research article image classification using pso-svm and an...

Research ArticleImage Classification Using PSO-SVM and an RGB-D Sensor

Carlos Loacutepez-Franco Luis Villavicencio Nancy Arana-Daniel and Alma Y Alanis

Computer Science Department CUCEI University of Guadalajara 44430 Guadalajara JAL Mexico

Correspondence should be addressed to Carlos Lopez-Franco clzfrancogmailcom

Received 10 February 2014 Revised 29 May 2014 Accepted 13 June 2014 Published 10 July

Academic Editor Francesco Ubertini

Copyright copy 2014 Carlos Lopez-Franco et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Image classification is a process that depends on the descriptor used to represent an object To create such descriptors we useobject models with rich information of the distribution of pointsThe object model stage is improved with an optimization processby spreading the point that conforms the mesh In this paper particle swarm optimization (PSO) is used to improve the modelgeneration while for the classification problem a support vector machine (SVM) is used In order to measure the performanceof the proposed method a group of objects from a public RGB-D object data set has been used Experimental results show thatour approach improves the distribution on the feature space of the model which allows to reduce the number of support vectorsobtained in the training process

1 Introduction

Over the past years there has been an increasing interestin object recognition Object recognition can be divided intwo major tasks object localization and image classificationObject localization detects instances of a given category inthe image Image classification can be defined as the task ofdefining labels to an image depending on the presence of anobject

In this paper we propose an image classification systembased on invariant moment descriptor that includes depthinformation The 3D data allows producing small and robustdescriptors that will improve the image classification Thesedescriptors are constructed using object models with richinformation of the distribution of points The model genera-tion stage requires that best points are selected therefore thisstage can be defined as an optimization problem

Mathematical optimization is the selection of the bestelement with regard to some criteria In the simplest case it isconsisted of maximizing or minimizing a fitness function [1]Metaheuristic designates a computational method that opti-mizes a problem by iteratively trying to improve a candidatesolution [2]Manymetaheuristics implement nature-inspiredstochastic optimization One of these algorithms is particle

swarm optimization (PSO) developed by [3] and inspired bysocial behavior of bird flocking or fish schooling It has beenapplied in many fields such as tremor analysis for biomedicalengineering trajectory planning [4] electric power [5] andimage processing [6] Optimization algorithms are often usedin computer vision tasks such as image classification whichfinds a relation between an input image and a set of previouslyknown models [7]

The sensor used in this work is a Kinect [8] which isan RGB-D sensor providing synchronized color and depthimages This sensor is widely used by the computer visioncommunity due to its capabilities

In this work we analyze the inclusion of 3D information(provided by an RGB-D sensor) and the use of the PSOalgorithm to create robust objectmodels Using this approachwe can construct small and robust descriptors that improveimage classification

The rest of the paper is organized as follows The nextsection will present the proposed image classification systemIn Section 4 we present the mesh optimization process witha brief introduction to the PSO algorithm In Section 5 theinvariant moment descriptor and classification are explainedIn Section 6 we show the results of the proposed approachFinally in Section 8 we give the conclusions

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 695910 17 pageshttpdxdoiorg1011552014695910

2 Mathematical Problems in Engineering

2 Related Work

In recent years image classification has become a very activeresearch field The goal of the classification task is to developan algorithm which will assign a class-label to each one ofthe samples in the training data set There are two mainapproaches for classification in the literature namely (a)supervised approach and (b) unsupervised approach wherethe former uses a set of samples to train the classificationand the latter performs the classification by exploring the data[9] Many popular image classification methods [10 11] arebased on local descriptor and support vector machines asbasic techniques

One of the most widely used detectors is the HarrisCorner detector [12] However the Harris detector is notinvariant to affine transformations and scale [13] In [14]an automatic scale detection was proposed Later in [15]the authors present an algorithm for interest point detectionwhich is invariant to important scale changes Then in [16]the SIFT detector is proposed the approach approximatesthe Laplacian of Gaussians (LoG) by difference of Gaussians(DoG) filter to identify the potential keypoints Later in[17] the SURF detector was presented and SURF uses anapproach similar to SIFT however the approach uses integralimages instead of DoGs this allows a fast computation ofapproximate LoG using a box filter

Many different feature descriptors have been proposed inthe literature steerable filters [18] moment invariants [19]phase-based local features [20] Gaussian derivatives [21] anddescriptors that represent the distribution of smaller-scalefeatures [10] The SIFT detector produces robust descriptorshowever the speed of the detector is not very high In [17]the SURF detector was presented this approach is fasterthan SIFT and it has a similar performance on real images[22]

Histogram of oriented gradients (HoG) was proposed in[23] The HoG local patters are based on gradient histogramThe HoG and SIFT detectors share the same concept ofimplementation in the sense that local histograms of ori-ented patches defined across the image are used to producethe descriptor In [23] the authors used HoG descriptorsas feature vectors for a linear SVM In [24] the authorspropose the use of HoG for object detection and show thatHoG descriptors have robust and effective features for suchapplication In [25] the authors build an efficient detectorusing AdaBoost to train a chain of progressively morecomplex region rejection rules based on Haar descriptors forpedestrian detection

Our Approach In the last decades SVMs have been provento be an effective classifier approach for small sample setsor high dimensional problems [26 27] Such result is veryimportant since both problems are too difficult to be solvedby classical paradigms

One of the problems encountered during the design of animage classification technique is data overfitting which ariseswhen the training samples are small in comparison with thenumber of features To overcome this problem we design adescriptor with an optimized 3D point distribution

The contribution of this work is the development of asmall and robust descriptor based on invariant moments and3D information in order to improve the classification processThe 3D information is incorporated fromdepth data obtainedfrom an RGB-D sensor The object model is optimizedthrough the use of a PSO algorithm this optimization allowsimproving the classification

3 Image Classification System

The proposed classification system uses 3D informationand is based on local features invariant moments contourgeneration and a reduction of depth information using amesh grid It consists of five main steps feature extractioncontour creation mesh reduction mesh optimization andinvariant moment descriptor formation see Figure 1

31 Feature Extraction The first step is the extraction ofSURF keypoints from the input image (Figure 5(b)) Thespeeded up robust features algorithm (SURF) [28] findsfeatures using integral images and Haar-like filters SURFfeatures provide robustness and speed and are able to bedetected despite scale translation and rotation of objectsor changes in illumination The following steps rely on thisstability to group points and create object models and theyare described in the following subsections

SURF and SIFT detectors employ slightly different waysof detecting features SIFT detector builds image pyramidsand filters each layer with a Gaussian of increasing sigmavalues and then it takes the difference SURF detector usesa box filter approximation A comparison of the SIFT andSURF detectors is presented in [22] In suchwork the authorsconclude that SURF is as good as SIFT on most tests exceptfor scaling large blur and viewpoint With respect to thiswe have to mention that the proposed descriptor only usesSURF detector in the first step of the classifier to detectinterest points Then the proposed descriptor is constructedusing additional 3D information and invariant moments Inthe experiments results show that the proposed descriptoris small and robust In [22] the authors also mention thaton real image data sets there is little to separate the differentSIFTs and SURF except for efficiency In this work we chooseSURF detector because we are interested in implementinga real time classifier algorithm that could be used by amobile robot In addition as we will see in the resultsrsquosection the proposed descriptor is improved thanks to thespreading step in conjunction with the SURF detector andthis provides good results that are comparable and overcomesimilar approaches

