 Research
 Open Access
A modified shape context method for shape based object retrieval
 Radhika Mani Madireddy^{1},
 Pardha Saradhi Varma Gottumukkala^{2},
 Potukuchi Dakshina Murthy^{3} and
 Satyanarayana Chittipothula^{4}Email author
https://doi.org/10.1186/219318013674
© Madireddy et al.; licensee Springer. 2014
 Received: 10 September 2014
 Accepted: 31 October 2014
 Published: 15 November 2014
Abstract
The complexity in shape context method and its simplification is addressed. A novel, but simple approach to design shape context method including Fourier Transform for the object recognition is described. Relevance of shape context, an important descriptor for the recognition process is detailed. Inclusion of information regarding all the contour points (with respect to a reference point) in computing the distribution is discussed. Role of similarity checking the procedure details regarding the computation of matching errors through the alignment transform are discussed. Present case of shape context (for each point with respect to the centroid) descriptor is testified for its invariance to translation, rotation and scaling operations. Euclidean distance is used during the similarity matching. Modified shape context based descriptor is experimented over three standard databases. The results evidence the relative efficiency of the modified shape context based descriptor than that reported for other descriptor of concurrent interests.
Keywords
 Object recognition
 Shape representation
 Feature extraction
 Distance measure
Introduction
Although significant progress is witnessed in the field of automated object recognition, it is still remains challenging task (Zhang and Lu2004; Iyer et al.2005) from the broad purview of machine learning and computer vision processes of contemporary requirements. The shape of an object contains (Forsyth and Mundy1999) an important, unique and characteristic features of the object. The shape based methods consider either the contour or the entire region of the object. The consideration of contour involves less representative points in comparison with the region based methods (Nixon and Aguado2002). The regionbased methods consider the global information (all the pixels within a shape) for the design of the descriptor which involves the geometrical moments (Hu1962; Flusser2000), Zernike moments (ZM) (Teague1980; Khotanzad1990), pseudo Zernike moments (Belkasim et al.1991), Legendre moments (Teague1980), and Tchebichef moments (Mukundan et al.2001), generic Fourier descriptor (FD) (Zhang and Lu2002), compounded image descriptor (Li and Lee2005), shape matrix (Goshtasby1985), the grid technique (Lu and Sajjanhar1999) and shock graph (Sebastian et al.2004; Siddiqi et al.1999) etc. However, the contour based representation is reported to be more efficient (Yang et al.2008). Several recently reported contour based methods rely on viz., Fourier transform (Zahn and Roskies1972; Wallace and Wintz1980; Kunttu et al.2006), curvature scale space (CSS) (Mokhtarian and Mackworth1986; Abbasi et al.1999,2000), wavelet transform (Chauang and Kuo1996; Yadav et al.2007), contour displacement (Adamek and O’Connor2004), chain codes (Junding and Xiaosheng2006), autoregressive (Dubois and Glanz1986), Delaunay triangulation (Tao and Wi1999), multiresolution polygonal (Day et al.2004) robust symbolic representation (Daliri and Torre2008), distance sets (Grigorescu and Petkov2003), elastic matching (Attalla and Siy2005) etc techniques for the design of the shape descriptor. Basing on the consideration of shape boundaries (Petrakis et al.2002; Arica and Vural2003; Bartolini et al.2005; Lateckia et al.2005; Alajlan et al.2007), dynamic programming (DP) technique is also adopted to achieve high accuracy rate. The DP based techniques suffer from being computationally expensive and get reduced to be impractical for large databases, despite the fact that they offer better performance.
