A New Version of Flusser Moment Set for Pattern Feature Extraction

Transcription

1 A New Version of Flusser Moment Set for Pattern Feature Extration Constantin-Iulian VIZITIU, Doru MUNTEANU, Cristian MOLDER Communiations and Eletroni Systems Department Military Tehnial Aademy George Cosbu Avenue 81-83, Buharest ROMANIA Abstrat: - The hoie of a suitable feature extration method is essential for the suess of the lassifiation or reognition proess. This paper proposes a design method for a new pattern desriptor set based on the Flusser moment lass, whih is invariant to elementary geometri transforms and has an inreased robustness to the ation of some perturbations. Experimental results based on the use of a real video image database onfirm the basi properties of this new desriptor set. Key-Words: - Pattern feature extration, Flusser moment lass, pattern reognition, neural network 1 Introdution To be useful in a pattern reognition proess, the objets or patterns resulting from the segmentation of an input image must be represented in a suitable manner [7], [8], [10]. This speifi pattern representation assumes some basi properties like: onision, redundant information disarding, invariane to elementary geometri (or affine) transforms and onservation of relevant information for pattern reognition appliations [11], [15]. Aordingly, this proess known as feature extration is in a straight onnetion with the struture hosen for the input data representation, and is strongly dependent on the implemented appliations [10], [12]. In the fundamental literature assigned to the theory of digital image proessing, a lot of suitable standard and neural methods for pattern feature extration are mentioned [2], [3], [6], [9]. Having as a starting point the high level of lassifiation performanes offered by the use of geometri moment lass, an interesting approah in the field of reognition systems is represented by the Flusser moment lass [4], [13]. The use of standard Flusser moments inside of a reognition system leads to very good lassifiation performanes [4], but this approah has some important disadvantages, suh us: average robustness to noisy fators, defiienies for symmetrial input images et. However, an improved version of the standard Flusser moments that orrets most of these diffiulties must be onsidered [5]. The aim of this paper is to provide a design proedure for a new invariant set belonging to the Flusser moment lass, whih represents an improved and more robust alternative solution to the referene standard moments. In the first part of the paper, a theoretial demonstration of the design algorithm, as well as basi properties of this new invariant set is presented. In the last part of the paper the experimental results that onfirm the theoretial properties of the proposed invariant set are shown. Finally, some important onlusions and future researh are also inluded. 2 Design Proedure for Proposed Invariant Set The design proedure for the proposed invariant set th p q - starts with the basi definition of the ( ) order omplex moment for an integral funtion f xy, [1]: ( ) ISSN: Issue 4, Volume 5, April 2008

2 p q ( i ) ( i ) (, ) dd (1). pq = x y x y f xy xy where pq, Z Aording to [4], it an be demonstrated that: p q p q q s ( 1) i p q k s k s (2), pq = k= 0s= 0 mk sp, q k s, pq, Z mpq is given by equation: mpq = x p y q f( xy, )dxy d, pq, Z (3), and represents the ( p q) th -order standard geometri moment [11]. In [4], the following signifiant theorem has been demonstrated (Flusser theorem): If the following onditions are met: n 1 n and kj( pj qj) 0, kj, pj, qj Z =, then the j= 1 n kj produt I = is rotation invariant. pq j j j= 1 The most important disadvantage of the above I is given by the fat that this set desriptor set { } k k is not invariant to all elementary geometri transforms. An immediate and effiient solution to this problem ould be the substitution mpq η pq in equation (2): p q (1) pq = k= 0s= 0 η k sp, q k s, pq, Z where ηpq p q q s ( 1) i p q k s k s (4), is given by equation: p q ( x x ) ( y y ) f ( xy, ) dxd y, (5), η pq = pq, Z th and represents the ( p q) geometri moment. Also, it is known that ( x, y ) -order entered are the oordinates of the input pattern entroid [7]. Consequently, onsidering this substitution as a starting point, the translation and saling invariane of η pq moments (aording to [7]), and applying the theoretial results from Flusser theorem, we an demonstrate that this new desriptor set { I (1) } k is k invariant to elementary geometri transforms (translation, saling, and rotation). For normal onditions, this pattern desription generates omplex values. Therefore, to obtain real values, we an retain, for example, either its real part, or imaginary part. To obtain the final form of the proposed invariant set, a new type of ( p q) th order omplex moment must be onsidered, aording to the following equation: p q ( x x x ) ( y y y ) (6), λ pq = s s f xy, ddy, x pq, Z ( ) where the shift fators x s and y s are hosen aording to [7], and are given by the equation: η20 η x, 02 s = ys = (7). m00 m00 Moreover, the new entral moment set an be normalized to ensure its translation and sale invariane using the following equation: λpq ξ pq =, pq, Z (8). p q 1 m 2 00 It has been demonstrated that, using the above shift fators, the most important advantage is the disarding of the standard entral moment defiienies when using symmetrial images as ISSN: Issue 4, Volume 5, April 2008

