## 3 Proposed Method  
The objective of our proposed method is to obtain an extended field of depth (EFD) image (i.e., an image with all the object points in focused, disregard of their geometry and distance from hologram \(H(x,y)\)). The process can be divided into 4 stages as shown in Figure 2, and outlined as follows. In stage 1, the hologram \(H(x,y)\) is back-propagated to a stack of regular spaced reconstruction planes along the axial direction, a process that is referred to as \"sectional reconstruction\". The image on each reconstruction plane is evenly partitioned into non-overlapping square blocks. In stage 2, the entropies of the image blocks in all the reconstruction planes are evaluated. The depth of each block is taken to be the distance of the reconstruction plane that exhibits the minimum entropy for that particular block position. It can be envisaged that the integration of the depth of all the constituting blocks (each contributed by one of the reconstructed images) will form a blocky depth map. Next, in stage 3, low-pass filtering is applied to smoothen the depth map. Finally, in stage 4, an extended depth of field image of the object scene is obtained by selecting each pixel from the reconstruction image plane that corresponding to the depth of the pixel in the depth map. Details of each stage are described as follows. _Stage 1: Sectional reconstruction_  
In this stage, a stack \(S\) of evenly spaced images is reconstructed from hologram \(H(x,y)\) through back-propagation. Let \(S_{j}(x,y)\left|{}_{0\leq j<N}\right.\) and \(Z\) denotes the magnitude of the reconstructed image on the \(j\)-th reconstruction plane section, and the maximum range of the depth of the object scene, respectively, we have  
\[S_{j}(x,y)=\left|{}H(x,y)\otimes{}F^{*}\big{(}x,y;z_{j}\big{)}\right| \tag{3}\]  
where \(F^{*}\big{(}x,y;z_{j}\big{)}\) is the conjugate of \(F\big{(}x,y;z_{j}\big{)}\). The separation between adjacent reconstruction planes is given by  
\[\Delta z=z_{j+1}-z_{j}=\frac{Z}{N} \tag{4}\]  
Subsequently, each reconstructed image \(S_{j}(x,y)\) is partitioned into a 2-D array of non-overlapping square image blocks of size \(M\times M\), as shown in Figure 3. Without loss of generality, we assume that both the hologram and its reconstructed images are square in size, and having the same size \(K\times K\), where \(K\) is an integer multiple of \(M\). As such, there are \(K/M\) blocks along the horizontal direction, and \(K/M\) blocks along the vertical direction. Each image block in \(S_{j}(x,y)\) is represented by the block function \(b_{j}(p,q)\left|{}_{0\leq pq<(K/M)}\right.\), which includes the pixels bounded by the square within the  
Figure 2: Proposed method. region \(pM\leq x<(p+1)M\) and \(qM\leq y<(q+1)M\). The EDF image \(I^{\prime}(x,y)\) to be determined is also partitioned in a similar way with each block of pixels denoted by \(b_{EFD}(p,q)\). _Stage 2: Non-overlapping block partitioned entropy minimization_  
In this stage, the entropy of all the image blocks that have been partitioned in stage 1 is computed as  
\[E_{j}(p,q)=-P_{j}(p,q)\log P_{j}(p,q) \tag{5}\]  
where  
\[P_{j}(p,q)=\sum_{x=pM}^{(p+1)M-1}\sum_{y=qM}^{(q+1)M-1}S_{j}(x,y) \tag{6}\]  
An image that is in focus generally exhibits minimum entropy. As such, the depth (or focused plane) of each block \(D(p,q)\) can be taken to be the position of the plane that exhibits minimum entropy of the block, i.e.,  
\[E_{D(p,q)}=MIN\big{|}E_{j}(p,q)\big{|}_{0\leq j<N} \tag{7}\]  
where the expression on the right-hand-side of Equation (7) denotes the minimum value of \(E_{j}(p,q)\) for \(0\leq j<N\). Collection of depth of each block results in a crude, blocky depth map of the object scene. Intuitively, we can obtain an EDF image \(I(x,y)\) by selecting each block of pixels from the corresponding block with minimum entropy from the stack of reconstructed images.