**PURPOSE**- Compute center of object in an image (not volume) using cross-correlation between original image and the image after a 180-degree rotation. Coordinates are listed with respect to the SPIDER image center: (NX/2 + 1, NY/2 + 1). Example.

**SEE ALSO****CG**[Center of Gravity] **CG PH**[Center of Gravity - Phase approximation ||] **SH**[Shift - using bilinear/trilinear interpolation ||] **SH F**[Shift - using Fourier interpolation ||] **CENT PH**[Center image/volume using phase approximation]

**USAGE**- .OPERATION: CG SYM [xi],[yi],[xr],[yr]

- [This operation can return up to four optional register variables:
Variable Example Receives First [xi] Integer approximation of X center of gravity Second [yi] Integer approximation of Y center of gravity Third [xr] Sub-pixel X center of gravity Fourth [yr] Sub-pixel Y center of gravity .INPUT FILE: IMG001

[Enter name of image.]

**NOTES**

- Register variables [xi],[yi] receive integer approximations of the
offset from the quasi-symmetry center. Registers [xr],[yr] receive sub-pixel
coordinates of the offset from the quasi-symmetry center.
To place the center of quasi-symmetry at the image center one has to use the
'SH' operation and reverse the signs of the shifts.
- Implemented by G.Kishchenko.
- The operation is noise- and fool-proof. It's based on
two-fold quasi-symmetry of objects. This operation usually
produces results similar to operation 'CG PH',
but it is preferred, since it does not report incorrect
center for dumbbell-shaped objects.
- The estimation of center of quasi-symmetry in this algorithm is based
on cross-correlation between original image and the image after a
180-degree rotation, implemented as follows:

i) The 2D Fourier transform of original image is computed.

ii) The square of a complex number in each point of 2D Fourier transform is computed. (Notes: it's an equivalent of multiplication of Fourier transform of original image and complex conjugate of image after a 180-degree rotation, because conjugate of image after a 180-degree rotation is equal to Fourier transform of original image.)

iii) The reversed 2D Fourier transform is computed to obtain 2D cross-correlation function. This is based on cross-correlation theorem that states that the Fourier transform of the cross-correlation of two functions is equal to the product of multiplication of the individual Fourier transforms, where one of them has been complex conjugated.

iv) The X,Y-coordinates corresponding to maximum of correlation function are divided by 2 to obtain the center of object.

**SUBROUTINES**- FINDCENT, CENT_SYM

**CALLER**- UTIL1

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