Generally, the descriptor relevant to the shape context (Belongie et al.2002) method for object recognition is developed with an established correspondence between the point sets. The procedure combines the shape context information with the information formatted by using thin plate spline (Bookstein1989) processing. Due to the proven simplicity and capability of discrimination, the shape context based methods proficiently proposed in the literature (Dubois and Glanz1986; Tao and Wi1999; Day et al.2004; Daliri and Torre2008; Grigorescu and Petkov2003; Attalla and Siy2005; Petrakis et al.2002; Arica and Vural2003; Bartolini et al.2005; Lateckia et al.2005; Alajlan et al.2007; Belongie et al.2002; Bookstein1989; Mori and Malik2003; Thayananthan et al.2003; Zhang and Malik2003; Salve and Jondhale2010). Recently, Xin Shu proposed Contour Points Distribution Histogram (CPDH) (Shu and Xiao Jun2011) for the shape context method. The shape matching process which speaks out the performance of a descriptor is dealt in different ways. The Zucker et al (Siddiqi et al.1999) has developed shock graph grammar and the relevant tree matching algorithm. The spectral distance (based on diffusion geometry, heat trace) estimated through the Laplacian transform is also used for matching (Bronstein and Kokkinos2010; Bronstein and Bronstein2011; Konukoglu et al.2013). On the other hand, the Fourier transform based matching procedures are is also popular (Cem Direko glu and Nixon2011; Xingyuan and Zongyu2013; Ghazal et al.2009; Ghazal et al.2012).
In the wake of the results reported in the area of shape context based object recognition techniques involving a wide variety of design of description and matching measures, it serves that the utility of the Fourier based descriptors for the shape context based recognition presents a superior method rather than the contour based methods. Hence, the authors propose for the design of a novel hybrid contour based shape descriptor which is constructed with respect to the centroid, while the feature vector is estimated by a 1D Fourier transform. The shape toning phase is involves the Euclidean Distance to enhance the quality.
The paper is organized in three sections. Introduction to the computerized object recognition method is presented in sectionIntroduction. Methodology adopted for the present shape context technique is presented in sectionMethodology along with the information for indices to evaluate its performance. The results obtained by adopting present method to the standard databases and their trends are presented in sectionResults and discussion along with the relevant discussions of performance.
Methodology
A multi staged novel and hybrid shape context based scheme for the object recognition process is proposed. The phase wise information during the processing is presented in sectionDesign of system, while the proposed indices to estimate the performance are presented in sectionPerformance.
Design of system
 (i)
Shape representation with contour
 (ii)
Computation of Shape Context
 (iii)
Construction of histogram for each bin of shape context
 (iv)
Shape description by using Fourier Transform
θ(x, y) is the angle measurement between two points x and y,
y_{2} is the y coordinate of the first point,
y_{1} is the y coordinate of the second point.
where:
θ(x,y) is the angle measurement of a point (x, y),
m_{1} is the slope of the line between first point and second point,
m_{2} is the slope of the line between first point and center point.
where:
s(t) represents the 1D contour signal,
N represents number of representative points of the contour,
n = 0,1,2,…,N1 and,
FD_{n} represents n^{th} Fourier descriptor.
Using Equation (3), the required Fourier Descriptors of size ‘N’ are generated. Further, the extracted features are testified for their invariance to translation, rotation and scaling operations (performed over the set of images). In the wake of the fact that the proposed descriptor is obtained with respect to the centroid, the obtained features are expected to be invariant to translation. The possible finite (and stipulated) magnitude of the values for the features vouches for the rotation invariance. For the present method, the scaling invariance is also presented by involving the process of dividing the features with the first feature value. In the third step, the feature vector is constructed, which describes the entire shape features of the object.
To further improve the quality of proposed methods, the global information of the object is also considered. For this, experiments are conducted with considering different global descriptors and identified that three global descriptors (GD) are efficient to represent the global shape information. The GD feature vector, viz., {S, C, A} contains the measures of solidity, circularity and aspect ratio is computed for the given object.
In the fourth step, the shape toning process is executed. In the shape toning process, the distance measures (Ghazal et al.2009,2012) used is viz., the Euclidean distance (ED). The distance measure between two objects shape context vectors is given by the Equation (4). In this, the average global distance of global feature vectors is directly added to the Euclidean distance of the Fourier descriptor feature vector.
