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loginorsinup
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Hello everyone.
I'm trying to better understand structured illumination microscopy and in the literature, I keep coming across bits of text like this.
Source: http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/SAS_JOSAA09_PhShiftEstSupRes.pdf
From Fourier analysis, if I take the Fourier transform ##X(f)## of a time-varying function ##x(t)## that is a cosine, I get a pair of delta functions (quick derivation below).
\begin{align*}
X(f) & = \int_{-\infty}^{\infty} x(t)\exp(-i2\pi f t)\; dt\\
x(t) &= \cos(2 \pi f_0 t) = \frac{1}{2}\left[\exp(i2\pi f_0 t) + \exp(-i2\pi f_0 t)\right]\\
X(f) &= \int_{-\infty}^{\infty} \frac{1}{2}\left[\exp(i2\pi f_0 t) + \exp(-i2\pi f_0 t)\right]\exp(-i2\pi f t)\; dt\\
&= \frac{1}{2}[\int_{-\infty}^{\infty} \exp(i2\pi f_0 t)\exp(-i2\pi f t)\; dt + \int_{-\infty}^{\infty} \exp(-i2\pi f_0 t)\exp(-i2\pi f t)\; dt\\
&= \frac{1}{2}[\mathcal{F}\left\{\exp(i2\pi f_0 t)\right\} + \mathcal{F}\left\{\exp(-i2\pi f_0 t)\right\}]\\
\mathcal{F}\left\{\exp(i2\pi f_0 t)\right\} &= \delta(f-f_0)\\
\mathcal{F}\left\{\exp(-i2\pi f_0 t)\right\} &= \delta(f+f_0)\\
\therefore X(f) &= \frac{1}{2}[\delta(f-f_0) + \delta(f+f_0)]
\end{align*}
But the paper says that I should be getting three impulses. One at the origin, and the two I have detailed above. Where does the one at the origin come from?
My only hunch so far is it might have something to do with the fact that this derivation I just did was in 1D, and what they are describing is in 2D (a surface). Of course, the other minor difference is that they are describing spatial frequency and I am describing temporal frequency, but replacing ##t## with ##x## and ##f## with a scaled ##k## (spatial frequency) isn't a big deal.
Any tips?
I'm trying to better understand structured illumination microscopy and in the literature, I keep coming across bits of text like this.
Source: http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/SAS_JOSAA09_PhShiftEstSupRes.pdf
From Fourier analysis, if I take the Fourier transform ##X(f)## of a time-varying function ##x(t)## that is a cosine, I get a pair of delta functions (quick derivation below).
\begin{align*}
X(f) & = \int_{-\infty}^{\infty} x(t)\exp(-i2\pi f t)\; dt\\
x(t) &= \cos(2 \pi f_0 t) = \frac{1}{2}\left[\exp(i2\pi f_0 t) + \exp(-i2\pi f_0 t)\right]\\
X(f) &= \int_{-\infty}^{\infty} \frac{1}{2}\left[\exp(i2\pi f_0 t) + \exp(-i2\pi f_0 t)\right]\exp(-i2\pi f t)\; dt\\
&= \frac{1}{2}[\int_{-\infty}^{\infty} \exp(i2\pi f_0 t)\exp(-i2\pi f t)\; dt + \int_{-\infty}^{\infty} \exp(-i2\pi f_0 t)\exp(-i2\pi f t)\; dt\\
&= \frac{1}{2}[\mathcal{F}\left\{\exp(i2\pi f_0 t)\right\} + \mathcal{F}\left\{\exp(-i2\pi f_0 t)\right\}]\\
\mathcal{F}\left\{\exp(i2\pi f_0 t)\right\} &= \delta(f-f_0)\\
\mathcal{F}\left\{\exp(-i2\pi f_0 t)\right\} &= \delta(f+f_0)\\
\therefore X(f) &= \frac{1}{2}[\delta(f-f_0) + \delta(f+f_0)]
\end{align*}
But the paper says that I should be getting three impulses. One at the origin, and the two I have detailed above. Where does the one at the origin come from?
My only hunch so far is it might have something to do with the fact that this derivation I just did was in 1D, and what they are describing is in 2D (a surface). Of course, the other minor difference is that they are describing spatial frequency and I am describing temporal frequency, but replacing ##t## with ##x## and ##f## with a scaled ##k## (spatial frequency) isn't a big deal.
Any tips?
Last edited:
