## 5 Object Depth Modelling  
In the discrimination of objects in a scene in relation to their distance from the camera, the contour of objects has a significant role. The projection of such contours on an image sensor results in edges in the captured digital image. The orthogonal projection of a contour, represented as an edge in an image, is generally modelled as:  
\[\mathrm{f_{1}(x)}=\mathrm{Au(x)}+\mathrm{B}, \tag{4}\]  
where \(\mathrm{u(x)}\) is a step function, and A and B are the amplitude and offset of the edge, respectively. The simplified model has no information about the object distance from the camera. If we consider light to be represented by a complex plane wave which is traveling from any object towards the lens and then the image sensor, then each point on each of these object planes acts like a spherical wave source interfering constructively or destructively with every other spherical wave in the planes beyond the object plane. According to the Huygens-Fresnel principle, the phases change in any given plane, i.e., an  
Figure 5: Relations between depth of field and distance to the in-focus plane for certain focus distance values of 850, 1000, and 1150 mm. Figure 6: Relations between depth of field and distance to the in-focus plane for different focal length values with a distance to the in-focus value of 1000.  
interference pattern is generated in any given plane through a lens, the Fourier transform of the eventual captured image on the image sensor is generated in which the phase contains the information of objects from their respective object planes. Here, it can be argued that as long as object planes are at different distances to the lens, then the phase of the Fourier transform is related to the object distances. [PERSON] et al. [24] showed that the sharpness of images edges is related to phases and their coherency, and by a decrease in such coherency, the blurring effect appears. They also showed that the degree of blurring effect can be modeled by phase incoherency, which has a Gaussian distribution. Thus, to improve the model in Equation (4), we assume the step edge can undergo a variational blurring effect in relation to the variation in the phase incoherency, which in its turn is related to the distance variation in the object plane. Let us assume such a function is:  
\[\mathrm{g}(\mathrm{x})=\frac{1}{\sqrt{2\pi\sigma_{\mathrm{m}}^{2}}}\mathrm{e} ^{\frac{-\gamma^{2}}{2\sigma_{\mathrm{m}}^{2}}}\, \tag{5}\]  
where \(\sigma_{\mathrm{m}}=\mathrm{f}(\mathrm{d}_{\mathrm{i}},\mathrm{d}_{\mathrm{f} })=\mathrm{d}_{\mathrm{i}}-\mathrm{d}_{\mathrm{f}}\), i.e., the standard deviation varies by two parameters of \(\mathrm{d}_{\mathrm{i}}\) and \(\mathrm{d}_{\mathrm{f}}\). Thus, the improved model can be written as:  
\[\mathrm{f}_{2}(\mathrm{x})=(\mathrm{Au}(\mathrm{x})+\mathrm{B})\otimes\mathrm{ g}(\mathrm{x}), \tag{6}\]  
where \(\otimes\) represents the convolution operation. Here, the origin of \(\mathrm{x}\) is assumed to be in the focus distance position, i.e., \(\mathrm{d}_{\mathrm{f}}\) in Figure 1. \(\mathrm{f}(\mathrm{d}_{\mathrm{i}},\mathrm{d}_{\mathrm{f}})\) represents the variation in the object distance from the focal plane. By increasing the object distance from the focal plane, the blurring effect is increasing; the \(\sigma_{\mathrm{m}}\) in \(\mathrm{g}(\mathrm{x})\) is increasing. When the point (i.e., a point of the object) is not in focus, see Figure 1, its image on the image plane is no longer a point but a circular patch with a certain radius that defines the amount of defocus associated with the depth of the point in the scene. On the other hand, the defocusing process can be modeled as \(\mathrm{I}(\mathrm{x})=\int\mathrm{f}(\mathrm{r})\mathrm{h}(\mathrm{x},\mathrm{ y})\mathrm{d}\mathrm{y}\), where \(\mathrm{x}\) is adopted to denote the 2D space coordinates, \(\mathrm{f}(\mathrm{x})\) is the focused image of the scene, and \(\mathrm{h}\) is the space-varying PSF.