32 Contour Creation The RGB-D sensor provides a hugeamount of data thus the information must be reduced intoa contour-mesh model to decrease computational cost Toconstruct the contour-mesh we use keypoints First they arescaled and translated then we compute the magnitude of thekeypoints with respect to their centroid (119909 119910) as

119898(119909 119910) = radic(119909 minus 119909)2+ (119910 minus 119910)

2 (1)

Mathematical Problems in Engineering 3

Feature extraction

Contour creation

Mesh reduction

Invariant momentdescriptor generation

Input image

SVM classification

Input

Mesh optimization

Figure 1 Image classification system

and its orientation with

120579 (119909 119910) = tanminus1 (119910 minus 119910

119909 minus 119909) (2)

After that we divide the 360 degrees into orientation bins anduse a slidingwindow to take one point for every bin accordingto their magnitude If any of the bins remains empty thenits value is linearly interpolated using the previous and nextknown points in the contour Examples for this stage of theprocess are shown in Figures 2 and 3

In the next sections we will explain the last two steps ofthe proposed approach namely the mesh reduction step andthe construction of the descriptor

33 Mesh Reduction In this step the 3D data that belongsto the object contour is segmented and then a cloud of 3Dpoints is obtained Due to the large number of points in thecloud a reduction of the data will be required in order tokeep a low computational cost Therefore in this step a meshthat covers the 3D point cloud is constructed which allows to

reduce the number of 3D points in the cloud without losingimportant information

First we proceed to extract depth information containedwithin the boundaries of the contour and reduce the pointsthat will be taken to compute the moments The reduction ofpoints aims to generate a smaller set with rich informationby adjusting a mesh grid over the object The initial positionof the points is obtained sectioning the bounding box intoequally separated cells generating (119909 119910) coordinates Thiscreates a set of bidimensional points that is

Grid = (1199091 1199101) (1199092 1199102) (119909

119899 119910119899) (3)

where 119899 is a predefined constant Besides

119909119894isin [119909min 119909max] 119910

119894isin [119910min 119910max]

1199091= 119909min 119910

1= 119910min

119909119899= 119909max 119910

119899= 119910max


Figure 2 Totem object with SURF keypoints

119909119894+1= 119909119894+ 119909inc 119910

119894+1= 119910119894+ 119910inc

119909inc =119909max minus 119909min

119899 119910inc =

119910max minus 119910min119899

(4)

minimum andmaximum values are defined by the boundingbox The remaining points are tested using the point inpolygon (PiP) algorithm if the points are inside the polygonthen they are considered valid

Then for each invalid point we take at random two validpoints and move the outlier to a position between themthis can be seen as a biased migration Later we attach the119911 coordinate (depth) to the (119909 119910) points Examples for thisstage of the process are shown in Figures 4 and 5(d)

It is important to mention that a point will be consideredvalid if and only if the coordinates (119909 119910)of the point are insidethe contour and the depth is not zero

These steps generate an object model for which we canextract information thatwe can use for classification howeverthe simple migration of points produces a model in whichpoints are not equally distributed This problem is solved byapplying evolutionary computation

4 Mesh Optimization

Themesh reduction step produces an object model howeverthe points are not equally distributed and thus an optimiza-tion step is required For the optimization step we havechosen an evolutionary computation (EC) technique due toall the constraints of the mesh optimization problem

Evolutionary algorithms (EA) are stochastic searchmeth-ods inspired by the behavior of swarmsrsquo populations ornatural biological behavior In general there are five popularalgorithms genetic algorithms [29] memetic algorithms[30] particle swarm optimization [3] ant-colony optimiza-tion [31] and shuffled frog leaping algorithm [32] In [33]the authors present a comparison study of these five EA theyconclude that the PSO algorithm performs better in general

with respect to the quality of the solution and the success rateFor these reasons we decided to choose the PSO algorithmamong the others

To solve the mesh optimization problem the PSO algo-rithm (Algorithm 1) was adapted in order to spread the set of(119909 119910)points that conform themesh and to obtain bettermod-els with rich information about the distribution of the pointsThepurpose is tomaximize the distance between pointswhilemaintaining them inside the boundaries conformed by theobject contour

In our approach each PSO particle represents a point inthe mesh The problem has multiple boundaries every pointof the contour is one Thus we have to check if particleslie inside the polygon and clamp them after every updateInstead of gathering particles to a global best position theytake positions separated uniformly from each other This isobtained through the fitness function and a modification inthe updating rules

41 Description of the Method The objective of this methodis the construction of a mesh with the best distributionof points The first step in the mesh construction is thedetermination of the object contour (Section 32) This stepuses features detectedwith SURF and a sampling technique tofind the object contourThe second step is themesh reductionstep (Section 33) in this step the 3D data is used to enhancethe object model However if we use all the 3D data thatbelong to the obtained contour then the computational costrequired to process it will be high Therefore the objectiveof this step is the construction of a mesh that covers the3D point cloud but with fewer points The third step isthe mesh optimization this process is required since themesh constructed in the previous step does not distributeefficiently For this purpose the PSO algorithm was adaptedto spread the points over the object The adapted PSOminimizes the distance of each particle and its neighbors withrespect to the mean distance of all the particles with respectto its neighbors The PSO particles are initialized with thecoordinates of the pointsThe number of particles is thereforeequal to the number of points Finally the object model isrecovered from the best local particle value (119875

119894)

42 Particle Swarm Optimization Algorithm PSO is astochastic search method inspired from the behavior ofswarm animals like bird flocking and fish schooling In PSOparticles or solution candidates move over the search spacetowards the zone with the best conditions using a cognitivecomponent of the relative particle and a social componentgenerated by the swarm (best local and global positions)Thislets PSO to evolve social behavior and relativemovement intoglobal optimum solutions [34 35]

In the iterative process the position119883119894and velocity 119881

119894of

particles are updated according to the cognition component119875119894and the social component 119866 with

119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119883119894) + 11988821205932(119866 minus 119883

119894)

119883119894(119905 + 1) = 119883

119894(119905) + 119881

119894(119905 + 1)

(5)


Figure 3 Sample contour for the totem object

Figure 4 Mesh creation example

where 1198881 1198882are positive constants 120593

1 1205932are two random

variables with uniform distribution between 0 and 1 and 120596is the inertia weight which balances the effect of the previousvelocity vector on the new one The cognitive component 119875

119894

is updated by each particle when a better position is obtainedThe social component119866 is updatedwhen a newbest positionwithin thewhole swarm is foundAfter initializing the swarmin each iteration the PSO basic steps are performed until thestop criterion is reached [36]

For more details on PSO the interested reader is referredto view [2 3 36 37]

43 PSO Fitness Function Instead of a single fitness functionwe undertake three steps to get a fitness value First wemeasure the distance of each particle to its nearest neighborthis measure gives information about how separated is everyparticle We only take the distance between the current 119894-particle and a neighbor 119895-particle with

