Computer Vision and Image Understanding invariant.pdf · 2017. 6. 14. · Adnan Fatih Kocamazb a Bingol University, Department of Informatics, Turkey ... Computer Vision and Image

Computer Vision and Image Understanding 132 (2015) 87–101

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Computer Vision and Image Understanding

journal homepage: www.elsevier .com/ locate/cviu

Continuous rotation invariant features for gradient-based textureclassification q

http://dx.doi.org/10.1016/j.cviu.2014.10.0041077-3142/� 2014 Elsevier Inc. All rights reserved.

q This paper has been recommended for acceptance by Yasutaka Furukawa.⇑ Corresponding author.

E-mail address: [email protected] (K. Hanbay).

Kazim Hanbay a,⇑, Nuh Alpaslan b, Muhammed Fatih Talu b, Davut Hanbay b, Ali Karci b,Adnan Fatih Kocamaz b

a Bingol University, Department of Informatics, Turkeyb Inonu University, Department of Computer Engineering, Turkey

a r t i c l e i n f o

Article history:Received 1 March 2014Accepted 14 October 2014Available online 22 October 2014

Keywords:HOGCoHOGHessian matrixEigen analysisRotation invarianceTexture classification

a b s t r a c t

Extracting rotation invariant features is a valuable technique for the effective classification of rotationinvariant texture. The Histograms of Oriented Gradients (HOG) algorithm has been proved to be theoret-ically simple, and has been applied in many areas. Also, the co-occurrence HOG (CoHOG) algorithm pro-vides a unified description including both statistical and differential properties of a texture patch.However, HOG and CoHOG have some shortcomings: they discard some important texture informationand are not invariant to rotation. In this paper, based on the original HOG and CoHOG algorithms, fournovel feature extraction methods are proposed. The first method uses Gaussian derivative filters namedGDF-HOG. The second and the third methods use eigenvalues of the Hessian matrix named Eig(Hess)-HOG and Eig(Hess)-CoHOG, respectively. The fourth method exploits the Gaussian and means curvaturesto calculate curvatures of the image surface named GM-CoHOG. We have empirically shown that theproposed novel extended HOG and CoHOG methods provide useful information for rotation invariance.The classification results are compared with original HOG and CoHOG algorithms methods on the CUReT,KTH-TIPS, KTH-TIPS2-a and UIUC datasets show that proposed four methods achieve best classificationresult on all datasets. In addition, we make a comparison with several well-known descriptors. Theexperiments of rotation invariant analysis are carried out on the Brodatz dataset, and promising resultsare obtained from those experiments.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction The pioneer works to achieve rotation-invariant texture classi-

Texture is an important characteristic of the appearance ofobjects and is a powerful visual cue, used in describing and recog-nizing object surfaces [1]. Texture analysis plays an important rolein image processing, pattern recognition, and computer vision[2–8]. Texture classification methods usually consist of two stepsof feature extraction and classification. Feature extraction involvessimplifying the amount of resources required to describe a largeset of data accurately. To enhance the overall quality of textureclassification, both the quality of the texture features and the qual-ity of the classification algorithm must be improved [9–14].

There has been intensive research in developing robust featuresfor texture classification with strong invariance to rotation, scale,translation, illumination changes [15–23]. Rotation invariant fea-ture extraction is a difficult problem, thus many algorithms wereproposed to achieve the rotation invariance [24,25].

fication include generalized co-occurrence matrices (GCM) [26],polarograms [27], texture anisotropy [28], the methods based onMarkov random field (MRF) [29] and autoregressive model. Thewavelet based algorithms achieved effective classification perfor-mance [30–37]. Recently, the statistical based approaches haveattracted considerable attention [38–40]. However, many of theseapproaches achieve the rotation invariance by shifting the discreteorientations. For example, the method of local binary pattern (LBP)[18] is proposed to achieve rotation invariance [41].

The gradient based features such as edges or orientation anglesare widely used as feature descriptors in image processing. In orderto identify objects in images effectively, gradient based edge fea-tures have been developed, which are edge orientation histogram[42], Histograms of Oriented Gradients (HOG) [43,44], co-occur-rence HOG (CoHOG) [45], multilevel edge energy features [46],shapelets [47], and edge density [48]. The HOG method distributesthe gradients into several orientation bins. HOG encapsulateschanges in the magnitude and orientation of contrast over a gridof small image patches. HOG features have shown satisfactory per-formance in their ability to recognize a range of different object

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88 K. Hanbay et al. / Computer Vision and Image Understanding 132 (2015) 87–101

types including natural objects as well as artificial objects. CoHOG(Co-Occurrence Histograms of Oriented Gradients), an extension ofHOG to represent the spatial relationship between gradientorientations, has been proposed and its effectiveness for pedestriandetection, human detection and medical image analysis has beendemonstrated in [49–51].

According to a review [52], the extraction of basic features inimages is based on two mathematical concepts: differential geom-etry and scale-space theory. The differential geometry approachuses the assumption that features can be obtained from the imagebased on local anisotropic variations of pixel intensities. This con-cept is strong and effective. Recently, the differential featureextraction approaches have attracted more attentions [53,54].Among the different feature extraction methods, gradient-basedmethods were widely used in the past. These methods are effectivein defining and describing significant image features. Hessianmatrix information is a robust differential method and it has beenwidely used in many publications to extract the image features[55,56]. Compared with the conventional gradient, the Hessianmatrix and its Eigen analysis are more reliable and robust inrevealing the fundamental directions in data. In this study, we pro-pose four novel methods to improve the classification performanceof the HOG and CoHOG algorithms. The proposed four methods arebased on Gaussian derivatives filters and Hessian matrix. Instead ofusing the conventional gradient operator in HOG and CoHOG algo-rithms, the second-order partial derivatives in Gaussian derivativesfilters and Hessian matrix are more proper and stable to calculatethe intensity and texture variations of image surface.

The rest of this paper is organized as follows. In Section 2, wewill give a review of the original HOG and CoHOG algorithms. Sec-tion 3 presents the proposed new HOG and CoHOG algorithmsbased on Gaussian derivatives filters, Hessian matrix and Gauss-ian–mean curvatures. In Section 4, we test the performance ofnovel feature extraction algorithms on four standard texture data-sets and discuss the effect of the normalization step on the classi-fication performance. In Section 5, a series of rotation analysisexperiments are performed. Furthermore, the characteristics ofproposed descriptors are discussed in detail. The comparisonresults with the state-of-the-art the texture classification methodsperformed on Brodatz and UIUC datasets are shown in Section 6.The conclusions are given in Section 7.

2. Related works

2.1. Histograms of Oriented Gradients (HOG)

This section gives an overview of the HOG feature extractionprocess. The basic concepts of the HOG are the local object appear-ance and shape, which can be characterized by the distribution ofthe local intensity gradients or edge directions [57,58]. The gradi-ents orientations are strong against lighting changes since theforming histogram provides rotational invariance.

For each key point, a local HOG descriptor from a block is com-puted. The block size is not restricted to construct an extensive setof texture features, which allow extracting high-discriminated fea-tures in order to improve classification accuracy and reduce com-putational time of classification algorithms [59,60]. HOG is awindow based algorithm computed local to a detected interestpoint. The window is centered upon the point of interest anddivided into a regular square grid (n � n) [43,44]. This method con-sists of several steps.

