# PSDFH: A Phase-Space-Based Depth from Hologram Extraction Method
## 1 Introduction  
Holography [1], introduced by [PERSON] in 1948, was originally a two-stage coherent imaging technology for recording and reconstructing the light wave scattered by an illuminated object through interference. While initially restricted to monochromatic still models captured in optics laboratories, its modern evolution into digital holography has recently allowed the processing of completely general content, such as color-animated hologram sequences.  
A hologram contains the entire light field of the recorded object, including intensity and phase information. It therefore requires larger storage space, but this additional information can be used for a wider range of applications. In most computer vision applications, only the intensity image and depth map of the scene are used to completely describe the 3D scene. Both of them are contained in the hologram but require a preprocessing step in order to extract them. The simplest and most effective approach for this purpose is to use the _depth-from-focus_ (_DFF_) method [2]. This technique starts from a reconstruction volume, built using several numerical reconstructions performed at different sampled depths. It then estimates a depth map of the recorded scene by detecting the reconstruction depth at which the focus is locally optimal using a _focus measure_ (_FM_) for each pixel. Once the depth map has been estimated, the intensity map is constructed by assigning to each pixel the intensity present at the same position in the numerical reconstruction performed at the estimated depth.  
Although the DFF approach gives convincing results, it has many limitations. First, it requires carefully choosing several hyperparameters, including the reconstruction interval, the number of reconstructions to be performed, the focus operator, and how it applies to thereconstruction volume. Each of these factors can greatly affect the estimation performance. Second, this approach is computationally expensive, since it requires computing a large number of numerical reconstructions and in some particular cases pre-processing the hologram to remove the zero-order, twin image, and speckle noise from the reconstructed images, resulting in additional computational overhead.  
In this work, we propose a PS-DEF (phase-space-based DFF), a _region of interest_ (ROI) extraction procedure based on the phase space representation of holograms to pre-localize the scene in 3D space. The extracted regions can be used to select an appropriate numerical reconstruction interval, as well as the spatial location where the focus measure should be applied. In addition, an iterative method is proposed to refine the extracted regions of interest and produce a finer depth estimate. A series of experiments are conducted to evaluate the quality of the extracted regions of interest, the gain in computation time with respect to the classical DFF method, and the accuracy of the sparse depth estimate produced by the iterative refinement method.  
The main contributions of this paper can be summarized as follows:  
* A novel methodology to estimate a complex and deep depth map with multiple values based on the phase space representation of the hologram.
* An iterative voting process to extract regions containing scene objects (called ROIs in the paper) from the hologram phase space transform support.
* An ROI-constrained DFF method to improve speed and accuracy with respect to classical DFF.  
The remainder of this article is organized as follows: Section 2 reviews previous works that use phase space to process holographic data. Then, Section 3 presents the proposed method. In Section 4, a series of experiments are conducted to validate the proposed approach. Finally, in Section 5, we discuss the advantages and limitations of the proposed approach.