1 Introduction
Breast cancer is the most common and leading cause of cancer death in women [1]. For breast cancer diagnosis, mammography, which uses Xrays to examine the human breast, is usually used for early diagnosis. However, even for skilled radiologists, the mammogram diagnosis between benign and malignant is difficult due to their similarity in the target shape and texture. In order to reduce the human bias as well as to automate the process, ongoing effort has been paid to developing ComputerAided Diagnosis (CAD) systems for mammogram classification . Typically, these systems rely on extraction of the texture and the shape information from the radio images [2], [3]. The mammogram classification performance depends heavily on precise segmentations of the target tumors (i.e., complex lesion boundary of the tumors) for reliable features. To avoid this dependency, the texture information obtained from local descriptors such as the LBP [4], the Local Binary Convolution (LBC) [5] and the HOG [6] are adopted [7], [8].
Our motivation for mammogram classification consists of the following: (i) The shape based feature extraction can be significantly affected by the irregular lesion boundaries. (ii) The texture based feature extraction such as LBP, HOG and LBC typically use the convolution based iterative scanning process that is timeconsuming and often produces a large feature dimension. (iii) To utilize the diversity of different local descriptors for performance improvement.
The main contributions of this work are as follows: (i) Matrix formulations of two descriptors, namely the LBP and the HOG, to avoid the iterative convolution process in local computation and to get away from the high dimensional feature size. (ii) An integrated formulation of the LBP and the HOG under linear matrix products.
The remainder of the paper is organized as follows: Section 2
includes a brief review of the LBC descriptor, the DMP (Difference Matrix Projection) method and the totalerrorrate based classifier. In Section
3, we propose two convolutionfree descriptors in matrix formulations. Then we construct an integrated matrix formulation of these two descriptors. Section 4 presents our experiments, observations and evaluations of the proposed method using a public mammogram database. Concluding remarks are given in Section 5.2 Preliminaries
2.1 Local Binary Convolution network (LBC)
The Local Binary Pattern (LBP) [4] is a simple and popular descriptor adopted in many applications (e.g., face and palm print recognitions [9, 10, 11]). The LBP scans each central pixel of an image and its local neighborhood pixels (
) within an odd size window determined by
(e.g., indicates a window and indicates a window). The computed output of the LBP can be expressed as , where and are respectively the intensity values of the center pixel and the neighborhood pixels. is the thresholding operation given by . Frequently, for images with high enough resolution, the, and are set at and .Based on the LBP, a Local Binary Convolution network (LBC) was proposed in [5]
to encode the LBP efficiently utilizing the sparse convolutional filters. The LBC was developed as an alternative to the standard Convolutional Neural Networks (CNN) layer to reduce its complexity by reducing the number of learnable parameters. For a
window size, the LBC uses a weighted sum of eight sparse convolutional filters to produce the binary map. The original LBP is consequently reformulated in [5] as the basis of LBC as follows:(1) 
where is the convolution operation, and is the sparse convolutional filter with two nonzero values
which convolves with the vectorized input image
, is the Heaviside step function. This functioncan be replaced by a Sigmoid or a ReLU activation function for differentiability.
contains predefined weight values. is the number of neighbor pixels (e.g., with window).2.2 Difference Matrix Projection (DMP)
The Difference Matrix Projection (DMP) was proposed for pedestrian detection in [12]. Compared with the pixelwise calculations of the Histogram of Oriented Gradients (HOG) [6] based on the first order gradients, the DMP realized an approximated HOG based on linear matrix products utilizing both the first and the second order gradients with precalculated projection matrices. The DMP process uses a cellbased pooling to substitute the histogram construction.
