J Med Syst (2014) 38:23 DOI 10.1007/s10916-014-0023-3

RESEARCH ARTICLE

Detection of Breast Abnormality from Thermograms Using Curvelet Transform Based Feature Extraction Sheeja V. Francis & M. Sasikala & S. Saranya

Received: 5 November 2013 / Accepted: 4 March 2014 / Published online: 23 March 2014 # Springer Science+Business Media New York 2014

Abstract Breast cancer is one of the leading causes for high mortality rates among young women, in the developing countries. Currently mammography is used as the gold standard for screening breast cancer. Due to its inherent disadvantages, alternative techniques are being considered for this purpose. Breast thermography is one such imaging modality, which represents the temperature variations of breast in the form of intensity variations on an image. In the last decade, several studies have been made to evaluate the potential of breast thermograms in detecting abnormal breast conditions, from an image processing view point. This paper proposes a curvelet transform based feature extraction method for automatic detection of abnormality in breast thermograms. Statistical and texture features are extracted from thermograms in the curvelet domain, to feed a support vector machine for automatic classification. The classifier detects abnormal thermograms with an accuracy of 90.91 %. The results of the study indicate that texture features have better potential to detect abnormality in breast thermograms, when extracted in the multiresolution curvelet domain. Keywords Thermography . Curvelet transform . Texture features . Support vector machine

Introduction Breast cancer is one of the most dangerous types of cancer that affects over 11 % of women during their lifetime. World Health Organization’s International Agency for Research on Cancer (IARC), has reported that more than 400,000 women die from this disease each year [1]. Early detection is the key S. V. Francis (*) : M. Sasikala : S. Saranya Anna University, Chennai, India e-mail: [email protected]

to reduce the high mortality rates due to breast cancer. A recent study evaluates the performance of various breast imaging modalities for early detection of breast cancer [2]. Currently, mammography is considered to be the ‘gold standard’ imaging technology for diagnosing breast cancer. Mammography is a structural imaging modality that uses Xray exposures on compressed breasts. Due to difficulty in imaging dense breast tissues, its performance is poor in younger women. It has been reported that the average size of tumors undetected by mammography is 1.66 cm [3]. Also, patients who repeatedly undergo mammography for suspected lesions are posed with the hazards of harmful exposure to Xray radiations. Hence, attempts are being made to develop new imaging techniques for early detection of breast cancer. Thermography is one such technique that has been documented to have the potential to detect breast cancer, 8–10 years earlier than Mammography [4]. Thermography is a non-invasive functional imaging method that senses and represents the variation of surface temperature of human skin in the form of color mapped images called thermograms. Developing cancers are characterized by high rates of cell division. New blood vessels (neo-angiogenesis) are recruited inorder to supply nutrients to such regions, resulting in hyper-vascularity. Thus, the surface temperature around cancerous cells will be slightly higher (hyperthermia) than the normal cells due to increased levels of blood perfusion. Such regions will be seen as hot spots on a thermogram [5]. B.B. Lahiri et al. have discussed various medical applications of infrared thermography [6]. A study by Kontos et al., evaluates the performance of digital infrared thermal imaging in breast cancer detection [7]. With the use of high resolution infrared cameras for medical applications, standardized image acquisition protocols and image interpretation tools, clinical utility of breast thermography may be improved [8, 9]. Several studies have demonstrated the ability of temperature features in distinguishing breast cancers [10, 11]. A recent survey, has

