## 1 Background
Albeit significant enhancement on the computation speed, the process is relying heavily on the edge analysis method and the manual setting of the parameter(s). The method has successfully demonstrated its capability in reconstructing binary objects from a hologram, but it is unlikely that it can be applied in a more complicated scene. Recently, a more reliable AFD method, known as entropy minimization, was adopted in [17] for detecting the focused image planes. In this approach, a reconstruction plane is taken as a focused section if it exhibits a local minimum on its entropy value amongst the sequence of reconstructed images. This method is autonomous and does not require manual setting of parameters. Afterwards, a hologram reconstruction method was reported, based on the entropy minimization AFD [18]. For each pixel in each reconstruction plane, a small block with the pixel at the center is defined. The entropy of the pixel is then computed based on the pixels within its block. Next, the depth of each pixel is taken to be the depth of the reconstruction plane that has a minimum entropy value for the pixel of interest. As such, a depth map of the scene is deduced, with which the scene image can be reconstructed. Despite the effectiveness of the method, the amount of computation is overwhelming as the entropy of all the pixels have to be determined for all the reconstruction planes. A faster method has been proposed in [19], whereby the entropy is evaluated for each constituting objects in the scene instead of individual pixels. Briefly, a hologram is back-propagated to a virtual diffraction plane (VDP) that is close to the object space. Segmentation is applied on the VDP to separate the hologram into sub-holograms [20] each representing the fringe patterns of an isolated object in the scene. AFD is then applied to each sub-hologram to locate the focused image plane, and reconstruct the image of the object. However, this method is only applicable if the objects on the \(x\)-\(y\) plane in the scene are spaced far enough to be separated on the VDP, a requirement which is not guaranteed in practice. Another fast method for hologram reconstruction has been proposed in [21]. The method is based on the assumption that the imaginary component (or phase angle) of a pixel is close to zero on its focused image plane. On this basis, a focused image of the object scene can be built by selecting, from each reconstruction plane, pixels with small imaginary values. In this paper we present, partly based on the work in [18], a method for fast hologram reconstruction with extended depth of field (EDF). By EDF, it means that a focused image of all the objects (which may differ in depth and geometrical shape) in the scene can be recovered simultaneously from the hologram. Our proposed method is particular suitable for optical scanned holograms that are unrestricted in size, complexity of object scene, and capturing modes (i.e., coherent and incoherent modes). Briefly the method can be divided into 4 stages. First, the hologram is back-propagated into a sequence of parallel, and uniformly spaced reconstruction planes. Second, each reconstruction plane is evenly partitioned into non-overlapping square image blocks. Third, entropy minimization AFD is employed to locate the focused section of each image block, from which the depth is determined, and further refined with low-pass filtering. Fourth, the collection of depth information from all the image blocks is utilized to reconstruct a focused image of the object scene. In our proposed method, we have assumed that in a typical scene, the depth of pixels within close neighborhood should be similar, an assumption that is generally applicable in practice. As such, the depth map of the scene image can be evaluated on a block-by-block, instead of a pixel-by-pixel basis, resulting in significant increase in the computation efficiency as compared with the method in [18].