## Abstract

In this paper, a statistical approach is presented for three-dimensional (3D) visualization and recognition of objects having very small number of photons based on a parametric estimator. A truncated Poisson probability density function is assumed for modeling the distribution of small number of photons count observation. For 3D visualization and recognition of photon-limited objects, an integral imaging system is employed. We utilize virtual geometrical ray propagation for 3D reconstruction of objects. A maximum likelihood estimator (MLE) and statistical inference algorithms are applied to small number of photons counted elemental images captured with integral imaging. We have demonstrated that the MLE using a truncated Poisson model for estimating the average number of photon for each voxel of a photon starved 3D object has a small estimation error compared with the MLE using a Poisson model. Also, we present experiments to investigate the effect of 3D sensing parallax on the recognition performance under a fixed mean number of photons.

©2009 Optical Society of America

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### Equations (7)

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(1)
$$\mathrm{Pr}(C|\rho ,T)=\genfrac{}{}{0.1ex}{}{{[\rho T]}^{C}{e}^{-\rho T}}{C!},$$
(2)
$$\mathrm{Pr}({C}_{p}^{t}={c}_{p})=\genfrac{}{}{0.1ex}{}{\mathrm{Pr}({C}_{p}={c}_{p})}{\mathrm{Pr}({C}_{p}<k)}=\genfrac{}{}{0.1ex}{}{\mathrm{exp}(-\tilde{N}{I}_{p}){\left(\tilde{N}{I}_{p}\right)}^{{c}_{p}}}{{c}_{p}!{\displaystyle \colorbox[rgb]{}{$\sum _{i=0}^{k-1}\genfrac{}{}{0.1ex}{}{\mathrm{exp}(-\tilde{N}{I}_{p}){\left(\tilde{N}{I}_{p}\right)}^{i}}{i!}$}}},$$
(3)
$$E[{C}_{p}^{t}]=aE[{C}_{p}],V[{C}_{p}^{t}]=aE[{C}_{p}]+(a-{a}^{2}){\left\{E[{C}_{p}]\right\}}^{2},$$
(4)
$$L(\tilde{N}{I}_{p}|{c}_{p}(1),\mathrm{...},{c}_{p}({N}_{x}{N}_{y}))=\mathrm{log}{\displaystyle \colorbox[rgb]{}{$\prod _{n=1}^{{N}_{x}{N}_{y}}\genfrac{}{}{0.1ex}{}{\mathrm{exp}(-\tilde{N}{I}_{p}){\left(\tilde{N}{I}_{p}\right)}^{{c}_{p}(n)}}{{c}_{p}(n)!{\displaystyle \colorbox[rgb]{}{$\sum _{i=0}^{k-1}\genfrac{}{}{0.1ex}{}{\mathrm{exp}(-\tilde{N}{I}_{p}){\left(\tilde{N}{I}_{p}\right)}^{i}}{i!}$}}}$}},$$
(5)
$$MLE(\tilde{N}{I}_{p})=\mathrm{arg}\underset{\tilde{N}{I}_{p}}{\mathrm{max}}L(\tilde{N}{I}_{p}|{c}_{p}(1),\mathrm{...},{c}_{p}({N}_{x}{N}_{y})).$$
(6)
$$MLE(\tilde{N}{I}_{p})=I{I}_{p}(x,y,{z}_{0})=\genfrac{}{}{0.1ex}{}{1}{{N}_{x}{N}_{y}}{\displaystyle \colorbox[rgb]{}{$\sum _{n=1}^{{N}_{x}{N}_{y}}{c}_{p}(n)$}}/\left(1-\genfrac{}{}{0.1ex}{}{1}{{N}_{x}{N}_{y}}{\displaystyle \colorbox[rgb]{}{$\sum _{n=1}^{{N}_{x}{N}_{y}}{c}_{p}(n)$}}\right)=\genfrac{}{}{0.1ex}{}{\overline{MLE}(\tilde{N}{I}_{p})}{1-\overline{MLE}(\tilde{N}{I}_{p})},$$
(7)
$$\left[{\widehat{I}}_{p}\pm {z}_{\alpha /2}{\left(\genfrac{}{}{0.1ex}{}{1}{{N}_{x}{N}_{y}\tilde{N}}\right)}^{2}\sqrt{\genfrac{}{}{0.1ex}{}{{\widehat{I}}_{p}}{\gamma}}\right],,$$