Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

with scatter separations of increasing odd integer multiples of =2. The simulated results show that the original ZAP helps to correct the speckled backscatter signal. However, the method is unable to fill in the right amount of energy for different degrees of destructive interference. One can see that the filling-in procedure is too weak in some cases but too strong in others. Figure 9b presents the ZAP-corrected envelope (solid line) by optimal rotation of ZAP zero. The broken line shows the profile of the ideal envelope. Figure 9c illustrates the zeros of the z-transformed demodulated A-lines before ZAP is applied. The asterisks show the locations of the ZAP zeros. Figure 9d gives the ZAP zeros after optimal rotation. Figure 9e shows the absolute value of the instantaneous phase change, which indicates that for backscatter signals formed with pairs of scatterers spaced at 0.5 fractional multiples of wavelength, there must be a phase change of where destructive interference occurs.

7.4.1Speckle Detection

In Eq. (12), the complex signal yðnÞ is represented as a convolution of the scatterer function xðnÞ with the PSF hðnÞ centered on a carrier with a period . The zeros of YðzÞ are the collection of the zeros of HðzÞ and XðzÞ. The signals from scatterers of amplitudes and suffer from destructive interference when the scatterers are separated by an integer distance of d ¼ ðN 1=2Þ for any integer N. For this case, the scatterer function is given by Eq. (11), which after z-transformation, yields

results from destructive interference of the signals generated by two scatterers andseparated by ðN 1=2Þ . We consider the zeros of xðnÞ first, which, from Eq. (14) are located in the z-domain at

There must exist an n ¼ ðN þ 1=2Þ 1=2 such that

z ¼ zZAP ¼ l=d ej2 =

The angular frequency of yðnÞ is ! ¼ 2 = , which is the same as the frequency of the zero given by Eq. (18). This derivation shows that destructive interference must produce (ZAP zero) at the signal angular frequency !. As zeros of the PSF do not lie close to its angular carrier frequency, zeros near ! must be from the scatterer sequence. Therefore, the presence of a zero at or near ! indicates the presence of speckle.

Optimization of the ZAP-Zero Rotation Angle

To correct the distortion caused by destructive interference of signals generated by pairs of scatterers, energy must be added back to the signal within the proper time interval to replace the lost signal energy. One way to accomplish this is to rotate the angle of the ZAP zeros. The closer the separation of the scatterers to 0.5 fractional multiples of the wavelength, the more rotation of the ZAP zeros is required. Rotation of the zeros by a constant angle according to the algorithm of the original ZAP does not properly correct destructive interference of A-lines of different integer and fractional wavelength multiples.

We carried out an extensive simulation of ZAP correction for various pairs of scatterer separations to find an optimal ZAP-zero rotation angle for each case. The ZAP zeros were rotated by an adjustable angle until the ZAP-corrected envelope matched the ideal incoherent envelope as closely as possible.

The phase angle of each zero is related to the separation of scatterers. In the present case, the separation distance is ðN1=2Þ , so the angle between any two adjacent scatterer zeros is 4 =ð2N 1Þ . In general, the adjacent scatterer zero-angle difference in terms of multiple of wavelength M is given by

where denotes the wavelength of the light source in number of samples. From Eq. (2), the exact scatterer separation of an A-line can be found given prior knowledge of the adjacent scatterer zero-angle separation. After this division, the angles trace out a triangular wave function related to the multiple of wavelength M and written as

where the symbols b c and d e mean that the quantity inside is rounded toward minus infinity and plus infinity, respectively. After normalizing the rotation angles with a best-fit line and taking in the square root, we obtain a normalized rotation function. This function has a simple profile that can be approximated by a triangular wave according to

Hence, once the distance between two scatterers is known, we can estimate how much a ZAP zero needs to be rotated to correctly fill in speckles.

