**Figure 8.**

*Relation between RMSE of recovered depth and lambda: (a) σ<sup>r</sup>* ¼ 0*:*06*; (b) σ<sup>r</sup>* ¼ 0*:*08*; (c) σ<sup>r</sup>* ¼ 0*:*1 *(reprinted from [16]).*

Local optimization algorithms (LOA) are computationally expensive, and global optimization algorithms (GOA) are slow to converge. Therefore, we considered a hybrid algorithm that uses LOA sparsely to obtain the initial value of GOA. For the initial value of LOA, use the plane corresponding to the background of **Figure 4(b)**. To make a rough estimate, LOA uses blocks of 41 � 41 pixels as L in Eq. (21) and applies LOA once to each block. On the other hand, the size of R in Eq. (22) was adaptively determined according to the depth value updated by the optimization process. Therefore, R took a different size at each position in the image. We assumed a square area for R. The length of that side was 10 times the larger of the two deviations of *gx*ð Þ �; *d* along *x* axis and *y* axis. These can be evaluated using

The depth restoration simulation was executed while changing the camera rotation size *σr*. The recovered inverse depth map is shown in **Figures 5**–**7**. GOA uses various values of *λ*. The relationship between the root mean square error (RMSE) of

**Figure 5**, we can see that it is not suitable for depth recovery because it is difficult to accurately measure the motion blur of the image position with a small rotation of the camera. From **Figure 8(a)**, small rotations do not provide sufficient measurement information, so to reduce the RMSE of the recovered depth map, the smoothness constraint indicated by *λ* is strongly required. On the contrary, when the rotation is large, the point image distribution function becomes wider than the spatial change of the target shape. Therefore, the Gaussian function with the variance–covariance matrix in Eq. (19) is inappropriate, the motion blur recognized by this model is smoother than the actual blur of the image, and a depth recovery error occurs. This can be confirmed from the RMSE value in **Figure 8(c)**. Since the

> 1.11 1.112 1.114 1.116 1.118 1.12 1.122 1.124 1.126

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Lambda

(c)

*Relation between RMSE of recovered depth and lambda: (a) σ<sup>r</sup>* ¼ 0*:*06*; (b) σ<sup>r</sup>* ¼ 0*:*08*; (c) σ<sup>r</sup>* ¼ 0*:*1

RMSE of recovered depth

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Lambda

(b)

the recovered depth map and the value of *λ* is also shown in **Figure 8**. From

Eq. (19).

*Applications of Pattern Recognition*

 2.6 2.605 2.61 2.615 2.62 2.625 2.63

**Figure 8.**

**16**

*(reprinted from [16]).*

RMSE of recovered depth

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Lambda

(a)

 1.19 1.2 1.21 1.22 1.23 1.24 1.25

RMSE of recovered depth

smooth depth tends to be recovered from the recognized smooth motion blur from **Figure 8(c)**, it can be confirmed that the smoothness constraint of Eq. (23) is an obstacle to the reduction of RMSE.
