## Abstract

A spatial filtering technique is proposed to sharpen the correlation peak for a joint transform correlator (JTC) by using the inverse reference power spectrum. Ways of handling the pole problems are discussed under various noise conditions. The minimum mean-square-error method is used to locate the optimum bias value and to estimate threshold level as applied to eradicate the poles. Applications to multitarget recognition and spectral fringe binarization are also studied. Computer-simulated results show that the compensated JTC performs better than the conventional JTC.

© 1993 Optical Society of America

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### Equations (17)

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(1)
$$t(x,y)=f(x-b,y)+f(x+b,y),$$
(2)
$$T(p,q)=F(p,q)\text{exp}(-jbp)+F(p,q)\text{exp}(jbp),$$
(3)
$$\mid T{\mid}^{2}=(2\mid F{\mid}^{2})+(\mid F{\mid}^{2})\text{exp}(-2jbp)+(\mid F{\mid}^{2})\text{exp}(2jbp).$$
(4)
$$\mid T{\mid}^{2}H=(2\mid F{\mid}^{2}H)+(\mid F{\mid}^{2}H)\times \text{exp}(-2jbp)+(\mid F{\mid}^{2}H)\text{exp}(2jbp).$$
(5)
$$\mid F{\mid}^{2}H(p,q)=k\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{or}\hspace{0.17em}\text{equivalently}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}H(p,q)=k\mid F{\mid}^{2},$$
(6)
$$H(p,q)=\{\begin{array}{ll}1,\hfill & {V}_{T}\le \hspace{0.17em}\mid F(p,q){\mid}^{2}\hfill \\ 0,\hfill & {V}_{T}>\hspace{0.17em}\mid F(p,q){\mid}^{2}\hfill \end{array}.$$
(7)
$$\text{min}{\Vert \hspace{0.17em}\mid F{\mid}^{2}H-k\Vert}^{2}.$$
(8)
$$H=k/(\mid F{\mid}^{2}+a),$$
(9)
$$\text{min}{\Vert {F}^{*}(F+N)H-k\Vert}^{2},$$
(10)
$$\underset{a}{\text{min}}{\Vert \frac{k{F}^{*}(F+N)}{\mid F{\mid}^{2}+a}-k\Vert}^{2}=\underset{a}{\text{min}}{\Vert k\frac{{F}^{*}N-a}{\mid F{\mid}^{2}+a}\Vert}^{2},$$
(11)
$$\underset{a}{\text{min}}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}\frac{\mid {F}^{*}N-a{\mid}^{2}}{{(\mid F{\mid}^{2}+a)}^{2}}\text{d}p\text{d}q=\underset{a}{\text{min}}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}\frac{\mid F{\mid}^{2}\mid N{\mid}^{2}+{a}^{2}}{{(\mid F{\mid}^{2}+a)}^{2}}\text{d}p\text{d}q.$$
(12)
$$\frac{d}{\text{d}a}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}\frac{\mid F{\mid}^{2}\mid N{\mid}^{2}+{a}^{2}}{{(\mid F{\mid}^{2}+a)}^{2}}\text{d}p\text{d}q={\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}\frac{\mid F{\mid}^{2}(\mid N{\mid}^{2}-a)}{{(\mid F{\mid}^{2}+a)}^{3}}\text{d}p\text{d}q=0.$$
(13)
$$a=\mathrm{\gamma}{\mathrm{\sigma}}^{2},$$
(14)
$$H(p,q)=\mathrm{\Sigma}{k}_{i}/\mid {F}_{i}(p,q){\mid}^{2},$$
(15)
$$\mid T{\mid}^{2}H=4\mid {F}_{1}\hspace{0.17em}\text{cos}(bp)+{F}_{2}\hspace{0.17em}\text{cos}(bp){\mid}^{2}H.$$
(16)
$$2\{\mid {F}_{1}{\mid}^{2}\text{cos}(2bp)+\mid {F}_{2}{\mid}^{2}\text{cos}(2bq)\}({k}_{1}/\mid {F}_{1}{\mid}^{2}+{k}_{2}/\mid {F}_{2}{\mid}^{2})=2\{({k}_{1}+{k}_{2}\mid {F}_{2}{\mid}^{2}/\mid {F}_{1}{\mid}^{2})\text{cos}(2bp)+\hspace{0.17em}({k}_{2}+{k}_{1}\mid {F}_{1}{\mid}^{2}/\mid {F}_{2}{\mid}^{2})\text{cos}(2bq)\}.$$
(17)
$$\mathrm{\Delta}\%=100({P}_{\text{auto}}-{P}_{\text{cross}})/{P}_{\text{auto}},$$