119863119894= radic(119909

119894minus 119909119895)2

+ (119910119894minus 119910119895)2

(6)

to keep a low computational load


(a) (b)

10

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

86420minus2minus4minus6minus8minus10

(c) (d)

Figure 5 An image through the different stages of the procedure (a) Original input image (b) Feature extraction (c) Contour created forthe object (d) Depth information attached in a mesh grid

(1) repeat(2) Calculate fitness value of each particle using the fitness function(3) Update local best if the current fitness value is better(4) Determine global best take the best fitness particle and compare it to current best(5) For each particle(6) Calculate particle velocity according to (1)(7) Update particle position according to (2)(8) until Stop criteria is met

Algorithm 1 PSO algorithm

Then we calculate the mean of distances

119863 =1

119873

119873

sum119896=1

119863119896 (7)

where 119873 is equal to the swarm size This can be seen as theglobal value to maximize since we want the points to beuniformly separated and yet lie inside the boundaries that isthey have the same distance to each other and cover the spaceuniformly

Next we compute the difference between the local dis-tance of the particle to its neighbor and the mean globaldistance Thus the fitness of 119894th particle is defined as

119891119894= radic(119863

119894minus 119863)2

(8)

This is the fitness value by minimizing (8) every particletries to minimize its own distance with respect to the meandistance of the swarm and by doing this we would obtainan approximation to a geometric uniform distribution of theswarm on the space that is we will obtain a swarm in which


every particle is separated from its neighbors by a distanceof 119863 Thus the distance of the particle to its neighbor is theclosest value to the mean estimated distance This value isstored in by the cognitive component 119875

119894

Since our function is multiconstrained we add a penaltyterm to the fitness function Particles that go out of theboundaries are penalized according to the distance that theyhave to the centroid We add a constant 120572 which determinesthe influence of the penalization in the fitness value compu-tation [36] the fitness function with the penalization term isdefined as

119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with

119879 (119883) = 1 Particle is out side contour0 Particle is inside contour

(10)

where119883 is a point (119909 119910) in themesh represented by a particle119883 is the centroid (119909 119910) and 119879(119883) is a point-in-polygonfunction added so only points that got outside the contourare penalized

Additional to the penalization term we also use a pre-serve feasibility approach as explained by [36] since all ourinitialization particles are feasible we want to preserve thefinal result like that thus 119875

119894is only updated when the particle

lies inside the contour

44 PSO Update Formulas The position update includes avariable 120573 that multiplies velocity

119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)

120573 is a term that defines how close are particles from bound-aries we want particles to be sufficiently spread but we alsowant some particles close to the boundaries Clamping andupdates may cause particles to go out or be repositionedIf the particle is close to boundaries 120573 will be small andthe update effect will be less This value is thresholded soafter certain value it becomes one and the effect of velocityapplies normally this way particles far from boundaries arenot affected In conclusion 120573 can be seen as a function of thedistance of the particle to the boundaries with

119863119887= radic(119909

119894minus 119909boundary)

2

+ (119910119894minus 119910boundary)

2 (12)

the distance of a point to the object boundary and

120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)

The threshold value depends on the range of the dataset it establishes how far to the border the data can bewithout being affected In our case feature points are scaledto [minus10 10] and 120573 values in the range [02 10] were testedresulting in 07 as the value that best performed

In the velocity update the global coefficient has the effecton particles to move towards the best position of the swarmSince in this application we do not need that effect the termis replaced by one that makes particles get closer or fartherfrom their neighbor as neededThis behavior is accomplishedby using the sign of the distance between particles and aconstant value 120588 to determine how fast particles are going tomove towards or away from others that is

119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)

where119883119894 119883119895are two particles (119909

119894 119910119894) (119909119895 119910119895) with119883

119895being

the closest particle to119883119894 Another option is the inclusion of a

scale factor of the current fitness value since it gives a valueof how much the particles are separated

45 Details of the Modified PSO The fitness function of thealgorithm is defined to minimize the distance between theparticle 119894 with respect to themean distance of all the particleswith respect to their corresponding neighbors therefore atthe end of the PSO iterations we obtain the best spreadparticles found by the algorithm Although many differentfitness functions can be defined for this particular applicationthe chosen fitness function shows a good performance as itwas demonstrated on the experimental results

With respect to the stop criteria of the algorithm wecannot force PSO to stop when only the best value 119866 is closeto zero since it does not guarantee that the distances betweeneach particle with respect to their neighbors are close to themean distance Instead the algorithm stops when the particlebest value 119875

119894of each particle is close to zero or in practice

smaller than a certain threshold In addition we can stop thealgorithm if a certain threshold of iterations has been met

46 Point Spreading Algorithm (PSA) The inclusion of PSOin the mesh reduction step of the image classification systemaims to spread points and create models that describe theentire object surface better After the initial generation ofpoints in a grid over the object and the migration of pointsthat lie outside the object the PSO variation is applied to themesh of points Particles are initialized with the coordinatevalues of points The number of particles is therefore deter-mined by the number of points that we want over the objectA maximum number of iterations is set and the algorithm isexecuted until the stop criterion is reached An object modelis recovered from the local best position (119875

119894) that each particle

generated The main steps of the algorithm are portrayed inAlgorithm 2

In Figures 6 and 7 we can see an example of this proc-edure The first picture shows the original image The secondimage shows the initial coordinates of the mesh The thirdimage showsmigration of points some points aremigrated tovery close places and thus they are covering the object poorlyThe fourth image shows the points after the PSO algorithmis applied we can see that points are moved and the objectsurface is covered in a better way


(1) Obtain object contour(2) Generate starting point coordinates from bounding box(3) Migrate non-valid points(4) PSO spreading

Algorithm 2 Mesh construction algorithm

5 Invariant Moment Descriptorand Classification

Moments provide useful and compact information of a dataset such as its spread or dispersion A pattern may berepresented by a density distribution function moments canbe obtained for a set of points representing an object and theycan then be used to discriminate between objects [38] Thefirst order moments can be used to locate the centroid of thepointsrsquo distribution If we compute the moments consideringa translation to the centroid we generate central momentswhich can be made scale invariant [39 40] The generalequation for three-dimensional central moment (for short3D moment) is defined as

120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)

where119891(119909 119910 119911) is a distribution function of the variables and(119909 119910 119911) is the centroid The scaling is performed using

120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)

In particular [38] defines seven values computed bynormalizing central moments through order three that areinvariant to object scale position and orientation Tests wereperformed using different combinations of these seven valuesand then using the moments of order one to four Similarresults were obtained using the three first values defined byHu and themoments of order one to fourTherefore invariantmoments can be computed from the reduced 3D set In theproposed method we use the first four moments over thethree dimensions then the values for 119901 119902 and 119903 are definedover the interval from 1 to 4 with these moments we definea descriptor of 12 elements to represent each object modelSuch vector has the form

119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)

Finally the descriptor is given as input to a SVM andwe get the result on whether the image contains the targetobject The SVM [41] is an algorithm to solve classificationand regression problems It defines a subset on the train datacomposed by those samples that are closer to the decisionarea It is based in the maximization of the margin ofseparation between classes The SVM algorithm is able toperform nonlinear classification thanks to the use of kernelfunctions

