In many applications it is desirable to have a precise definition of the term “correlation time”. Such a definition can be made in terms of

*γ*(

*t*), but there are different expressions of

*γ*(

*t*) reported in the literature [

23]. We employed Mandel’s definition of the correlation time:

It is straightforward to show that

Eq 2 satisfies

Eq 8, but

Eq 5 does not. To address this discrepancy, we propose an alternate expression for the Gaussian normalized autocorrelation function:

which satisfies

Eq. 8. Substituting

Eq. 9 into

Eq. 1, we obtain:

which is similar to

Eq. 4. Furthermore, the argument of the error function in

Eq. 10 is similar to that described previously by Goodman (Eq. 6.1–20 in [

20]) for a Gaussian spectral profile.

For

*T*/

*τ*_{c} > 2,

Eq. 10 becomes:

where

*τ*_{cga} is the correlation time for the re-derived Gaussian-based speckle imaging equation. Note that this equation is identical to that derived using the more common Lorentzian velocity distribution assumption (

Eq. 6, top row).

Thus, when the normalized autocorrelation function for the Gaussian and Lorentzian velocity distributions satisfy the same definition for the correlation time (

Eq. 8), then both approximations predict the same SFI values for low

*C* values (, 0 <

*C* < 0.6).

For

*T*/

*τ*_{c} 1,

Eqs. 3 and

10 can be approximated as

demonstrating that use of either the Lorentzian or rederived Gaussian velocity distribution assumption predict different

*τ*_{c} (and hence SFI) values. Although T/τ

_{c}1 is not encountered in typical LSI experiments,

Eq. 12 demonstrates that only in this range of ratios (T/τ

_{c}1) will the velocity distribution assumption affect the mapping between speckle contrast and τ

_{c}.

From Goodman’s theory on integrated intensity [

20], which takes into account the

*triangular averaging* of the correlation function [

19], it is straightforward to obtain:

and

where

*τ*_{clg} and

*τ*_{cgg} are the correlation times for the Lorentzian and Gaussian approximations, respectively. For

*T*/

*τ*_{c} > 2,

Eqs. 13 and

14 can be simplified to the following expressions:

The relationship between τ

_{clg} and C is similar to that derived by Cheng and Duong [

18].

Once again,

Eq. 15 suggests that for

*T*/

*τ*_{c} > 2 (i.e., 0 <

*C* < 0.6, see ), Goodman’s theory predicts the same SFI range for the Lorentzian and Gaussian velocity distributions. Moreover, from

Eqs. 6 (top row),

11 and

15, the SFI values predicted by Goodman’s model are directly proportional to the Lorentzian and the rederived Gaussian-based speckle imaging equations.

In Ref [

20], the signal-to-noise ratio associated with measurement of (1/

*C*^{2}) is given by:

The (

*S/N*)

_{rms} associated with

Eq. 15 is greater than

. At higher (

*C* > 0.6) speckle contrast values, (

*S/N*)

_{rms} is less than

, which is unacceptably low for practical application.

Eq. 16 is valid for both the Briers and Goodman models.

Briers et al. [

6] first noted that experimental

*C* values did not reach the theoretical limit of unity for completely stationary objects; they instead observed a maximum value of 0.6. Experimental data from Yuan et al. [

10] also achieved a maximum

*C* value of 0.6. Dunn et al. [

1] and Bolay et al. [

12] presented experimental data taken from cortical tissue with maximum

*C* values of ~0.15. In experimental LSI data that we acquire from rodent dorsal window chamber models [

15–

17], we typically observe

*C* values greater than 0.6 in less than 1% of the pixels (). It is important to note that measured

*C* values may differ among LSI instruments due to differences in parameters such as quality of imaging optics and camera, coherence length of incident light source, etc. Nevertheless, we believe these studies collectively justify the rationale for other researchers employing LSI to utilize the proposed simplified speckle imaging equation (

Eq. 15). An advantage of

Eq. 15 over either use of approximate solutions or look-up tables to extract

*τ*_{c} from the speckle imaging equation is that it represents an exact analytical solution for

*C* < 0.6.

Cheng and Duong [

18] stated that typically-encountered ratios of T/τ

_{c} are 100 to 400. Values greater than 100 are encountered in clearly defined blood vessels, but the ratio is much lower for pixels that map to poorly-perfused regions of tissue. For example, a speckle contrast of 0.6, which is encountered experimentally, maps to a ratio of two. Our analysis demonstrates that, even for such a low ratio, the simplified imaging algorithm can be used with high accuracy.