ED (TE, TR) represents the Euclidean Distance between the test and trained shapes and,
D_{X} (TE,TR) represents the Global distance between the test and trained shapes.
where:
X represents the GD vector {S, C, AR},
X^{TE} represents the GD feature of the test shape and,
X^{TR} represents the GD feature of the trained shape.
According to the specificity of the data of distance measurement, the distances are further rearranged in ascending order and are assigned with ranks. However, the system is also enabled to recognize and register the top ranked images.
The standard databases (Sikora2001; Sebastian et al.2001) used for the evaluation of shape descriptors presently are Kimia {K99, K216} and MPEG CE1 Set B. It is noticed that the Set B database with 70 groups and each group with 20 images. It characteristically includes rotated, scaled, skewed and defected shapes. However the K99 database which consists of 9 groups, each group with 11 images. It is known to include the partially occluded shapes. The K216 database with 18 groups, each group with 12 images, it represents a sub database of Set B, and contains partially occluded shapes.
Performance
x denotes the true recognition results,
y denotes the total recognized result and,
P denotes the precision.
where:
R denotes the Recall,
x denotes the true recognition results and,
group size denotes the maximum true recognition result.
where:
FDR denotes the False Detection Rate,
z denotes the false recognition result and,
y denote the total recognized result.
The average FDR (AFDR) value of all test images corresponding to each database is estimated. Apart from the usual recognition rate, the Average Processing Time (APT) is also estimated for each query in the shape toning stage. The proposed descriptor is compared with 4 standard descriptors viz., Angular Radial Transform Descriptor (ARTD) (Zhang et al.2008), Moment Invariant Descriptor (MID) (Zhang et al.2008), Zernike Moment Descriptor (ZMD) (Tiagrajah and Razeen2011) and CurvatureScaleSpaceDescriptor (CSSD) (Tiagrajah and Razeen2011). A specific feature size of 35 for ARTD (n < 3, m < 12), 6 for MID, 34 for ZMD (order from 2 to 10) is adopted. The CSSD feature size is varying from that for one image to another image since number of peaks is varying. All the cited metrics viz., APLR, APHR, AFDR and APT are evaluated to estimate the performance for the proposed descriptor (with inclusion of GD), ARTD, MID, ZMD and CSSD.
Results and discussion
Shape context based object recognition is estimated as detailed in sectionDesign of system for the input of standard databases. The trends of the results that follow various approaches are presented in the following sub sectionProcessing of modified shape context based object recognition. The relative performance of the proposed descriptor is also analyzed in the sectionPerformance evaluation in the wake of the other reported methods.
Processing of modified shape context based object recognition
Performance evaluation
The yield of APLR and APHR values for the descriptor with the currently proposed distance measure is analyzed. In the wake of the four other standard descriptors, the aspect of compatibility (with three databases) is also explored, while the results are presented in Tables 1,2,3 respectively. From these results, it is clearly evident that the presently proposed descriptor out performs the other descriptors regarding all the three standard databases. However, among the presently considered descriptors, the CSSD descriptor is found to accompany with a lower performance, followed by that of MID, ARTD. However, the case of ZMD resulted for the next higher performance. But, for Set B database, the ZMD is yielding the highest result. From Table 1, it is found that the proposed MSC + GD happen to be influential to increase the APLR value of ZMD. It is also found to significantly increase the APHR value of ARTD. For K99 and K216 databases, the ZMD is giving distinctly improved APLR and APHR values than with the other descriptors. From the Tables 2 and3; it is evident that the proposed MSC + GD is accompanied with an improved performance in terms of enhancement of APLR and APHR.