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compared various image processing approaches for breast cancer detection from thermograms [12]. Texture features have been used to train artificial neural networks (ANN) for classification of breast thermograms [13]. Asymmetry analysis of selected texture features has been employed to automatically classify thermograms with an accuracy of 85.19 % [14]. The significance of texture features as indicators of surface temperature variation has been explained with thermal oculograms [15]. Acharya U et al. extracted texture features to detect abnormal thermograms using support vector machine (SVM) with an accuracy of 88.10 % [16]. First order statistical features [17], higher order statistical features like skewness, kurtosis and entropy [18] have been extracted for automatic classification of abnormal breast conditions. Wiecek et al. [19] used features based on Discrete Wavelet Transform (DWT) with biorthogonal and Haar mother wavelets, and neural networks to classify thermograms. Recently, higher order spectral (HOS) features [20] and bispectral invariant features [21] have been used for classifying thermograms. Fuzzy classifier [22], independent component analysis [23] and decision trees [24] have been used for the classification purpose in addition to neural networks. Case studies have compared thermal images with ultrasound, mammography and fine needle aspiration cytology to establish that temperature profiles of various pathologies were different [25]. This has led to the possibility of identifying various breast pathologies from thermograms. Texture features extracted in the wavelet domain have also been used for classifying thermograms [26]. In this paper the multiresolution capability of curvelet transforms has been explored for feature extraction. The aim of this work is to evaluate the scope of curvelet transform based features for automatic detection of abnormal breast thermograms using a support vector machine classifier. The block diagram of the proposed work is shown in Fig. 1. First, breast thermograms are converted to gray scale and regions of interest (ROIs) are segmented. These ROIs are then decomposed using discrete curvelet transform. Features are extracted in the curvelet domain for analysis and classification. These features are used to train a support vector machine classifier to automatically detect the abnormality. The performance of the system is tested and validated. The paper is organized as follows: “Data Acquisition” provides a detailed outline on the camera system used, imaging protocol and patient characteristics. The segmentation method employed has been presented in “Preprocessing and Segmentation” Section. A theoretical background on curvelet

Fig. 1 Proposed block diagram INPUT THERMOGRAM

transform, statistical and texture features and SVM classifier is given in “Feature Extraction in Curvelet Domain”. The results of the work are presented in “Results” and discussed in detail in the. “Discussion” section. Finally, the paper concludes in “Conclusion”.

Data acquisition Breast thermograms acquired by a med2000™ IRIS digital infrared camera were used for the study. This camera with WIN TES software has a thermal resolution of 0.01 °C and spectral response of 7–14 μm (30frames/s) (http://www. meditherm.com/). It is an uncooled camera with an amorphous silicon micro bolometer focal plane array detector. It was ensured that the camera was calibrated against black body reference as per the manufacturer’s recommendations. The camera was kept in a closed room maintained at 25 °C and images were acquired 3 min after switch-on according to the protocol given. The image capture process was performed by certified thermographer and lasted for 10–15 min for each subject. Thus, drift over time was assumed to be negligible. Protocol & & & & & & &

Subjects were asked to avoid use of lotion, cream, powder or deodorant. Subjects were asked to stop smoking 2 h before imaging. Examination was done in a temperature controlled room at a temperature of 25 °C. All windows were shielded to minimize IR Interferences with the source. The subjects were required to rest for at least 15 min to stabilize. This would reduce the basal metabolic rate, resulting in minimal surface temperature changes. The patients were asked to wear a loose cotton gown that does not restrict airflow. Images were captured in three positions, namely, frontal, oblique and lateral by a certified thermographer.

Patient characteristics The temperature profile of breasts changes with age and also within a menstrual cycle. It has been reported that mean temperature of breasts reduces in older women [27]. This has been attributed to the lower rates of metabolism and

PREPROCESSING & SEGMENTATION

CURVELET TRANSFORM

FEATURE EXTRACTION & ANALYSIS

CLASSIFICATION

NORMAL/ ABNORMAL

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& & &

(a) Normal

(b) Abnormal

Fig. 2 Sample breast thermograms. a Normal. b Abnormal

vascularization due to the natural process of ageing. In normal healthy women, vascularization is reported to be at basal level during the 5th to 12th and 21st day after the onset of menstrual cycle. It is reported that the temperature of breasts stabilizes during these periods [28]. Hence, it was ensured that all subjects underwent a thermal scan during this crucial period. Menopausal women were excluded from the study. In this work, thermograms were obtained from 22 women with their consent. The subjects include 11 biopsy proven cancer patients (age: 42 ±8 years) and 11 normal volunteers (age: 37 ± 10 years). The abnormal class includes, five - ductal carcinoma in situ cases, five - stage II cancer and one - stage III inflammatory cancer. Breast thermograms were acquired according to protocol, in contra lateral, oblique right, oblique left, lateral right and lateral left positions. Only contra lateral images were used for the present study.

