## Abstract

In this paper, we present a method to recover the complex amplitude of speckle fields from measurements performed by a shear interferometer. It is based on the optimization of an objective function using the steepest descent gradient technique in combination with a heuristic initial guess. In contrast to already existing methods, the algorithm finds a local minimum least-squares solution even in the presence of Poissonian and Gaussian noise.

© 2013 Optical Society of America

Full Article |

PDF Article
### Equations (18)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$u=a\xb7\mathrm{exp}(i\varphi -i\omega t),$$
(2)
$${|u+{u}_{R}|}^{2}={|u|}^{2}+{|{u}_{R}|}^{2}+2a{a}_{R}\text{\hspace{0.17em}}\mathrm{cos}(\varphi -{\varphi}_{R}).$$
(3)
$${I}_{S}(\mathrm{x\u20d7})={|u(\mathrm{x\u20d7})|}^{2}+{|u(\mathrm{x\u20d7}+\mathrm{s\u20d7})|}^{2}+2\mathfrak{R}\{{u}^{*}(\mathrm{x\u20d7})u(\mathrm{x\u20d7}+\mathrm{s\u20d7})\}.$$
(4)
$${M}_{n}={u}^{*}(\mathrm{x\u20d7})\xb7u(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})$$
(5)
$$L=\sum _{n}{\parallel {M}_{n}(\mathrm{x\u20d7})-{f}^{*}(\mathrm{x\u20d7})f(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})\parallel}^{2},$$
(6)
$$\tilde{f}(\mathrm{x\u20d7})=\mathrm{arg}\text{\hspace{0.17em}}\underset{f}{\mathrm{min}}\{L\}.$$
(7)
$${f}^{(m+1)}(\mathrm{x\u20d7})={f}^{(m)}(\mathrm{x\u20d7})-{\alpha}^{(m)}\xb7\nabla {L}^{(m)}(\mathrm{x\u20d7}),$$
(8)
$$\nabla {L}^{(m)}(\mathrm{x\u20d7})=\frac{\partial L}{\partial {f}^{(m)}(\mathrm{x\u20d7})}.$$
(9)
$$\nabla {L}^{(m)}(\mathrm{x\u20d7})=-2\sum _{n}{f}^{(m)}(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})\xb7{\psi}_{n}^{(m)*}(\mathrm{x\u20d7})+{f}^{(m)}(\mathrm{x\u20d7}-{\mathrm{s\u20d7}}_{n})\xb7{\psi}_{n}^{(m)}(\mathrm{x\u20d7}-{\mathrm{s\u20d7}}_{n}),$$
(10)
$$\tilde{f}(\mathrm{x\u20d7})=\frac{\sum _{n}\tilde{f}(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n}){M}^{*}(\mathrm{x\u20d7})+\tilde{f}(\mathrm{x\u20d7}-{\mathrm{s\u20d7}}_{n})M(\mathrm{x\u20d7}-{\mathrm{s\u20d7}}_{n})}{{\sum}_{n}\tilde{I}(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})+\tilde{I}(\mathrm{x\u20d7}-{\mathrm{s\u20d7}}_{n})}.$$
(11)
$${g}^{*}(\mathrm{x\u20d7})g(\mathrm{x\u20d7}+\mathrm{s\u20d7})=M(\mathrm{x\u20d7})[{\alpha}^{2}+{(1-\alpha )}^{2}+\alpha (1-\alpha )\xb7({f}_{\varphi}(\mathrm{x\u20d7}+\mathrm{s\u20d7})+{f}_{\varphi}^{*}(\mathrm{x\u20d7}))],$$
(12)
$${L}_{S}=\sum _{n}{\parallel {M}_{n}(\mathrm{x\u20d7})-{f}^{*}(\mathrm{x\u20d7})f(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})\parallel}^{2}+\gamma {\parallel {\mathrm{\Delta}}_{D}f(\mathrm{x\u20d7})\parallel}^{2}.$$
(13)
$$\nabla {L}_{S}^{(m)}(\mathrm{x\u20d7})=\frac{\partial {L}_{S}}{\partial {f}^{(m)}(\mathrm{x\u20d7})}=\nabla {L}^{(m)}(\mathrm{x\u20d7})+2\gamma {\mathrm{\Delta}}_{D}\{{\mathrm{\Delta}}_{D}{f}^{(m)}(\mathrm{x\u20d7})\},$$
(14)
$${f}^{(0)}(\mathrm{x\u20d7};{\mathrm{x\u20d7}}_{0})=\{\begin{array}{ccc}1& \text{for}& \mathrm{x\u20d7}={\mathrm{x\u20d7}}_{0}\\ 0& \text{elsewhere}& \end{array}.$$
(15)
$$\nabla L(\mathrm{x\u20d7})=\frac{\partial L}{\partial f(\mathrm{x\u20d7})}=\frac{\partial L}{\partial r(\mathrm{x\u20d7})}+i\frac{\partial L}{\partial q(\mathrm{x\u20d7})}.$$
(16)
$$\nabla L({\mathrm{x\u20d7}}_{0})=(\frac{\partial}{\partial r({\mathrm{x\u20d7}}_{0})}+i\frac{\partial}{\partial q({\mathrm{x\u20d7}}_{0})})\sum _{n}\sum _{\mathrm{x\u20d7}\in R}[{M}_{n}(\mathrm{x\u20d7})-{f}^{*}(\mathrm{x\u20d7})f(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})]{[{M}_{n}(\mathrm{x\u20d7})-{f}^{*}(\mathrm{x\u20d7})f(\mathrm{x\u20d7}+{\mathrm{s\u20d7}}_{n})]}^{*}.$$
(17)
$$\nabla L({\mathrm{x\u20d7}}_{0})=(\frac{\partial}{\partial r({\mathrm{x\u20d7}}_{0})}+i\frac{\partial}{\partial q({\mathrm{x\u20d7}}_{0})})\sum _{n}[{\psi}_{n}({\mathrm{x\u20d7}}_{0}){\psi}_{n}^{*}({\mathrm{x\u20d7}}_{0})+{\psi}_{n}({\mathrm{x\u20d7}}_{0}-{\mathrm{s\u20d7}}_{n}){\psi}_{n}^{*}({\mathrm{x\u20d7}}_{0}-{\mathrm{s\u20d7}}_{n})],$$
(18)
$$\nabla L({\mathrm{x\u20d7}}_{0})=\sum _{n}[-2{\psi}_{n}^{*}({\mathrm{x\u20d7}}_{0})f({\mathrm{x\u20d7}}_{0}+{\mathrm{s\u20d7}}_{n})-2{\psi}_{n}({\mathrm{x\u20d7}}_{0}-{\mathrm{s\u20d7}}_{n})f({\mathrm{x\u20d7}}_{0}-{\mathrm{s\u20d7}}_{n})].$$