3 system inputs, simultaneously with the enhanement of the sensibility to the noise ation [11]. A similar replaement as the one from equation (3) leads to an improved struture of the proposed moments: p q (2) pq = k= 0s= 0 ξk sp, q k s, pq, Z p q q s ( 1) i p q k s k s (9). Therefore, the final form of the proposed invariant set an be rewritten as it follows: (2) ζ 1 = 11 (2) (2) ζ 2 = (2) (2) ζ 3 = Re 20 2 (2) (2) ζ 4 = Im 20 3 (2) (2) ζ 5 = Re 30 3 (2) (2) ζ 6 = Im 30 (2) ζ 7 = 22 2 (2) (2) ζ 8 = Re 31 2 (2) (2) ζ 9 = Im 31 4 (2) (2) ζ 10 = Re 40 4 (2) (2) ζ 11 = Im 40 (10). The moment set given by equation (10) has the following four important properties: - elementary geometri transforms (rotation, translation and saling) invariane; - elimination of the standard entral moment defiienies in ase of symmetrial image use; - robustness to the ation of noisy fators inside of a reognition system, better than the use of standard Flusser moments; - independent base of pattern invariants (in the sense of the Flusser theorem). The theoretial properties of the proposed invariant set are onfirmed by the experimental results presented in the next setion of the paper. 3 Experimental Results The main objetives of these experiments are: demonstration of basi properties of the proposed moment set, aording to the theoretial bakground presented in the previous paragraph; a omparative study with the standard Flusser moments as referene, to evaluate the performane level of a reognition system that will use these feature extration methods (in fat, it is important to evaluate the robustness to digital image proessing speifi noise). The input video database used in the first part of the paper onsists of two prototype-images of modern military airrafts, as it an be seen in Fig.1. Eurofigther Mirage 2000 Fig.1: Prototype-images of video database used in the first experimental setion (Top view) Eah prototype-image has a resolution of pixels, in an unompressed BMP format. To test the elementary geometri transforms invariane of the proposed invariant set, eah prototype-image was translated, saled and rotated aording to [11]. The results obtained after the moment alulus given by equation (10) are shown in Table 1. Therefore, the invariane property of the proposed invariant set is obtained (the relative lose values from the shady rows in Table 1). One explanation for the small differenes that appear between these values is the omputing errors aused by the MATLAB internal working algorithm. To analyze the behavior of the proposed ISSN: Issue 4, Volume 5, April 2008