Determination of Scatterer Separation

We now have a procedure for correcting speckled A-lines with an optimal amount of energy based on a modified form of ZAP. With the help of the original ZAP-zero angle and the sum of the absolute phase change, we can estimate the spacing between adjacent scatterers. The problem now is that we do not know the exact multiple of the wavelength by which a given pair of scatterers is separated because the adjacent

194 Schmitt et al.

scatterer zero separation is still unknown. However, according to Eq. (20), it is possible for us to extract an exact distance of scatterer separation in terms of multiples of wavelength from adjacent zero angle differences. The difference in angle between adjacent zeros can be calculated directly by finding the difference of the phase angles between that of a ZAP zero and the closest zero of the same magnitude within the angular bandwidth of the A-line. The zeros beside the asterisks at the months of two Y-structures in the bottom three plots of Fig. 9c illustrate such adjacent zeros. One might ask why we still need to calculate the fractional multiple from either the original ZAP-zero phase angle or the sum of phase change when both integer and fractional multiples can be obtained readily from adjacent zero angles. The reason is that the change of the adjacent zero separation angle becomes negligible for the same integermultiple of separation distance of widely separated scatterers. To avoid error in calculating the exact scatterer separation, we break the procedure down into three steps. First, an estimate of the exact scatterer spacing is found from the adjacent scatterer zero-angle difference by using Eq. (20). For an A-line for which the angle of ZAP zero is , the angle difference of adjacent scatterer zeros is and the summation of phase changes is P, the estimate of scatterer separation in terms of the multiple of wavelength can be found by evaluating 1ð Þ, where y ¼ f ðxÞ and x ¼ f 1ðyÞ are equivalent. Next, two sets of scatterer spacing are found by computing s 1ð=Þ andsum1 ðPÞ. Finally, an element is selected from each set of separations that are closest to the estimate in the first step. The two values are averaged to give a more accurate estimate of the scatterer separation than either of them alone.

A problem arises, however, when the scatterer spacing is between 0 and 6 integer multiples. Referring to the top two plots of Fig. 9c, we observe that no adjacent zeros can be found within the signal bandwidth. Adjacent zeros are hidden by PSF zeros. Without this information, it is impossible for us to know the estimated scatterer spacing. Therefore, when this situation arises, we simply approximate the estimated exact scatterer separation by the median of integer multiples with the problem and then go through the three steps above.

Figure 10 shows OCT images of tissue before and after processing with the original and modified ZAP. A weak low-pass filter was applied on all images in the lateral dimension to smooth the transitions between A-lines. The speckle fluctuations in both of the processed images are clearly smaller than in the original image; however, some features in the original ZAP-processed image are blurred. For reference, the image created by angle compounding using a four-element detector [11,13] is shown in Fig. 10d.

Our preliminary investigation of the ZAP suggested its potential to suppress speckle noise. The disadvantage with the traditional ZAP is that its smoothing effect is sometimes excessive. It is caused by the difficulties with ZAP in filling in speckles with the right amount of energy according to the local signal properties. The modified ZAP we presented in this section is more adaptive to speckles and linear in response. However, applied to real OCT signals, the modified ZAP does not perform as well as our theoretical analysis. One likely reason is that noise in weak OCT signals introduces false phase changes. Another problem is the inability of ZAP to cope with the speckle generated by very large numbers of scatterers in the sample volume. This problem also appears to be a major stumbling block in deconvolution and filtering of OCT images. A better phase-noise filtering procedure needs to be developed as an alternative to the thresholding function now used in our OCT

where f 0ðh; iÞ denotes the real OCT image pixels that need to be recovered: n0ðh; iÞ denotes independent, identically distributed x2 random variables with two degrees of freedom; and y0ðh; iÞ represents the observed noisy image data [24]. When the spatial resolution of the imaging system is small in comparison with the details of the imaged object, speckle can be modeled as Gaussian noise provided that the image is sampled coarsely enough that the degradation at any pixel is independent of the degradation at all other pixels [9].

