Why the amplitude of Ram-Lak filter is multiplied by 2 in iradon transform?

The FBP reconstrcution algorithm tells us to convolve the sinogram using Ram-Lak Filter (Ramp Filter) along detector channels for every view. Since the Ramp Kernel is generated using the well-known form:

1/4              n = 0
0                n even
-1/(pi*n)^2      n odd

MATLAB iradon function performs filtering in Fourier Domain (Frequency Domain), but the amplitude is multiplied by 2:

filtImpResp = [filtImpResp filtImpResp(end-1:-1:2)]; 
filt = 2*real(fft(filtImpResp)); % Here is a factor 2, why?
filt = filt(1:(order/2)+1);

For other implementations like in ASTRA Toolbox, they perform filtering in Spatial Domain (Time Domain), there is also a factor 2 to adjust the amplitude:

pfFilt[iDetectorIndex] = 2.0f * pfData[2*iDetectorIndex];

Without this factor, the value of the reconstructed image is HALF of the ground-truth.

[Matt J]

Without this factor, the value of the reconstructed image is HALF of the ground-truth.

I think you've answered your own question. The factor of 2 is there, because you get the wrong values without it.

[z0gSh1u]

Yeah.. But why? filtImpResp is the impulse response of Ramp in Spatial domain. Convolution with it is equal to multiplying fft(filtImpResp) in Fourier domain. The multiplier 2 just comes from nowhere. @Matt J

[Matt J]

Well, I gave you a possiblity in my answer below.

Answer 1

[Matt J]

Perhaps it is different from what you expect because the backprojection sums only from 0 to 180 degrees. If you summed from 0 to 360, you would not include the factor of 2.