The DMP process can be written as:
(2) 
where
(3) 
are the predefined projection matrices for cellbased nonoverlapping pooling, and is the cell size. Based on the first and the second order gradients from four orientations (, , and ), the gradient can be expressed as:
(4) 
where to are the firstorder gradients and to are the secondorder gradients. is the input image. The predefined horizontal and vertical shifting matrices are given respectively by
(5) 
where
is the identity matrix.
indicates the number of shifting pixels which determine the first and the second order gradients when and respectively. Similar to HOG in [6], the block normalization is subsequently performed on the cellbased pooling to obtain the final DMP features.2.3 TotalErrorRate (TER) Minimization
The TotalErrorRate (TER) based classification [13]
utilized the sum of the type I and type II errors. The type I error (also known as False Positive Rate (FPR)) is the ratio of falsely recognized positive samples over the negative sample size given by
. The type II error (also known as False Negative Rate (FNR)) is the ratio of falsely recognized negative samples over the positive sample size given by . Then, TER can be written as . For a classifier which is linear in its parameters, the TER parameters can be optimally determined by(6) 
where , and are respectively the transformed samples of each category, is the total number of samples and is a classspecific weight matrix where , . is the learning target vector, with and for a given threshold and offset . is a vector of element ones. is the regularization factor with being the identity matrix that matches the dimension of .
3 A ConvolutionFree LBPHOG Descriptor
3.1 Overview
In this section, we propose a ConvolutionFree LBPHOG descriptor (CFLBPHOG) in matrix form. Specifically, we propose a ConvolutionFree LBP descriptor (CFLBP) in matrix form in the first step. This is followed by a ConvolutionFree HOG descriptor (CFHOG). The proposed CFLBP and CFHOG are then integrated into a single matrix product form.
3.2 ConvolutionFree LBP in Matrix Form
The original LBP and the LBC iteratively compute the pixel differences in eight directions at angle difference ( to ). The proposed CFLBP can be formulated as the weighted sum of eight directional difference matrices as follows:
(7) 
where
(8) 
with and being the predefined shifting matrices from equation (5).
3.3 ConvolutionFree HOG in Matrix Form
In [12], the DMP method utilized the window based iterative scanning process for block normalization based on the norm. The proposed CFHOG as an approximated HOG is based on the norm block normalization in matrix form including overlapping. A cellbased overlapping pooling is also proposed to obtain the local connection between the cell groups. The cellbased overlapping pooling can be written as follows:
(9) 
where
(10) 
(11) 
and is the cellbased nonoverlapping pooling from equation (2) using a cell size of in which and are the predefined projection matrices for the cellbased overlapping pooling from equation (10) and (11) using a cell size of and an overlapping size of .
In [6], the normalization is applied to each block which consisted of a cell group. The blockbased overlapping normalization in matrix form can be written as follows:
(12) 
where denotes the Hadamard elementwise operation. and are the predefined projection matrices for the blockbased overlapping normalization. is the sum of the squared elements of each block based on the pooling technique. and are for upsampling to retain the original matrix size. The superscript is the elementwise inverse of the squared root.
3.4 Integrating LBP and HOG into One Matrix Form
Based on the CFLBP and CFHOG, the proposed CFLBPHOG in matrix form can be written as follows:
(13) 
where , , , , is the input matrix utilized by in equation (7) and is obtained by using equation (4) with . For the final CFLBPHOG features, each are concatenated together into one feature vector.
3.5 A Case Study of The LBP Based Descriptors
In this study, we compare the proposed CFLBP with the original LBP and with the LBC using a mammogram image as the input image. Based on a typical LBP setting at and , the LBP based descriptors, namely, LBP, LBC and CFLBP, are performed to obtain the LBP based texture image. Fig. 1 shows the same output values within the borders from each descriptor whereas the borders of each descriptor have different output values due to different techniques to perform the LBP.
4 Experiments
In this section, we evaluate the proposed descriptor for mammogram classification. The experimental goals are as follows: 1) Observing the effect of overlapping pooling among the HOG, the DMP and the proposed CFHOG; 2) Performance comparison of the proposed CFLBPHOG with stateofthearts descriptors.
