## 1 Introduction  
Lung cancer is the most common cause of death among all cancers worldwide [1]. To cope with this serious situation, mass screening for lung cancer has been widely performed by simple X-ray films with sputum cytological tests. However, it is known that the accuracy of this method is not sufficient for the early detection of lung cancer [2, 3]. Therefore, a lung cancer screening system by computedtomography (CT) for mass screening is proposed [4]. This system improves the accuracy of the cancer detection considerably [5], but has one problem that the number of the images is increased to over dozens of slice sections per patient from 1 X-ray film. It is difficult for a radiologist (_i.e._, a medical doctor who specializes in reading radiographs) to interpret all the images in a limited time. In order to make the system more practical, it is necessary to build a computer-aided diagnosis (CAD) system that automatically detects abnormal regions suspected to comprise pulmonary nodules that are the major radiographic indicators of lung cancers, and informs a radiologist of their positions in CT scans as a second opinion. Extensive research has been dedicated to automated detection of pulmonary nodules in thoracic CT scans [6]. Morphological [7] image filters [8; 9; 10] are conventional approaches. In the work [11], a multiple-thresholding technique was used to detect nodules that had peaked intensity distribution. Hessian-based image filters [12; 13] individually enhanced blob- and ridge-shaped regions that corresponded to nodules and blood vessels, respectively. These methods were often used for initial detection of nodules, and were intentionally adjusted to minimize the number of misdetection. Consequently, they yielded many false candidates called false positives (FP) that corresponded to normal pulmonary structures such as blood vessels. In order to reduce false positives, feature-based discrimination methods between nodules and false positives have been also developed [14; 15; 16; 17; 18; 19]. [PERSON] _et al._ reported a classification method [20] of nodules based on differences in shape indexes, which were computed from two principal curvatures of intensity profiles in images, between nodules and false positives. [PERSON] _et al._ proposed a method [21] that suppressed false positives by using voxel values in regions of interest as input for a novel artificial neural network [22]. Model-based methods are the promising approaches as well. Several works with nodule models were reported. [PERSON] _et al._[23] proposed a template-matching method using nodular models with two-dimensional (2-D) Gaussian distribution as reference images to CT scans. The method had an advantage of being able to use the geometrical features of nodules as well as gray level features. [PERSON] _et al._[24] designed a 3-D prismatic nodule template that was composed of several layered matrices. [PERSON] _et al._[25] developed 3-D deformable nodule models such as spherical models of various radii. These methods can make use of the characteristics of the 3-D relation between a suspicious region in a slice section and the other regions in the adjacent slice sections. Not only nodules but also normal pulmonary structures were modeled in a study reported by [PERSON] _et al._[26]. A ribbon model, for example, was used for representing a blood vessel region in a slice section. Based on the [PERSON] theorem, the probabilities of the models were calculated from image features such as step-edges extracted by the Canny edge detector. The most likely model was determined by the maximum a posteriori (MAP) estimation. In this study, by using the models, the anatomical knowledge on the normal structures of human organs can be introduced into recognition of diseases, but only 2-D models and image features were used. In the present paper, we propose a novel recognition method of pulmonary nodules in thoracic CT scans by use of 3-D deformable spherical and cylindrical models that represent nodules and blood vessels, respectively.