First, the grayscale image was filtered to obtain x and y deriva-tives of pixels. The filter kernels were used to compute discretederivative in the x and y direction. The gradient values at everyimage pixel were computed as follows:

f xðx; yÞ ¼ Iðxþ 1; yÞ � Iðx� 1; yÞf xðx; yÞ ¼ Iðx; yþ 1Þ � Iðx; y� 1Þ

ð1Þ

where fx and fy denotes x and y components of image gradientrespectively. I(x, y) denotes the pixel intensity at position (x, y).The magnitude and orientation is computed in Eqs. (2) and (3):

mðx; yÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif xðx; yÞ

2 þ f yðx; yÞ2

qð2Þ

hðx; yÞ ¼ tan�1 f yðx; yÞf xðx; yÞ

� �ð3Þ

Second, the image intensity gradients are divided into layersbased on their orientation. The original HOG descriptor usesunsigned gradients in conjunction with 9 bins (a bin correspondsto 20�) to construct the histograms of oriented gradients. There-fore, there are 9 layers of orientated gradient.

Finally, orientation histogram of every cell and larger spatialblocks n �m are normalized. To normalize the cells’ orientationhistograms, they should be grouped into blocks. Since a cell has korientations, the feature dimension of each block is n �m � k foreach block. v denotes feature vector in a block h(i,j) denotes unnor-malized histogram of the cell in the position (i, j) in a block.Although there are three different methods for block normaliza-tion, L1-Norm normalization is implemented as:

h0ði;jÞ ¼h0ði;jÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikvk1 þ e

p ðe ¼ 1Þ ð4Þ

where e is the small constant [43]. Here, e is set to 1 empirically.

2.2. Co-Occurrence Histograms of Oriented Gradients (CoHOG)

The CoHOG feature descriptor is based on a co-occurrencematrix which is obtained from a 2D histogram of pairs of gradientorientations [45]. It performs on grayscale images. The co-occu-rence matrix expresses the distribution of gradient orientationsat given offset over an image as shown in Fig. 1.

The combinations of neighbor gradient orientations providereliable features of objects in images and this is very advantageousfor object classification problems. The co-occurrence matrix C isobtained from n �m image of gradient orientations, and formu-lated in Eq. (5);

Ci;j ¼Xn�1

p¼0

Xm�1

q¼0

1 if Iðp; qÞ ¼ i and Iðpþ x; qþ yÞ ¼ j

0 otherwise

�ð5Þ

where I indicates a gradient orientation image, i and j indicates gra-dient orientations and x, y denotes vertical and horizontal offsets.The gradient orientations from I are calculated in Eq. (6);

h ¼ tan�1 vh

� �ð6Þ

where v and h are the vertical and the horizontal components ofgradient calculated by appropriate filters. Then, the orientations inthe range (0, 2p) are quantized into eight labels. Each label is usedfor representing an orientation. Thus, the size of the co-occurrencematrix C becomes 8 � 8. Six offsets are used in experiments. The co-occurrence matrix contains information on the local textures byusing short-range offsets and the global textures by using long-range offsets [45,61].

The co-occurrence matrices are computed for each tiled regionswith all offsets. Hence, the number of CoHOG descriptor features ism � n � d2 where d is the number of gradient orientation bins, m isthe number of tiled regions and n is the number of offsets. Finally,the CoHOG descriptor is determined as a vector by concatenatingthe components of all the co-occurrence matrices. The size of theoriginal CoHOG descriptor is 2 � 2 � 6 � 82 = 1536.

Tiled Regions Offsets C

Fig. 1. Process diagram of CoHOG algorithm.

K. Hanbay et al. / Computer Vision and Image Understanding 132 (2015) 87–101 89

3. The proposed novel HOG and CoHOG methods

In this section, we introduce four novel derivative methods toimprove the HOG and CoHOG feature extraction algorithms. Inoriginal HOG algorithm, the gradient orientations of the pixels arecalculated by using gradient information of the input image. In ori-ginal CoHOG algorithm, in the same way, the labeling process of thepixels is performed in the specified offset by using pixel orientationsobtained from gradient information. This labeling process directlyaffects the quality and distinctiveness of the information containedin the co-occurrence matrix created in the next step. Therefore, thecalculation of gradient information is the most basic and importantstep in both algorithms. In this study, we propose an efficient globalderivative scheme that uses Gaussian derivative filter and Hessianmatrix for feature extraction. Thus, by obtaining distinctive androbust characteristics of the images, significant improvement hasbeen provided in the classification performances of algorithms. Inthis paper, the first gradient calculation method proposed toimprove HOG algorithm uses x–y separable Gaussian derivative fil-ters. In addition, the other proposed method uses the eigenvalues ofthe Hessian matrix. The proposed first descriptor to strengthen thegradient step of CoHOG algorithm uses eigenvalues of the Hessianmatrix. Instead of directly using the responses of gradient, oursecond descriptor uses the Gaussian curvature and mean curvatureinformations. Furthermore, for the developed new CoHOGalgorithm, a new formulation is used in labeling process. For brevity,novel HOG algorithm using proposed separable Gaussian derivativefilters is named as GDF-HOG. Also, novel HOG algorithm using theeigenvalues of the Hessian matrix is named as the Eig(Hess)-HOG.The proposed novel Co-HOG algorithm using the eigenvalues ofthe Hessian matrix is named as the Eig(Hess)-CoHOG. Finally, thenovel Co-HOG algorithm using Gaussian and mean curvatureinformation is named as GM-CoHOG.

3.1. The proposed GDF-HOG algorithm

HOG is a well-known feature descriptor which computes thefeatures through a gradient orientation histogram within the localregion. The original HOG algorithm uses a gradient computationmethod which is based on calculation difference of neighboringpixels. The primary weak point of original HOG is that it cannotdescribe the characteristics of textures efficiently and distinctively.Another weak point of HOG is that it is mathematically weak andsensitive to noise since the label of a local histogram is easy to

change in terms of the definition of uniform histogram. To over-come the most prominent shortcoming of the definition of ‘‘uni-form’’ histogram in HOG, we propose a novel extension in whichthe local texture patterns are subjected to further treatment andthen computed in Gaussian derivative filters way. We can findthe continuous rotation invariant features via Gaussian function.Therefore, we employ the Gaussian derivative filters approach torepresent and classify texture images. Since the rotation, scalingand translation operations are each a linear transformation, theyretain the shape of the image. So, even though the coordinates ofthe image change, the exact state of shape does not change. Gauss-ian function is a continuous and linear-dependent function. Thusthis function can be used in the calculation of first and second-order rotation invariant derivative. The Gaussian first and secondderivative filters could be rotated at any angle by linear combina-tion of two basis filters [62].

