# Review of 3D Imaging by Coded Aperture Correlation Holography (COACH) +
## 1 Introduction  
Holography is a tool for recording a visual scene and reproducing it as close as possible to reality. In holography, even though only intensity sensors are used, it is possible to record the phase pattern of light, and thus the depth information, along with the intensity variation. The phase is detected indirectly by recording an interference pattern of the object wave with a reference wave. Typically, the interference pattern is obtained by combining coherent waves [1]. However, most of the imaging tasks in optics are performed with natural incoherent light. This is true for most microscopes, telescopes, and many other imaging devices. Thus, holography is not widely applied to general incoherent natural light imaging because usually, creating holograms with incoherent light requires uncommon designs. This review concentrates on the more challenging case of interference with spatially incoherent light. More precisely, the holograms herein are of objects in which there is no statistical correlation between the waves emitted from various points of these objects. The historical roots of incoherent holograms are planted in the mid-nineteen-sixties [2; 3; 4; 5; 6; 7], where some of these pioneering systems made use of the self-interference principle, a principle that is extensively used in some of the systems mentioned herein. The self-interference principle indicates that any two beams that originate from the same source point and then split to two waves are mutually coherent and hence they can be mutually interfered. In the case of incoherent illumination, where any two different source points are mutually incoherent, the self-interference property becomes the only way to obtain any interference pattern, and thus enables us to record a hologram. The use of the self-interference principle has been continuously developed beyond the sixties by implementing several interesting systems [8; 9; 10; 11; 12]. Other methods of recording incoherent holograms like optical scanning holography [13; 14] and multiple view projection methods [15; 16] do not make use of the self-interference property and are out of the scope of thisreview. Other reviews which include some of the self-interference methods, together with some of the scanning techniques, can be found in Refs. [17; 18]. While the term incoherent holography refers to the method of forming holograms, the term digital holography refers to the recording of the holograms by a digital camera and to the digital reconstruction of holograms. Explicitly, digital holograms recorded by digital cameras are reconstructed in digital computers by digital algorithms. The history of the digital holography also started in the nineteen-sixties [19] and is still an active research field today. Only the topic of incoherent digital holography with coded apertures is discussed herein. Nowadays, we are in the middle of the era of digital imaging, in which images are recorded by digital cameras and processed by computer software. Digital imaging has accelerated the field of indirect imaging in which a non-image pattern of the observed scene is first recorded in the computer as an intermediate pattern. In the computer, the image of the scene is recovered from the intermediate pattern by digital processing. Digital holography is a typical example of indirect imaging in which the digital camera records one or more holograms, rather than a direct image of the scene. The indirect imaging and digital holography techniques are more complicated than direct imaging, and hence should be justified by as many as possible advantages in comparison to the much simpler direct imaging. The main benefit of digital holography and the initial motivation to begin the research in this field is the ability to image a three-dimensional scene with a single, or very few, camera shots [20]. Other advantages of digital holography have accumulated along the years of research and are discussed in the later sections of this review.