## 1 Introduction  
Thyroid nodules are a common clinical problem, particularly in iodine-deficient areas [1; 2]. Though many people have nodules, they are asymptomatic, only 3-7% nodules harbor thyroid cancer [3; 4]. However, the prevalence discovered by ultrasound (US) diagnosis increases significantly to 10-67% [4]. The majority of nodules should be benign [5], while clinical purpose of diagnosis is to detect malignancy, which only happens to 3-7% patients with thyroid nodules. Normally, suspected nodules must be assessed with fine needle aspiration (FNA) by collecting cells for cytological examination [6; 7]. Although currently FNA is the reference standard to identify thyroid nodules, previous studies showed FNA cytology should be indeterminate in 10-30% of thyroid nodules [8; 9]. To improve diagnostic accuracy, molecular biomarkers and gene expression, combined with FNA, have been explored in recent years [10; 11; 12; 13; 14]. Though FNA can differentiate benign from malignant nodules, it can cause physical and psychological discomfort due to its invasive procedures. Therefore, there is an urgent need to improve ultrasound diagnostic accuracy, so as to reduce unnecessary FNA biopsies.  
Although US is commonly used to identify malignant from benign nodules according to some ultrasonic characteristics, no single ultrasound feature is reported to be adequately accurate to identify malignant nodules. It has been indicated that an increased risk of thyroid cancer could be marked by numerous ultrasonographic findings, i.e., calcifications, intranodular vascularity, absent halo sign,irregular contour, and hypo-echogenicity [15; 16]. However, the mentioned features are not enough to identify malignant nodules, since the echogenicities of thyroid nodules are complex [15]. In recent years, ultrasound elastography has attracted much attention in detecting malignant thyroid nodules, which is developed based on the hypothesis that a malignant lesion is stiffer than a benign one [17; 18; 19; 20]. Although this method shows good capability for identifying malignancy in the thyroid gland, it has been indicated that this technique is unsuitable for detecting small nodule and malignant nodule in low-risk population [21; 22; 23; 24].  
Since backscattered US signals are affected by the shapes and spatial distribution of scatters in tissues, it is suggested that ultrasound images follow a random dynamic pattern related to random feature of underlying structures. Regarding the pixel intensity as the height above a plane, B-mode images are regarded as rough surfaces. Fractal analysis is a method to assess textural patterns according to the visualization of ultrasound images, which can reveal the randomness of tissues or tumors. Referring to the fractal geometry, the surface intensities of ultrasound images are regarded as fractal objects, and their characteristics can be quantitatively evaluated by using fractal dimension (FD). Fractal dimension is a non-integer number between 2-3. Previous studies have proven that FD assessment shows promising results in characterizations of the breast, liver, prostate lesions, bones, and carotid plaque [25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38]. In particular, [PERSON] et al. have developed a computer-aided diagnostic tool to characterize benign and malignant thyroid nodules, using some texture-based feature extraction approaches (e.g., local binary pattern, FD, Laws texture energy, and Fourier spectrum descriptor) [39]. They reported that an accuracy higher than 98% for 20 patients could be achieved by combing texture features with different classifiers (e.g., support vector machine, Sugeno Fuzzy, and Gaussian mixture model). However, the specificity and sensitivity of FD parameter for thyroid nodules had not been examined. Therefore, the present study attempts to identify malignant and benign thyroid nodules with only fractal-geometry-based textural technique.