The gradient computation which is calculated via the Gaussianfunction and two-dimensional convolution provides more promi-nent texture and intensity information than conventional gradient.Thus the Gaussian derivative filters are usually an appropriatemodel to extract the basic features of the texture patterns. TheGaussian function can be formulated as:

Gðx; yÞ ¼ e�ðx2þy2 Þ

r2 ð7Þ

The primary first-order differential quantity for an image is thegradient. Gradient is a 2-D vector quantity. It has both directionand magnitude informations which vary at each point [63]. Imagederivatives can be calculated as follows:

rI ¼I � Gx

I � Gy

� ð8Þ

where ⁄ denotes the convolution, Gx and Gy are the first partialderivatives of G with regard to x and y respectively. The responsesof the oriented Gaussian first derivative filter Gh in Eq. (9):

Ih ¼ I � Gh ¼ I � ðcosðhÞGx þ sinðhÞGyÞ¼ cosðhÞI � Gx þ sinðhÞI � Gy ð9Þ

Since the two basic filters are x–y separable, the response ofeach filter could be calculated by twice one-dimensionalconvolutions [41]. Also, the computation cost of two-dimensionalconvolution is more than twice one-dimensional convolutions.The four basic one-dimensional Gaussian derivative filters aregiven by


f 1 ¼�2tr2 e

�t2

r2 ;

f 2 ¼ e�t2

r2 ;

f 3 ¼�2tr2

2t2

r2 � 1� �

e�t2

r2 ;

f 4 ¼2tr2 e

�t2

r2

ð10Þ

where t controls the length of the derivative filter.The generation step of five basic Gaussian filter responses is

shown in Table 1. In this section, we focus on novel gradient com-putation method for the HOG algorithm. So, we only used Gx and Gy

filters for novel gradient computation. The other filters are used forthe second-order derivative calculation. The standard deviation (r)of the Gaussian derivatives filters is selected as 1.

For the calculation of the gradient in the proposed GDF-HOGalgorithm, we used Gx and Gy filters based on the convolution withthe derivatives of the Gaussian according to the Eq. (10). In GDF-HOG algorithm, gradient information is calculated as follows:

Ix ¼ I � Gx

Iy ¼ I � Gyð11Þ

From Eq. (11), we can obtain the gradient magnitude of image Iwith regard to h in Eq. (12):

Ih ¼ cosðhÞI �Gxþ sinðhÞI �Gy

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðI �GxÞ2þðI �GyÞ2

qcosh

I�Gx

ðI �GxÞ2þðI �GyÞ2þ sinh

I �Gy

ðI �GxÞ2þðI �GyÞ2

!

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðI �GxÞ2þðI �GyÞ2

qsinðhþuÞ

ð12Þ

where u ¼ arctan I�Gy

I�Gx. Thus, when h ¼ p

2 �u; gradient magnitude of

input image Ih obtains

Imag�gradient ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðI � GxÞ2 þ ðI � GyÞ2

qð13Þ

To calculate gradient magnitude in proposed GDF-HOG algo-rithm, Eq. (13) is used. It should be noted that there are majormathematical differences between classical gradient computationand the proposed gradient computation. The proposed gradientcomputation is invariant to image rotation at any angle. In addi-tion, the proposed gradient method is mathematically strong androbust to noise. Classification results of the GDF-HOG obtainedon various datasets are higher than the original HOG method.

3.2. The proposed Eig(Hess)-HOG and Eig(Hess)-CoHOG algorithms

Instead of using the Gaussian derivative filters in original HOGand CoHOG algorithms, our second method uses the Hessianmatrix to calculate the eigenvalues of image surface. In order toanalyze the local behavior of an image Ix, we apply local analysisof an image by using Hessian matrix (second fundamental formrepresented in Appendix A). The Hessian matrix is a square andsymmetric matrix, and consists of second-order partial derivativesof the function [64]. According to differential geometry concepts,

Table 1Generation step of separable two-dimensional Gaussian filters using the Gaussianone-dimensional filters.

Basic filters Filter in x Filter in y

Gx f1 f2

Gy f2 f1

Gxx f3 f2

Gxy f4 f4

Gyy f2 f3

the information about maximum, minimum and saddle points ofthe function have been obtained by looking at the minor determi-nants of Hessian matrix. The Hessian of an image is defined as sec-ond-order partial derivative matrix of gray level image. TheHessian matrix H, as a real-valued matrix, has real-valued eigen-values. Hessian matrix of one point in a gray image I for a scaler is computed as

Hrðx; yÞ ¼Dxx Dxy

Dyx Dyy

� ¼

I � Gxx I � Gxy

I � Gxy I � Gyy

� ð14Þ

where ⁄ denotes the convolution, Dxx, Dyy, Dxy are the second-orderderivative of the image along direction of x, y, xy respectively. Gxx,Gyy, Gxy are the second-order derivative filter of the image alongdirection of x, y, xy respectively. The generation of these three filtersis detailed in the former section. In Eq. (14), r is implicitly includedin the calculation of second-order derivatives.

The Hessian matrix contains more differential information thanthe gradient computation. Especially, first order operations (i.e.gradient) are insufficient to describe the behavior of nonlinearfunctions. But the second order differential operations (i.e. Hes-sian) enable a more accurate analysis in detail about functioncurves [64]. From this viewpoint, the state-of-the-art geometry-based operators try to estimate the shape of the underlying imageregion by estimating second-order partial derivatives such asLaplacian and Hessian approaches [41,65]. In many state-of-the-art studies, the object edges, corners, and shape information areobtained by the Hessian matrix [65,66].

The basic idea behind eigenvalue and eigenvector analysis ofthe Hessian matrix is to extract the principal directions and princi-pal curvatures in image surface. Thus, the local second-order struc-ture of the image can be examined. Since this directly gives thedirection of the smallest curvature in image. The eigenvalues ofthe Hessian are called principal curvatures and invariant underrotation.

The eigenvalues of the Hessian matrix can be defined as follows[41]:

k ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðI � Gxx � I � GyyÞ2

4þ ðI � GxyÞ2

sþ I � Gxx � I � Gyy

2ð15Þ

where k are the eigenvalues of the Hessian matrix. Fig. 2 presentsthe procedure for computing eigenvalues of the Hessian matrix ata specific Gaussian standard deviation. The Eq. (15) allows theextraction of the principal directions in image. So, the local sec-ond-order structure of the image is examined. In this paper, wefocus on the novel derivative method for the HOG and CoHOG algo-rithms. Thus, k1 and k2 can be used instead of gradient calculationstep of the original algorithms. In original HOG algorithm, gradientorientation and magnitude of the pixels are calculated by using con-ventional gradient. Instead of using the conventional gradient, gra-dient magnitude and orientations can be calculated by using theeigenvalues k1 and k2.

In proposed Eig(Hess)-HOG algorithm, the gradient magnitudeis calculated as:

Ih gradient ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1Þ2 þ ðk1Þ2

qð16Þ

h ¼ arctank2

k1

� �ð17Þ

Then we calculated the gradient orientations of proposed algo-rithm as in Eq. (17). The obtained gradient orientations are labeledin the range (0–180�) with eight different labels as mentioned inSection 2. In the experimental analysis, the novel Eig(Hess)-CoHOGalgorithm performs this labeling process for seven groups, whilethe original method pixels using the gradient often label eightdifferent groups. Discarding high order derivatives can simplify

Fig. 2. Procedure for computing eigenvalues of the Hessian matrix. The original texture image is taken from the KTH-TIPS2-a dataset. The Gaussian standard deviation r = 1.


computation, but loss of high order information leads to reductionof classification accuracy. Since the first order derivatives cannotcomprehensively characterize a complex texture image, moreorder derivatives should be included. Thus, the co-occurrencematrix in novel Eig(Hess)-CoHOG has more distinctive textureinformation. Owing to discriminative capability of co-occurrencematrix, the novel CoHOG algorithm reduces the size of feature vec-tor from 1536 to 1176. The extraction of a high-dimensional fea-ture descriptor, such as the original CoHOG and LBP, is timeconsuming, especially when the operation is applied to a wholeimage with an exhaustive block-based search [18,41]. The novelCoHOG approach helps reduce the computational load especiallywhen it is of interest to carry out real-time operations. Despitethe lower feature vector dimensionality, the classification perfor-mance of the algorithm is substantially increased.