The APLR and APHR values for various descriptors with Set B database
Avg. precision  

Descriptor  Recall < =50%  Recall >50%  Average 
BSC  82.65  49.37  66.01 
MSC  84.02  49.56  66.79 
BSC + GD  84.48  52.85  68.67 
MSC + GD  88.83  55.58  72.21 
ARTD  82.10  45.69  63.90 
MI  79.54  44.50  62.02 
ZMD  82.56  45.62  64.09 
CSSD  78.61  41.81  60.21 
The APLR and APHR values for various descriptors with K99 database
Avg. precision  

Descriptor  Recall < =50%  Recall >50%  Average 
BSC  86.01  51.25  68.63 
MSC  89.11  56.87  72.99 
BSC + GD  87.21  57.82  72.52 
MSC + GD  90.77  62.43  76.60 
ARTD  84.26  45.72  64.99 
MI  81.96  44.74  63.35 
ZMD  89.61  61.37  75.49 
CSSD  82.32  44.11  63.22 
The APLR and APHR values for various descriptors with K216 database
Avg. precision  

Descriptor  Recall < =50%  Recall >50%  Average 
BSC  88.50  57.59  73.05 
MSC  90.16  61.18  75.67 
BSC + GD  91.01  64.00  77.50 
MSC + GD  91.50  66.00  78.75 
ARTD  81.35  44.67  63.01 
MI  80.14  46.04  63.09 
ZMD  88.94  61.71  75.33 
CSSD  80.12  44.97  62.55 
The AFDR for various descriptors with three databases
Descriptor  AFDR  

Set B  K99  K216  
MSC + GD  0.70  0.72  0.73 
ARTD  0.90  0.85  0.86 
MID  0.91  0.81  0.84 
ZMD  0.78  0.76  0.76 
CSSD  0.84  0.88  0.89 
APT of various descriptors with set B database
Descriptor  APT 

MSC + GD  0.0011 
ARTD  0.0017 
MID  0.0302 
ZMD  0.0017 
CSSD  2.1640 
Bull’s eye score for various descriptors with set B database
Descriptor  Score % 

CSS (Mokhtarian et al.1997)  75.44 
Visual Parts (Latecki and Rolf2000)  76.45 
SC + TPS (Belongie et al.2002)  76.51 
Aligning Curves (Sebastian et al.2003)  78.16 
SSC (Xie et al.2008)  79.92 
CPDH + EMD (Cem Direko glu and Nixon2011)  76.56 
General Model (Tu and Yuille2004)  80.03 
MSC + GD  81.64 
IDSC + DP (Ling & Jacobs2007)  85.40 
Conclusions

Shape context based description is proved to be efficient when compared with various other standard descriptors with respect to various performance measures viz., APLR, APHR, AFDR and APT.

The proposed descriptor improves the precision measures at high recalls when compared with the low recalls thus enabling more relevant objects to be recognized.

With less feature vector size, the proposed descriptor enables the object recognition system to be efficient with less APT and AFDR measures.