4 invariant set to the ation of different symmetries existing in the input images, a olletion of standard testing images has been used [11]. The results after the alulus of the moments given by equation (10) are shown in Table 2. We an observe that the defiienies of the standard entral moments for symmetrial input images (odd rank moments are zero) are eliminated (see, for example, the non-zeros values from shady rows in Table 2). Table 1 Transform Referene Translation Saling Rotation (pixel number) (saling fator) (degree) Moment images o 180 o ζ 1 Eurofighter -9.88e e e003-10e e003-9,84e003-9,88e003 F e e e e e003-4,87e003-4,96e003 ζ 2 Eurofighter -8.62e e e e e e e010 F e e e e e e e005 ζ 3 Eurofighter 8.52e e e e e e e014 F e e e e e e e009 ζ 4 Eurofighter -5.53e e e e e e e006 F e007 1,48e007 1,47e e e e e007 ζ 5 Eurofighter 7.43e e e e e e e021 F e e e e e e e012 ζ 6 Eurofighter 3.21e e e e e e e008 F e e e e e e e009 ζ 7 Eurofighter 2.60e e e e e e e008 F e e e e e e e007 ζ 8 Eurofighter -2.25e e e e e e e019 F e e e e e e e013 ζ 9 Eurofighter 5.34e e e e e e e011 F e e e e e e e008 ζ 10 Eurofighter 1.94e e e e e e e030 F e e e e e e e019 ζ 11 Eurofighter 6.74e e e e e e e002 F e e e e e e e004 Experimental results obtained for invariane testing Moment Symmetry referred at both axes Symmetry referred at x axis The type of symmetry Symmetry referred at y axis Table 2 Symmetry referred at entroid ζ ζ e e e e e e e e-005 ζ e e e e e e e e-006 ζ e e e e e e e e-006 ζ e e e e e e e e-010 ζ e e e e e e e e-010 ζ ζ e e e e e-006 2,19e e e-006 ζ e e e e e e e e-006 ζ e e e e e e e e-012 ζ e e e e e e e e-011 Experimental results obtained for standard entral moment defiieny disarding ISSN: Issue 4, Volume 5, April 2008

5 Another important aspet is a omparative study between the invariant set given by equation (10) and the standard Flusser invariant set used as a referene to reveal the noise robustness. The logial diagram used to generate the input video database for neural lassifier training and testing is presented in Fig.2. As we an see from Figure 2b, the video database used in this appliation was obtained from a (digital) photographial survey of five modern military airraft models (F117, Mirage 2000, Mig 29, F16 and Tornado) saled at 1:48. The survey was taken using a 5 0 inrement in the azimuthal plane, using a range of 0 0,180 0 justified by the geometri airraft shape symmetry. Mirage 2000 Examples of alulus proedures from GUI (Aquisition and preproessing GUI) b) Aquisition and preproessing step F 16 a) Examples of military airraft models used in video database design Classifier training substep Basi aquisition and preproessing diagram Noise adding and lassifier testing substep ISSN: Issue 4, Volume 5, April 2008

6 Variable resolution and lassifier testing substep Examples of alulus proedures from GUI (Classifier learning GUI) ) Perturbation adding and lassifiation step Fig.2: The testing proedure of robustness Eah image from the input video database has a resolution of pixels, in an unompressed BMP format. A supervised ART artifiial neural network (SART) was used for lassifiation purposes [12]. The SART lassifier uses the priniple of prototype generation similar to ART neural network but, unlike this one, the prototypes are generated in a supervised manner. It has the apability of fast learning using loal approximations of the lass and its operation does not depend on any hosen parameter. The idea is to reate a new prototype for a lass whenever the atual set of prototypes is not apable anymore to lassify the training data set satisfatorily. The prototypes are updated using the mean of the samples whih are orretly lassified by eah of them. The updating proess is repeated as long as there are lassifiation errors on the training samples and as long as it dynamially hanges the loation of the prototypes. Speaking in terms of probability distribution, the prototypes beome the entroids of the modes of the multimodal lass probability distribution. The lassifier struture is similar to that of an RBF or LVQ neural network. The algorithm desribed above is then used to train hidden layer, while a MADALINE network implements the output layers (see Fig.3). An important property of the SART neural algorithm is that it needs no initial system parameter speifiations and no prespeified number of odebook vetors. Aording to [12], a short version of the standard SART algorithm is indiated in Table 3. Also, the neural lassifier struture is shown in Fig.3. Table 3 Step 1: Initialization a randomly seletion of a vetor as prototype for eah input lass initialization of the vetor list assoiated to eah lass prototype Step 2: Learning while there are hanged prototypes and the error rate is higher than a preset value For eah vetor belong to an input lass the vetor is ompared with the atual prototypes and lassified If the vetor is orret lassified the vetor is added to the list assigned to the winner prototype If the vetor is not orret lassified the vetor is delared as new prototype and the initial vetor list is updated ISSN: Issue 4, Volume 5, April 2008

