The signal-to-noise ratio (SNR) relates the mean power reflectance, I
, to the standard deviation of the power reflectance, σI
: SNR = I
, where the average is performed over pixels from an optically homogeneous region [7
]. We note that the SNR differs from the system sensitivity, which is the minimum detectable reflectance in the absence of speckle. The dominant contribution to signal variation within scattering regions is typically speckle. When angular compounding is performed, the SNR is increased, and the magnitude of the increase is determined by the extent to which angular samples are correlated. We emphasize that the SNR differs from the system sensitivity, as the latter is a measure of the minimum detectable reflectance in the absence of speckle.
In this section, we present a computational framework that relates the SNR improvement resulting from angular compounding to the system optical parameters. It is constructed from more detailed analyses of OCT speckle statistics that do not incorporate angular measurements [13
]. We employ a linear systems framework, because the measured backscattered field S
and the actual backscattered field G
corresponding to a particular spatial volume element are related by the convolution operator
The parameter x is related to the angular backscattering angle θ and the focal length of the lens f as x = f tan(θ). Simply put, it is the radial displacement of a beam that has been backscattered at angle θ, as measured on the side of the imaging lens L1 opposite to the sample (). The angular response kernel K is the field magnitude of the collimation beam, and it is responsible for the correlations between angular samples. K operates separately on the real and imaginary parts of G, so that S is complex-valued. It was measured by directing light from the collection collimator at an area camera, and calculating the square root of the measured radial intensity distribution.
Fig. 2 Scattering geometry employed in the speckle model showing the incident beam traversing the center of the lens L1 (focal length f) and light backscattered at angle θ. After traversing L1, the backscattered light is radially displaced by a distance (more ...)
Using the linear systems framework above, SNR improvements can be determined numerically using a numerical simulation consisting of the following steps:
- Simulating the speckle field G by drawing n independent and identically-distributed (I.I.D.) random values from a standard normal distribution, independently for real and imaginary parts of G, where n is the number of measured angular data points.
- Calculating the effect of the angular response by convolving G with K;
- Calculating the power reflectance for each angular data point by taking the magnitude-squared of the n values obtained from step b), so that the resulting probability distribution for a given backscattering angle is exponential;
- Calculating the SNR improvements for averages over different numbers of angular data points, ranging from 1 to n. Averages are performed with the results of part c);
- Repeating steps a) to d) and averaging the results. The number of repetitions was chosen to be 10,000.
The SNR is known to increase in proportion to the square root of the number of uncorrelated angular samples [1
]. For the case where angular correlations are present, we define the effective
number of uncorrelated angles as the square root of the SNR improvement obtained when all angular data points are included in angular compounding. In this simulation, the number of effective angles is determined by two factors: the diameter of the lens, which determined the minimum and maximum values of x
, and the width of the collection beam, as determined by K
. We note that there were no arbitrary parameters in the simulation.