Imaging-Notes

Incoherent Image formation¶

The essence of incoherent image formation is captured by the following equation:

$$I_{em}(x,y) = [I_{ex}(x,y) \times S(x,y)]*P\!S\!F_{em}(x,y) + N(x,y)$$

where,

$I_{em}(x,y)$ is the emission intensity observed at (x,y)

$I_{ex}(x,y)$ is the excitation intensity

$S(x,y)$ is the distribution of source (eg., fluorophores, emitters)

$P\!S\!F_{em}(x,y)$ is the blurring function for incoherent detection (convolution kernel)

$N(x,y)$ is additional noise

* denotes 2D convolution operation

This term, $[I_{ex}(x,y) \times S(x,y)]$, is the underlying object under observation. As seen here, it first gets blurred by a convolution kernel (PSF), then there are various noises that come for a ride, and this is the signal that we measure. Deconvolution is an attempt to solve the inverse-problem, given the measured image, an estimate of the convolution kernel and the noise

In frequency-domain, the above equation takes the following form:

$$I_{em}(k_x,k_y) = [I_{ex}(k_x,k_y) * S(k_x,k_y)] \times OT\!F_{em}(k_x,k_y)$$

where, $OT\!F_{em}(k_x,k_y)$, optical transfer function, is a Fourier transform of the point-spread function (PSF), so technically, it is a complex-valued function of spatial frequencies ($k_x$ and $k_y$), although for a PSF that is symmetric about its center, it is real-valued. It estimates how an imaging system handles various spatial frequencies as light travels from object space to image space. For a diffraction-limited optical system, the spatial frequency has a detection cutoff at $k_{em} = 2 N\!A/\lambda_{em}$. In other words, spatial frequencies higher than this don't make it through a diffraction-limited imaging system. Note that this cutoff here is taken as the inverse of Abbe lateral resolution

1-photon process¶

The density of flourophores excited in time $\Delta t$ during a single-photon fluorescence imaging can be written as follows (assuming fluorophores are below saturation):

$$\frac {\rho_{ex}}{\Delta t} = \rho_{o} \ \sigma_{1p} \Big[ \frac{n}{\Delta A \ \Delta t} \Big] $$

where,

$\rho_{ex}$: Density of excited fluorophores; [Unit: Volume$^{-1}$]

$\rho_{o}$: Density of fluorophores; [Unit: Volume$^{-1}$]

$\sigma_{1p}$: Fluorophore's absorption cross-section [Unit: Area]

$n$: Number of excitation photons in $\Delta t$ across an area of $\Delta A$

Note that: $\Big[ \frac{n}{\Delta A \ \Delta t} \Big] h \nu $ is beam energy per unit time per unit area, or beam power ($P_{\nu}$) per unit area, aka beam Intensity ($I_{\nu}$) [Unit of Watt/Area$^2$]

So, the above can be re-written as:

$$\frac {\rho_{ex}}{\Delta t} = \rho_{o} \ \sigma_{1p} \Big[ \frac{P_{\nu}}{h \nu \ \Delta A} \Big] = \rho_{o} \ \sigma_{1p} \Big[ \frac{I_{\nu}}{h \nu} \Big] $$

Next, not all excited fluorophores return to ground state and emit a photon (Stokes shift). Quantum efficiency ($\phi$) quantifies this as a ratio of excited fluorophores that emit a photon and the total number of fluorophores in the excited state.