6 Results

In the following we validate the proposed approach andcompare it with histograms of oriented gradients (HOG)[42] scale invariant feature transform (SIFT) detector [10]and a detection system using cascades of HAAR-like features[43]

First a data set composed by 5 objects was defined (cupshair dryers irons cereal boxes and soda cans) with around50 images for each object see Figure 8 The images includedchanges in the scene conditions such as illumination objectorientation and position besides partially occluded objects

In addition to the house made data set the RGB-D objectdata set from the University of Washington [44] was usedto perform tests We used a set consisting of eight differentobjects (bowl cap flashlight coffeemug cereal box soda cancamera and pitcher) similar to [45] (Figure 9) This data setwas selected since it contains similar objects to our set andthey were acquired through the same range sensor For eachof the classes the set includes a group of around four objectsthat belong to the same class with three different views and200 images for each viewThe bounding box cropped imageand object mask are provided

61 Algorithm Parameters We worked with images contain-ing a single object with a discriminative background Asexplained before we start by extracting key points from theimageThe contours were formed by 72 bins of 5 degrees eachThemesh was composed by a 9times9 grid generating 81 pointsThe algorithmic control parameters of PSO coefficient ofcognition (119888

1) and inertia weight (120596) (14) were set to 120596 =

0729844 and 1198881= 149618 as suggested in [46] Different

values for the parameters 120572 (9) and 120588 (14) were tested the bestresults were obtained with 120572 = 1230 and 120588 = 3751 thoughthis might be application dependent

In our case for 120588 values greater than 5 resulted tooaggressive bringing neighbor particles too close rapidly andvalues smaller than 2 had little effect on the movement ofparticles Values for 120572 within the range [05 15] caused theexpected penalty result while other values caused fitnessvalues to boost or to be decreased abnormally Finally asexplained before 120573 (11) was set to 07 defining that particlescloser to the boundaries will be less affected by the velocityupdate

62 Training The cross-validation method was used to vali-date the training process [47] Using a 4-fold approach in thefive-object set and a 5-fold approach in the six-object set thedata was randomly separated in different subsets with equalnumber of elements three of the subsets were used to traindifferent SVMs and the rest were used for validation Theclassifiers were trained and tested 5 times rotating the subsetsused for training the estimate of accuracy is the overallnumber of correct classifications divided by the number ofinstances in the dataset finally the classifier performance ismeasured with the average of the accuracy throughout therotation process


(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)

Figure 6 Different stages of the mesh grid procedure for the hair dryer object (a) Original input image (b) Feature extraction (c) Contourcreated for the object (d) Depth information attached in a mesh grid

(a) (b)

(c) (d)

Figure 7 Different stages of the mesh grid procedure for the earth globe object (a) Original input image (b) Feature extraction (c) Contourcreated for the object (d) Depth information attached in a mesh grid


Figure 8 Objects conforming the data set

To define the SVM model different SVMs with linearx1015840119894sdot x119895 and polynomial (120572x1015840

119894sdot x119895+ 119887)119889 kernel functions were

tested varying the cost parameter 119862 using values in therange (102 108) and the degree of the polynomial 119889 usingvalues in the range (3 10) cross-validationwas used to definewhich value is the best A supervised learning approach wasconsidered in thewhole training and validation processes theclassifiers were trained using a labeled dataset of images ineach image the information about which object is present andthe bounding box of the object was also provided Thereforethe class to which each descriptor belongs is known Toclassify each object 100 tests were performed

63 Classification The first experiment consisted of binaryclassification of one object being discriminated from another1 versus 1 classification see Table 1 The results are summa-rized in a result table Each row indicates the percentage of(a) correct recognitions (CR) and (b) class one errors (C1E)objects of class one are classified as objects of class two and(C2E) class two errors objects of class two are classified asobjects of class one

The second test consisted of binary classification usingthe 5 objects (an object being discriminated from the rest)see Table 2 In this case result table contains the percentagesof (a) correct recognitions (CR) (b) false positives (FP) adifferent object is classified as the target object and (c) false

Table 1 Binary classification using 2 objects

1 versus 1 classification percentagesTest PSA-moment descriptorName CR C1E C2EDryer and box 100 0 0Can and irons 94 3 3Cup and can 86 10 4Iron and dryer 88 4 8Box and cup 90 0 10


1 versus all classification percentagesTest PSA-moment descriptorName CR FP FNDryer 96 2 2Iron 88 0 12Cup 92 2 6Can 90 10 0Box 96 2 2

negatives (FN) the target object is classified as a differentobject


Figure 9 Objects from the RGB-D data set of the University of Washington [44]

Finally a test was made in multiclass classification whereobjects were classified all at once Each object was set asa different class and the classification delivered the class towhich the objects belongThis test generated a 79 of correctclassification An 8 of incorrect classification was present inclass 1 (cups) and 9 in class 2 (irons) all objects of classes3 and 4 (irons and cans) were correctly classified and 4 ofincorrect classification was in class 5 (box)

After testing the descriptors with the small data setclassification tests were performed using a group of objectsfrom the RGB-D object data set of the University of Wash-ington All objects were discriminated from the rest in binaryclassification Tests were performed using first the momentdescriptor obtained when points in the net are simplymigrated then the same tests were done using the point

spreading approach Later these results were compared withhistograms of oriented gradients (HOG) [42] scale invariantfeature transform (SIFT) detector [10] and a detection systemusing cascades of HAAR-like features [43]

The classification has been made for each descriptor(momentmoment-PSAHOG and SIFT) using SVMs Fromcross-validation the best results obtained were a linear kernelfunction with a cost parameter 119862 = 103 Results of the testsfor the four objects are summarized in Table 3 object one isthe bowl object two is the pitcher object three is the cap andobject four is the soda can

We can see from these results that spreading the pointsover the object surface improves the descriptor computedand then the classification of the objects With this approachwe have similar results to the well-known HOG descriptors


Table 3 Tests using objects from the Washington University data set

Classification percentagesObject Object 1 Object 2 Object 3 Object 4Result CR FN FP CR FN FP CR FN FP CR FN FPMoment descriptor 80 8 12 96 0 4 92 4 4 79 7 14Moment-PSA descriptor 85 5 10 98 0 2 97 1 2 82 8 10HOG descriptor 99 1 0 98 2 0 90 8 2 85 10 5SIFT descriptor 74 22 4 85 8 7 76 10 14 70 27 3HAAR cascade 90 9 1 89 7 4 75 10 15 72 27 1

Table 4 Average and minimum number of support vectors

Support vectorsObject Object 1 Object 2 Object 3 Object 4Support vectors avg min avg min avg min avg minMoment descriptor 20 15 25 17 101 73 27 20Moment-PSA descriptor 8 5 24 12 73 52 12 10HOG descriptor 300 23 400 60 370 43 418 70SIFT descriptor 310 22 404 65 401 84 450 72

Table 5 Descriptor construction time Where the steps are featureextraction (FE) contour creation (CC) mesh reduction (MR) meshoptimization (MO) and invariant moments (IM)

Step FE CC MR MO IMTime 00929 s 00419 s 00202 s 00566 s 785 times 10minus4

which are used in many of the state-of-the-art recognitionsystems These two descriptors achieved the highest correctrecognition percentages in all the tests