3.3. The proposed GM-CoHOG algorithm

In Section 3.2, to improve gradient calculation of original HOGand CoHOG algorithms, the Hessian matrix is used instead of gra-dient computation. These second-order differential operationsenable us to obtain distinctive information of pixel and texture ori-entation about the image surfaces. In addition, Hessian matrix andits eigenvalues give us edge informations in the image as well ascorner point details of objects in the image. Edge and corner points

are important to obtain important image features accurately.Therefore, very high classification ratio is obtained in the devel-oped Eig(Hess)-HOG and Eig(Hess)-CoHOG algorithms. In rotationanalysis on Brodatz dataset, obtained positive results by the use ofHessian matrix and eigenvalues revealed the strength of second-order differential analysis techniques. The success of second-orderdifferential analysis has urged us to use different second-order dif-ferential analyses. Starting from this point, the Gaussian curvatureand mean curvature data used in the calculation of the curvature ofthe functions in differential geometry have been integrated intothe gradient computation step of the original CoHOG algorithm.

In the original CoHOG algorithm, the gradient orientations arecalculated based on the conventional gradient information. Morerobust gradient orientations are obtained with the use of Gaussianand mean curvature of image. In Section 3.1, first and second-orderpartial derivatives of the image are calculated. Since the Gaussianderivatives filters could be treated as special difference operators,the first and second partial derivatives of the image I are calculatedas:

Ix ¼ I � Gx

Iy ¼ I � Gy

Ixx ¼ I � Gxx

Ixy ¼ I � Gxy

Iyy ¼ I � Gyy

ð18Þ


where Ix and Iy are the first partial derivatives of I with regard to xand y respectively, and Ixx, Iyy and Ixy are the second partial deriva-tives of I with regard to x and y. The Gaussian curvature K and themean curvature H of the image surface I can be calculated in Eqs.(19) and (20) [64]:

H ¼Ixxð1þ I2

yÞ þ Iyyð1þ I2x Þ þ 2IxIyIxy

1þ I2x þ I2

y

� �32

ð19Þ

K ¼IxxIyy � I2

xy

ð1þ I2x þ I2

yÞ2 ð20Þ

Labeling process of the pixels is performed by using the Gauss-ian and mean curvature data. In the labeling process, Eq. (21) isused.

Ol ¼ 1þ 3ð1þ signðHÞ þ 1þ signðKÞÞ ð21Þ

Through these detailed and advanced geometric derivative cal-culations, the most detailed pixel analysis is made by calculatingthe Gaussian and mean curvature data of each image. Thanks tothe two curvature information of each pixel, even the most distin-guishing characteristics of the two neighboring pixels could bedetected. The proposed method has the best discriminating abilityfor texture images. The classification results of GM-CoHOG algo-rithm reveal the power of the proposed method. Fig. 3 presentsthe procedure for computing Gaussian and mean curvatures ofan image at a specific Gaussian standard deviation.

4. Experiments on the standard material datasets

In this section, the novel HOG and CoHOG methods using ourproposed four novel derivative computation model are testedand compared with the original HOG and CoHOG methods in termsof classification rate and feature vector size. Our algorithms arecoded by Matlab. All the experiments in this section are run in arelease version of our code on an Intel(R) Core(TM) 2 Duo CPU2.93 GHz, 8 GB RAM personal computer.

4.1. Dissimilarity metric

The dissimilarity of training and test histogram is a test of good-ness-of-fit, which can be measured with a nonparametric statistictest. There are many metrics for measuring the fit between twoimage histograms, such as Euclidean distance, log-likelihood ratio,and chi-square statistic. In this paper, a test sample S is assigned tothe class of model M that minimizes the chi-square distance:

DðS;MÞ ¼XL

k¼1

ðSk �MkÞ2

Sk þMkð21Þ

where L is the number of bins. Sk and Mk are the values of the sam-ple and model images at the k th bin, respectively. To evaluate theeffectiveness of the proposed new HOG and CoHOG methods, weuse the nearest neighborhood (NN) classifier with chi-square dis-tance kernel rather than other classifiers such as artificial neuralnetwork and support vector machines which have been obtainedto produce perfect results [41,67,68].

4.2. Datasets

We tested the new HOG and CoHOG methods by classifyingimages from CUReT dataset [69], KTH-TIPS dataset [17,70], KTH-TIPS2-a [70,71], UIUC [15] and Brodatz datasets [72]. Brodatz data-set is only used to show the rotation invariance properties of theproposed methods.

The CUReT dataset contains 61 different texture classes, andeach texture class includes 205 images of a physical texture sam-ple photographed under a range of viewing and illuminationangles. For fair comparison with other HOG and CoHOG studiesusing CUReT dataset, the 92 images per class are chosen withthe size of 200 � 200. Some examples of CUReT dataset are givenin Fig. 4a.

The KTH-TIPS dataset includes 10 different texture classes ofsandpaper, crumpled aluminum foil, linen, sponge, corduroy, sty-rofoam, cracker, brown bread, orange peel and cotton. This datasetconsists of texture images that variation in scale as well as poseand illumination. Images were captured at nine different scalesspanning two octaves, viewed under three different illuminationdirections and different poses, thus giving a total of nine imagesper scale. Sample images from the KTH-TIPS dataset are given inFig. 4b.

The KTH-TIPS2-a ensures a considerable extension to previousKTH-TIPS dataset. The KTH-TIPS2 includes four physical samplesof 11 different materials. The images of each sample texture havevarious poses, illumination conditions and scales. The imageswhich are not 200 � 200 pixels in size are deleted, thus there are4395 images in all. Sample images from the KTH-TIPS2-a datasetare given in Fig. 4c.

The UIUC dataset includes 25 classes, as shown in Fig. 4d and 40images in each texture class. The resolution of each image is640 � 480. The dataset includes materials imaged under significantview-point variations.

The Brodatz dataset includes 112 different textures images. It iscomposed of 112 grayscale images representing a large variety ofnatural grayscale textures. This dataset has been widely used withdifferent levels of complexity in texture classification and texturesegmentation. In this paper, a rotation invariant version of the Bro-datz dataset is used for texture classification. Some examples ofBrodatz datasets are given in Fig. 4e.

4.3. The effect of normalization process

The normalization of texture images is a considerable pre-pro-cessing step for texture classification methods. The textureimages are generally normalized to have zero means and unitstandard deviations. This normalization process provides invari-ance to global affine transformations in the illumination intensity[19]. For all methods with the normalization step, higher classifi-cation accuracies are obtained on the CUReT, KTH-TIPS, KTH-TIPS2-a, UIUC datasets. Table 2 shows the classification resultsfor the proposed GDF-HOG, Eig(Hess)-HOG, Eig(Hess)-CoHOGand GM-CoHOG algorithms with normalization and withoutnormalization.

