It is advantageous to perform a noise analysis of the fixed pattern after Fourier transform since each noise source is located at an axial point while its effect is spread over the entire spectral range in the OCT spectrum. Due to the linear property of Fourier transform (denoted by F
} in this report), the Fourier transform of an averaged spectrum is equal to the average of the transformed A-line spectra:
is the number of A-lines involved with the averaging. Thus, the mean-spectrum subtraction can be replaced by its transformed version. The mean A-line
or mean line
, can be defined in the same way as Eq. (2)
after Fourier transform:
) is an A-line that consists of complex-reflectance data points along the axial coordinate z
. In the mean-line subtraction
method, a final A-line is obtained by subtracting the mean line from each A-line (gl
to remove the fixed-pattern noise. This method produces a mathematically equal result to that of the mean-spectrum subtraction as Eq. (3)
suggests. Only the mean-spectrum subtraction has been used in the OCT signal processing so far, probably because of its conceptual simplicity and computational ease of processing real numbers. Even though the mean-line subtraction is mathematically equivalent to the mean-spectrum subtraction, this approach gives more insight on the characteristic of the fixed-pattern noise. It becomes easier to isolate the cause of the error found in those methods after the transform.
In this study, it was found that a statistical characteristic of the OCT image can result in residual noise lines as a consequence of averaging complex reflectance signals of a non-monotonic probability distribution. In addition, the noise source can be differentiated in the depth-resolved domain of OCT with ease. As a case study, the 162nd horizontal line which is highlighted by a blue arrow in was investigated in detail. Since the raw A-line data were given as a set of complex numbers, a statistical analysis could be performed with a complex plane that represents a distribution of complex data points (amplitude and phase) in a Cartesian-coordinate space.
shows the complex-plane representation of the A-line data (a), their histogram on the real axis (b), histogram on the imaginary axis (c), and the magnified version ( × 20) of the complex plane for the central area (d). The blue dots in the complex-plane representation stand for complex reflectance data (x + iy) obtained from the horizontal line of at the axial position of observation.
Fig. 2 Complex-plane representation of the A-line data (a), their histogram on the real axis (b), their histogram on the imaginary axis (c), and the magnified version ( × 20) of the complex plane for the central area (d). The data are from the 162nd (more ...)
The overall distribution of the data points resembled a two-dimensional Gaussian distribution centered at a certain complex number as depicted by . This distribution was shifted by a systematic bias that a reference signal gives to the measured interference spectra. This shift resulted in the fixed horizontal pattern visible in the final image. The mean-spectrum subtraction method, or the mean-line subtraction, is understood as a simple way to find the center of the data point distribution by taking the mean vector (i.e., the mean complex number). In this method, each dot is shifted back by this mean vector for the final distribution to be relocated around the origin in an even and symmetric distribution manner. However, the mean vector may not coincide with the effective center of the distribution because of the statistical oddness. As observed in , some distinguished points have distinctly high amplitudes while the others are densely located around the origin with low amplitudes. These high-amplitude points are distributed in an odd and asymmetric manner so that their distribution does not look purely random. These were generated by a different reflectance source from those of the central area and obeyed a distinguished distribution function as a consequence. The high-amplitude points came from the high-contrast interface of the tissue and the air shown by a bright horizontal curve in . Their amplitudes were far higher than the others in one or two orders of magnitude, i.e., 20 to 40 dB above the reflectivities of the other ordinary points. Thus, the average of the high-amplitude points has a chance to produce a statistical fluctuation that is stronger than the low amplitudes of the ordinary data points.
Such a residual noise can be produced by two mechanisms. First, the phase of such a high-amplitude point may not be random. As seen in , the high-amplitude points did not exhibit completely random phases but rather was concentrated to the lower right corner (x>0, y<0) with a certain average phase. It was obvious that the smooth tissue-air interface of the sample could make a certain correlation of those phases. In , the black dot represents the mean complex number which was averaged over the whole data for the 162nd line. It was offset to the lower right with respect to the effective center of the distribution as anticipated by the odd phase distribution of the high-amplitude points.
Second, even under an assumption of nearly random phases without an apparent correlation, a mean value calculated from finite sampled information deviates as a result of sampling fluctuation. A morphological roughness of the tissue-air interface determines the phase of the reflectance. As the deterministic motion of a roulette wheel looks random in its result, the phase of the sample surface can be understood as a sampled value of an apparently random variable. However, a statistical sampling does not exactly conserve the statistical moment in a stochastic process. For example, an expectation value of a numbered cubic die after a finite number of measurement samples is not exactly 3.5 (1/6 + 2/6 + 3/6 +
+ 6/6) but fluctuates around 3.5 because of the sampling fluctuation. Thus, even if the surface morphology provides no apparent phase correlation, a mean of apparently random complex reflectance cannot be zero. The standard deviation of such a statistical fluctuation decreases in an inversely proportional rate to the square root of the number of the statistical samples which are involved with the mean. Hence, for the number of high-amplitude points, M
, their mean produces a sampling error of a standard deviation to be (1/M
of the initial deviation of the samples which roughly equals the mean amplitude. In a rough estimation for the given case of , the sampling error in the mean of the high-amplitude points must have been larger than 1/10 of their average amplitude for M
<100. This error is still higher than the low amplitudes of a majority of data points located around the center. Therefore, even under the most favorable assumption of the random phase distribution, a mean can produce an error in finding the distribution center when a part of the data points have extraordinarily high amplitudes beyond the magnitude expected by a monotonic normal distribution.