The accuracy of the classification system has beenassessed in terms of receiving operating characteristics curves(ROC) which relate the positive and false acceptance ratesaccording to an acceptance threshold 120575 varying in the rate[0 1] Figure 10 shows the ROC curves for the classification ofthe bowl and Figure 11 the ROC curves for the classificationof the coffee mug

With the ROC curves we can also note that themoments-PSA classifiers show improvement over the one with the gridthat is not optimized and also it shows similar performanceto the classification through HOG descriptors

The efficiency of the descriptors using a SVM classifierwas also evaluated in terms of the number of support vectorsobtained by the SVM algorithmThis was done in the cases ofmoment moment-PSA HOG and SIFT descriptors Table 4shows the average and minimum number of support vectorsfor 100 tests on the classification of the four objects used inTable 3

With these results we can see that although HOG andour approach have similar classification results the numberof support vectors needed in the training process is smallerfor our approach and when the distribution of the points isoptimized it also results in patterns that are better distributedin the feature space so we need even less support vectors

In this paper we propose the design of a new smalland robust feature descriptor based on SURF features 3Ddata obtained by RGB-D sensor an optimization based ona modified PSO and moment invariants The experimentalresults presented in this section show that even that the newdescriptors are small the addition of 3D data is advantageousand provides robust features The importance of the meshoptimization step lies on the reduction of points and moredistributed points over the object Table 4 shows that theproposed approach requires less support vectors thereforethe classification is faster From the ROC curves we canalso note that the proposed approach has a performancecompared with the HOG approach but as we mention withfewer support vectors

In order to determine the time required to constructthe descriptor we have tested each step 100 times for eachobject to estimate its average processing time a resume of thisresults is presented in Table 5 In this table we can note thatthe most time consuming process is the feature extractionprocess and that the computation of the invariants momentsof the optimized mesh is insignificant With respect to theclassification time we use the same process used to estimatethe descriptor construction time and we obtained an averageof 0007565 s The implementation of the algorithm has beenmade in Matlab and therefore it can be improved by a C orC++ implementation

7 Discussion

71 Difficulties of Each Step of the System In the previoussection the different steps of the proposed approach hadbeen presented The approach produces a descriptor basedon moment invariants which has been computed using 3Ddata provided by an RGB-D sensor A new method has beenintroduced for the extraction of object models from the 3Ddata


1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate10908070605040302010

(b)

Moments-PSA bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)

Figure 10 ROC plots for the classification of the bowl (a) HOG ROC (b) moments ROC (c) moments-PSA ROC (d) HAAR ROC

The first step of the procedure is the feature extractionin this approach the SURF detector was used and thedifficulties of obtaining the SURF features are the traditionalof any image feature detector The second step is the contourcreation this step takes as input the features detected in theprevious step and extracts the contour of the object In thisstep we must take into account that some interest points donot belong to the object they are close to the contour but theyare outside of it This point can be seen as noise and they are

eliminated if their distance to the centroid is the double of themean distance

After the construction of the object contour the depthinformation is added If the area occupied by the object isbig then there would be a lot of data thus the next stepswill require a lot of computational process For this reasonthe third step of the proposed approach tries to optimize themesh in order to reduce the computational time In this step amodified version of PSO has been used to optimize the points


HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)

Figure 11 ROC plots for the classification of the coffee mug (a) HOG ROC (b) moments ROC (c) moments-PSA ROC (d) HAAR ROC

on the mesh One of the challenges encountered on this stepwas the stop criteria we cannot force PSO to stop when thebest value is zero and therefore the algorithm stops whenthe best value of each particle 119894 (119875

119894) is smaller than a certain

threshold With respect to the SVM the only problem wasthe estimation of the cost parameter119862 but this was solved byusing cross-validation

72 Computation Complexity of the System Taking intoaccount the steps illustrated in Figure 1 next a complexityanalysis of the classification system per step is presented

With respect to the feature extraction step as we men-tioned before it is made by using a SURF detector Thetheoretical complexity of SURF was determined and vali-dated through experimentation in [48] this is 119874(119872119873 + 119870)


where119872 and119873 are the image height and width respectivelyand 119870 stands for the number of detected points It wasproven that SURF is a very attractive algorithm concerningits performance when usual size and high rates of frames byseconds are used this is our case the imagesrsquo size is around640 times 480 pixels This step produces a dense 3D point cloudthat is greatly reduced in the next steps

The contour creation step is a linear process with respectto the number of points of the cloud produced by the SURFalgorithm nevertheless when the contour is created thisnumber is reduced and only the points which belongs tothe silhouette of the object are taken to be processed by themesh reduction step which is also linear with respect to thisreduced number of points

The mesh optimization step is performed by using thePSO algorithm The number of computations to complete aPSO run is the computations of the cost function and theposition and velocity updates which are directly proportionalto the number of particles and iterations of the algorithmThe computational complexity of evaluating the cost functiondepends basically on the Euclidean distance of the particleswhich requires 2119873 additions 2119873 multiplications and 119873square roots where 119873 represents the swarm size Whereasthe position update requires 2119873multiplications and 2119873 addi-tions and the velocity update requires 7119873 multiplicationsand 4119873 additions Therefore in total the PSO requires 6119873additions 11119873 multiplications and 119873 square roots Againthis step reduces even more the number of points whichbelong to the contour of the object

The invariant moment descriptor generation seems to bethe more expensive step which is executed in real time by thesystem because it is a process with a computation complexityof 119874(1198733) with 119873 the number of points of each contour butthis number is small (because it was reduced by the abovethree steps of the system) and the descriptors are very fastcomputed as it is shown in Table 5

The last step of the system is the classification of thedescriptors obtained it is made using a SVM A SVM solvesa quadratic programming problem (QP) and therefore itscomputation complexity depends on the QP solver usedThecomputation complexity of the traditional algorithm of theSVM classifier is 119874(1198733sv + 119871119873

2

sv + 119889119897119871119873sv) [49] where 119889119897 isthe dimension of the input space 119871 is the number of trainingpoints the number of the samples in the support vector set is119873sv Nevertheless there are several efficient implementationssuch as incremental learning algorithm [50] which reports acomputation complexity of 119874(1198733sv + 119889119897119873sv2) or sequentialminimal optimization (SMO) [51] which for a real-world testsets are 1200 times faster for linear SVMs and 15 times fasterfor nonlinear SVMs against another quadratic programmingsolver techniques SMO was the training method used in oursystem It is important to note that even when the systemincludes a training step with the complexity that was alreadydetermined for the SVM algorithm this step is made offlineand in the real time process the classification step consistsof evaluating the classification function obtained with thetraining of the SVM and this is a linear process with respectto the number of support vectors119873sv which is usually relatedwith the number of training data119873 as follows119873sv ≪ 119873

8 Conclusions

In this paper we have presented an image classificationtechnique based on an invariant moment descriptor thatincludes depth informationThe inclusion of 3D data enablesinvariant moments to produce small and robust descriptorsimproving image classification To create such descriptors weused object models with rich information of the distributionof points The application of the optimization algorithmPSO to the model generation stage improved the computeddescriptor and the object recognition From experimentalresults it is clear that these descriptors have achieved highcorrect recognition percentages Furthermore the number ofsupport vectors obtained in the training process is smaller forour approach due to the fact that the points are optimizedand thus the patterns are better distributed in the featurespace