4.1 Database and Experimental Setup
The most commonly used database in mammography is the Digital Database of Screening Mammography (DDSM) [14]. Recently, in [15], a Curated Breast Imaging Subset of the DDSM (CBISDDSM) has been released in view of the segmentation difficulty for a standarized evaluation. In this experiment, we used the CBISDDSM database. The database includes ROI images which are categorized into two classes (malignant, benign) for patients and the images are resized to a resolution. By following the data split setting in [15], a training set ( images) and a test set ( images) are obtained.
For stateoftheart descriptors, the original LBP [4], the LBC [5], the HOG [6] and the DMP [12] are implemented for comparison. The LBPHOG is also implemented based on a cascade of the LBP image and the HOG feature representation. For a stateoftheart mammogram classification system, the VGGNet [16] is additionally implemented. For the parameter settings of each descriptor, we followed the general settings [4], [5], [6], [12] for LBP, HOG, DMP, LBPHOG and the proposed CFLBP, CFHOG and CFLBPHOG: , , Heaviside step function, , , and . The histogram bin sizes of LBP and HOG are respectively fixed at and
. The TER and SVM classifiers are utilized with a radial basis function (RBF). The regularization parameter
of the TER is fixed at . According to [16], the parameters settings of the VGGNet is set.4.2 Observing the effect of the overlapping pooling among the HOG based descriptors
To compare the proposed CFHOG descriptor to the HOG and DMP descriptors, classification test accuracies at different cell and block sizes ( and ) were acquired utilizing CBISDDSM database. Fig. 2 shows the effect of the proposed overlapping pooling at different cell size in CFHOG compared to HOG and DMP where there is no overlapping pooling. The accuracy performances are observed according to increasing the nonoverlapping pooling size . According to the increment, the performance of the CFHOG is increasing while that of HOG and DMP are unstable. The proposed CFHOG outperformed the other stateoftheart methods when is and . The best performance is observed in CFHOG when the overlapping pooling is included.
Method  LBP  HOG  DMP  LBPHOG  CFLBPHOG 

VGGNet  63.35  
SVM  60.79  60.80  60.51  63.49  62.36 
TER  56.25  59.52  60.23  62.07  64.35 
Method 






VGGNet  150,528    174,460  209.88  
LBP  19,116  53.53  24.40  0.0069  
HOG  10,404  42.50  14.59  0.0038  
DMP  9,248  62.00  12.97  0.0034  
LBPHOG  10,404  43.21  14.25  0.0038  
CFLBPHOG  1,800  31.13  04.00  0.0008 
4.3 Performance Comparison and Summary
In terms of classification performance, Table 1 shows the classification accuracies for the compared descriptors, namely LBP, HOG, DMP, LBPHOG and CFLBPHOG. The proposed CFLBPHOG based on the TER classifier showed the best performance compared to the stateofthearts while the LBPHOG based on the SVM classifier and the VGGNet respectively showed the second and the third bests.
In terms of computational performance, Table 2 shows the CPU processing time in seconds. The averaged CPU times are reported over 10 runs. The proposed CFLBPHOG based on the TER classifier showed the best CPU times in the training and test phases due to the predefined projection matrices under global computation form. In addition, the CFLBPHOG produced the smallest feature dimension which is 5 times less than the other descriptors.
In summary, we have shown that 1) the overlapping pooling step of the proposed CFHOG is effective compared with the HOG and the DMP, 2) the proposed CFLBPHOG achieved better performance with a small feature dimension than that of the other stateoftheart methods in terms of classification accuracy and CPU time.
5 Conclusion
Different from the convolution based LBP, we have presented a convolutionfree LBP (CFLBP) in matrix form. In addition, we have shown a convolutionfree HOG based on Difference Matrix Projection (DMP). The integrated form of these two proposed descriptors, CFLBPHOG, was then proposed in a matrix formulation. The proposed descriptors were evaluated using the CBISDDSM database for mammogram classification where the results show promising performance comparing with stateoftheart descriptors.
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