&

Use Sobel Edge detection algorithm, to obtain breast outline and edges from background. Clear the borders and remove weak edges, to find candidate abnormal regions if any. Perform morphological operations such as dilation, hole filling and erosion to segment the abnormal area on the breast regions. Multiply this binary image with the gray scale input image to obtain the intensity variations of the original abnormal region.

Sample breast thermograms for one normal and one abnormal case are shown in Fig. 2a and b respectively. The preprocessed abnormal image is shown in Fig. 3a and the abnormal region segmented by the algorithm is shown in Fig. 3b.

Feature extraction in curvelet domain Feature extraction is the process of defining the set of features which will most efficiently represent information that is important for analysis and classification. Statistical and texture features have been widely employed in breast cancer detection studies as they are found to represent thermal variations in breasts quite effectively. In this work a series of statistical and texture features are extracted from breast thermograms in the curvelet domain.

Preprocessing and segmentation

Curvelet transform

Breasts have been segmented from other regions on the thermogram by Canny edge detection methods followed by gradient operators [29] and Hough transform [30, 31] for boundary detection. The abnormal regions have been segmented from the breasts using k-means clustering methods [32]. In this work regions of abnormality have been segmented from breast thermograms using morphological image processing method. The preprocessing and segmentation algorithm is given below.

Continuous Curvelet transform introduced by Candes and Donoho [33] has been discretized [34], using a “wrapping” algorithm. The transform consists of four steps: obtaining a 2dimensional fast Fourier transform of the image, formation of a product of scale and angle windows, wrapping this product around the origin, and computing a 2-dimensional inverse fast Fourier transform. The approximate scales and orientations are supported by a generic ‘wedge’. Two parameters are involved in the digital implementation of the curvelet transform: number of resolutions (scales) and number of angles (orientations) at the coarsest level. The image is decomposed into sub bands at different scales. This results in multiresolution sub bands comprising of curvelet coefficients. An Increase in scale and/or orientation results in many sub bands that may carry redundant

& & &

Convert input image into gray scale. Enhance the contrast of the image. Crop the normally hot regions such as neck carotid regions, armpits and infra-mammary folds to obtain the breast area.

Fig. 3 Segmentation results for an abnormal thermogram. a Preprocessed image. b abnormal region

23, Page 4 of 9 Table 1 Statistical analysis on statistical features extracted based on normal and abnormal groups

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Statistical Features

Average value of the normal group

Average value of the abnormal group

p

Mean (M) Median (MED) Mode (MOD) Standard Deviation (SD) Variance (V)

0.000±0.000 0.000±0.016 −1.378±0.798 0.116±0.051 0.020±0.033

0.000±0.00 0.000±0.05 −7.504±4.32 0.671±0.45 26.943±52.02

0.9602 0.9620 1.6213e-004 5.5040e-004 0.1015

information. Hence it is necessary to select significant sub bands for extracting features. Features Spatial statistical features represent the distribution of intensity variations in a given image. Simple first order statistical features, such as Mean, Median, Mode, Variance and Standard Deviation are extracted from the thermograms. Texture of an image may be examined by constructing the gray-level co-occurrence matrix (GLCM), also known as the gray-level spatial dependence matrix [35]. The GLCM characterizes the texture of an image by calculating how often pairs of pixels with specific intensities occur in a predefined spatial proximity. GLCM of the curvelet sub bands are found. Texture features proposed by Haralick [36] are then extracted from the GLCM in the curvelet domain. These features include, energy, contrast, correlation, sum of squares- variance, inverse difference moment, sum variance, sum entropy, entropy, difference variance, difference entropy, information measures of correlation 1 and 2. The Haralick’s texture features are listed in Appendix I. Classifier Features extracted in the curvelet domain from normal and abnormal images are used to train a Support Vector Machine