Abbreviations

119883119894 Position of PSO particle 119894

119875119894 Best position found by PSO particle 119894119881119894 Velocity of PSO particle 119894119866 Best position found by the swarm120596 PSO inertia weight1205931 1205931 Uniform random numbers [0 1]

1198881 1198882 PSO acceleration constants

119891119894 PSO fitness function120572 Fitness penalization gain120573 Particle closeness to boundaries120588 Particle acceleration with respect to closest particle119889 Moment descriptor119863119894 Distance of particle 119894 with respect to its neighbors

119863 Mean of particles distances

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CONACYT and the Univer-sity of GuadalajaraThis work has been partially supported bythe CONACYT projects CB-156567 CB-106838 CB-103191and INFR-229696

References

[1] M Minoux Mathematical Programming Theory and Algo-rithms Wiley-Interscience 1986

[2] K Parsopoulos and M Vrahatis Particle Swarm Optimizationand Intelligence Advances and Applications IGI Global 2010

[3] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[4] M Jin and D Wu ldquoCollision-free and energy-saving trajectoryplanning for large-scale redundantmanipulator using improved


psordquo Mathematical Problems in Engineering vol 2013 ArticleID 208628 8 pages 2013

[5] A Y Alanis E Rangel J Rivera and C Lopez-Franco ldquoParticleswarm based approach of a real-time discrete neural identifierfor linear induction motorsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 715094 9 pages 2013

[6] J Kennedy R C Eberhart and Y Shi Swarm IntelligenceMorgan Kaufmann 2001

[7] R Jain R Kasturi andBG SchunckMachineVisionMcGraw-Hill 1995

[8] Kinect sensor httpwwwxboxcomen-USKinect[9] A K Jain M N Murty and P J Flynn ldquoData clustering a

reviewrdquo ACM Computing Surveys vol 31 no 3 pp 316ndash3231999

[10] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004

[11] K Grauman and T Darrell ldquoThe pyramid match kerneldiscriminative classification with sets of image featuresrdquo in Pro-ceedings of the 10th IEEE International Conference on ComputerVision (ICCV rsquo05) vol 2 pp 1458ndash1465 October 2005

[12] C Harris and M Stephens ldquoA combined corner and edgedetectorrdquo in Proceedings of the 4th Alvey Vision Conference pp147ndash151 1988

[13] C Schmid R Mohr and C Bauckhage ldquoEvaluation of interestpoint detectorsrdquo International Journal of Computer Vision vol37 no 2 pp 151ndash172 2000

[14] T Lindeberg ldquoFeature detectionwith automatic scale selectionrdquoInternational Journal of Computer Vision vol 30 no 2 pp 79ndash116 1998

[15] K Mikolajczyk and C Schmid ldquoIndexing based on scaleinvariant interest pointsrdquo in Proceedings of the 8th InternationalConference on Computer Vision vol 1 pp 525ndash531 July 2001

[16] D G Lowe ldquoObject recognition from local scale-invariantfeaturesrdquo inProceedings of the 7th IEEE International Conferenceon Computer Vision (ICCV rsquo99) pp 1150ndash1157 Kerkyra GreeceSeptember 1999

[17] H Bay T Tuytelaars and L Gool ldquoSurf speeded up robustfeaturesrdquo in Proceedings of the Computer Vision Conference(ECCV rsquo06) A Leonardis H Bischof and A Pinz Eds vol3951 ofLectureNotes inComputer Science pp 404ndash417 SpringerBerlin Germany 2006

[18] W T Freeman and E H Adelson ldquoThe design and use ofsteerable filtersrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 13 no 9 pp 891ndash906 1991

[19] J Flusser B Zitova and T Suk Moments and Moment Invari-ants in Pattern Recognition JohnWiley amp Sons Chichester UK2009

[20] G Carneiro and A D Jepson ldquoPhase-based local featuresrdquoin Proceedings of the European Conference on Computer Vision(ECCV rsquo02) pp 282ndash296 2002

[21] L M J Florack B M ter Haar Romeny J J Koenderinkand M A Viergever ldquoGeneral intensity transformations anddifferential invariantsrdquo Journal of Mathematical Imaging andVision vol 4 no 2 pp 171ndash187 1994

[22] N Y Khan B McCane and G Wyvill ldquoSIFT and SURFperformance evaluation against various image deformations onbenchmark datasetrdquo in Proceedings of the International Confer-ence on Digital Image Computing Techniques and Applications(DICTA rsquo11) pp 501ndash506 December 2011

[23] N Dalal and B Triggs ldquoHistograms of oriented gradients forhuman detectionrdquo in Proceedings of the IEEE Computer SocietyConference on Computer Vision and Pattern Recognition (CVPRrsquo05) vol 1 pp 886ndash893 San Diego Calif USA June 2005

[24] P F Felzenszwalb R B Girshick D McAllester and DRamanan ldquoObject detection with discriminatively trained part-based modelsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 9 pp 1627ndash1645 2010

[25] P Viola M J Jones and D Snow ldquoDetecting pedestrians usingpatterns of motion and appearancerdquo in Proceedings of the 9thIEEE International Conference on Computer Vision vol 2 pp734ndash741 October 2003

[26] C J C Burges ldquoA tutorial on support vector machines forpattern recognitionrdquo Data Mining and Knowledge Discoveryvol 2 no 2 pp 121ndash167 1998

[27] N Cristianini and J Shawe-Taylor An Introduction to SupportVector Machines and Other Kernel-Based Learning MethodsCambridge University Press New York NY USA 2000

[28] H Bay A Ess T Tuytelaars and L Van Gool ldquoSpeeded-uprobust features (surf)rdquoComputer Vision and Image Understand-ing vol 110 no 3 pp 346ndash359 2008

[29] J H HollandAdaptation in Natural and Artificial Systems MITPress Cambridge Mass USA 1992

[30] P Merz and B Freisleben ldquoA genet ic local search approach tothe quadratic assignment problemrdquo in Proceedings of the 7thInternational Conference on Genetic Algorithms pp 465ndash472Morgan Kaufmann 1997

[31] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics B Cybernetics vol 26 no 1 pp29ndash41 1996

[32] M Eusuff and K Lansey ldquoOptimization of water distributionnetwork design using the shuffled frog leaping algorithmrdquoJournal of Water Resources Planning and Management vol 129no 3 pp 210ndash225 2003

[33] E Elbeltagi T Hegazy and D Grierson ldquoComparison amongfive evolutionary-based optimization algorithmsrdquo AdvancedEngineering Informatics vol 19 no 1 pp 43ndash53 2005

[34] P Civicioglu and E Besdok ldquoA conceptual comparison of theCuckoo-search particle swarmoptimization differential evolu-tion and artificial bee colony algorithmsrdquo Artificial IntelligenceReview vol 39 no 4 pp 315ndash346 2013

[35] Q Ni and J Deng ldquoAnalysis of population diversity of dynamicprobabilistic particle swarm optimization algorithmsrdquo Mathe-matical Problems in Engineering vol 2014 Article ID 762015 9pages 2014