For all algorithms with the normalization step, higher classifica-tion accuracies are obtained on the on the CUReT, KTH-TIPS, KTH-TIPS2-a and UIUC datasets. Here, following question may arise:Why does image normalization affect the classification perfor-mance positively on the four texture datasets? The question mightbe answered in the following explanations [41]:

� The images are normalized to have unit standard deviationwhich adjusts the global contrast of the images to a standardlevel. The normalization process is only invariant to globaltransformation of the illumination intensity. Therefore, inthe dataset in which the intensity of illumination changedlocally, the impact of the normalization process on the clas-sification accuracy may not be at the expected level. As, inKTH-TIPS2-a dataset, the illumination intensity oftenchanges locally, positive effect of the normalization processis limited compared to other datasets.

Fig. 3. Procedure for computing Gaussian and mean curvatures of a texture image. The original texture image is taken from the KTH-TIPS2-a dataset. The Gaussian standarddeviation r = 1.


� The use of Gaussian local derivative filters to obtain the firstand second derivative information already has the strengthof local illumination invariance. For all the proposed meth-ods, feature vectors are normalized to have an average inten-sity of 0 and a standard deviation of 1. This process provideslocal brightness and contrast invariance. Global normaliza-tion process is thus unnecessary.

Different features of the dataset show that a single normaliza-tion process cannot be applied to them all. In this study, althougha positive influence is obtained for the entire datasets with thenormalization operation on classification accuracy, suitablenormalization process should be applied for different datasetcharacteristics. KTH-TIPS, KTH-TIPS2-a and UIUC datasets containmaterials imaged under significant pose variations, various poses,illumination conditions and scales. Therefore, the differencesbetween the above-mentioned histogram of the images in thethree datasets and histogram of the images in CUReT dataset arenoticed. Fig. 5 demonstrates the texture images and theirhistogram distributions for the CUReT and KTH-TIPS2-a datasets.All of the texture images are from the same class in the CUReTand KTH-TIPS2-a datasets. The histograms denote that the imagesin the CUReT dataset usually have different mean values andstandard deviations. However, images histograms from the KTH-TIPS2-a dataset do not have such features. Thus, the normalizationstep eliminates such effects, and it provides higher classificationperformance on various datasets.

4.4. Comparison of classification performance

In this paper, to evaluate the effects of the proposed first andsecond-order derivatives on the original HOG and CoHOG algo-rithms, a series of experiments are conducted on large and compre-hensive texture datasets. The first experiment is conducted on theproposed GDF-HOG, Eig(Hess)-HOG algorithms and the originalHOG algorithm. The experiments are conducted on CUReT, KTH-TIPS, KTH-TIPS2-a and UIUC datasets. In order to show the bestclassification results, the image normalization step is adopted forthe all datasets. Table 3 shows that the proposed GDF-HOG andEig(Hess)-HOG algorithms achieve best classification results onCUReT, KTH-TIPS, KTH-TIPS2-a and UIUC datasets. The GDF-HOGalgorithm uses the magnitudes of Gaussian first derivatives butachieving continuous rotation invariance. On the CUReT dataset,the classification accuracy of original HOG algorithm is 89.39%,and GDF-HOG achieves 94.52%. The development is considerable.On the KTH-TIPS dataset, the classification accuracy of originalHOG algorithm is 82.75%, and the proposed Eig(Hess)-HOGachieves 95.58%. The improvement in the classification accuracyis utterly incredible. The similar results are true for KTH-TIPS2-aand UIUC datasets as well. Since the UIUC dataset contains bigaffine and scale variation, GDF-HOG and Eig(Hess)-HOG algorithmscannot get very high classification results. Classification resultsshow that the pair of eigenvalues of image surface is a significantproperty which describes local structures distinctively and it ismore robust on all datasets. Furthermore the major advantage of

Fig. 4. Image examples from standard datasets. (a) CUReT dataset. (b) KTH-TIPS dataset. (c) KTH-TIPS2-a dataset. (d) UIUC dataset. (e) Brodatz album.


our proposed Eig(Hess)-HOG algorithm is its continuous rotationinvariance. Generally, the rotation invariant feature extractionmay not achieve superior performance than the original HOG with-out rotation invariance. The Hessian matrix-based feature extrac-tion achieves best classification accuracy on all datasets.According to experimental results, it is also seen that the proposedGaussian derivative filters and eigenvalues of Hessian matrix

achieve excellent results compared with the original HOGalgorithm.

The second experiment is done between the novel CoHOG algo-rithms and the original CoHOG algorithm. Pixel labeling process ofthe proposed Eig(Hess)-CoHOG algorithms and the original CoHOGalgorithm is made with the same formula. So, though feature vec-tor is expected to be the same size in both algorithms, the lower

Table 2Effect of image normalization on the CUReT, KTH-TIPS, KTH-TIPS2-a and UIUC datasets. I means the images are normalized and II means the normalization is omitted. The boldedvalues represent the best classification results.

Methods CUReT KTH-TIPS KTH-TIPS2-a UIUC

I (%) II (%) I (%) II (%) I (%) II (%) I (%) II (%)

Original HOG 89.39 75.00 82.75 77.77 84.98 83.72 66.66 60.00GD-HOG 94.52 85.18 94.02 91.80 91.08 89.42 88.63 85.18Eig(Hess)-HOG 94.08 93.33 95.58 91.80 92.40 84.84 85.71 70.00Original CoHOG 87.63 80.00 97.93 82.35 97.74 97.73 77.41 76.47Eig(Hess)-CoHOG 99.67 90.00 99.00 98.18 99.28 98.34 96.82 91.66GM-HOG 99.38 92.30 99.02 96.62 99.09 98.20 98.41 93.54

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dimensional feature vector is obtained in the proposedEig(Hess)-CoHOG algorithm. Because images are characterizedwith the second-order differential analysis more detailed,co-occurrence matrix is composed of distinctive and sensitivederivative information. Therefore, the distinctive informationcontained in the co-occurrence matrix led to reducing in the size

of the feature vector. Another important aspect to be emphasizedhere is that: since Eig(Hess)-CoHOG algorithm has low-dimensional feature vector and a very high classification rate, itis suitable for real-time applications. Pixel labeling process in theproposed GM-CoHOG algorithm is performed with Gaussian andmean curvature information and a new formula that enabled

Table 3Classification accuracy on the CUReT, KTH-TIPS, KTH-TIPS2-a and UIUC datasets for the proposed novel HOG algorithms. The bolded values represent the best classificationresults.

Method Feature size CUReT (%) KTH-TIPS (%) KTH-TIPS2-a (%) UIUC (%)

Original HOG 128 89.39 82.75 84.98 66.66GDF-HOG 128 94.52 94.02 91.08 88.63Eig(Hess)-HOG 128 94.08 95.58 92.40 85.71

Table 4Classification accuracy on the CUReT, KTH-TIPS, KTH-TIPS2-a and UIUC datasets for the proposed novel CoHOG algorithms. The bolded values represent the best classificationresults.

