## 3 Relation between Blurring and DOF Based on Geometrical Optics  
To understand the origin of the blurring problem, we need to focus on the most crucial part of the capturing process: the projection of a scene by a lens on a camera sensor. Figure 1 shows the principle of such projection using a thin lens model ([PERSON] et al., 2017) where the camera has a focal length of f. When an object stands at the focus distance of \(\mathrm{d}_{\mathrm{f}}\), i.e., the shooting distance, the image sensor capture the object as it is at the distance of \(\mathrm{d}_{0}\), see the top sketch in Figure 1. However, at any other distances, e.g., \(\mathrm{d}_{1}\) or \(\mathrm{d}_{2}\), the image sensor captures a deformed and blurred shape of the object. In the thin lens model, a circular imprint of the blurring effect, known as the circle of confusion (CoC), is used to measure the blurriness. The distance C in Figure 1b,c represents the diameter of such a CoC, and it is calculated by  
\[\mathrm{D}_{\mathrm{CoC}}=\frac{|\mathrm{d}_{\mathrm{i}}-\mathrm{d}_{\mathrm{ f}}|}{\mathrm{d}_{\mathrm{i}}}\frac{\mathrm{f}^{2}}{\mathrm{N}(\mathrm{d}_{ \mathrm{f}}-\mathrm{f})}. \tag{1}\]  
where \(\mathrm{d}_{\mathrm{i}}\) is the object distance, and N is the relative aperture or f-stop number. Figure 2 shows the \(\mathrm{D}_{\mathrm{CoC}}\) as a function of the object distance \(\mathrm{d}_{\mathrm{i}}\), where distance \(\mathrm{d}_{\mathrm{f}}\) is 800 mm, f-stop number N is 8, 16, or 22, and the focus f is \(\mathrm{f}_{0}\) and \(\mathrm{f}_{1}\) at 50 and 52 mm, respectively. This figure shows that the diameter of \(\mathrm{D}_{\mathrm{CoC}}\) is a non-linear function of the object distance. The function monotonically decreases or increases for object distances less or bigger than the focus distance, respectively. The expansion tolerance of the CoC while capturing images is a subjective issue, and the conjugate of the CoC in object space is typically bigger than the CoC due to the lens magnification, as shown in Figure 3.  
According to [PERSON]'s ray construction, there are two distance regions around the CoC in image space and around the conjugate of the CoC in object space where the image plane and object can change their positions and still make it possible to capture a non-blurred image. These distance regions in image space and object space are asymmetrical and known as the depth of focus and depth of field (DOF), respectively; see the blue regions in Figure 3. The \(\mathrm{D}_{\mathrm{CoC}}\) is shown by C and the diameter of the conjugate of the CoC is presented by \(\frac{\mathrm{C}}{\mathrm{M}}\) in Figure 3, where M is the magnification from image space to object space, e.g., \(\mathrm{M}\triangleq-\frac{\mathrm{image\;plane\;distance}}{\mathrm{ Object\;distance}}\). When the conjugate of CoC is small in relation to the lens aperture size, it yields:  
\[\mathrm{DOF}\approx\frac{2\mathrm{ND}_{\mathrm{CoC}}\mathrm{d}_{\mathrm{i}}^{2 }}{\mathrm{f}^{2}}. \tag{2}\]Using Equation (1) in Equation (2) yields:  
\[\text{DOF}\approx\frac{2\text{d}_{i}|\text{d}_{i}-\text{d}_{f}|}{(\text{d}_{f}- \text{f})}. \tag{3}\]  
Figure 1: Scene projection using a thin lens model with object distance (**a**) equal to \(\text{d}_{f}\), (**b**) less than \(\text{d}_{f}\), (**c**) greater than \(\text{d}_{f}\). Figure 2: Relation between \(\text{D}_{CoC}\) and object distance. Thus, a successful non-blurred image capturing is constrained by Equation (3); a more detailed calculation of the DOF can be found in Appendix A. In unintentional image degradation, i.e., caused by motion or defocus blur, the focal length f does not change, i.e., the appropriate focal length is chosen intentionally and as part of the camera setting. However, the focus distance \(\mathrm{d}_{\mathrm{f}}\), as a result of the internal camera setting, can conflict with scene changes.