7 Table 3 (ont.) Step 2: Learning Update the prototypes assigned to eah lass The prototype is realulated as average between the vetors that are assoiated with its own list Chek if some prototypes have hanged Eliminate the prototypes for that assoiated vetor list ontains only a vetor Exit if the number of maxim interations or prototypes is reahed the neuronal weigths of the output layer are alulated using the pseudo-inverse method or Widrow-Hoff algorithm SART lassifier standard algorithm After the aquisition and preproessing step (see Fig.2b), a number of 37 video images per lass is obtained. Using an interlaing spliting method on input video database, a number of 19 images were used for SART neural lassifier training, while for testing 18 images were used. After the SART lassifier training and testing steps, it is important to quantify and analyze the perturbation fator influene on the reognition system performanes by moment alulus. Therefore, the GRS (good reognition sore) has been omputed, as it is the most important performane indiator. The GRS represents, in %, Unnoisy input database image ( σ=0 ) Input database image after noise adding ( σ=0.1) a) Salt and pepper noise adding substep Fig.3: SART neural lassifier struture To quantify the reognition system (or ATR system) performanes, eah database image was mixed in testing substep with salt and pepper noise (it is known that this noise is harateristi for a digital image proessing hain), with σ dispersion as 0,0.1 range a ontrol parameter, modified into [ ] with a 10-2 inrement (see Fig.2 and 4a). To test the robustness of the proposed invariant set for input images with digital variable resolution (it is known that, for example, for a video homing missile system, the variable digital resolution simulates the variation of the relative distane between the missile video sensor and the urrent target from its field of view), the real dimensions of eah database image were also divided in testing substep with a suitable ratio, with r as a ontrol 1,2,4,8,10 set (see parameter, modified into { } Fig.2 and 4b). Usually, a serious derease of lassifiation performanes is expeted when the input database images are affeted by these perturbation fators [8]. b) Influene of variable resolution on pattern shape Fig.3: Influene of tested perturbation fators on input database image quality ISSN: Issue 4, Volume 5, April 2008

8 the ratio between the number of orret lassified input patterns and the total number of patterns used for lassifiation. In Fig. 4 a graphial representation of the resulted GRS = f σ dependene is shown. ( ) As we an see, the moments given by equation (10) have a higher value of noise robustness than the standard Flusser moments. Another interesting experimental result of the above appliations is the possibility to draw the GRS surfaes for the invariant set give by equation (10) and respetively, standard Flusser moments. In the ontext of the present work, a GRS surfae represents a graphial representation of GRS = f σ, r dependene (see Fig.6). ( ) Fig.4: GRS f ( ) = σ dependene a) GRS surfae for invariant set give by equation (10) In Fig.5, using a proper interpolating spline algorithm, a graphial representation of the resulted GRS = f r dependene is shown. ( ) As we an see, only for a ratio r less than 5.5, the proposed invariant set has a higher value of variable resolution robustness than the referene Flusser moments. However, this aspet is not very important beause, in pratie, we an onsider an effetive or real reognition system (ATR system) if its GRS is more than 50 % (see Fig.5). b) GRS surfae for standard Flusser moments GRS = f σ, r dependene Fig.6: ( ) Fig.5: GRS = f ( r) dependene As we an see from Fig.6, the GRS surfae shape is relatively similar, but in ase of standard Flusser moments use, this is more stirring and its slope is more slanted than invariant give by equation (10) use. Generaly speaking, this observation based on GRS surfae shape analysis is an additional demonstration of proposed invariant set superiority to perturbations omparing to the referene set. To prove that the proposed invariant set forms an independent base (aordingly to Flusser theorem), a proper feature seletion method was applied on the video image database obtained after the aquisition and preproessing step. The seletion method is an ISSN: Issue 4, Volume 5, April 2008

