**1. Introduction**

The reactor pressure vessel (RPV) of a nuclear power plant (NPP) requires periodic inspection to ascertain current conditions. Any defects on the internal surfaces may undermine the safe operation of the NPP. Moreover, exposure to irradiation and corrosive coolants, or damage caused by manufacturing and outage activities, could accelerate the growth of these defects [1]. Thus, inspection systems and implementation practices must be capable of detecting small flaws, to prevent them from growing to a size that could compromise the leak tightness of the pressure boundary.

Visual inspection is the main method for detecting defects, structural integrity issues, or leakage traces on the surface of key components in an NPP. Owing to its advantages, the demand for more advanced visual inspection techniques is increasing. The U.S. Nuclear Regulatory Commission (NRC) has approved the use of high-resolution cameras for inspecting specific areas of key NPP components instead of ultrasonic examination [2]. In addition, machine vision technology has been applied to measure fuel assembly deformation [3].

Visual inspection systems capture images of the surfaces of objects by using an image sensor, a charge-coupled device (CCD), or a complementary metal-oxide semiconductor (CMOS), with appropriate optical tools and lighting conditions. The visual module is typically composed of light sources and image sensing units, and completes the inspection with the aid of automated tools. Companies such as AREVA and CYBERIA in France, DEKRA in Germany, Ahlberg in Sweden,

DIAKNOT in Russia, and Westinghouse in the United States have been actively developing such devices. The majority of inspection systems are equipped with high-definition cameras, include a choice of light sources, from halogen to light-emitting diode (LED) lights, and boast anti-irradiation and waterproof properties. The types of automated tools are diverse. Some products have data processing functions, which are becoming increasingly popular. For example, AREVA's latest RPV device, SUSI 420 HD, is equipped with a high-definition camera and four adjustable high-power LEDs, but the sizing of indications is limited to the length measurement [4].

Existing NPP visual inspection methods still use two-dimensional (2D) images to identify defects. Occasionally, the lack of three-dimensional (3D) observations makes it difficult to evaluate certain observations—specifically, the potential size of the defect. Some inspection tasks can only be performed using 3D analysis methods, because surface defects may only appear with changes in the shape of the surface [5]. Therefore, visual inspections can be improved by detecting changes in the 3D surface. Three-dimensional shape reconstruction methods based on visible light include structured light and stereo vision technology. For example, the laser 3D scanner of the Newton Laboratory can map NPP fuels and check the size of defects [6]. Karl Storz laser technology, called MULTIPOINT, is a 3D laser system with 49 laser points that enables cooperation between the camera and the software to detect the surface structure of the subject [7]. However, few of these devices can operate individually without additional overheads. Therefore, to save time and money, it is preferable to use conventional devices to extract and analyze 3D data for defects. One such method achieves 3D reconstruction of the inner surfaces of boreholes or cavities using conventional endoscopy equipment [5]. However, this method does not provide system calibration results or evaluation criteria of the results; thus, further improvement is required. Furthermore, the 3D visualization function obtained through the shape from motion (SFM) method can be used to inspect the advanced, air-cooled core in an NPP, but a lack of surface features limits the application of this method [8].

The photometric stereo (PS) technique recovers 3D shapes from multiple images of the same object, taken under different illumination conditions. As a result of the pioneering work of Woodham, it has been widely applied to 3D surface reconstruction [9]. This technique features two advantages: low hardware costs and low computation costs. In the field of industrial inspection, PS has improved the detection of very small surface defects [10,11]. The Lambertian model assumption is commonly used where albedo is assumed to be constant. Although this does not necessarily correspond to the actual conditions, there are approaches available to realize the normal calculations [12–14]. However, for models with non-Lambertian reflection properties, highlight and shadow processing requires additional images [13].

The assumption of the light source and the demand for extensive calibration procedures in conventional PS limit its applicability [15]. Some previous studies have established illumination models that conform to actual conditions, such as near-field light models [16–19]. Light calibration, which aims to estimate the light direction and intensity, often requires a specific equipment or a dedicated process [20]. However, equipment such as precise calibration spheres or positioning devices are unlikely to be available in actual applications. Some studies have proposed fully uncalibrated or semi-calibrated PS methods. A fully uncalibrated, near-light PS method achieves the calculation of the light positions, light intensities, normal, depth, and albedo without making any assumptions about the geometry, lights, or reflectance of the scene [21]. Regarding semi-calibrated PS, various approaches achieve light intensity calibration [22]. However, additional information is required to solve the high-dimensional ambiguity, and more importantly, at least 10 images are typically required [15]. To apply PS to an NPP environment, a fully automatic calibration method should be designed.