Method Feature size CUReT (%) KTH-TIPS (%) KTH-TIPS2-a (%) UIUC (%)

Original CoHOG 1536 87.63 97.93 97.74 77.41Eig(Hess)-CoHOG 1176 99.67 99.00 99.28 96.82GM-CoHOG 1536 99.38 99.02 99.09 98.41


labeling. Feature vector size of this algorithm is in the same sizewith the original CoHOG algorithm and more larger than Eig(H-ess)-CoHOG we proposed. However, it has the highest classificationsuccess on UIUC and KTH-TIPS datasets. The classification resultsare listed in Table 4. Table 4 shows that the proposed Eig(Hess)-CoHOG and GM-CoHOG algorithms achieves best classificationaccuracies on all datasets. On the CUReT dataset, the classificationaccuracy of original CoHOG algorithm is 87.93%, the proposedEig(Hess)-CoHOG achieves 99.67%, and the proposed GM-CoHOGachieves 99.38%. The highest classification rate is obtained for theproposed CoHOG algorithms. Eig(Hess)-CoHOG algorithm achieves99.34% classification success, while the GM-CoHOG algorithmachieves 99.09% on the KTH-TIPS2-a dataset on which a lot of clas-sification methods achieved an accuracy rate of only 70%. The fun-damental challenge of the KTH-TIPS and KTH-TIPS2-a texturedatasets lies on its large intra-class differences, so Eig(Hess)-CoHOGand GM-CoHOG algorithms which have powerful intra-class con-gregate capability could achieve best classification performance.However, the CUReT dataset has opposite properties. The textureimages from the same texture class are very alike, while the textureimages from different class are also similar. That means the CUReTdataset has both small inter-class and intra-class differences. Eig(H-ess)-CoHOG and GM-CoHOG have presented their superiority onCUReT dataset due to their good inter-class distinguish capability.In the experiments carried out on UIUC and KTH-TIPS datasets, veryhigh classification results have been obtained compared to the ori-ginal CoHOG algorithm. In particular, it must be emphasized that;the KTH-TIPS2-a dataset is obtained under different lighting, posesand scaling. The classification success of the Eig(Hess)-CoHOG algo-rithm is an indication of how powerful eigenvalues of Hessianmatrix are in feature extraction process. On the other hand, theexcellent classification rate achieved by GM-CoHOG algorithmshows the value added to the original CoHOG algorithm by theGaussian and mean curvature calculations.

Moreover the important advantage of our proposed Eig(Hess)-CoHOG and GM-CoHOG algorithms are their continuous rotationinvariance. This property of both of the developed algorithms isanalyzed in detail in the following sections.

Table 5Classification accuracy on the rotated datasets of Brodatz album for the original HOGand Eig(Hess)-HOG algorithms. The bolded values represent the best classificationresults.

Method Feature size Dataset1 (%) Dataset 2 (%) Dataset 3 (%)

Original HOG 128 5.05 4.10 1.30Eig(Hess)-HOG 128 95.81 94.31 99.37

5. Rotation analysis

In order to show the rotation invariant performance of the pro-posed algorithms clearly, we carry out various implementations onthree datasets based on Brodatz dataset [72]. To make an objectivecomparison with the other methods, the production of testing andtraining images are performed as described in the literature [35].

Each texture image with the size of 512 � 512 in Brodatz albumis divided into four 256 � 256 non-overlapping image regions. The

center 128 � 128 subimage from each region is used for trainingprocess. For the generate test set, each 256 � 256 image region isrotated at angles of 10–160� with 10� increments and, from eachrotated image, a 128 � 128 subimage is selected. In this way, thetraining set is 4 � 25 subimages and the test set is 4 � 16 � 25subimages.

The dataset 2 includes 60 texture of from Brodatz album alsoused in [14]. The 60 texture images are D01, D04, D05, D06, D08,D09, D10, D11, D15, D16, D17, D18, D19, D20, D21, D22, D23,D24, D25, D26, D27, D28, D34, D37, D46, D47, D48, D49, D50,D51, D52, D53, D55, D56, D57, D64, D65, D66, D68, D74, D75,D76, D77, D78, D81, D82, D83, D84, D85, D86, D87, D92, D93,D94, D98, D101, D103, D105, D110 and D111. The training and testsets are generated in the same way as dataset 1. Therefore, thetraining set is 4 � 60 subimages and the testing set is4 � 16 � 60 subimages. In this way, an objective comparison isperformed with other methods in the literature.

The dataset 3 contains 12 Brodatz texture classes, and each tex-ture class included seven rotation images of size 512 � 512 rotatedby different angles (0�, 30�, 60�, 90�, 120�, 150�, and 200�).

Table 5 shows the classification results for the original HOG andEig(Hess)-HOG algorithms. The proposed GDF-HOG algorithmobtained poor performance, thus its classification results are notpresented. For dataset 1, 2 and 3, the Eig(Hess)-HOG algorithmachieves remarkable classification results. Because the originalHOG algorithm does not have the property of rotation invariance,it has a very low classification rate on three of the datasets.

The very high classification rate of Eig(Hess)-HOG algorithmthat we proposed can be explained as follows:

� Eig(Hess)-HOG algorithm is a method developed by modifyingthe gradient calculation step of the original HOG algorithm. Inthis method, firstly the Hessian matrix of the image is calcu-lated. Secondly, the k1 and k2 eigenvalues of the Hessian matrixare calculated. The main characteristic of eigenvalues of Hessianis invariant under rotation. Therefore, the Eig(Hess)-HOGalgorithm developed by using the magnitude of the eigenvalueshas the property of rotation invariance. Therefore, theEig(Hess)-HOG algorithm achieved high classification accuracyin all datasets. Each image from the Brodatz dataset is rotated

Table 6Classification accuracy on the rotated datasets of Brodatz album for the originalCoHOG, Eig(Hess)-CoHOG and GM-CoHOG algorithms. The bolded values representthe best classification results.

Method Featuresize

Dataset 1(%)

Dataset 2(%)

Dataset 3(%)

Original CoHOG 1536 31.51 15.99 9.69Eig(Hess)-

CoHOG1176 83.29 79.75 94.36

GM-CoHOG 1536 78.28 78.48 98.63


at angles of 0–1600 with 100 increments. The proposed methodachieves a remarkable classification rate for all angle values.Figs. 6 and 7 show the classification accuracy of subimages withdifferent rotated angles in dataset 1, 2 and 3. The result of theoriginal HOG algorithm is too low along with the rotationangles, while the Eig(Hess)-HOG algorithm with continuousrotation invariance achieves more stable and accurate classifi-cation results.

The second experiment about rotation analysis is done amongoriginal CoHOG, Eig(Hess)-CoHOG and GM-CoHOG algorithms.Table 6 shows the classification results for the original CoHOG,Eig(Hess)-CoHOG and GM-CoHOG algorithms. For the datasets 1,2, and 3, the classification results of the original CoHOG algorithmvary from 9% to 68%. These results are very low. The proposedEig(Hess)-CoHOG and GM-CoHOG algorithms achieve incrediblesuccess. In both methods, a high and stable classification resultsare obtained on the all the datasets. Fig. 7 shows the classificationaccuracy of subimages with different rotated angles in dataset 1,2 and 3 for all CoHOG algorithms. The result of the original CoHOGalgorithm is unsuccessful along with the rotation angles, while theEig(Hess)-HOG and GM-CoHOG algorithms with continuous rota-tion invariance obtain more robust and high classification results.