9 improved version of Sammon projetion algorithm on a R 11 R n form. The new form used for normalized error or stress definition assigned to standard Sammon algorithm an be rewritten as it follows (aording to [9], using this modified version of standard Sammon algorithm, a serious inreasing of feature seletion step performanes is expeted): N 1 N 1 p E new = d 1 ij d N N δ p 2 d i= 1 j= i 1 ij i= 1 j= i 1 p Z ( ij ij ) (11), where dij,, i j = 1, N desribes the distane between either points of n dimensional input spae (in our ase, this value is 11), δ ij,, i j = 1, N desribes the distane between either points of m dimensional output spae, N is the point number from input map and, p is a parameter used for Sammon normalized error minimization. Aording to [14], using the parameter p, the intralass variane an be minimized and respetively, the interlass variane an be maximized. Also, in order to inrease the omputing speed, this nonlinear projetion algorithm was initialized with the final projetion solution given by PCA algorithm [7], [13]. A short version of the generalized Sammon projetion algorithm is presented in Table 4. Table 4 Step 1: Initialization using PCA solution, the vetors from output spae are initialized Step 2: Projetion alulus the distanes between either points of n dimensional input spae are alulated using previous alulus, the distanes between either points of m dimensional output spae are alulated in order to minimize the Sammon stress, the optimal value for parameter p is determined aording to equation (11), the Sammon stress is alulated if the Sammon stress is more than a preset value stop else 2 Table 4 (ont.) Step 2: Projetion alulus the output vetor map is realulated go to substep used for Sammon stress minimization end Generalized Sammon projetion algorithm This late experiment searhes for informational redundany inside the moment set given by equation (10). If by preserving (or inreasing) the lassifiation performanes (see first row in Table 5), the projetion spae dimension an be dereased (less than 11) then the proposed invariant set does not form an independent base for pattern reognition. The lassifiation results using the same lassifiation diagram as the one from Fig.2 are presented in Table 5. Table 5 Feature Classifiation performanes seletion GRS: 95% SART training parameters: is not eradm= 10-2 ; applied nepmax=10; ( m = 11 ) nprmax=10 Training/testing time: 3.5 s GRS: 78% SART training parameters: is applied eradm= 10-2 ; ( m = 10 ) nepmax=10; nprmax=10 Training/testing time: 3.1 s GRS: 52% SART training parameters: is applied eradm= 10-2 ; ( m = 3 ) nepmax=10; nprmax=10 Training/testing time: 1.8 s Classifiation results after Sammon projetion As we an see from Table 5, trying to redue the feature spae dimension an lead to muh lower GRS values omparing to the ase with no feature seletion. Fig.7 shows the 3D input data projetion. The struture of the output data does not allow a very good lassifiation. Consequently, it results that the invariants given by equation (10) form an independent base for pattern reognition [4]. ISSN: Issue 4, Volume 5, April 2008