[36] N A A Aziz M Y Alias A W Mohemmed and KA Aziz ldquoParticle swarm optimization for constrained andmultiobjective problems a brief review inrdquo in Proceedingsof the International Conference on Management and ArtificialIntelligence (IPEDR rsquo11) pp 146ndash150 Bali Indonesia 2011

[37] M-P Song and G C Gu ldquoResearch on particle swarm opti-mization a reviewrdquo in Proceedings of International Conferenceon Machine Learning and Cybernetics pp 2236ndash2241 August2004

[38] M-K Hu ldquoVisual pattern recognition by moment invariantsrdquoIRE Transactions on Information Theory vol 8 no 2 pp 179ndash187 1962

[39] M Mercimek K Gulez and T V Mumcu ldquoReal objectrecognition using moment invariantsrdquo Sadhana vol 30 pp765ndash775 2005


[40] M Rizon H Yazid P Saad et al ldquoObjectdetection usinggeometric invariant momentrdquo American Journal of AppliedSciences vol 2 pp 1876ndash1878 2006

[41] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995

[42] N Dalal and B Triggs ldquoHistograms of oriented gradients f orhuman detectionrdquo in Proceedings of the IEEE Computer SocietyConference on Computer Vision and Pattern Recognition (CVPRrsquo05) vol 1 pp 886ndash893 IEEE Computer Society WashingtonDC USA 2005

[43] P Viola and M Jones ldquoRapid object detection using a boostedcascade of simple featuresrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo01) vol 1 pp I511ndashI518 December 2001

[44] K Lai L Bo X Ren and D Fox ldquoA large-scale hierarchicalmulti-view RGB-D object datasetrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo11) pp 1817ndash1824 May 2011

[45] K Lai L Bo X Ren and D Fox ldquoDetection-based objectlabeling in 3D scenesrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo12) pp 1330ndash1337 Saint Paul Minn USA May 2012

[46] L Mussi F Daolio and S Cagnoni ldquoEvaluation of parallelparticle swarm optimization algorithms within the cuda archi-tecturerdquo Information Sciences vol 181 no 20 pp 4642ndash46572011

[47] R Kohavi ldquoA study of cross-validation and bootstrap foraccuracy estimation and model selectionrdquo in Proceedings ofthe 14th International Joint Conference on Artificial Intelligence(IJCAI rsquo95) vol 2 pp 1137ndash1143 Morgan Kaufmann PublishersSan Francisco Calif USA 1995

[48] P Drews R de Bem and A de Melo ldquoAnalyzing and exploringfeature detectors in imagesrdquo in Proceedings of the 9th IEEEInternational Conference on Industrial Informatics (INDIN 11)pp 305ndash310 Lisbon Portugal July 2011

[49] V N Vapnik Statistical LearningTheory JohnWiley amp Sons 1stedition 1998

[50] Y Wang F Zhang and L Chen ldquoAn approach to incrementalSVM learning algorithmrdquo in Proceedings of the ISECS Interna-tional Colloquium on Computing Communication Control andManagement (CCCM rsquo08) vol 1 pp 352ndash354 August 2008

[51] J C Platt ldquoSequential minimal optimization a fast algorithmfor training support vector machinesrdquo Tech Rep Advances inKernel Methods-Support Vector Learning 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Related Work

In recent years image classification has become a very activeresearch field The goal of the classification task is to developan algorithm which will assign a class-label to each one ofthe samples in the training data set There are two mainapproaches for classification in the literature namely (a)supervised approach and (b) unsupervised approach wherethe former uses a set of samples to train the classificationand the latter performs the classification by exploring the data[9] Many popular image classification methods [10 11] arebased on local descriptor and support vector machines asbasic techniques

One of the most widely used detectors is the HarrisCorner detector [12] However the Harris detector is notinvariant to affine transformations and scale [13] In [14]an automatic scale detection was proposed Later in [15]the authors present an algorithm for interest point detectionwhich is invariant to important scale changes Then in [16]the SIFT detector is proposed the approach approximatesthe Laplacian of Gaussians (LoG) by difference of Gaussians(DoG) filter to identify the potential keypoints Later in[17] the SURF detector was presented and SURF uses anapproach similar to SIFT however the approach uses integralimages instead of DoGs this allows a fast computation ofapproximate LoG using a box filter

Many different feature descriptors have been proposed inthe literature steerable filters [18] moment invariants [19]phase-based local features [20] Gaussian derivatives [21] anddescriptors that represent the distribution of smaller-scalefeatures [10] The SIFT detector produces robust descriptorshowever the speed of the detector is not very high In [17]the SURF detector was presented this approach is fasterthan SIFT and it has a similar performance on real images[22]

Histogram of oriented gradients (HoG) was proposed in[23] The HoG local patters are based on gradient histogramThe HoG and SIFT detectors share the same concept ofimplementation in the sense that local histograms of ori-ented patches defined across the image are used to producethe descriptor In [23] the authors used HoG descriptorsas feature vectors for a linear SVM In [24] the authorspropose the use of HoG for object detection and show thatHoG descriptors have robust and effective features for suchapplication In [25] the authors build an efficient detectorusing AdaBoost to train a chain of progressively morecomplex region rejection rules based on Haar descriptors forpedestrian detection

Our Approach In the last decades SVMs have been provento be an effective classifier approach for small sample setsor high dimensional problems [26 27] Such result is veryimportant since both problems are too difficult to be solvedby classical paradigms

One of the problems encountered during the design of animage classification technique is data overfitting which ariseswhen the training samples are small in comparison with thenumber of features To overcome this problem we design adescriptor with an optimized 3D point distribution

The contribution of this work is the development of asmall and robust descriptor based on invariant moments and3D information in order to improve the classification processThe 3D information is incorporated fromdepth data obtainedfrom an RGB-D sensor The object model is optimizedthrough the use of a PSO algorithm this optimization allowsimproving the classification

3 Image Classification System

The proposed classification system uses 3D informationand is based on local features invariant moments contourgeneration and a reduction of depth information using amesh grid It consists of five main steps feature extractioncontour creation mesh reduction mesh optimization andinvariant moment descriptor formation see Figure 1

31 Feature Extraction The first step is the extraction ofSURF keypoints from the input image (Figure 5(b)) Thespeeded up robust features algorithm (SURF) [28] findsfeatures using integral images and Haar-like filters SURFfeatures provide robustness and speed and are able to bedetected despite scale translation and rotation of objectsor changes in illumination The following steps rely on thisstability to group points and create object models and theyare described in the following subsections

SURF and SIFT detectors employ slightly different waysof detecting features SIFT detector builds image pyramidsand filters each layer with a Gaussian of increasing sigmavalues and then it takes the difference SURF detector usesa box filter approximation A comparison of the SIFT andSURF detectors is presented in [22] In suchwork the authorsconclude that SURF is as good as SIFT on most tests exceptfor scaling large blur and viewpoint With respect to thiswe have to mention that the proposed descriptor only usesSURF detector in the first step of the classifier to detectinterest points Then the proposed descriptor is constructedusing additional 3D information and invariant moments Inthe experiments results show that the proposed descriptoris small and robust In [22] the authors also mention thaton real image data sets there is little to separate the differentSIFTs and SURF except for efficiency In this work we chooseSURF detector because we are interested in implementinga real time classifier algorithm that could be used by amobile robot In addition as we will see in the resultsrsquosection the proposed descriptor is improved thanks to thespreading step in conjunction with the SURF detector andthis provides good results that are comparable and overcomesimilar approaches

