Laser speckle contrast imaging

For a fully developed speckle pattern which formed when the complete coherent and polarized light illuminate a static medium, the contrast (K) range from 0 to 1 is defined by the ratio between the standard deviation and mean intensity:[6][7]

${\displaystyle K={\frac {\sigma }{\langle I\rangle }}}$

The intensity distribution of the speckle pattern will be used to compute the contrast value.

Autocorrelation functions of electric field are used to measure the relationship between contrast and the motion of scatterers because the intensity fluctuations are produced by electric field changes of scatterers. E(t) is the electric field over time, E* is the complex conjugate of electric field and ${\displaystyle \tau }$ is the autocorrelation delay time.[7][6]

${\displaystyle g_{1}(\tau )={\frac {\langle E(t)\cdot E^{*}(t+\tau )\rangle }{\langle E(t)\cdot E^{*}(t)\rangle }}}$

Bandyopadhyay et al. showed that the reduced intensity variances of speckle pattern are related to ${\displaystyle g_{1}}$. Therefore, the contrast can be written as[7][8][9]

${\displaystyle K(T)^{2}={\frac {2\beta }{T}}\int \limits _{0}^{T}|g_{1}(\tau )|^{2}\left(1-{\frac {\tau }{T}}\right)\mathrm {d} \tau }$

where T is the exposure time. The normalization constant ${\displaystyle \beta }$ takes into account the loss of correlation due to the detector pixel size, and depolarization of the light through the medium.

Dynamic scatterers' motion can be classified into two categories, one is the ordered motion and the other one is disordered motion. The ordered motion is the ordered flow of scattered while the disordered motion is caused by the temperature effects. The total dynamic scatterers' motions were thought of as Brownian motion historically, the approximate velocity distribution of Brownian motion can be considered as the Lorentzian profile. However, the ordered motion in dynamic scatterers follows Gaussian distribution. When considering the motion distribution, the contrast equation related to the autocorrelation can be updated. The updated equations are as follows, ${\displaystyle K^{L}}$ is the contrast equation function in Lorentzian profile and ${\displaystyle K^{G}}$ is the contrast equation function in Gaussian profile. ${\displaystyle \tau }$ is the decorrelation time. Both equations can be used in contrast measurement, some scientists also use contrast equations with the combination of them. However, what the correct theoretical contrast equation should be is still under investigation.[7][10][11]

${\displaystyle K^{L}(T)^{2}=\beta {\begin{bmatrix}{\frac {\tau _{cl}}{T}}+{\frac {(\tau _{cl})^{2}}{2T^{2}}}(e^{-2T/\tau _{cl}}-1)\end{bmatrix}}}$

${\displaystyle K^{G}(T)^{2}=\beta {\begin{bmatrix}{\frac {\tau _{cg}}{T}}{\sqrt {\frac {\pi }{2}}}\mathrm {erf} ({\sqrt {2}}T/\tau _{cg})+{\frac {\tau _{cg}^{2}}{2T^{2}}}(e^{-2T^{2}/\tau _{cg}^{2}}-1)\end{bmatrix}}}$

${\displaystyle \beta }$ is the normalization constants that vary in different LSCI systems, the value of it is ${\displaystyle \leq }$ 1, the most common method to determine the value of it is using the following equation. ${\displaystyle \beta }$ is account for the instability and maximum contrast of each LSCI system.

${\displaystyle \beta =\lim _{T\to 0}K(T)}$

Static scatterers are present in the assessed sample, speckle contrast produced by static scatterers remains constant. By adding statics scatterers, the contrast equation can be updated again.

${\displaystyle K(T)^{2}={\frac {2\beta }{T}}\int \limits _{0}^{T}P_{1}^{2}|g_{1}(\tau )|^{2}\left(1-{\frac {\tau }{T}}\right)P_{2}^{2}\mathrm {d} \tau }$

*The above equation did not account for the motion distributions.

P1 and P2 are two constants that range from 0 to 1, they are determined by fitting this equation to the actual experimental data.[7]

The relationship between the velocity of scatterers and decorrelation time is as follows, velocity of scatterers such as the blood flow is proportional to the decorrelation time, ${\displaystyle \lambda }$ is the laser light wavelength.[7][11]

${\displaystyle V={\frac {\lambda }{2\pi \tau _{c}}}}$

The method to compute the contrast of speckle patterns can be classified into three categories: s-K (spatial), t-K (temporal), and st-K (Spatio-temporal). To compute the spatial contrast, raw images of laser speckle will be separated into small elements, and each element corresponds to a ${\displaystyle n\times n}$ pixels. The value of ${\displaystyle n}$ is determined by the speckle size. The intensity of all the pixels in each element will be summed and averaged to return a mean intensity value (μ), the final contrast value of this element will be calculated based on the mean intensity and actual intensity of each pixel. To improve the resolution limitation, scientists also compute the temporal contrast of the speckle pattern. The method is the same as how to compute spatial contrast but just in temporal. The combination computation of spatial contrast and temporal contrast is Spatio-temporal contrast processing algorithm and this is the most commonly used one.[7][11]

• Several parameters should take into considerations to maximum contrast and signal to noise ratio (SNR) of LSCI. The size of individual speckle is essential and it will determine the requirement of the photodetector. The size of each speckle pattern should smaller than the photodetector's pixel size to avoid the decrease of contrast. The minimum speckle diameter for an LSCI system depends on the wavelength of light, imaging system magnification, and imaging system f-number: ${\displaystyle d_{\min }\thickapprox 1.2(1+M)\lambda f/\#}$[7]
• Measurement of normalization constant ${\displaystyle \beta }$, static scatters is necessary, as they can determine the maximum contrast the LSCI system can obtained.[7]
• Both too short or too long exposure time (T) can decrease the efficiency of the LSCI system as too short exposure can not ensure the adequate photons to be accumulated while too long exposure time can reduce contrast. Suitable T should be analyzed in advance.[7]
• The illumination angle should be considered to achieve higher light transmittance efficiency.[7]
• Suitable laser source should be chosen to get rid of a decrease in contrast and SNR.[12]