**PURPOSE**- Computes phase residual and Fourier shell correlation measures of proximity between two volumes. Calculates the differential phase residual over a shell with of specified ring width and the Fourier shell correlation between shells of the specified widths and stores the values in a document file. Takes real or Fourier input volumes. Example.

**SEE ALSO****FSC**[Fourier shell correlation and resolution determination ||] **RF**[Phase Residual & Fourier ring correlation, 2D ||] **RF SN**[Spectral SNR of images, resolution determination & integral SNR ||] **FT**[Fourier Transform ||]

**USAGE**- .OPERATION: RF 3 [half],[sf]

- [Optional output registers receive:

First register -- Interpolated frequency (voxel) at which FSC falls below a value of 0.5

Second register -- Approximate resolution (in spatial fequency units) at this level.

If there are multiple downward crossings of the FSC = 0.5 cutoff line then values are reported for the**last**crossing. If there is no crossing the final values from the FSC document file output are reported.].FIRST INPUT VOLUME: SUB001

[Enter the name of the first input volume. File can be real or Fourier format. ].SECOND INPUT VOLUME: SUB002

[Enter the name of the second input volume. File must have same dimensions as first input file.].SHELL WIDTH (Reciprocal space sampling units): 0.5

[Enter the shell thickness in reciprocal space sampling units.].SCALE FACTORS (LOWER,UPPER): 0.2, 2.0

[Give the range of scale factors by which the second Fourier must be multiplied for the comparison. Only affects the phase residual calculation.].MISSING CONE/WEDGE ANGLE (C/W): C

[Use 'C' if you have a missing cone and 'W' if you have a missing wedge. For volumes with no missing data enter 'C'.].MAXIMUM TILT ANGLE: 90

[Give the angle of maximum tilt angle in degrees. For volumes with no missing data enter '90'. The inclination angle theta starts from the Z*=0 plane.].NOISE COMPARISON FACTOR: 3.0

[The factor given here determines the FSCCRIT. Here 3.0 corresponds to the three sigma criterion i.e., 3/SQRT(N), where N is number of voxels for a given shell. You could use 2, 1,4 or anything.].FSC OUTPUT DOCUMENT FILE: DOC001

[Enter name of doc. file in which results are to be saved.]

**NOTES**

- The following measures are reported for each shell:

a) Normalized Frequency.

|NORM-FREQ| ---- Doc. file register: #1

b) Phase residual

|DPH| ---- Doc. file register: #2

c) Fourier Shell Correlation = [SUM(F1 * CONJ(F2))] / [SQRT{SUM(ABS(F1)**2) * SUM(ABS(F2)**2)}]

where 'CONJ' denotes complex conjugate. FSC should be larger than the Critical Fourier Shell Correlation (FSCCRIT) provided for comparison.

|FSC| ---- Doc. file register: #3

d) Critical Fourier Shell Correlation.

|FSCCRIT| ---- Doc. file register: #4

e) Number of voxels in shell.

|VOXELS| ---- Doc. file register: #5

- If you do not want a results file listing use
'MD' operation with option
'VERBOSE OFF'.
- References

Saxton and Baumeister,*J. of Microscopy*, 127,(1982) 127-138.

M. van Heel,*Ultramicroscopy*, 21, (1987) 95-100.

Unser, et. al.,*Ultramicroscopy*, 23, (1987) 39-52.

- Scale search is done separately for each shell. This will NOT lead
to sensible results if one of the transforms falls off rapidly in a
frequency range where the other transform is strong. The range specified
by the user is divided into 20 steps and searched for the lowest
value The value:

R(McPherson) = 2*SUM(ABS(F1)-ABS(F2))/SUM(ABS(F1)+ABS(F2))

is calculated at each step within a shell and its minimum is used to determine the correct scale factor for the second Fourier transform. - The 'NO SOLUTION' warning message relates to DPR normalization
factor between two volumes, which is searched using a brute
force method. This message is
**NOT**relevant to FSC results. - The "LACKS MAXIMUM" warning pertains to the DPR only. If
you do not use DPR, you can ignore it.

If you use DPR, you should be aware that DPR, unlike FSC, depends on the relative multiplicative scaling of two Fourier transforms . Simply speaking, FSC is a correlation coefficient, so it does not depend of either F or G being multiplied by anything. At the same time, DPR contains sums of |F| and |G|, so the result will depend on 'S', as in |F|+S|G|. Thus, the proper definition of DPR should state that it is a minimum over 'S', the fact rarely brought up. The range of S has to be specified by the user and one is prompted for it. If there is no minimum within the range specified, the warning will appear. If the two structures compared originated from the same data set, the minimum should be found somewhere between 0.2, 2. If it does not, it is likely that one of the structures is wrong. If, on the other hand, 'RF 3' is used to calculate cross-resolution, i.e., to compare EM and X structures, the warning will almost always appear, as two structures are usually in completely different scales. - A correlation coefficient 'R' implies a SNR=2*R/(1-R). (Factor 2 comes from the fact that for the purpose of comparison the whole data set is divided into halves). SNR=4.0 implies R=0.67 SNR=2.0 implies R=0.5 Why these two special cases of SNR? According to Unser FSC=0.67 should correspond to DPH=45.

**SUBROUTINES**- PR3D, PR3DB, RFACTSD2, FMRS_3

**CALLER**- FOUR1

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