Suggested reasons underlying the success of both methods canbe expressed substantially as follows:

(1) The gradient calculation of the original CoHOG algorithmheavily relies on simple pixel difference. However, theeigenvalues of the Hessian matrix are used in the Eig(H-ess)-CoHOG algorithm that we proposed. The eigenvaluesof an image surface represent the principal curvature detailsin the image and are rotation invariant. These eigenvaluesare also used in the analysis of the non-linear function cur-vatures in differential geometry. Orientation information ofpixels calculated by k1 and k2 eigenvalues of the matrixobtained in the Eig(Hess)-CoHOG algorithm directly affectsaccuracy and quality of the pixel labeling process in the nextstep of the algorithm. Eig(Hess)-CoHOG algorithm,

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Fig. 6. Classification accuracy of different rotated angles. (a) Is for dataset 1, the variancefor dataset 2, the variances of the classification accuracies are VARHOG = 0.0061, VAREig(He

VARHOG = 0.0050, VAREig(Hess)-HOG = 0.00007.

containing higher order differential analysis, reaches highclassification rate by pixel labeling process independent ofthe rotation angle. In addition to all these, Eig(Hess)-CoHOGalgorithm that we proposed has the highest classificationrate in spite of having smaller feature vector size than twothe other methods.

(2) There are two fundamental novelties in GM-CoHOG algo-rithm that we proposed. Firstly, in the pixel labeling process,the second-order horizontal, vertical and diagonal deriva-tives of image are used instead of the first-order derivative.Instead of using first-order derivative, Gaussian and meancurvatures are used. In the literature, there are almost nostudies depending on curvature information in gradient ori-entations and texture classification. Finally, novel labelingformulation is used as the best strategy to label texture pix-els. Another important novelty developed in this algorithmis the labeling formula. This formulation used in proposedalgorithms is also used in the analysis of function curves indifferential geometry. Therefore, precise measurementsand calculations are performed on images.

6. More experiments and discussions

6.1. Comparison with texton dictionary-based descriptors

In this section, the novel developed HOG and CoHOG algorithmsare compared with the texton dictionary-based methods in terms

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Fig. 7. Classification accuracy of different rotated angles. (a) Is for dataset 1, the variances of the classification accuracies are VARCoHOG = 0.0265, VAREig(Hess)-CoHOG = 0.0028, andVARGM-CoHOG = 0.0019. (b) Is for dataset 2, the variances of the classification accuracies are VARCoHOG = 0.0399, VAREig(Hess)-CoHOG = 0.0036, and VARGM-CoHOG = 0.0041. (c) Is fordataset 3, the variances of the classification accuracies are VARCoHOG = 0.0926, VAREig(Hess)-CoHOG = 0.0070, and VARGM-CoHOG = 0.0002.

Table 7Classification accuracy based on the Brodatz dataset. The results of CMR and PC [41],and Guo et al. are from the Ref. [13]. The bolded values represent the bestclassification results.

Methods Feature size Brodatz (%)

GDF-HOG 128 88.75Eig(Hess)-HOG 128 85.91Eig(Hess)-CoHOG 1176 98.21GM-CoHOG 1535 97.25CMR [41] 4480 93.71PC [41] 4480 92.83Joint_Sort [13] 11,200 86.71


of their classification accuracy and feature vector size. The textondictionary methods are learned from the training images by clus-tering the local descriptors, and the representation of each imageis the frequency histogram of the textons. Guo et al. [13] proposetwo types of local descriptors based on Gaussian derivatives filters,both of them have the property of continuous rotation invariance.The first local descriptor directly uses the maximum of the filterresponses named continuous maximum responses (CMR). The sec-ond local descriptor rectifies the Gaussian filter responses to calcu-late principal curvatures (PC) of the image surface. The other studyto which we compare the algorithms we developed is Joint_Sorttexton dictionary-based techniques developed by Guo et al. [13].The idea of the Joint_Sort method could be extended to other localpatch based methods. In order to compare them with the proposedfour new HOG and CoHOG algorithms, we conducted the same tex-ture classification experiments of paper [41]. Classification resultsof CMR, PC and Joint_Sort methods were taken from their ownpaper [13,41]. The classification results for the texton number ofCMR and PC methods are considered as 40. In addition, the textonnumber of Joint_Sort method is considered as 100, because thehighest classification results in the related articles are achievedwith these numbers of textons. The feature vector sizes used bythe texton dictionary-based methods are calculated with the texton number � class number formulation used in articles. Thus, dur-ing the test process of PC and CMR methods, feature vector sizesare found to be 1000 for UIUC dataset, and 4480 for Brodatz album.In the same way, as in Joint_Sort method texton number is 100,they are calculated as 2500 for the UIUC dataset, and 11,200 forBrodatz album.

Tables 7 and 8 show the classification results for UIUC and Bro-datz datasets. The first experiment is carried out on the Brodatzalbum of 112 textures, each of which is divided into nine equallysized regions, giving 999 texture samples. The proposed Eig(H-ess)-CoHOG algorithm obtained the highest classification rate witha ratio of 98.21% on Brodatz dataset. GM-CoHOG algorithm has asuccess rate of 97.25%, leaving behind CMR, PC and Joint_Sorttechniques. The second experiment is performed on the UIUC data-set. Our proposed Eig(Hess)-CoHOG and GM-CoHOG algorithms

achieve better classification results on the UIUC dataset, andCMR, PC and Joint_Sort methods achieve better performance onthe UIUC dataset.

Here, there are two important differences between the methodswe proposed and other texton dictionary-based methods. First, intexton dictionary-based methods, the number of textons used dur-ing the training process significantly affects classification perfor-mance. For example, in experiments on the UIUC dataset, CMRmethod has a success rate of 91.51% for 10 texton. Also, it has aclassification success rate of 93.03% for 40 texton. This situationis also true for the PC and Joint_Sort methods. The second impor-tant difference is K-means algorithm used in the training stage ofthe texton dictionary-based methods. During the training stageof the texton dictionary methods, feature vectors obtained in origi-nal image size are divided into subsets equal to the number of tex-tons determined using K-means algorithm. This process increasesthe computational time of the algorithm in a very significant rate.To analyze this situation in a more concrete form.

Let us briefly examine only the training phase of PC methodregarding texton number to be 40. This method performs the firstand second derivative calculations for each of r = 1, 2, 4, 8 valueswith the purpose of extraction of the feature vector of a singleimage with 200 � 200 sizes within CUReT dataset. It calculatesprincipal curvature information in the same size with two originalimages by using the derivative information obtained for each of r

Table 8Classification accuracy based on the UIUC dataset. The results of CMR and PC [41], andGuo et al. are from the Ref. [13]. The bolded values represent the best classificationresults.

Methods Feature size UIUC (%)

GDF-HOG 128 88.63Eig(Hess)-HOG 128 85.71Eig(Hess)-CoHOG 1536 96.82GM-CoHOG 1176 98.41CMR [41] 1000 93.03PC [41] 1000 90.69Joint_Sort [13] 2500 92.73

Table 9Classification accuracies (%) of LBP, RTRID, V2-RTRID and the proposed CoHOGalgorithms on Brodatz album (rotated and noisy texture dataset from Brodatz album).The bolded values represent the best classification results.