Then, the emitted photon flux ($\Phi_{em}$, number of photons emitted per sec) from a small sub-volume ($\Delta x \Delta y \ \Delta z$) can be written as:

$$ \Phi_{em} = \phi \ \frac {\rho_{ex}}{\Delta t} \ \Delta x \ \Delta y \ \Delta z = \phi \ \rho_{o} \ \sigma_{1p} \frac{P_{\nu}}{h \nu} \ \Delta y $$

From this, we see that the number of photons emitted scales linearly with excitation power (only true before saturation of excited state population is reached; you can't drive more fluorophores into excited state than what's available). Note that Y is the beam propagation direction and the beam cross-sectional area is taken as $\Delta x \ \Delta z$

During imaging, only a portion of these emitted photons are captured by the detection objective (limited by the $N\!A$). Assuming no further loss in the beam path, some loss finally occurs during the photon-to-electron conversion at the detector chip (dictated by its quantum efficiency)

Estimates for Illumination (Gaussian-beam)¶

Note that there is no universally agreed metric to characterize focal spot of a Gaussian beam, so different metrics are in use:

$1/e^2$ width (Gaussian beam radius, $\omega_{o}$), is the American National Standard and is the most widely used metric to characterize focal-spot radius. $1/e^2$ thickness, $2 \ \omega_{o}$, includes light-cones with beam intensity down to 13.5% of the peak. The electric field strength at this location only drops to $1/e$ or 37% of the peak
For an aberration-free Gaussian beam (free-space wavelength of $\lambda_{LS}$) that is limited only by diffraction, the product of its far-field divergence angle $\theta$ and $\omega_{o}$ is a constant and is given by the following expression (also assuming beam quality factor, $M^2$ = 1): $$\omega_{o} \times \theta = \frac{\lambda}{\pi} = \frac{\lambda_{LS}}{n\pi}$$

which, in the paraxial limit is: $$\omega_{o} n \sin\theta = \frac{\lambda_{LS}}{\pi}$$ $$\omega_{o} = \frac {\lambda_{LS}} {\pi N\!A_{LS}}$$
$1/e$ width ($s_{o}$), is also used (less frequent) to characterize focal-spot radius. $1/{e}$ thickness, $2 \ s_{o}$, includes light-cones with beam intensity down to 37% of the peak. The electric field strength at this location only drops to $\sqrt{1/e}$ or 61% of the peak

$$s_{o} = \frac{\omega_{o}} {\sqrt{2}}$$

$D4\sigma$ is the ISO international standard to estimate beam diameter. It is 4 times the standard deviation of the intensity distribution along the major and minor axis of the beam. For a perfect Gaussian beam with zero background, this is equal to $1/e^2$ thickness.

Light-sheet thickness (FWHM)

Full-width-half-maximum (FWMH), also called half-power beam-width, has been adopted by the light-sheet community to estimate the thickness of light-sheet at the focal spot. FWHM only captures a portion of light-cone with beam intensity down to 50% of the peak. The electric field strength at this location only drops to $\sqrt{1/2}$ or 71% of the peak

$$ t_{FWHM} = \sqrt{2 \ln2} \ \omega_{o} = 2\sqrt{\ln2} \ s_{o} $$

In terms of focal spot diameter, $t_{FWHM} \ \ t_{1/e} \ \ t_{1/e^2}$ : $$= \Bigg[\frac{\sqrt{2\ln2}}{\pi} \ \ \ \ \frac{\sqrt{2}}{\pi} \ \ \ \ \frac{2}{\pi}\Bigg] \times \frac {\lambda_{LS}} {N\!A_{LS}} $$ $$\approx [0.375 \ \ \ 0.45 \ \ \ 0.64] \times \frac {\lambda_{LS}} {N\!A_{LS}}$$

$$t_{1/e} = 1.2 \times t_{FWHM}$$$$t_{1/e^2} = 1.7 \times t_{FWHM}$$

Light-sheet length (Confocal parameter)

$$ \Delta y = \frac {2 n \lambda_{L\!S}}{\pi N\!A^2_{L\!S}}$$

where, $N\!A_{LS}$ is the numerical aperture of the light-sheet excitation with a wavelength of $\lambda_{LS}$. A practical field of view (along beam propagation direction) is set by the confocal parameter of a focused Gaussian beam, which is twice the Rayleigh length

Estimates for Detection¶

$\lambda$ here refers to emission wavelengh, and $N\!A$ refers to the numerical aperture of the detection optics

Lateral resolution (wide-field) or Airy disk radius ($r_o$)

$$ r_o = P\!S\!F_{xy} \Bigg|_{Rayleigh} = \frac {1.22 f \lambda}{n D} = \frac {0.61 \lambda} {N\!A} $$

$f$ is the focal length of the lens, and $D$ is the aperture diameter.

