## 1 Introduction  
Three-dimensional (3D) functional imaging of targets in deep live tissues is a fundamental challenge in biophotonics. For the functional imaging of deep tissues, X-ray projection and computed tomography (CT) [1], ultrasonography [2; 3], positron-emission tomography (PET), and magnetic resonance imaging (MRI) [4] are useful techniques in biomedical fields. However, each technique has some limitations, such as ionizing radiation exposure of X-rays, low spatial resolution in PET and ultrasound, and the need for complex equipment for MRI. Near-infrared (NIR) imaging is an additional method for observing deep biological tissues. For example, indocyanine green (ICG), the working wavelength which is called NIR-I (700\(-\)950 nm), is used to visualize blood flow and liver function for surgical guidance [5]. NIR wavelengths longer than 1000 nm, over-thousand-nanometer (OTN) NIR, also called NIR-II and NIR-III [6], have the advantages of minimizing scattering by biological tissues [7] and, thus, expanding the observation depth to 1-2 cm [8], as compared to several millimeters in the NIR wavelength of ICG. Thus, optical imaging in the NIR wavelength range, especially OTN-NIR, enables observation of biological deep tissues transparently [9]. Recently, the distinction of submucosal tumors through the mucosal layer using spectral features in NIR images has also been reported [10]. For detection, cameras for visible and NIR-I light should be replaced with cameras with indium gallium arsenide (InGaAs) sensitive to OTN-NIR [6]. Nevertheless, in this wavelength range, fluorescence imaging of deep tissues can be performed using a simple device composed of a laser light source for excitation, optical mirrors and filters, and cameras. We aimed to perform 3D mapping of NIR phosphors with multiplexity based on the fluorescence lifetime, which corresponds to the decay rate of emission. Tomography of the decay rate of signals from inside the tissues is established in the imaging of the nuclear spin relaxation under a controlled magnetic field for MRI. Instead of nuclear spin relaxation, the decay rate of optical emission is used for tomographic imaging. The decay rate of the emitted light from phosphors provides the fluorescence lifetime, which is a time constant independent of the absolute fluorescence intensity or power density of the laser for excitation. Actually, fluorescence lifetime has attracted attention for realizing multiplex imaging in the OTN-NIR wavelength range [11]. In microscopic imaging, 3D mapping of fluorescence lifetimes using fluorescence lifetime imaging microscopy (FLIM) with a single-plane illumination (recently named light-sheet) setup [12; 13], or with a confocal system [14], has been reported. A methodology for fluorescence lifetime tomography has also been proposed [15] and experimentally validated by microscopic tomography using FLIM 3D fluorescence lifetime imaging using high-speed cameras [16; 17; 18]. The NIR fluorescence (NIRF) lifetime images can be captured by a time-gated imaging (TGI) system. It is composed of a synchronized image acquisition system with a conventional NIR camera, and a pulsed laser for excitation to collect the decay images of fluorescence immediately (at several tens of microseconds) after cutting off the laser [19]. Contrasting the fluorescence lifetime also reduces noise, such as autofluorescence in fluorescence imaging in the ultraviolet, visible [20], and NIR wavelength ranges [21; 22]. Multiplexity also realizes contactless temperature imaging of deep tissues using NaYF\({}_{4}\) co-doped with Nd\({}^{3+}\) and Yb\({}^{3+}\)[23]. Because NIR fluorescence makes deep tissues observable, its cross-sectional images can be obtained from two-dimensional projected images acquired from various angles by reconstruction with CT, based on the back-projection algorithm [24; 25; 26].