32 Contour Creation The RGB-D sensor provides a hugeamount of data thus the information must be reduced intoa contour-mesh model to decrease computational cost Toconstruct the contour-mesh we use keypoints First they arescaled and translated then we compute the magnitude of thekeypoints with respect to their centroid (119909 119910) as

119898(119909 119910) = radic(119909 minus 119909)2+ (119910 minus 119910)

2 (1)


Feature extraction

Contour creation

Mesh reduction


Input image

SVM classification

Input

Mesh optimization



120579 (119909 119910) = tanminus1 (119910 minus 119910

119909 minus 119909) (2)






Grid = (1199091 1199101) (1199092 1199102) (119909

119899 119910119899) (3)


119909119894isin [119909min 119909max] 119910

119894isin [119910min 119910max]

1199091= 119909min 119910

1= 119910min

119909119899= 119909max 119910

119899= 119910max



119909119894+1= 119909119894+ 119909inc 119910

119894+1= 119910119894+ 119910inc

119909inc =119909max minus 119909min

119899 119910inc =

119910max minus 119910min119899

(4)





4 Mesh Optimization







119894)



119894of


119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119883119894) + 11988821205932(119866 minus 119883

119894)

119883119894(119905 + 1) = 119883

119894(119905) + 119881

119894(119905 + 1)

(5)







119894




119863119894= radic(119909

119894minus 119909119895)2

+ (119910119894minus 119910119895)2

(6)



(a) (b)

10

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)





119863 =1

119873

119873

sum119896=1

119863119896 (7)



119891119894= radic(119863

119894minus 119863)2

(8)




119894


119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with


(10)






119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)


119863119887= radic(119909


2

+ (119910119894minus 119910boundary)

2 (12)


120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)



119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)


119894 119910119894) (119909119895 119910119895) with119883

119895being
















120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Feature extraction

Contour creation

Mesh reduction


Input image

SVM classification

Input

Mesh optimization



120579 (119909 119910) = tanminus1 (119910 minus 119910

119909 minus 119909) (2)






Grid = (1199091 1199101) (1199092 1199102) (119909

119899 119910119899) (3)


119909119894isin [119909min 119909max] 119910

119894isin [119910min 119910max]

1199091= 119909min 119910

1= 119910min

119909119899= 119909max 119910

119899= 119910max



119909119894+1= 119909119894+ 119909inc 119910

119894+1= 119910119894+ 119910inc

119909inc =119909max minus 119909min

119899 119910inc =

119910max minus 119910min119899

(4)





4 Mesh Optimization







119894)



119894of


119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119883119894) + 11988821205932(119866 minus 119883

119894)

119883119894(119905 + 1) = 119883

119894(119905) + 119881

119894(119905 + 1)

(5)







119894




119863119894= radic(119909

119894minus 119909119895)2

+ (119910119894minus 119910119895)2

(6)



(a) (b)

10

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)





119863 =1

119873

119873

sum119896=1

119863119896 (7)



119891119894= radic(119863

119894minus 119863)2

(8)




119894


119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with


(10)






119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)


119863119887= radic(119909


2

+ (119910119894minus 119910boundary)

2 (12)


120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)



119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)


119894 119910119894) (119909119895 119910119895) with119883

119895being
















120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119894+1= 119909119894+ 119909inc 119910

119894+1= 119910119894+ 119910inc

119909inc =119909max minus 119909min

119899 119910inc =

119910max minus 119910min119899

(4)





4 Mesh Optimization







119894)



119894of


119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119883119894) + 11988821205932(119866 minus 119883

119894)

119883119894(119905 + 1) = 119883

119894(119905) + 119881

119894(119905 + 1)

(5)







119894




119863119894= radic(119909

119894minus 119909119895)2

+ (119910119894minus 119910119895)2

(6)



(a) (b)

10

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)





119863 =1

119873

119873

sum119896=1

119863119896 (7)



119891119894= radic(119863

119894minus 119863)2

(8)




119894


119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with


(10)






119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)


119863119887= radic(119909


2

+ (119910119894minus 119910boundary)

2 (12)


120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)



119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)


119894 119910119894) (119909119895 119910119895) with119883

119895being
















120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119894




119863119894= radic(119909

119894minus 119909119895)2

+ (119910119894minus 119910119895)2

(6)



(a) (b)

10

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)





119863 =1

119873

119873

sum119896=1

119863119896 (7)



119891119894= radic(119863

119894minus 119863)2

(8)




119894


119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with


(10)






119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)


119863119887= radic(119909


2

+ (119910119894minus 119910boundary)

2 (12)


120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)



119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)


119894 119910119894) (119909119895 119910119895) with119883

119895being
















120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

10

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)





119863 =1

119873

119873

sum119896=1

119863119896 (7)



119891119894= radic(119863

119894minus 119863)2

(8)




119894


119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with


(10)






119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)


119863119887= radic(119909


2

+ (119910119894minus 119910boundary)

2 (12)


120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)



119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)


119894 119910119894) (119909119895 119910119895) with119883

119895being
















120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894


119891119894= radic(119863

119894minus 119863)2

+ 120572119879 (119883)10038161003816100381610038161003816119883 minus 119883

10038161003816100381610038161003816 (9)

with


(10)






119883119894(119905 + 1) = 119883

119894(119905) + 120573119881

119894(119905 + 1) (11)


119863119887= radic(119909


2

+ (119910119894minus 119910boundary)

2 (12)


120573 = 119863119887119863119887ltThreshold

1 Otherwise(13)



119881119894(119905 + 1) = 120596119881

119894(119905) + 119888

11205931(119875119894minus 119909119894) + 120588 sign (119883

119894minus 119883119895)

(14)


119894 119910119894) (119909119895 119910119895) with119883

119895being
















120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120583119901119902119903= sum119909

sum119910

sum119911

(119909 minus 119909)119901(119910 minus 119910)

119902

(119911 minus 119911)119903119891 (119909 119910 119911) (15)


120578119901119902119903=120583119901119902119903

120583120574

000

120574 = [119901 + 119902 + 119903

3] + 1 (16)


119889 = [120578100 120578200 120578300 120578400 120578010 120578020 120578030 120578040

120578001 120578002 120578003 120578004]

(17)


6 Results











(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(b)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10


(c) (d)


(a) (b)

(c) (d)



































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































7 Discussion



1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate

False positive rate

HOG ROC bowl ROC

10908070605040302010

(a)

Moments bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(b)


09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(c)

HAAR bowl ROC1

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


(d)






HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










HOG mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(a)

Moments-ROC mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(b)

Moments-PSA mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(c)

HAAR mug ROC

09

08

07

06

05

04

03

02

01

0

True

pos

itive

rate


1

(d)













2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















2


8 Conclusions


Abbreviations








Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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