Table 2 Statistical analysis on texture features extracted based on normal and abnormal groups

classifier. The SVM is a supervised learning method that performs well in pattern recognition problems. Using SVM for classification, the input data is transformed to high-dimensional feature space with the use of kernel functions. So the transformed data becomes more separable compared to the original input data. Leave-one-out method has been used for validation of the classifier. This method involves using a single observation from the original sample as the validation data, and the remaining observations as the training data. This is repeated such that each observation in the sample is used once as the validation data. This is the same as a K-fold cross-validation with K being equal to the number of observations in the original sampling.

Results The ROI was decomposed using Curvelet transform at level 2 and orientation 8. This step produces one approximate and eight detail coefficient sub bands. The approximate sub band contains the low-frequency components and the rest capture the high-frequency details along the different orientations. Maximum variance criterion was used to select the optimum sub band for feature extraction. Statistical features were extracted from the selected curvelet sub band as listed in Table 1. Further, normalized GLCM of this sub band was computed at an inter-pixel distance of d=1,

Texture features

Average values in the normal group

Average values in the abnormal group

p

Energy (E) Contrast (C) Correlation(COR)

0.154±0.029 4.900±3.593 91.175±52.680

0.132±0.03 21.646±5.56 318.363±65.93

0.0753 5.4969e-008 2.0534e-008

Sum of squares: variance (SOSV) Inverse difference moment (IDM) Sum variance (SV) Sum entropy (SE) Entropy (ENT) Difference variance (DV) Difference entropy (DE) Information measure of correlation 1 (IMC1) Information measure of correlation 2 (IMC2)

5.871±3.060 0.547±0.061 8.357±4.994 1.729±0.251 2.406±0.316 0.019±0.007 1.554±0.208 −0.048±0.009 0.333±0.041

21.459±4.79 0.362±0.06 40.544±11.70 2.018±0.18 2.399±0.79 0.017±0.01 1.695±0.17 −0.061±0.02 0.386±0.05

3.7467e-005 4.0215e-007 5.5145e-008 0.0058 0.0686 0.5926 0.0943 0.0592 0.0142

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Table 3 Linear correlations among features of the normal group (only features with ρ>0.5 are shown)

E C COR SOSV IDM SV SE ENT DV DE IMC2 MOD SD V

E

C

COR

SOSV

IDM

SV

SE

ENT

DV

DE

IMC1

IMC2

M

MOD

SD

1.00 −0.95 −0.97 −0.96 0.97 −0.95 −0.97 −0.98 0.98 −0.97 −0.68 0.90 −0.86 −0.90

1.00 0.99 1.00 −0.98 1.00 0.95 0.95 −0.97 0.96 0.78 −0.91 0.80 0.89

1.00 0.99 −0.99 0.99 0.95 0.96 −0.97 0.95 0.75 −0.92 0.82 0.93

1.00 −0.98 1.00 0.96 0.97 −0.97 0.97 0.77 −0.92 0.82 0.90

1.00 −0.98 −0.95 −0.95 0.98 −0.96 −0.77 0.93 −0.82

1.00 0.95 0.96 −0.97 0.96 0.79 −0.92 0.82

1.00 1.00 −0.99 0.99 0.68 −0.92 0.92

1.00 −0.99 0.99 0.67 −0.91 0.90

1.00 −0.99 −0.72 0.93 −0.88

1.00 0.70 −0.92 0.90

−0.87 – –

1.00 −0.81 0.61

– –

1.00 −0.93

1.00

−0.93

0.91

0.87

0.87

−0.90

0.87

0.64

−0.91

0.84

thus representing the intensity variations between horizontally adjacent pixels in the input image. Texture features listed in Table 2 were extracted from the normalized GLCM. All the features extracted from thermograms of normal and abnormal groups were analyzed using hypothesis testing on means of two independent samples (student’s t test). The average values of these features are tabulated, for both the groups, along with their p values in Tables 1 and 2. Among the statistical features, Mode and Standard deviation were found to be statistically significant with p0.5 were considered to be strongly associated with each other. Such features, have been listed, for the normal group, in Table 3. It was found that almost all the features were weakly

