RF 3 - Phase Residual & Fourier shell correlation, 3D ||

PURPOSE

Calculate the differential 3-D phase residual and the Fourier Shell Correlation between two volumes. The Differential Phase Residual over a shell with thickness given by shell width and the Fourier Shell Correlation between shells of specified widths are computed and stored in the document file. Does not need powers of two dimensions (for exclusions see 'FT' operation) and takes real or Fourier input volumes. NSAM, NROW and NSLICE need to be the same in both volumes.

SEE ALSO

RF	[R Factor ||]
RF SN	[Spectral SNR of images ||]
FT	[Fourier Transform ||]

USAGE

.OPERATION: RF 3

<or>
RF 3 x11,x12

.FIRST INPUT FILE: SUB001
[Enter the name of the first input volume. File can be real or Fourier. ]

.SECOND INPUT FILE: SUB002
[Enter the name of the second input volume. File must be of same type and dimensions as first input file.]

.RING WIDTH: 1.
[Enter the shell thickness in reciprocal space sampling units.]

.SCALE FACTOR (LOWER,UPPER): 0.6,1.2
[Give the range of scale factors by which the second Fourier must be multiplied for the comparison.]

.MISSING CONE/WEDGE ANGLE (C/W): C
[Use 'C' if you have a missing cone and 'W' if you have a missing wedge.

.MAXIMUM TILT ANGLE: 30
[Give the angle of maximum tilt angle in degrees. (When the missing cone is covered angle = 90.0)]

.FACTOR FOR NOISE COMPARISON: 3.0
[The factor given here determines the FSCCRIT. Here 3.0 corresponds to the 3 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.]

.DOCUMENT FILE: DOC001
[Enter name of doc. file in which results are to be saved.]

NOTES

1. The inclination angle theta starts from the Z*=0 plane.

2. 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 ring and its minimum is used to determine the correct scale factor for the second Fourier transform.

3. 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 FRC results.

4. The "LACKS MINIMUM" 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 appear almost always, as two structures are usually in completely different scales.

5. COMMENTS and REFERENCES with regard to FSC Ref: Saxton and Baumeister, J.of.Micr., 127,(1982) 127-138. M. van Heel, Ultramicroscopy., 21, (1987) 95-100. Unser,et.al., Ultramicroscopy, 23, (1987) 39-52. A correlation coefficient "r" implies a SNR=2*r/(1-r). (factor 2 comes from the fact that for the purpose of comparison 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.

6. If registers were specified on the operation line then the first register receives the interpolated frequency (pixel) at when FSC falls below a value of 0.5, and the second register receives the interpolated spatial frequency at this point. If there are multiple downward crossings of the FSC cutoff line then the values are reported for the last crossing. If there is no such crossing then the last values from the doc. file output are reported.

7. The following measures are computed:

    
    a) PHASE RESIDUAL =   
         SQRT(SUM[(ABS(F1)+ABS(F2))*DPHI**2]/ SUM(ABS(F1)+ABS(F2)))  
         where DPHI = The phase difference between corresponding Fourier, 
         coefficients which should be < 45 DEG.  
    b) SHELL CORRELATION =  
         [SUM(F1*CONJ(F2))]/ [SQRT{SUM(ABS(F1)**2)*SUM(ABS(F2)**2)}]  
         where CONJ implies complex conjugate. 
      

   8. Contents of DOC and RESULTS file:

    
      COLUMN:       #1          #2      #3        #4          #5   
     |NUMBER|   |NORMALIZED    |DPH|   |FSC|   |FSCCRIT|    |VOXELS| 
                |FREQUENCY| 
      

SUBROUTINES

PR3D, PR3DB, RFACTSD2, FMRS_3

CALLER

FOUR1