Declarations
Authors’ Affiliations
References
 Abbasi S, Mokhtarian F, Kittler J: Curvature scale space image in shape similarity retrieval. Multimed Syst 1999, 7: 467476.View ArticleGoogle Scholar
 Abbasi S, Mokhtarian F, Kittler J: Enhancing CSSbased shape retrieval for objects with shallow concavities. Image Vis Comput 2000, 18: 199211.View ArticleGoogle Scholar
 Adamek T, O’Connor NE: A multi scale representation method for non rigid shapes with a single closed contour. IEEE Trans Circuits Syst Video Technol 2004, 14: 742753.View ArticleGoogle Scholar
 Alajlan N, Rube IE, Kamel MS, Freeman G: Shape retrieval using trianglearea representation and dynamic space warping. Pattern Recogn 2007, 40: 19111920.View ArticleGoogle Scholar
 Arica N, Vural FTY: BAS: a perceptual shade descriptor based on the beam angle statistics. Pattern Recogn Lett 2003, 24: 16271639.View ArticleGoogle Scholar
 Attalla E, Siy P: Robust shape similarity retrieval based on contour segmentation polygonal multi resolution and elastic matching. Pattern Recogn 2005, 38: 22292241.View ArticleGoogle Scholar
 Bartolini I, Ciaccia P, Patella M: WARP: accurate retrieval of shapes using phase of Fourier descriptors and time warping distance. IEEE Trans Pattern Anal Mach Intell 2005, 27: 142147.View ArticleGoogle Scholar
 Belkasim SO, Shridhar M, Ahmadi M: Pattern recognition with moment invariants: a comparative study and new results. Pattern Recogn 1991, 24: 11171138.View ArticleGoogle Scholar
 Belongie S, Malik J, Puzicha J: Shape matching and object recognition using shape contexts. IEEE Trans Pattern Anal Mach Intell 2002, 24(4):509522.View ArticleGoogle Scholar
 Bookstein FL: Principal warps: thinplatesplines and decomposition of deformations. IEEE Trans Pattern Anal Mach Intell 1989, 11(6):567585.View ArticleGoogle Scholar
 Bronstein MM, Bronstein AM: Shape recognition with spectral distances. IEEE Trans Pattern Anal Mach Intell 2011, 33(5):10651071.View ArticleGoogle Scholar
 Bronstein MM, Kokkinos I: Scale invariant heat kernel signatures for non rigid shape recognition. IEEE Conf Comput Vision Pattern Recogn (CVPR) 2010, 17041711. doi:10.1109/CVPR.2010.5539838Google Scholar
 Nixon MS, Cem Direko glu: Shape classification via imagebased multi scale description. Pattern Recogn 2011, 44: 21342146.View ArticleGoogle Scholar
 Chauang G, Kuo C: Wavelet descriptor of planar curves: theory and applications. IEEE Trans Image Process 1996, 5: 5670.View ArticleGoogle Scholar
 Daliri MR, Torre V: Robust symbolic representation for shape recognition and retrieval. Pattern Recogn 2008, 41: 17821798.View ArticleGoogle Scholar
 Day AM, Arnold DB, Havemann S, Fellner DW: Combining polygonal and subdivision surface approaches to modeling and rendering of urban environments. Comput Graph 2004, 28(4):497507.View ArticleGoogle Scholar
 Dubois SR, Glanz FH: An autoregressive model approach to two dimensional shape classification. IEEE Trans Pattern Anal Mach Intell 1986, 8: 5565.View ArticleGoogle Scholar
 Flusser J: On the independence of rotation moment invariants. Pattern Recogn 2000, 33: 14051410.View ArticleGoogle Scholar
 Forsyth D, Mundy J: Shape, contour and grouping in computer vision. Lect Notes Comput Sci 1999, 1681: 13.Google Scholar
 Ghazal AE, Basir O, Belkasim S: Farthest point distance: a new shape signature for Fourier descriptors. Signal Process Image Comm 2009, 24: 572586.View ArticleGoogle Scholar
 Ghazal AE, Basir O, Belkasim S: Invariant curvaturebased Fourier shape descriptors. J Vis Commun Image R 2012, 23: 622633.View ArticleGoogle Scholar
