## Appendix A  
For a thin lens, the DOF is at a limited distance \(Z\), which is determined by the aperture stop diameter \(A\). When we have an acceptable circle of confusion \(c\), the front and rare DOFs are \(z_{1}\) and \(z_{2}\), see Figure 11.  
Using similar triangles yields  
\[\frac{z_{1}}{\frac{c}{M}}=\frac{x-z_{1}}{A},\text{and}\ \frac{z_{2}}{\frac{c}{M}}= \frac{x+z_{2}}{A}, \tag{11}\]  
where \(x\) is the distance to the in-focus plane, and \(M\) is the lens magnification. Using the definition of numerical aperture, \(N\), the aperture diameter is \(A=\frac{f}{N}\). By approximating the magnification as \(M\approx\frac{f}{x}\), \(z_{1}\) and \(z_{2}\) in Equation (11) are  
\[z_{1}=\frac{Ncx^{2}}{f^{2}+Ncx}, \tag{12}\]  
\[z_{2}=\frac{Ncx^{2}}{f^{2}-Ncx}. \tag{13}\]According to Equation (1), \(c=\frac{|\mathrm{x}-\mathrm{y}|}{\mathrm{x}}\frac{\mathrm{f}}{\mathrm{y}-\mathrm{f}}\), where \(\mathrm{y}\) is the focus distance. By replacing Nc in Equations (A2) and (A3), we obtain  
\[\mathrm{z}_{1}=\frac{\mathrm{x}|\mathrm{x}-\mathrm{y}|}{(\mathrm{y}-\mathrm{f}) +|\mathrm{x}-\mathrm{y}|},\] (A4)  
\[\mathrm{z}_{2}=\frac{\mathrm{x}|\mathrm{x}-\mathrm{y}|}{(\mathrm{y}-\mathrm{f} )-|\mathrm{x}-\mathrm{y}|},\] (A5)  
and subsequently the DOF is  
\[\mathrm{Z}=\frac{2\mathrm{x}|\mathrm{x}-\mathrm{y}|}{(\mathrm{y}-\mathrm{f}) -\frac{(\mathrm{x}-\mathrm{y})^{2}}{\mathrm{y}-\mathrm{f}}}.\] (A6)  
According to Equation (A6), when the distance to the in-focus plane, \(\mathrm{x}\), is the same as the focus distance, \(\mathrm{y}\), then the DOF, \(\mathrm{Z}\), is zero, and when \(\mathrm{x}\) remains in a limited range, the DOF is limited; otherwise the DOF extends to infinity. The constraint to obtain a limited range of \(\mathrm{x}\) is  
\[(\mathrm{y}-\mathrm{f})-\frac{(\mathrm{x}-\mathrm{y})^{2}}{\mathrm{y}-\mathrm{ f}}>0,\] (A7)  
which gives  
\[\mathrm{x}>\mathrm{f}\,\mathrm{and}\,\mathrm{x}<2\mathrm{y}-\mathrm{f}.\] (A8)  
From similar triangles, we have  
\[\frac{\mathrm{x}-\mathrm{y}}{\mathrm{y}-\mathrm{f}}=\frac{\mathrm{\frac{c}{ \mathrm{M}}}}{\mathrm{A}}\frac{\mathrm{y}}{\mathrm{y}-\mathrm{f}}\;,\] (A9)  
when the conjugate of the circle of confusion, \(\mathrm{\frac{c}{\mathrm{M}}}\), is smaller than the lens aperture, \(A\); then,  
\[\frac{z_{1}}{\frac{c}{M}}=\frac{x-z_{1}}{A},\text{and}\ \frac{z_{2}}{\frac{c}{M}}= \frac{x+z_{2}}{A}, \tag{11}\]0 (A10)  
Then, from Equation (A6), we obtain  
\[\frac{z_{1}}{\frac{c}{M}}=\frac{x-z_{1}}{A},\text{and}\ \frac{z_{2}}{\frac{c}{M}}= \frac{x+z_{2}}{A}, \tag{11}\]1 (A11)  
According to Figure A2, by an unintentional change in the focus distance, a defocus blurring effect is developed, and as a result, when the distance to the in-focus plane remains at the same distance, the DOF is scaled by a factor. According to Figure A3, by an unintentional change in the distance to the in-focus plane, the motion blurring effect is the result of varying the distance to the in-focus plane; consequently, it is the result of shifting the DOF. The feature vector represents the relative estimated depth of the objects in the related image. Two dominant object planes are detected in each feature vector whose positions are related to the DOF. By comparison between the sharp and blurred feature vectors, the two types of blurring classes are detected as shown in Figure A4. According to expert description, our class I of blurring effects is caused by unintentional defocusing, that is, a mistake in having the right focus distance when capturing the image. The blurring effect in this class has partially or entirely affected the image, i.e., the left or right side of the image is partially affected. Our class II is caused by motion blur according to the expert description. The comparison between sharp and blurred images verifies the hypothesis in Section 4. Thus, defocus and motion blurs are caused by a scaling and shifting of the DOF, respectively. It should be noted that the results are independent of the hypothesis from the methodology in Section 5.