Positional, rotational and scale invariant optical correlation method and apparatus
First Claim
1. A method for correlating two functions f1 (x,y) and f2 (x,y) which are scaled and rotated versions of each other, comprising the steps ofobtaining |F1 (ω
-
x,ω
y)|, and |F2 (ω
x,ω
y)|, the magnitudes of the Fourier transforms of these functions;
performing a polar coordinate conversion on |F1 (ω
x,ω
y)| and |F2 (ω
x,ω
y)| thereby to obtain the functions F1 (r,θ
) and F2 (r,θ
);
logarithmically scaling the coordinate r in the functions F1 (r,θ
) and F2 (r,θ
) thereby to obtain the functions F1 (e.sup.ρ
,θ
) and F2 (e.sup.ρ
,θ
);
Fourier transforming F1 (e.sup.ρ
,θ
) and F2 (e.sup.ρ
,θ
) thereby to obtain the Mellin transforms M1 (ω
.sub.ρ
,ω
.sub.θ
) and M2 (ω
.sub.ρ
,ω
.sub.θ
);
obtaining the conjugate Mellin transform M1 *(ω
.sub.ρ
,ω
.sub.θ
);
producing the product M1 *M2 ;
Fourier transforming said product, all of the aforementioned steps being performed by optical or electro-optical means; and
recording on film the results of said last-mentioned Fourier transformation.
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Abstract
A method and electro-optical apparatus for correlating two functions, f1 (x,y) and f2 (x,y) which are shifted, scaled and rotated versions of each other without loss of signal-to-noise and signal-to-clutter ratios as compared to the autocorrelation case. The coordinates of two correlation peaks provide an indication of the scale and orientation differences between the two functions. In performing the method, the magnitudes of the Fourier transforms of the functions are obtained, |F1 (ωx, ωy)| and |F2 (ωx,ωy)| and then a polar coordinate conversion is performed, and the resultant functions F1 (r,θ) and F2 (r,θ) are logarithmically scaled in the r coordinate. The functions thus produced F1 (e.sup.ρ,θ) and F2 (e.sup.ρ,θ) are Fourier transformed to produce the Mellin transforms M1 (ω.sub.ρ, ω.sub.θ) and M2 (ω.sub.ρ,ω.sub.θ). The conjugate of one of these Mellin transforms is obtained, and the product of this conjugate with the other Mellin transform is produced and, subsequently, Fourier transformed to complete the correlation process.
39 Citations
10 Claims
-
1. A method for correlating two functions f1 (x,y) and f2 (x,y) which are scaled and rotated versions of each other, comprising the steps of
obtaining |F1 (ω -
x,ω
y)|, and |F2 (ω
x,ω
y)|, the magnitudes of the Fourier transforms of these functions;performing a polar coordinate conversion on |F1 (ω
x,ω
y)| and |F2 (ω
x,ω
y)| thereby to obtain the functions F1 (r,θ
) and F2 (r,θ
);logarithmically scaling the coordinate r in the functions F1 (r,θ
) and F2 (r,θ
) thereby to obtain the functions F1 (e.sup.ρ
,θ
) and F2 (e.sup.ρ
,θ
);Fourier transforming F1 (e.sup.ρ
,θ
) and F2 (e.sup.ρ
,θ
) thereby to obtain the Mellin transforms M1 (ω
.sub.ρ
,ω
.sub.θ
) and M2 (ω
.sub.ρ
,ω
.sub.θ
);obtaining the conjugate Mellin transform M1 *(ω
.sub.ρ
,ω
.sub.θ
);producing the product M1 *M2 ; Fourier transforming said product, all of the aforementioned steps being performed by optical or electro-optical means; and recording on film the results of said last-mentioned Fourier transformation. - View Dependent Claims (2, 3, 4, 5, 6)
-
x,ω
-
7. A method for correlating two functions f1 (x,y) and f2 (x,y) which are scaled and rotated versions of each other, comprising the steps of
producing an optical representation of |F2 (ω -
x,ω
y)|, the magnitude of the Fourier transform of f2 (x,y);performing a polar coordinate conversion on this representation of |F2 (ω
x,ω
y)| thereby to obtain an optical representation of the function F2 (r,θ
);logarithmically scaling the coordinate r in the function F2 (r,θ
) thereby to obtain an optical representation of the function F2 (e.sup.ρ
,θ
);Fourier transforming F2 (e.sup.ρ
,θ
) thereby to obtain a light distribution pattern corresponding to the Mellin transform M2 (ω
.sub.ρ
,ω
.sub.θ
);pouring a film transparency having an interference pattern recorded therein which contain a term that is proportional to the conjugate Mellin transform M1 *(ω
.sub.ρ
,ω
.sub.θ
);illuminating said film transparency with said light distribution pattern thereby to produce a light distribution pattern corresponding to the produce M1 *M2 ; Fourier transforming said last-mentioned light pattern; and
recording the results thereof.
-
x,ω
-
8. Apparatus for correlating two functions f1 (x,y) and f2 (x,y) which are shifted, scaled and rotated versions of each other, comprising in combination
an optical correlator having an input plane P0, a frequency plane P1 and an output plane P2 ; -
a film transparency having an interference pattern recorded therein which contain a term proportional to the conjugate Mellin transform M2 *(ω
.sub.ρ
,ω
.sub.θ
), said film transparency being positioned at plane P1 ;means for creating a light pattern leaving plane P0 that corresponds to F1 (e.sup.ρ
,θ
),said light pattern being Fourier transformed by lens means within said correlator located between planes P0 and P1 and the illumination of said film transparency by the resultant light pattern producing a light pattern leaving plane P1 that corresponds to the product M2 *(ω
.sub.ρ
,ω
.sub.θ
) M1 (ω
.sub.ρ
,ω
.sub.θ
),said last-mentioned light pattern being Fourier transformed by other lens means within said correlator located between planes P1 and P2 ; and means positioned at plane P2 for recording the results of said last-mentioned Fourier transformation.
-
-
9. Apparatus for correlating two functions f1 (x,y) and f2 (x,y) which are shifted, scaled and rotated versions of each other, comprising
optical means for Fourier transforming f1 (x,y) so as to obtain |F1 (ω -
x,ω
y)|, the magnitude of the Fourier transform of this function;means for performing a polar coordinate conversion on |F1 (ω
x,ω
y)| so as to obtain the function F1 (r,θ
);
means for logarithmically scaling the r coordinate in the function F1 (r,θ
) so as to obtain F1 (e.sup.ρ
,θ
), |F1 (ω
x,ω
y)|, F1 (r,θ
) and F1 (e.sup.ρ
,θ
) occurring as optical images;optical means for Fourier transforming F1 (e.sup.ρ
,θ
) so as to obtain a light pattern corresponding to the Mellin transform M1 (ω
.sub.ρ
,ω
.sub.θ
);a film transparency having an interference pattern recorded therein which contains an optical representation of the conjugate Mellin transform M2 *(ω
.sub.ρ
,ω
.sub.θ
);means for illuminating said film transparency with said light pattern so as to obtain a light pattern corresponding to M1 M2 *; optical means for Fourier transforming said last-mentioned light pattern; and means for recording the results thereof. - View Dependent Claims (10)
-
x,ω
Specification