 Goshtasby A: Description and discrimination of planar shapes using shape matrices. IEEE Trans Pattern Anal Mach Intell 1985, 7: 738743.View ArticleGoogle Scholar
 Grigorescu C, Petkov N: Distance sets for shape filters and shape recognition. IEEE Trans Image Process 2003, 12(10):12741286.View ArticleGoogle Scholar
 Hu M: Visual pattern recognition by moment invariants. IRE Trans Inform Theor IT 1962, 8: 115147.Google Scholar
 Iyer N, Jayanti S, Lou K, Kalyanaraman Y, Ramani K: Three dimensional shape searching: state of the art review and future trends. Comput Aided Des 2005, 37: 509530.View ArticleGoogle Scholar
 Junding S, Xiaosheng W: Chain Code DistributionBased Image Retrieval. International Conference on Intelligent Information Hiding and Multimedia Signal Processing, China; 2006:139142.Google Scholar
 Khotanzad A: Invariant image recognition by Zernike moments. IEEE Trans Pattern Anal Mach Intell 1990, 12: 489497.View ArticleGoogle Scholar
 Konukoglu E, Glocker B, Criminisi A, Pohl KM: WESDweighted spectral distance for measuring shape dissimilarity. IEEE Trans Pattern Anal Mach Intell 2013, 35(9):22842297.View ArticleGoogle Scholar
 Kunttu I, Lepisto L, Rauhamaa J, Visa A: Multi scale fourier descriptors for defect image retrieval. Pattern Recogn Lett 2006, 27: 123132.View ArticleGoogle Scholar
 Latecki LJ, Lakamper R: Shape similarity measure based on correspondence of visual parts. IEEE Trans Pattern Anal Mach Intell 2000, 22(10):11851190.View ArticleGoogle Scholar
 Lateckia LJ, Lakaempera R, Wolter D: Optimal partial shape similarity. Image Vis Comput 2005, 23: 227236.View ArticleGoogle Scholar
 Li S, Lee MC: Effective invariant features for shapebased image retrieval. J Am Soc Inf Sci Technol 2005, 56: 729740.View ArticleGoogle Scholar
 Ling H, Jacobs DW: Shape classification using the inner distance. IEEE Trans Pattern Anal Mach Intell 2007, 29(2):286299.View ArticleGoogle Scholar
 Lu G, Sajjanhar A: Regionbased shape representation and similarity measure suitable for content based image retrieval. Multimed Syst 1999, 7: 165174.View ArticleGoogle Scholar
 Mokhtarian F, Mackworth A: Scalebased description and recognition of planar curves and twodimensional shapes. IEEE Trans Pattern Anal Mach Intell 1986, 8: 3443.View ArticleGoogle Scholar
 Mokhtarian F, Abbasi F, Kittler J, Smeulders AWM, Jain R: Efficient and robust retrieval by shape content through curvature scale space. In Image Databases and MultiMedia Search. Singapore: World Scientific Publishing; 1997:5158.Google Scholar
 Mori G, Malik J: Recognizing objects in adversarial clutter: breaking a visual CAPTCHA. IEEE Conf Comput Vision Pattern Recogn 2003, 1: 10636919.Google Scholar
 Mukundan R, Ong SH, Lee PA: Image analysis by Tchebichef moments. IEEE Trans Image Process 2001, 10: 13571364.View ArticleGoogle Scholar
 Nixon MS, Aguado AS: Feature Extraction and Image Processing. 1st edition. Newnes Publishers, Burlington MA; 2002:247287.View ArticleGoogle Scholar
 Peter J, Otterloo V: A contour Oriented Approach to Shape Analysis. Prentice Hall, Hertfordshire UK; 1991.Google Scholar
 Petrakis EGM, Diplaros A, Milios E: Matching and retrieval of distorted and occluded shapes using dynamic programming. IEEE Trans Pattern Anal Mach Intell 2002, 24: 15011516.View ArticleGoogle Scholar
 Salve SG, Jondhale KC: Shape matching and object recognition using shape contexts. In Computer Science and Information Technology (ICCSIT), vol 9. 3rd IEEE International Conference; 2010:471474.Google Scholar
 Sebastian TB, Klein PN, Kimia BB: Recognition of shapes by editing shock graphs. ICCV 2001, 1: 755762.Google Scholar
 Sebastian T, Klein P, Kimia B: On aligning curves. IEEE Trans Pattern Anal Mach Intell 2003, 25(1):116125.View ArticleGoogle Scholar
