## 1 Introduction  
Three-dimensional shape measurement systems that use contactless methods are necessary in many industrial fields [1]. A grating projection method is often used as a high accuracy, high speed, and real-time 3D shape measurement method [2; 3; 4]. It can also be effectively applied to a wide and smooth curved surface object, such as a deformed sheet, a press formed object, and a space deployable structure. The evaluated distribution such as height distribution, deformation distribution, and an image of contour lines can be obtained from 3D shape measurement results. They are generally displayed on a monitor as a 2D distribution or a 3D view. However, it is difficult for an operator to recognize the correspondence between a point on the monitor and the corresponding point on the actual measured object. Therefore, there is a demand for technology that supports the visual inspection of flat objects such as sheet metal. Projection mapping is an effective method used to show the evaluated distribution. An operator can easily recognize the point to be noticed if the evaluated distribution is projected onto the measured object. The object is wider, and the effect becomes greater. A real-time projection mapping system with a shape measurement was developed in 1999 [5]. Here, height distribution was measured using a phase-shifted grating projection method with an infrared light. The height distribution was analyzed with a real-time phase analysis board that employed a field-programmable gate array [6]. The measured height distribution was projected onto the actual measured object by a digital light processing (DLP) projector with visible lights. In this system, each camera pixel and the corresponding mapping projector pixel must be adjusted to the same optical axis using a half mirror. This is one of the difficulties in producing the device. Figure 1 shows an overview of our projection mapping system. In the first phase, a grating pattern is projected onto an object with a measurement projector to measure the 3D shape, as shown in Figure 1a. In the second phase, the measured results and the evaluation values are projected onto an object from a projector different with a measurement projector, as shown in Figure 1b. In this configuration, the evaluation image should be deformed according to the 3D shape of the object because the camera and projector positions are different. Many studies of projection mapping combined with the 3D shape measurement have been performed in the fields of art, augmented reality, and 3D displays [7; 8; 9; 10; 11; 12]. In conventional studies, the deformed projected image is produced with a coordinate transform calculation. In contrast, in this study, a method used to produce a deformed evaluation image using a coordinate transform table to convert a camera pixel coordinate into a projector pixel coordinate is proposed. This is a key idea of this study. The computing time necessary to produce the deformed evaluation image can be reduced with the tabulation. The table can produced at the same time when a phase and 3D coordinate table can be produced based on a whole-space tabulation method (WSTM) [13; 14]. The WSTM makes it possible to produce a real-time 3D shape measurement system. The authors also developed a light-source-stepping method (LSSM) using linear light-emitting diode (LED) arrays [13; 14; 15; 16]. A real-time and wide-range 3D shape measurement unit for a large object using the WSTM in combination with the LSSM was developed [16]. This method has an advantage of stability because of its lack of mechanical moving parts for phase-shifting as compared to several individualistic fringe projection methods that have been proposed by other researchers [17; 18; 19]. An image deformed according to a 3D shape projected by a mapping projector can be quickly generated using this table. We developed a quick 3D shape measurement projection mapping system (hereinafter referred to as the 3 DM-PM system) using the coordinate transform table.