This study employs the PS method with a conventional NPP visual inspection device to reconstruct the 3D shapes of defects from visual inspection images. Additional contributions include the development of an auto-calibrated, near-field light calibration method that can easily and accurately calibrate the light source to meet the demands of practical applications. Moreover, depth information

can be extracted from the captured images, which enables better and more reliable visualization of surface defects.

The structure of this paper is as follows: The PS formulation is briefly introduced in Section 2. The algorithm details for extracting 3D information from captured images are described in Section 3, as well as the method for estimating light direction and intensity. The experimental setup and results are discussed in Section 4, and the conclusions are presented in Section 5.

#### **2. Photometric Stereo Technique**

PS techniques use multiple images taken from the same viewpoint, but under illumination from different directions, in order to recover the surface orientation from a known combination of reflectance and lighting values. The depth and shape of the surface can be obtained via the reconstruction algorithms. The objects in the scene are Lambertian, and the illumination is a distant point light; the measured image intensity at a point *P*(*x*, *y*, *z*) can be written as:

$$I = \rho \langle l, n \rangle E \tag{1}$$

where ρ is the albedo at *P*; *l* is the light direction; *n* is the surface normal; and *E* is the light irradiance. The image intensity *I* can be measured per-pixel.

At least three independent light sources are required.

Suppose we have *M* ≥ 3 images under varying light directions, which we denote as direction vectors *l1,* ... *.., lM* <sup>∈</sup> <sup>R</sup>3. By assuming equal light irradiance; i.e., <sup>ρ</sup>*<sup>E</sup>* <sup>=</sup> <sup>ρ</sup>*E*<sup>1</sup> <sup>=</sup> ······ <sup>=</sup> <sup>ρ</sup>*EM*, we can estimate the normal vector *n* on a surface point *P* by solving the pseudo-inverse matrix of *L***:**

$$m = \frac{\left(L^T L\right)^{-1} L^T I}{\left\| \left(L^T L\right)^{-1} L^T I \right\|}\tag{2}$$

where *L* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *l*1 . . . *lM* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , *I* = [*I*1, ··· , *IM*] T.

The surface normal can be computed using Equations (1) and (2). Then, the recovery of the surface from the computed normal can be achieved using algorithms, including optimization iterative methods and pyramid reconstruction algorithms. The entire procedure of PS-based 3D shape reconstruction includes: calibration, surface normal computation, and shape reconstruction from the normal.

#### **3. Three-Dimensional Shape Reconstruction of Defects**

#### *3.1. Existing Two-Dimensional Image Capture and Data Analysis Method*

The RPV generally consists of a cylindrical part, a spherical part, and nozzles. Although its size varies, the diameter of the cylinder is always approximately 4000 mm. An examination of the entire internal surface of the reactors is required to detect surface disorders, deformation, or other important defects; i.e., (1) mechanical defects, such as scratches or impact damages caused by foreign bodies; (2) metallurgical defects, such as cracks or arc strikes; and (3) corrosion pitting or deposits. Moreover, the defects may need to be evaluated both qualitatively and quantitatively.

During the inspection process, tools are used to position the cameras close to the different areas requiring inspection. Scanning is performed using tools to record videos of the internal surface, using cameras facing the wall within a field limited to the zone under examination. Technicians observe the video and images throughout the entire process to identify defects. If an abnormality or suspicious observation is recorded, the movement of the tool can be paused to allow more detailed information to be observed manually.

Because of its beneficial features, a pan/tilt/zoom (PTZ) setup, which normally consists of a camera, a number of surrounding light sources, lasers (optional), and a built-in pan/tilt unit, has been widely used as the visual module. The pan and tilt functions allow adjustment of the camera position to capture better images. Data analysis is performed by inspectors; therefore, suitable frames are required for observation and evaluation, along with other information recorded during the inspection, to make a qualified evaluation [23]. As the existing analysis method is based on 2D images, it is often difficult to evaluate defects due to a lack of depth information; thus, some inspection tasks cannot be solved using 2D analysis methods.
