HowManyReferenceProjections

How many reference projections

How many reference projections are needed to cover the whole projection sphere (no mirrors and no symmetry are considered) with an approximately uniform distribution of samples?

Let us assume that the angular step is deltaAng.

The number of samples in tilt is Ntilt=round(180/deltaAng). The number of samples in rot at the equator is Nrot=round(360/deltaAng). At each tilt sample (tilt_i=180i), the deltaRot_i is modified to deltaRot_i=deltaAng/abs(sind(deltaAngi)), and the number of samples at that circle is round(360/deltaRot_i). Mathematically, the number of samples is approximately

sum_{i=1}^{Ntilt} {Nrot*abs(sind(tilt_i))}

Here is a MATLAB code that computes it

function Nsamples=even_samples_sphere(deltaAng)
    Ntilt=round(180/deltaAng);
    Nrot=round(360/deltaAng);
    Nsamples=2; % Corresponding to tilt=0 and tilt=180
    for i=1:Ntilt-1
        deltaRoti=deltaAng/abs(sind(deltaAng*i));
        Nsamples=Nsamples+round(360/deltaRoti);
    end
end

This value can be approximated by the following closed formula

function Nsamples=even_samples_sphere2(deltaAng)
    Nsamples=2*round(1+round(360/deltaAng)*...
        sind(0.5*(90/deltaAng+1)*deltaAng)*...
        sind(0.5*(90/deltaAng)*deltaAng)/...
        sind(deltaAng/2));
end

The formula comes simply from summing the number of samples in tilt from 0 to 90 and multiplying by 2. Here we have used the formula

sum_{i=1}^n {sin(nx)}=(sin(0.5*(n+1)*x)*sin(0.5*n*x))/sin(x/2)

Here is a table of the exact values computed with the first function

 deltaAng=  0.5: 165016
 deltaAng=  1.0: 41258
 deltaAng=  1.5: 18338
 deltaAng=  2.0: 10318
 deltaAng=  2.5: 6600
 deltaAng=  3.0: 4586
 deltaAng=  3.5: 3367
 deltaAng=  4.0: 2586
 deltaAng=  4.5: 2042
 deltaAng=  5.0: 1652
 deltaAng=  5.5: 1367
 deltaAng=  6.0: 1148
 deltaAng=  6.5: 977
 deltaAng=  7.0: 843
 deltaAng=  7.5: 732
 deltaAng=  8.0: 643
 deltaAng=  8.5: 569
 deltaAng=  9.0: 510
 deltaAng=  9.5: 460
 deltaAng= 10.0: 412
 deltaAng= 11.0: 340
 deltaAng= 12.0: 288
 deltaAng= 13.0: 245
 deltaAng= 14.0: 211
 deltaAng= 15.0: 184
 deltaAng= 16.0: 161
 deltaAng= 17.0: 146
 deltaAng= 18.0: 128
 deltaAng= 19.0: 113
 deltaAng= 20.0: 106
 deltaAng= 25.0: 66
 deltaAng= 30.0: 46
 deltaAng= 35.0: 35
 deltaAng= 40.0: 28
 deltaAng= 45.0: 22
 deltaAng= 50.0: 19
 deltaAng= 55.0: 13
 deltaAng= 60.0: 12
 deltaAng= 65.0: 11
 deltaAng= 70.0: 10
 deltaAng= 75.0: 7
 deltaAng= 80.0: 6
 deltaAng= 85.0: 6
 deltaAng= 90.0: 6

-- Main.CoSS - 17 Jan 2008