Table 4 Correlations among features between normal and abnormal groups E

C

COR

SOSV IDM

SV

SE

ENT

DV

DE

IMC1 IMC2 M

MED

MOD SD

V

E −0.22 0.09 0.07 −0.03 C 0.16 −0.12 0.02 0.02 COR 0.19 −0.10 0.01 0.04 SOSV 0.18 −0.12 −0.01 0.02 IDM −0.22 0.08 −0.06 −0.09 SV 0.17 −0.14 −0.01 0.00 SE 0.22 −0.18 −0.11 −0.05 ENT 0.21 −0.16 −0.10 −0.04 DV −0.24 0.14 0.05 −0.01 DE 0.22 −0.18 −0.10 −0.04

−0.37 −0.10 0.15 0.24 −0.13 0.20 −0.39 0.41 0.11 −0.09 −0.18 0.10 −0.17 0.42 0.39 0.12 −0.12 −0.21 0.12 −0.19 0.43 0.41 0.10 −0.11 −0.19 0.11 −0.18 0.42 −0.39 −0.18 0.13 0.26 −0.15 0.23 −0.44 0.43 0.09 −0.10 −0.19 0.10 −0.17 0.42 0.48 0.05 −0.15 −0.21 0.13 −0.20 0.47 0.44 0.05 −0.15 −0.21 0.12 −0.19 0.45 −0.45 −0.11 0.15 0.25 −0.15 0.23 −0.44 0.48 0.06 −0.15 −0.22 0.13 −0.21 0.45

0.50 −0.52 −0.54 −0.52 0.56 −0.52 −0.57 −0.56 0.56 −0.56

−0.08 0.08 0.11 0.07 −0.09 0.09 −0.08 −0.06 −0.01 −0.05

0.51 −0.35 −0.39 −0.37 0.34 −0.35 −0.37 −0.42 0.36 −0.36

0.32 −0.30 −0.51 −0.32 0.35 0.50 −0.35 0.39 0.56 −0.31 0.34 0.50 0.42 −0.42 −0.58 −0.32 0.35 0.51 −0.30 0.27 0.47 −0.29 0.26 0.47 0.37 −0.34 −0.53 −0.33 0.30 0.49

−0.16 0.08 0.10 0.04 0.24 −0.14 −0.14 −0.05 0.21 0.07 0.11 0.19 0.22 0.18 −0.03 0.13 −0.12 0.34 0.17 0.19 0.18 −0.40 −0.36 −0.30 0.22 −0.14 −0.06 0.00

−0.21 −0.01 0.08 0.18 −0.12 0.15 0.21 0.38 0.04 −0.15 −0.25 0.16 −0.23 0.06 0.15 0.25 −0.20 −0.29 0.21 −0.26 0.35 −0.22 0.12 −0.25 −0.20 0.22 −0.21 −0.15 −0.64 0.08 0.04 0.12 −0.02 0.11 −0.49 0.68 −0.18 −0.14 −0.14 0.06 −0.14 0.50 0.46 0.10 −0.16 −0.26 0.15 −0.23 0.46

−0.21 −0.12 −0.49 0.13 0.57 −0.59 −0.59

−0.36 0.22 0.18 −0.27 −0.12 −0.07 0.22

−0.08 −0.14 0.03 0.20 0.22 −0.25 −0.28

−0.02 −0.08 −0.04 −0.14 0.20 0.27 −0.68 0.80 0.84 0.01 −0.14 −0.10 0.24 −0.28 −0.47 −0.21 0.22 0.45 −0.48 0.55 0.73

IMC1 IMC2 M MED MOD SD V

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correlated with each other, when analysed across the normal and abnormal groups. This is evident from Table 4. These features were fed to SVM for classification of normal and abnormal breast conditions. The classifier was trained and tested using the leave one out method using 22 images (11 normal and 11 abnormal). This method of cross validation was used as the sample size used for the study was small. The performance of classifier was evaluated by computing measures such as accuracy, sensitivity and specificity from its confusion matrix as tabulated in Table 5. The proposed system was able to classify abnormal and normal thermograms with accuracy of 90.91 % with a high sensitivity and specificity of 81.82 % and 100 % respectively.