 Sebastian TB, Klein PN, Kimia BB: Recognition of shapes by editing their shock graphs. IEEE Trans Pattern Anal Mach Intell 2004, 26(5):550571.View ArticleGoogle Scholar
 Shu X, Xiao Jun W: A novel contour descriptor for 2D shape matching and its application to image retrieval. Image Vis Comput 2011, 29: 286294.View ArticleGoogle Scholar
 Siddiqi K, Shokoufandeh A, Dickinson SJ, Zucker SW: Shock graphs and shape matching. Int J Comput Vis 1999, 35(1):1332.View ArticleGoogle Scholar
 Sikora T: The MPEG7 visual standard for content description an overview. IEEE Trans Circuits Syst Video Technol 2001, 11(6):696702.View ArticleGoogle Scholar
 Tao Y, Wi G: Delaunay Triangulation for Image Object Indexing: A Novel Method for Shape Representation. Seventh SPIE Symposium on Storage and Retrieval for Image and Video Databases San Jose, CA; 1999:631642.Google Scholar
 Teague M: Image analysis via the general theory of moments. J Opt Soc Am 1980, 70: 920930.View ArticleGoogle Scholar
 Thayananthan A, Stenger B, Torr PHS, Cipolla R: Shape context and chamfer matching in cluttered scenes. IEEE Conf Comput Vision Pattern Recogn 2003, 1: 127133.Google Scholar
 Tiagrajah VJ, Razeen AASM: An enhanced shape descriptor based on radial distances. IEEE Int Conf Signal Image Process Appl (ICSIPA) 2011, 472477. doi:10.1109/ICSIPA.2011.6144073Google Scholar
 Tu Z, Yuille A: Shape matching and recognitionusing generative models and informative features. Proc Eur Conf Comput Vis 2004, 3: 195209.Google Scholar
 Wallace TP, Wintz PA: An efficient three dimensional aircraft recognition algorithm using normalized Fourier descriptors. Comput Graph Image Process 1980, 13: 99126.View ArticleGoogle Scholar
 Xie J, Heng P, Shah M: Shape matching and modeling using skeletal context. Pattern Recogn 2008, 41(5):17561767.View ArticleGoogle Scholar
 Xingyuan W, Zongyu W: A novel method for image retrieval based on structure elements’ descriptor. J Vis Commun Image R 2013, 24: 6374.View ArticleGoogle Scholar
 Yadav RB, Nishchal NK, Gupta AK, Rastogi VK: Retrieval and classification of shapebased objects using Fourier, generic Fourier, and waveletFourier descriptors technique: a comparative study. Opt Lasers Eng 2007, 45: 695708.View ArticleGoogle Scholar
 Yang M, Kpalma K, Ronsin J: A Survey of shape feature extraction techniques. In Pattern Recognition Edited by: Pen Y. 2008, 4390.Google Scholar
 Zahn T, Roskies RZ: Fourier descriptors for plane closed curves. IEEE Trans Comput 1972, 21: 269281.View ArticleGoogle Scholar
 Zhang D, Lu G: Shapebased image retrieval using generic Fourier descriptor. Signal Process Image Commun 2002, 17: 825848.View ArticleGoogle Scholar
 Zhang D, Lu G: A comparative study of curvature scale space and Fourier descriptors for shape based image retrieval. J Vis Commun Image Represents 2003, 14(1):3957.View ArticleGoogle Scholar
 Zhang D, Lu G: Review of shape representation and description techniques. Pattern Recogn 2004, 37(1):119.View ArticleGoogle Scholar
 Zhang D, Lu G: Study and evaluation of different fourier methods for image retrieval. Image Vis Comput 2005, 23(11):3349.View ArticleGoogle Scholar
 Zhang H, Malik J: Learning a discriminative classifier using shape context distances. IEEE Conf on Comput Visi Pattern Recogn 2003, 1: 242247.Google Scholar
 Zhang G, Ma ZM, Tong Q, He Y, Zhao T: Shape Feature Extraction using Fourier Descriptors with Brightness in ContentBased Medical Image Retrieval. International Conference on Intelligent Information Hiding and Multimedia Signal Processing; 2008:7174.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.