Quick detour: $f/D$ is called the F-number. F-number tells us the size of the focal spot at the back focal plane in units of wavelength, when the front focal plane of the lens is evenly illuminated (note: not a Gaussian shaped illumination that falls of radially, but a uniform illumination i.e., a circ function). So, an F-number of 1 means the Airy disk radius (focal spot) will be on the order of 1 wavelength, whereas an F-number of 10 means the focal spot will be approx. 10x larger than the wavelength. In other words, F-number = 1 is a powerful lens whereas F-number = 10 is a lower power lens.

Also note that the Airy disk radius ($r_o$) approximately matches the expression for $1/e^2$ diameter of a Gaussian beam, $t_{1/e^2}$. So, the focal spot diameter of a Gaussian beam is also sometimes estimated by using the Airy disk radius expression, although the Airy disk itself has nothing to do with a Gaussian beam

Axial resolution (wide-field)

$$ P\!S\!F_{z} \Bigg|_{Rayleigh} = \frac {2 n \lambda} {N\!A^2} $$

Note that the above estimates are following the Rayleigh criterion, which is a widely-used criterion for the minimum resolvable detail. According to this, the imaging process is said to be diffraction-limited when the first Airy disk minimum of one point source coincides with the maximum of another nearby point source, and when the two point sources are brought closer than this limit, they can't be resolved unambiguously in the image

Resolution estimates are commonly carried out by measuring intensity signal from sub-diffraction-sized beads and measuring the FWHM along the lateral and axial dimensions.
Abbe's resolution estimates are also widely used and are slightly different:

$$ P\!S\!F_{xy} \Bigg|_{Abbe} = \frac {\lambda} {2 N\!A} $$$$ P\!S\!F_{z} \Bigg|_{Abbe} = \frac {2 \lambda} {N\!A^2} $$

As a rule of thumb, confocal resolution (with a really tiny pinhole) is approx. $\sqrt{2}$ better than wide-field estimates (both lateral and axial), whereas 2-photon is approx. $\sqrt{2}$ worse than wide-field. See this figure from Zipfel et. al., Nature Biotech, 2003 for 2-photon $1/e$ width estimates. The corresponding $1/e^2$ width will $\sqrt{2}\ \times$ of these expressions. Also, notice that the wavelength in these expressions is the 2-photon excitation wavelength, which is typically $2\times$ larger than 1-photon excitation wavelength

Axial resolution in light-sheet imaging

$$ P\!S\!F_{z} \Bigg|_{LS}= \Bigg[\frac {2 N\!A_{LS}} {\lambda_{LS}} + \frac {n\ (1 - \cos \alpha)} {\lambda}\Bigg]^{-1}$$

where, $\alpha = \sin^{-1}(N\!A/n)$ is the detection half-angle of collected light

The first term inside the brackets is the inverse of light-sheet thickness and the second term is the inverse of the detection instrument's axial resolution.^[1],[2] This equation serves as an upper-limit since all spatial frequencies are assumed to get through the instrument. As can be seen, for thin light-sheets, the first term dominates, whereas for thick light-sheets, the second term dominates. Thin or Thick with respect to what? That'd be in comparison to the depth of field (d) of the detection objective

Conceptually, imagine painting with two wet brushes-- the first brush stroke (excitation) creates a line and a much broader second brush (detection) then smears the paint in the orthogonal direction. From this picture, it's easy to see how we can reduce the smear caused by our broad second brush by using a fine brush stroke with the first one. This is essentially what lattice light-sheet does by engineering a thin Bessel beam light-sheet. Similarly, a fine brush stroke can be achieved by making a Gaussian light-sheet really thin. When the thickness starts to approach the wavelength of light though, the beam angles at focus become large and the paraxial approximation no longer applies.^[3] That means, we can no longer characterize the thickness and confocal parameter of light-sheets as described above- technically non-paraxial Gaussian light-sheets at this point.