Discussion Several feature extraction methods have been employed for detecting abnormality from breast thermograms. The features extracted include, simple first order statistical features, cooccurrence based texture features, wavelet features and higher order spectral features. The motivation to use curvelet transform based features, comes from the various breast tumor modeling studies reported in literature. As a tumor causes local rise in tissue temperature, it can be modeled as a heat source embedded deep within the breast tissues [37]. The heat radiated from this source reaches the skin surface in approximately concentric circles with hot spot at its centre. Umadevi et al., have demonstrated such a surface distribution of heat by using agar and water melon models of abnormal breasts [38]. This study is made on an underlying assumption that abnormal regions on breast thermograms may exhibit curvilinear properties in spatial distribution or texture. As curvelet transform is best suited to represent the edges and singularities along curves in the image, [39], feature extraction has been proposed in this domain. Hence simple statistical features and GLCM based texture features have been extracted from the curvelet transformed breast thermograms. Average values of statistical features extracted from thermograms in the normal and abnormal groups are shown in Table 1. In a normal thermogram the distribution of heat is more or less homogeneous. Thus its mean value may be lower than an abnormal thermogram. Variance and standard deviation are features that indicate the measure of deviation from the mean. Table 5 Performance evaluation of SVM Performance measure

Accuracy Sensitivity Specificity

Curvelet features Statistical

Textural

86.36 % 81.82 % 90.91 %

90.91 % 81.82 % 100 %

Fig. 4 Regression plots. a Energy versus Entropy in the normal group. b Energy in abnormal group vs. Entropy in normal group. c Entropy in abnormal group vs. Energy in the normal group

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Due to the abrupt changes in pixel intensities, these features are expected to be higher for an abnormal image. From Table 1, it can be inferred, that the average values of standard deviation and variance features are higher in the abnormal group and are also found to be statistically significant. But no such inference could be made in the case of the other features. Average values of texture features extracted from the normal and abnormal groups are shown in Table 2. Texture features help in identifying the properties in an image, based on the interpixels relationships. A normal thermogram is more homogenous than an abnormal one. Hence contrast and entropy features may be expected to be higher for an abnormal image. Also its energy value may be expected to be lower. The average values of texture features such as contrast, correlation, sum of squaresvariance, sum variance, sum entropy, difference entropy, information measures of correlation 1 and 2 are observed to be higher in the abnormal group. A substantial inference could not be drawn in the case of entropy feature which was also found to be statistically insignificant. From Table 3, it is observed that the Median feature does not correlate with any other feature and is also found to be statistically insignificant. Information measure of correlation 1 was found to correlate only with information measure of correlation 2 and was statistically insignificant. It is observed, that most of texture features are highly correlated with each other in the normal group. A deviation from this behaviour has been observed in abnormal thermograms. Thus it is inferred that curvelet based texture features are able to represent variations in a thermogram better than the statistical features. This inference has been substantiated, with the classifier’s performance. The classification accuracy is found to be higher when the SVM was trained with texture features. Table 6 Comparison with other works

