dup1 and dup2 encompass photos that have been taken at unique instances. 1% and three. Five% lower than slf-rkr_l2 in not unusual. It is also interesting that the collaborative example based definitely classifiers (as an instance sparse demonstration primarily based classifier, crc, sparse demonstration based totally classifier, kcrc, and rkr) regardless of the fact that have masses better recognition costs than nn and hisvm inside the case that every state of affairs has handiest one education pattern. 23 Table 10: face reputation fees (%) on feret database.

Approach fb fc dup1 dup2 slf+nn ninety-four. 1 sixty 3. 2 slf+sparse demonstration primarily based classifier 98. 2 80 five. 2 slf+ksparse demonstration based classifier ninety eight. three 68. It indicates that the proposed slf-rkcr_l2 not terrific outperforms slf+nn and slf+svm in all instances, however moreover has better ordinary regular average overall performance than the great strategies suggested in the literature. Mainly, slf-rkcr_l2 has popularity accuracies of 96. three% and 90 four. Four% in dup1 and dup2, respectively, which can be the exceptional effects thus far. Desk eleven: face recognition fees (%) of slf-rkr and fantastic modern-day-day strategies on the feret database. zero seventy nine. 5 Dies [38] ninety-nine. Zero ninety-nine. Zero ninety 3. zero 90 3. The photo is normalized to 168×128. The characteristic dimensionality extracted via bfld in each block is about to 220 and gaussian kernel function is utilized in slf-rkr.

23 3 checks with five, 10 and 15 purpose samples for every problem are made in the experiments. The popularity fees of slf+nn, slf+lrc, slf+hksvm, slf+crc, slf+sparse demonstration based totally definitely classifier, slf+kcrc, slf+ksparse demonstration based classifier, and the proposed slf-rkr are indexed in desk 12. Once more, slf-rkr performs the exquisite, regardless of the truth that the development is not large because of the reality there aren't any occlusion, misalignment and pose versions of the question set. 977 slf+crc zero. 967 zero. 976 slf+kcrc 0. 932 zero. 973 slf+sparse demonstration based totally classifier 0. (a) (b) Decide 7: samples of lfw. (a) and (b) are samples in training and locating out gadgets, respectively. 25 Desk thirteen lists the PR outcomes of competing techniques with the mpmp based totally clearly slf.

The photograph is normalized to 127×116. We are capable of seeing that slf-rkr, however, achieves the incredible regular standard ordinary overall performance. 788 zero. Conclusion On this paper, we proposed a statistical close by characteristic based robust kernel function instance (self-rkr) model for face popularity. A strong example version to picture outliers (for instance occlusion and real hide) have come to be constructed within the kernel function region, and a multi-divided max pooling technology changed into proposed to enhance the invariance of community pattern function to picture misalignment and pose version. We evaluated the proposed approach to specific conditions, which embody variations of illumination, expression, misalignment and pose, further to dam occlusion and coil occlusion. One huge benefit of slf-rkr is its excessive face popularity prices and robustness to numerous occlusions.

, Pentland, A. P. , Eigenfaces for recognition, 1991, Cognitive Neurosci. , V. 3, no. , Phillips, P. J. , Rosenfeld, A. , 2003, Face recognition: A literature survey, ACM Computing Surveys (CSUR), V. 35, Issue 4, pp. V. et al. , 1998, Hidden Markov models for face recognition, In Proceedings, International Conference on Acoustics, Speech and Signal Proceeding, 2721 2724 [9] Face recognition: eigenface, elastic matching and neural nets Jun Zhang; Yong Yan; Lades, M. Proceedings of the IEEE Volume 85, Issue 9, Sep 1997 Page(s):1423 – 1435 [10] Thomas Fromherz, Peter Stucki, Martin Bichsel "A Survey of Face Recognition," MML Technical Report, No 97. 01, Dept. Tian, T. Kanade, and J. Cohn. Recognizing action units for facial expression analysis. IEEE Trans. of the 5th International Workshop on Image Analysis for Multimedia Interactive Services, WIAMIS 2004, 2123 April 2004, Lisboa, Portugal [18] FRVT 2006 and ICE 2006 LargeScale Results [19] The FERET Evaluation Methodology for Facerecognition Algorithms, by P.

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