Methods SNR Average accuracy (%)

LBP 4 86.50RTRID 4 89.70V2-RTRID 4 91.4Eig(Hess)-CoHOG 4 93.54GM-CoHOG 4 96.08


values. For four different r values, a total of eight basic curvaturedata is calculated. Eventually, for each training set image, a featurevector of 8 � 40,000 size is calculated. When the number of train-ing images for each class is taken 20 for CUReT dataset, traininginformation in size of 160 � 40,000 shall be given to K-means algo-rithm. When the histogram of this information is prompted to sep-arate into a set of 40 textons with K-means algorithm, significanttime requirements will occur. Training image histograms obtainedin the training phase by such processes and labeled as 40 textonsare used in the testing phase. Texton dictionary-based methodsare time-consuming and awkward with the K-means algorithmand they also require sufficient training texture images to con-struct texton dictionary. When considering the computational costof the each texture class in the datasets, it will be understood howslow the texton dictionary-based methods work. Note that theabove-mentioned texton-dictionary based methods do not provideus with information about the computation time. Therefore, anycomputational comparison could not be made with our algorithms.However, K-means and any other similar algorithms are not usedin the four novel texture classification algorithms we proposed.The obtained feature vectors are directly given to the NN classifier.Especially owing to the principal curvature information of eachpixel computed by eigenvalues in Eig(Hess)-CoHOG, timeperformance of the algorithm is improved algorithm throughconsiderably minimizing feature vector size. Although proposedalgorithms have lower dimensional feature vector, they generallyhave higher classification performance.

6.2. Comparison with several descriptors

In this experiment we make a brief comparison between novelCoHOG algorithms and the several rotation invariant descriptorsincluding rapid-transform based rotation invariant descriptor(RTRID) [11], V2-RTRID [11] and LBP [18]. The RTRID and V2-RTRIDare based on the local circular neighborhood and the local featurevector is obtained by means of Rapid-transform. This experimentis conducted on the Brodatz album. This set consists of 16 Brodatztexture classes, and each texture class includes seventeen rotationimages of size 128 � 128 rotated by different angles (0�, 10�,20�, . . . ,160�). The single pattern (16,2) is used for the three neigh-borhood based methods LBP, RTRID, V2-RTRID. To evaluate the

robustness of methods under the condition of additive Gaussiannoise, we tested the performance of classifying texture imageswhich were added Gaussian noise with zero mean and a variancedependent on specific texture to obtain required signal-to-noiseratio (SNR). Table 9 lists the comparative results. It can be seen thatthe proposed CoHOG algorithm achieves excellent classificationperformance under rotation conditions. From Table 9, it can be seenthat single descriptors RTRID and V2-RTRID obtain better classifica-tion accuracy than LBP method. Their good performance should becontributed to feature selection processing. The RTRID methodobtains poor performance due to its global rotation invariance.The textures of the same texture class have large intra-class dissim-ilarity, thus the directions selected by Radon transform are nolonger stable. In many image processing applications we need todeal with noisy texture images. As a result, robustness to noise isconsidered as one of the most significant factors to assess textureclassification methods. It is obvious that LBP method has a goodresult under ideal condition; however it is not as robust as the othermethods under noise and rotation conditions. Our CoHOG algo-rithms also have the best correct classification rates of 93.54% and95.08% among them. Due to usage of Hessian and principal curva-tures, novel CoHOG descriptors have a good average classificationrate. The experimental results show that higher order directionalderivatives can obviously improve classification accuracy. So it isimportant to take into account the higher order directional deriva-tives (i.e. principal curvatures) for the improvement of accuracy.

7. Conclusion

Designing an effective and robust feature extraction algorithmfor texture classification under non-ideal conditions is a challeng-ing task. In this paper we have proposed rotation invariant featureextraction algorithms which are robust to illumination and posevariations. The developed four novel feature extraction algorithmsare based on the strengthening of the calculation of the gradientorientations in the original HOG and CoHOG algorithms. The clas-sical HOG and CoHOG algorithms directly use the gradient infor-mation, which will discard the useful information in textureimages. Therefore, we have presented four novel extensions ofHOG and CoHOG, the key idea is to make efficient and reasonableuse of derivative information in the local texture patterns, espe-cially in the nonuniform patterns. In four novel algorithms wedeveloped, Gaussian derivative filters, eigenvalues of the Hessianmatrix and Gaussian–mean curvatures are used. Thanks to robustand powerful mathematical methods, rotation invariant and qual-ified texture features are obtained. Very high classification perfor-mance is obtained on well-known and widely used texturedatasets without increasing feature vector dimension of the algo-rithms. Moreover, in the Eig(Hess)-CoHOG algorithm that wedeveloped, higher classification performance is obtained by reduc-ing the size of feature vector of the original CoHOG algorithm tothe ratio of 23.43%. Rotation invariant classification experimentsare carried out on the Brodatz dataset and promising results areobtained from experiments. Finally, the superiority of the proposedalgorithms is analyzed by a comprehensive comparison with thestate-of-the-art the texture classification methods.

Appendix A

The image behavior in a local neighborhood of a point can beexpressed by the Taylor expansion. Therefore, the local neighbor-hood can be simplified as an expansion of functions. The Taylorexpansion of a local neighborhood of a point~x0 is defined in Eq. (A1),

f ð~x0 þ~hÞ ¼ f ð~x0Þ þ rf ð~x0Þ~hþ12!

h*

Tr2f ð~x0Þ~hþ rð~hÞ ðA1Þ


where rð~hÞ is a residual.Differential geometry, in general, allows extracting properties

and features of points by means of first and second-order termsof the Taylor expansion. While first order terms in Taylor expan-sion contain information about gradient distribution in a localneighborhood, second-order terms include information about thelocal shape.

G.1. First fundamental form (gradient distribution)

Given a point p, the first-order terms of the Taylor expansioncorrespond to its so-called Jacobian matrix is defined in Eq. (A2),

J ¼@p@x@p@y

!¼

1 0 f x

0 1 f y

!ðA2Þ

where first partial derivatives fx and fy are estimated applyingGaussian derivative kernels on the image. The first fundamentalform I for the point p on an image f is defined in Eq. (A3):

I ¼ J � JT ¼1þ f 2

x f xf y

f yf x 1þ f 2y

!ðA3Þ

G.2. Second fundamental form (Hessian matrix)

The normal vector N to the point p on an image f is defined inEq. (A4),

N ¼@p@x �

@p@y

@p@x �

@p@y

¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ f 2x þ f 2

y

q �f x

�f y

1

0B@

1CA ðA4Þ

Second-order terms of the Taylor expansion conform theHessian matrix H, which is similar to the second fundamental form.Eq. (A2) is derived from the normal curvature estimation and it iscalculated using the second-order partial derivatives and the nor-mal vector N for each point p. So the second fundamental formfor images is described in Eq. (A5),

H ¼@2p@x2 N @2p

@x@y N

@2p@x@y N @2p

@y2 N

0@

1A ¼ ð0 0 f xxÞN

ð0 0 f xyÞNð0 0 f xyÞNð0 0 f yyÞN

!

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f 2

x þ f 2y

q f xx f xy

f yx f yy

!¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

detðIÞp H

ðA5Þ

where second partial derivatives fxx, fxy, fyy are estimated applyingGaussian derivative kernels on the image and det (I) is the determi-nant of the first fundamental form.

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