Because y0ðh; iÞ and f 0ðh; iÞ represent image intensities and are therefore nonnegative, n0ðh; iÞ is also nonnegative. Applying the logarithmic operation to Eq. (23), we obtain

The logarithmic operation transforms the multiplicative noise n0ðh; iÞ into additive noise nðh; iÞ, which is amenable to restoration algorithms based on linear noise models [24]. Because the dynamic range of OCT interference signals is high (80– 110 dB), logarithmic compression is generally required before display. For displayed OCT images, the additive white Gaussian noise model is a reasonable approximation for speckle modeling.

7.5.2Maximization of the Signal-to-Noise Ratio

To deal further with Eq. (24), we now apply the separable dyadic multiresolution analysis developed by Mallat [34]. The basic concept of multiresolution decomposition is to split an image dataset into components of different resolutions. To achieve optimal noise reduction, the localized signal components are reduced if their magnitudes exceed a certain threshold.

Multiresolution analysis is carried out in a series of spaces Vm L2ðRÞ, m 2 Z, such that

V2 V1 V0 V 1 V 2

with

[m2ZVm ¼ L2ðRÞ and \m2Z ¼ f0g

which describes a successive approximation sequence to the space L2ðRÞ [34]. The symbol means ‘‘in’’; for example, V2 V1 means space V2 in space V1; m 2 Z means m in the integer space. More precisely, for a given energy-limited OCT image

Wsðj; k; lÞ for the signal coefficients and Wnðj; k; lÞ for the noise coefficients at different levels.

:

we obtain denoised coefficients sðj; k; l; Þ and nðj; k; l; Þ for the signal and noise, respectively.

To reconstruct the image signal foðh; i; Þ, which contains the residual image noise noðh; i; Þ, the inverse wavelet transform (IWT) is applied. By defining the SNR as

k l

represent the energy of the signal and the energy of the noise in the wavelet domain, respectively. The steps involved in simplifying Eq. (27) to Eq. (28) are given in Ref. 31. In Eq. (28), ½0; 1& is a constant that represents the signal inhomogeneity in the wavelet domain. When the image data are transformed into the wavelet domain, the noise projects into the whole wavelet space in ðj; k; lÞ, but the most significant coefficients of the signal, in general, project into a limited subspace ðj 0; k0; l 0Þ. The

j;k;l

where is the mean of signal coefficients Ws. Since, according to Eq. (31), is proportional to the ratio of the total number of wavelet coefficients whose absolute values lie above the mean of Ws and the dimension of the total wavelet space, a smooth image will give a lower value of than a sharp image containing more features of high spatial frequency.

trates the processing result obtained after applying a threshold 2.5 times the optimum. Although the speckle noise was reduced the most in this case, considerable blurring of the image is evident.

The ONWT technique presented in this section incorporates optimal nonlinear thresholding in multiple spatial frequency bands of OCT interference signals. Using this approach, the SNR is maximized with minimal loss of high spatial frequency information.

7.6CONCLUSION AND REMAINING PROBLEMS

Speckle is an undesirable consequence of all coherent or partially coherent imaging systems. Any overview of a topic as complex as speckle in OCT inevitably unearths a number of problems for future research. The following is a short list of problems that deserve special attention.

Current understanding of the classes of speckle and their origins is sketchy. More experimental work is needed to understand the relationship between the microscopic scattering properties of tissue and the statistical properties of speckle in OCT images, particularly the second-order properties that specify the distribution of the correlation spot sizes in the projected speckle patterns.

As a fundamental manifestation of coherent noise, speckle is a natural consequence of the limited spatial frequency bandwidth of an interferometric measurement system. To enable effective suppression of speckle effects in OCT, techniques for simultaneously widening the bandwidth of the light source and the light collection aperture must be developed.

Too few studies have focused on ways of adapting spatialand frequencycompounding methods employed in synthetic aperture radar and medical ultrasound to optical coherence tomography. More work in this area is needed.

The applications of complex-domain processing in OCT merit further investigation, especially those techniques that exploit the phase of the OCT signal.

Topics for further investigation in wavelet-based speckle reduction include methods for optimum selection of mother wavelets and for increasing execution speed.

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