Effect of Pickup Position Uncertainty in Three-Dimensional Computational Integral Imaging

Abstract

This chapter will review the relevant investigations analyzing the performance of Integral Imaging (II) technique under pickup position uncertainty. Theoretical and simulation results for the sensitivity of Synthetic Aperture Integral Imaging (SAII) to the accuracy of pickup position measurements are provided. SAII is a passive three-dimensional, multi-view imaging technique that, unlike digital holography, operates under incoherent or natural illumination. In practical SAII applications, there is always an uncertainty associated with the position at which each sensor captures the elemental image. In this chapter, we theoretically analyze and quantify image degradation due to measurements’ uncertainty in terms of Mean Square Error (MSE) metric. Experimental results are also presented that support the theory. We show that in SAII, with a given uncertainty in the sensor locations, the high spatial frequency content of the 3-D reconstructed images are most degraded. We also show an inverse relationship between the reconstruction distance and degradation metric.

Appendix

The error spectrum follows the expression (5.18). Note that in the Eq. (5.18) only Δp has a random nature. Thus, the expected value for the error spectrum depends on the behavior of this variable, i.e., the distribution governing the spatial dislocation of the sensors during pickup.

$$\begin{aligned} \left| {Err({\bf{f}},z)} \right|^2 & = \frac{1}{{R^4 }}\left| {\sum\limits_i^{K \times K} {{\bf{\tilde I}}_i (1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} )} } \right|^2 \\ & = \frac{1}{{R^4 }}\left[ {\sum\limits_i^{K \times K} {\sum\limits_j^{K \times K} {{\tilde{\bf I}}_i {\tilde{\bf I}}_j^{\ast} (1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} )(1 - e^{ + jf_x \frac{{\Delta p_j }}{M}} )} } } \right]. \end{aligned}$$

((5.18))

Since expectation is a linear operator, one can break down the error expectation as follows:

$$\begin{aligned} E\left\{ {\left| {Err({\bf{f}},z)} \right|^2 } \right\} & = \frac{1}{{R^4 }}\left[ {\sum\limits_i^{K \times K} {\sum\limits_j^{K \times K} {{\tilde{\bf I}}_i {\tilde{\bf I}}_j^{\ast} E\left\{ {(1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} )(1 - e^{ + jf_x \frac{{\Delta p_j }}{M}} } \right\}} } } \right] \\ & = \frac{1}{{R^4 }}\left[ {\sum\limits_i^{K \times K} {\sum\limits_i^{K \times K} {{\tilde{\bf I}}_i {\tilde{\bf I}}_i^{\ast} E\left\{ {2 - e^{ - jf_x \frac{{\Delta p_i }}{M}} - e^{ + jf_x \frac{{\Delta p_i }}{M}} } \right\}} } } \right] + ... \\ & \quad\ \frac{1}{{R^4 }}\left[ {\sum\limits_{i \ne j}^{K \times K} {\sum\limits_{j \ne i}^{K \times K} {{\tilde{\bf I}}_i {\tilde{\bf I}}_j^{\ast} E\left\{ {(1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} )(1 - e^{ + jf_x \frac{{\Delta p_j }}{M}} )} \right\}} } } \right]{\rm{ }} \end{aligned}$$

((5.19))

Now, let $\gamma = E\left\{ {\exp ( - jf_x \Delta p/M)} \right\}$ which is the moment generating function of random variable Δp [31]. Equation (5.19) reduces to:

$$\begin{aligned} E\left\{ {\left| {Err({\bf{f}},z)} \right|^2 } \right\} & = \frac{1}{{R^4 }}\left[ {\sum\limits_i^{K \times K} {\sum\limits_i^{K \times K} {{\tilde{\bf I}}_i {\tilde{\bf I}}_i^{\ast} (2 - \gamma - \gamma ^{\ast} )} } } \right] + \\ & \quad\ \frac{1}{{R^4 }}\left[ {\sum\limits_{i \ne j}^{K \times K} {\sum\limits_{j \ne i}^{K \times K} {{\tilde{\bf I}}_i {\tilde{\bf I}}_j^{\ast} E\left\{ {(1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} )(1 - e^{ + jf_x \frac{{\Delta p_j }}{M}} )} \right\}} } } \right] \end{aligned}.$$

((5.20))

Note that ∆p _i denotes the sensor location error associated with ith sensor which is a random variable and the expected value of the random variable or a function of the random variable is constant which can come outside of the summation. In addition the sensor location errors are assumed to be independent, thus we have:

$$\begin{aligned} E\left\{ {(1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} )(1 - e^{ + jf_x \frac{{\Delta p_j }}{M}} )} \right\} & = E\left\{ {1 - e^{ - jf_x \frac{{\Delta p_i }}{M}} } \right\}E\left\{ {1 - e^{ + jf_x \frac{{\Delta p_j }}{M}} } \right\} \\ & = (1 - \gamma )(1 - \gamma ^{\ast} ) = \left| {1 - \gamma } \right|^2. \\ \end{aligned}$$

((5.21))

Consequently, the error spectrum expectation can be written as in Eq. (5.9).