10 Fig.7: Database 3D Sammon projetion (Aordingly with the last row from table 5) More details regarding experimental aspets treated in this setion an be found in [14]. 4 Conlusion and Future Work The theoretial and experimental results presented in this paper leads to the following remarks onerning the basi properties of the proposed moment set: the moment set given by equation (10) is elementary geometri transforms invariant, it eliminates the entral moments defiienies for symmetrial input images, it is more robust to perturbations omparing to standard Flusser moments, and it forms an independent base for pattern reognition; when this set is used as a feature extration method in a lassifiation system (ATR system), and in ase of noise absene, the GRS are greatly improved, generally 4% more than the standard Flusser moments. For noise adding ase, the GRS are also very good, generally 7% more than the referene moment set (see Fig.4). Finally, for input images with variable resolution use and respetively, for divide ratio less than 5.5, the GRS are good, generally 2% more than the standard Flusser moment (see Fig.5); the omputing resoures required are relatively lower than those for the standard Flusser moment alulus. As a final onlusion, the design of the proposed moment set as a feature extration method inside of reognition system (ATR system) is feasible. Also, the invariants given by equation (10) represent an important and robust alternative as well as performane, not only for Flusser moment lass, but for all geometri moments. In a future improvement, the omparative study of the proposed invariant set and the standard Flusser moments will be extended to a larger lass of pattern desription methods, [11], [15]. Aordingly, it will be interesting to make a omprehensive omparision as well as performane, between invariants given by equation (10) and other well-known desriptor lasses (e.g., Zernike moment lass, Hu moment lass, improved Flusser moments et.). Also, it will be interesting to analyze and quantify the behavior (in terms of GRS and robustness) of the proposed invariants when the perturbation fators are hanged. For example, it is important to known what is the robustness level of this invariant set for other types of noises whih are speifi to a digital proessing hain (partiularly, the noises by multipliative type). Another interesting point for a future development refers to integration of the invariats given by equation (10) into a new lassifiation sheme. For example, some possible lassifiation shemes an be: {neural version of the generalized Sammon algorithm and a geneti optimized feedforward neural network} or {ICA tehnique and a neuro-fuzzy network}. In the same sientifi researh diretion, is will be useful to make a more serious experimental analysis onerning the GRS surfae shape properties in onnetion with lassifiation and/or robustness performanes of reognition system. Aordingly, as a result of GRS surfae analysis, it will be possible to identify the optimal lassifiation shemes for different types of real ATR systems, [14]. Referenes: [1] A. Abo-Zaid, O. Hinton, and E. Horne, About moment normalization and omplex moment desriptors, Proeedings of ICPR Conferene, pp , 1988 [2] J. Adams, E. Curtis, Feature Extration Methods for Form Reognition Appliations, WSEAS Transations on Information Siene and Appliations, vol.3, issue 3, pp , ISSN , 2006 [3] C. Aviles, A. Garia, Image Searhing based on Prinipal Components Analysis and Invariant Moments, WSEAS Transations on Information Siene and Appliations, vol.3, issue 10, pp , ISSN , 2006 [4] J. Flusser, On the Independene of Rotation Moments Invariants, Pattern Reognition Journal, no.33, pp , 2000 [5] J. Flusser, T. Suk, Rotation Moment Invariants for Reognition of Symmetri Objets, IEEE Transation on Image ISSN: Issue 4, Volume 5, April 2008

11 Proessing, vol.15, pp , ISSN: , 2006 [6] A. Foulonneau, P.Charbonnier, Affine- Invariant Geometri Shape Priors for Region- Based Ative Contours, IEEE Transations on Pattern Analysis and Mahine Intelligene, vol.28, no.8, pp , 2006 [7] R. Gonzales, R.E. Woods, Digital Image Proessing, Seond Edition, Prentie Hall, 2002 [8] A. Jain, Fundamentals of Digital Image Proessing, Pretine-Hall International In., Englewood Cliffs, 1990 [9] R.I. Maheb, Appliation of Neural Networks Analysis in Image Feature Extration, WSEAS Transations on Information Siene and Appliations, vol.2, issue 7, pp , ISSN , 2005 [10] M.S. Nixon, A.S. Aguado, Feature Extration in Computer Vision and Image Proessing, ELSEVIER, 1st Edition, USA, 2002 [11] M. Rodrigues, Invariants for Pattern Reognition and Classifiation, World Sientifi Publishing Co., Danvers, 2000 [12] I.C. Vizitiu, Neural Networks used in Pattern Reognition, MTA Publishing House, Buharest, 2005 [13] I.C. Vizitiu, A New Approah to Inrease the Video Pattern Reognition Performanes, International Symposium on Defense Tehnology, ZMNDU, pp , Budapest, Hungary, 2006 [14] I.C. Vizitiu and olletive, Smart Multisensor System for Automati Air Target Traking and Reognition, Sientifi Researh Report of CNCSIS grant, AT type, no. 88 GR, pp , [15] D.Xu, H.Li, Geometri moment invariants, Pattern Reognition Journal, No.41, pp , 2008 ISSN: Issue 4, Volume 5, April 2008