The correlation between features of the normal and abnormal groups was analyzed and is listed in Table 4. Almost all features are found to be uncorrelated, which is indicative of their discriminative nature. The relationships between features were analyzed with the help of regression graphs. For example, Fig. 4a, shows regression plot for the features, energy and entropy, in the normal group. It is observed that these features are strongly correlated with ρ=− 0.97863 in the normal study. It can be seen from Fig. 4b, that the energy of abnormal group is weakly correlated with entropy of the normal group (ρ=0.20576). Figure 4c, shows that the entropy of abnormal group is weakly correlated with energy of the normal group (ρ=0.24407). The strong negative correlation between these two features (in the normal study), is found to change into a weak positive correlation when observed across the groups. Such a change from normal behaviour is observed in most of the features. This is evident from the change in magnitude and direction of slope of the respective regression lines. Hence all features were used for training the classifier. The SVM classifier exhibited an accuracy of 86.36 % for statistical features and 90.91 % for Haralick features respectively, as given in Table 5. Though the sensitivity obtained was same for both feature sets, specificity and accuracy were found to be better when texture features were employed. Thus, it has been demonstrated that curvelet based texture features can effectively represent breast surface temperature variations in a malignant breast. Exploring the multiresolution capability of curvelet transforms for feature extraction promises a novel image processing approach for the detection of abnormality from breast thermograms. The results of the study are comparable with recent studies listed in Table 6.

Paper

Features

Classifier

Performance measures

Schaefer et. al. [22]

Cross co-occurrence texture features DWT features Co-occurrence texture features

Fuzzy

Accuracy 80 %

Neural Network SVM

Accuracy 86.6 % Accuracy 88.1 % sensitivity 85.71 % specificity 90.48 % Accuracy 85.19 %

Wiecek B et al. [19] Acharya U.R et al.[16]

Sheeja V Francis et al. [14]

U. RajendraAcharya et al. [20]

Co-occurrence texture features

Back Propagation Neural Network

HOS features

ANN SVM

Results of present study

Curvelet based Co-occurrence texture features

SVM

Sensitivity 88.89 % specificity 77.78 % Sensitivity 92 % specificity 88 % Sensitivity 76 % specificity 84 % Accuracy 90.91 % Sensitivity 81.82 % specificity 100 %

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Information measure of correlation 1 : f 12 ¼

Conclusion In the present work, curvelet based feature extraction method has been proposed for classification of breast thermograms. It is found that use of curvelet based texture features improves the classification efficiency when compared to statistical feature set. The classification accuracy achieved by the proposed system is comparable with previous feature extraction studies conducted in wavelet and spatial domains. The accuracy of the system may be further improved by increasing the number of images for training the classifier and by implementing an effective feature selection method. To conclude, curvelet based texture features may be used to improve the efficiency of automatic detection of abnormality in breast thermograms.

Declaration of Interest The authors report no declarations of interest.

Appendix I Haralick’s Texture Features {p(i,j)} is the Normalized GLCM. N is the number of gray levels in {p(i,j)}. Angular second moment ðEnergyÞ : f 1 ¼

Contrast : f 2 ¼

XN XN i¼1

j¼1

fpði; jÞg2

o XN−1 nXN XN 2 n p ð i; j Þ f g i¼1 i¼1 j¼1 XX i

Correlation : f 3 ¼

j

ðijÞpði; jÞ−μx μy σx σy

Sum of squares−variance : f 4 ¼

XX i

Inverse Difference Moment : f 5 ¼

X2N

Sum Entropy : f 8 ¼ − Entropy : f 9 ¼ −

i¼2

j

1 1 þ ði− jÞ2

pði; jÞ

ði− f s Þ2 pðxþyÞ ðiÞ

n o p ð i Þlog p ð i Þ ð xþy Þ ð xþy Þ i¼2

X2N

XX i

j

XX i

Sum Variance : f 7 ¼

ði−μÞ2 pði; jÞ

j

pði; jÞlogfpði; jÞg

Difference variance : f 10 ¼ variance of pðx−yÞ Difference Entropy : f 11 ¼ −

N −1 X i¼0

n o pðx−yÞ ðiÞlog pðx−yÞ ðiÞ

HXY −HXY 1 maxðHX ; HY Þ

Information measure of correlation 2 : f 13 ¼ ð1−exp½−0:2ðHXY 2−HXY ފÞ1=2 Where HXY ¼ −

XX i

j

pði; jÞlogfpði; jÞg

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