Depth of field

$$ d = \frac {1} {2} \times P\!S\!F_{z}\Bigg|_{Rayleigh} = \frac {n \lambda} {N\!A^2} $$

This is the z-range in object space (sample space) over which the object is reasonably sharp

Depth of focus

$$D = d \times \frac {M^2}{n} = \frac {M^2 \lambda}{N\!A^2}$$

This in contrast is the z-range in image space over which the image remains reasonably sharp

Optics¶

Human eye:

Pupil diameter ($D$) of 3 mm (day) to 9 mm (dark) and a focal length of 22 mm. So,
- NA of $\frac{1.5 \ to \ 4.5 }{22} = 0.07 - 0.2$
- Rayleigh spatial resolution of 5 microns (day) and 2 microns (dark)
- Angular resolution ($\Theta$) of 1.22 x $\frac{\lambda}{D}$ of approx. 15 arcsecond (day) and 45 arcsecond (dark)
- In contrast, Galileo's microscope: 3 arcsecond, Hubble telescope: 0.05 arcsecond
- Note: 1 arcsecond is 1/3600 of a degree

Image Contrast, the different in signal between the image and the overall background, plays a critical role in live light-sheet imaging. Advanced light-sheet profiles that achieve higher axial resolution often do so at the expense of image contrast. No free lunch! Good-old Gaussian beam is the undisputed champion in the image contrast division!

Image Brightness is influenced by the photon flux (photons per unit area per unit time) detected by the camera chip. It is directly proportional to the detection NA and inversely proportional to the square of the lateral magnification.

Perceived Image Brightness is determined by the sensitivity and linearity of the sensors. The relationship between displayed ($I'$) and actual ($I$) pixel brightness is captured by a variable called gamma as $I' = \Big[ \frac{I-Min}{Max-Min} \Big] ^{\gamma}$. Scientific detectors (CCDs and sCMOS) perform this power-law correction using a built-in digital signal processing circuitry (eg, FPGAs). A $\gamma < 1$ means dark regions are accentuated, whereas a $\gamma > 1$ means bright regions are accentuated. A $\gamma = 0.7$ approximates the response of human eyes^[5] Understanding Gamma correction

Mirrors: Protected metallic mirrors (single reflective surface) are ideal for large angles of incidence and large cone angles-- expect light loss of >2-4%; Also, metallic mirrors tarnish over time and are also more susceptible to the environment. Broadband dielectric mirrors (layers of reflective surfaces) have excellent reflectivity (>99.9%) across a broad wavelength range, and are excellent choices for many applications. However, not a good fit at large angle of incident and large cone angles. Semrock's White paper on Mirrors

Optical tools¶

GFP is 28 kDa in size with a length of 4.2 nm and a diameter of about 2.4 nm; Fluorescence lifetime of GFP is about 2.4 ns; Quantum yield of 0.79

JF549 is 0.5 kDa in size; Fluorescence lifetime of 3.8 ns; Quantum yield of 0.88

Genetically-encoded Calcium Indicators (GECO) ^[4] Wide variety, each tailored to detecting neuronal activity in specific situation

To detect individual spikes, jGCaMP7s/f
- Need high sensitivity (SNR) for single action potential
- jGCaMP7s (single AP SNR of 17) and jGCaMP7f (signel AP SNR of 9) are ideal
- Kinetics (single AP): Similar half-rise time (70-75 ms), half-decay time of 1.7 sec/520 ms for jGCaMP7s/jGCaMP7f

To image neurites and neuropil or sparsely labeled neurons, jGCaMP7b
- Need brighter baseline fluorescence to pick out weak signals from small features
- Kinetics (single AP): half-rise time of 80 ms, half-decay time of 850 ms