## Abstract

Linear-correlation amplitude changes when the intensity level of the input image is modified. As recognition is often based on the correlation-peak level, a change of the input illumination may result in a false recognition. We propose an illumination-change compensation by a post processing of the correlation distribution that is based on statistical measures of the correlation histograms. The theoretical background and simulation results are provided in the frame of an actual application in biology.

© 2002 Optical Society of America

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### Equations (21)

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(1)
$$x\left(s,t\right)=\left(n\otimes p\right)\left(s,t\right),$$
(2)
$$E\left[n\left(s,t\right)\right]=\mathrm{\mu}E\left[n\left(s,t\right)n\left(s\prime ,t\prime \right)\right]={\mathrm{\sigma}}^{2}\mathrm{\delta}\left(s-s\prime ,t-t\prime \right).$$
(3)
$$X\left(U,V\right)=N\left(U,V\right)P\left(U,V\right).$$
(4)
$$N\left(U,V\right)=B\left(U,V\right)+{B}_{0}\mathrm{\delta}\left(U,V\right),$$
(5)
$$E\left[N\left(U,V\right)\right]={B}_{0}\mathrm{\delta}\left(U,V\right),$$
(6)
$$E\left[B\left(U,V\right)B\prime \left(U\prime ,V\prime \right)\right]={\mathrm{\sigma}}^{2}\mathrm{\delta}\left(U-U\prime ,V-V\prime \right).$$
(7)
$$\mathrm{\mu}=\frac{1}{N}P\left(0,0\right){B}_{0}.$$
(8)
$$c\left(s,t\right)=\left(x\otimes h\right)\left(s,t\right)=\frac{1}{N}\sum _{U,V-0}^{N-1}exp\left[\frac{2i\mathrm{\pi}}{N}\left(\mathit{Us}+\mathit{Vt}\right)\right]H*\left(U,V\right)\times \left[N\left(U,V\right)P\left(U,V\right)\right].$$
(9)
$$c\left(s,t\right)=\mathrm{\mu}H*\left(0,0\right)+\frac{1}{N}\sum _{U,V=0}^{N-1}W\left(s,t,U,V\right)B\left(U,V\right),$$
(10)
$$W\left(s,t,U,V\right)=exp\left[\frac{+2i\mathrm{\pi}}{N}\left(\mathit{Us}+\mathit{Vt}\right)\right]\times H*\left(U,V\right)P\left(U,V\right).$$
(11)
$$P\left[c\left(s,t\right)=a\right]\approx \frac{1}{\mathrm{\Sigma}\sqrt{2\mathrm{\pi}}}exp\left(-\frac{{a}^{2}}{2{\mathrm{\Sigma}}^{2}}\right),$$
(12)
$$E\left[{c}^{2}\left(s,t\right)\right]=E\left[\frac{1}{{N}^{2}}\sum _{{U}_{1},{V}_{1}=0}^{N-1}W\left(s,{\mathit{tU}}_{1},{V}_{1}\right)B\left({U}_{1},{V}_{1}\right)\times \sum _{{U}_{2},{V}_{2}=0}^{N-1}W*\left(s,t,{U}_{2},{V}_{2}\right)B*\left({U}_{2},{V}_{2}\right)\right],$$
(13)
$$E\left[{c}^{2}\left(s,t\right)\right]=\frac{1}{{N}^{2}}\sum _{{U}_{1},{V}_{1},{U}_{2},{V}_{2}=0}^{N-1}W\left(s,t,{U}_{1},{V}_{1}\right)\times W*\left(s,t,{U}_{2},{V}_{2}\right){\mathrm{\sigma}}^{2}\mathrm{\delta}\times \left({U}_{1}-{U}_{2},{V}_{1}-{V}_{2}\right),$$
(14)
$${\mathrm{\Sigma}}^{2}=\frac{1}{{N}^{2}}\sum _{U,V=0}^{N-1}|H\left(U,V\right)P\left(U,V\right){|}^{2}{\mathrm{\sigma}}^{2}.$$
(15)
$$P\left[{c}^{2}\left(s,t\right)=I;I0\right]\approx +\frac{1}{I\mathrm{\Sigma}\sqrt{2\mathrm{\pi}}}exp\left(-\frac{{I}^{2}}{2{\mathrm{\Sigma}}^{2}}\right).$$
(16)
$$I\u2033\left(s,t\right)=\frac{\mathrm{Per}\_\mathrm{Ref}}{\mathrm{Per}\_\mathrm{Obj}}I\prime \left(s,t\right),$$
(17)
$$x\prime \left(s,t\right)=\mathrm{\alpha}x\left(s,t\right)$$
(18)
$$c\prime \left(s,t\right)={\mathrm{\alpha}}^{2}\left(x\otimes h\right)\left(s,t\right)={\mathrm{\alpha}}^{2}c\left(s,t\right),$$
(19)
$${r}_{\mathit{kq}}=\sum _{U,V=0}^{n-1}H\left(U,V\right){X}_{\mathit{kq}}^{*}\left(U,V\right)$$
(20)
$$F=\sum _{U,V=0}^{n-1}\sum _{k,q,i}|H\left(U,V\right)\left[{X}_{\mathit{kq}}^{*}\left(U,V\right)-{Y}_{\mathit{kqi}}^{*}\left(U,V\right)\right]{|}^{2},$$
(21)
$$G=\frac{{I}_{1}-{I}_{0}}{{\mathit{SD}}_{1}+{\mathit{SD}}_{0}},$$