1.

Introduction

Optical coherence tomography (OCT) is a noninvasive optical imaging technique used to perform high-resolution cross-sectional in vivo and in situ imaging of microstructure in biological tissues.¹ OCT imaging of the cavernous nerves in the rat and human prostate has recently been demonstrated.^{2, 3, 4, 5} However, improvements in the quality of the images for identification of the cavernous nerves during prostate cancer surgery would aid preservation of the nerves and improve postoperative sexual function.

Speckle occurs in the OCT image when the particles composed in the underlying tissue structures are smaller than the coherence length of the light source.⁶ Because the speckle pattern is uncorrelated at different positions, angles, and optical wavelengths, speckle reduction using spatial and frequency compounding techniques cannot remove speckle from images acquired with standard OCT systems. Recently, wavelet techniques have been employed successfully in speckle noise reduction for OCT images.^{7, 8} Wavelet shrinkage denoising is denoising by nonlinear soft thresholding in the wavelet transform domain. Adler ⁷ applied the discrete wavelet transform (DWT) which provides the most compact representation; however, it has several limitations. The dual-tree complex wavelet transform (CDWT) overcomes these limitations because it is nearly shift-invariant and is oriented in 2-D.^{8, 9} The complex wavelet transform was used for wavelet denoising in OCT images.^{10, 11} Forouzanfar ¹⁰ applied the directional Bayesian estimator to remove noise from the wavelet coefficients, and we¹¹ applied a single threshold globally for all relevant parts of the transform in OCT images of experimental systems.

In this study, we have gone one main step further to apply the wavelet shrinkage technique, using CDWT, to an FDA approved clinical endoscopic OCT system. For this purpose, a bivariate shrinkage with local variance estimation algorithm is applied,¹² because the performance of the wavelet shrinkage algorithm for OCT image denoising can be improved by considering the statistical dependencies among wavelet coefficients. We compare the results of the DWT and CDWT algorithms for wavelet shrinkage denoising of in vivo OCT images of the rat prostate.

2.

Wavelet shrinkage denoising

Denoising attempts to remove noise and retain signal regardless of the frequency content of the signal. Wavelet shrinkage is denoising by shrinking (nonlinear soft thresholding) in the wavelet transform domain.

The observed image, $X$ , is modeled as an uncorrupted image, $S$ , and multiplicative speckle noise, $N$ . On a logarithmic scale, speckle is converted to additive noise, $X=S+N$ . The wavelet shrinkage denoising algorithm requires the following four-step procedure,¹³

3.

Wavelet transform

Operator $W(.)$ relates to the wavelet transform, which is the representation of a function by scaled and translated copies of a finite-length or fast-decaying oscillating waveform, which can be used to analyze signals at multiple scales. Wavelet coefficients carry both time and frequency information, as the basis functions vary in position and scale.

3.1.

2-D DWT

The DWT converts a signal to its wavelet representation. In a one-level DWT, a signal $c_0$ is split into an approximation part $c_1$ and a detail part $d_1$ . In a multilevel DWT, each subsequent $c_i$ is split into an approximation $c_{i+1}$ and detail $d_{i+1}$ . For 2-D images, each $c_i$ is split into an approximation $c_{i+1}$ and three detail channels $d_{i+1}^1$ , $d_{i+1}^2$ and $d_{i+1}^3$ for horizontally, vertically, and diagonally oriented details, respectively, Figures 1a and 1b . The inverse DWT reconstructs each $c_i$ from $c_{i+1}$ and $d_{i+1}$ .

Fig. 1

(a) Ordering of the approximation and detail coefficients of a two-level 2-D DWT; (b) a three-level DWT; (c) a three-level CDWT wavelet filter banks.

4.

Shrinkage denoising

Operator $d(.)$ in Eq. 1 selects a data-adaptive threshold, and $D(.,\lambda )$ denotes the denoising operator with threshold $\lambda$ . Bivariate shrinkage with local variance estimation algorithm¹² is applied for shrinkage denoising. After estimating the signal components of the noisy coefficients in the wavelet domain, the inverse wavelet transform, $W^{-1}$ , is taken to reconstruct the noise-free image.

5.

Results

Figure 2 shows time-domain OCT images of the cavernous nerves at different orientations (longitudinal, oblique, and cross-sectional) coursing along the surface of the rat prostate. OCT images were taken in vivo using a clinical endoscopic OCT system (Imalux, Cleveland, OH) which is based on a novel, low-maintenance, low-cost, lightweight OCT system using an all single-mode fiber common-path interferometer-based scanning system (Optiphase, Van Nuys, CA). Histologic sections of the cavernous nerves were processed for comparison, Fig. 2. Figure 3 shows the images after denoising using DWT and CDWT. Image-quality metrics were used to assess performance by measuring the average contrast-to-noise ratio, ${\mathrm{CNR}}_n=10\times \log ({\mu }_n-{\mu }_b∕\sqrt{{\sigma }_n^2+{\sigma }_b^2})$ ,⁶ and the average equivalent number of looks, ${\mathrm{ENL}}_n={\mu }_n^2∕{\sigma }_n^2$ ,⁷ over the region of interest which is the location of the cavernous nerve in each image. ${\mu }_n$ is the mean of the pixel values in the location of the cavernous nerve, ${\sigma }_{\mathrm{n}}$ is the pixel standard deviation, and ${\mu }_{\mathrm{b}}$ and ${\sigma }_{\mathrm{b}}$ are the pixel mean and standard deviation of a background region of the image, respectively. All calculations are made after logarithmic transformation of the image. In addition, the global SNR is calculated as $\mathrm{SNR}=10\times \log [\max {\left(X_{\mathrm{lin}}\right)}^2∕{\sigma }_{\mathrm{lin}}^2]$ , where $X_{\mathrm{lin}}$ is the 2-D matrix of pixel values in the OCT image and ${\sigma }_{\mathrm{lin}}^2$ is the noise variance, both on linear intensity scales.⁶

Fig. 2

OCT and corresponding histologic images of the rat cavernous nerve: (a,b) longitudinal section, (c,d) oblique section, and (e,f) cross section. In each section, the cavernous nerve corresponds to a relatively high signal intensity (shown in red) and lies superficial to the prostate. (color online only.)

Fig. 3

OCT images of the rat cavernous nerve after wavelet shrinkage denoising: (a,c,e) using DWT and (b,d,f) using CDWT. In comparison to DWT, CDWT has the advantages of being nearly shift-invariant and oriented in two dimensions.

The values of CNR, ENL and SNR for nine sample images show significant improvement, Table 1 . A paired t-test was performed for comparison of original and denoised images, with statistical significance given by values of $P<0.05$ . Comparison of original and CDWT denoised images results in $P$ values for CNR, ENL, and SNR of 0.0003, 0.07, and 0.0001, respectively. The values for CNR and SNR are considered to be statistically significant. Comparison of DWT and CDWT results in $P$ values for CNR, ENL, and SNR of 0.0001, 0.05, and 0.9, respectively. The values for CNR and ENL are considered to be statistically significant.

Table 1

Image quality values.

DWT lacks the shift-invariance property, and in multiple dimensions it performs poorly for distinguishing orientations, which is important in image processing. In other words, this lack of orientation selectivity greatly complicates modeling and processing of geometric image features like the ridges and edges in which the cavernous nerves belong.¹⁶ Therefore, denoising using CDWT, which is more selective with respect to orientation and provides approximate shift-invariance, overcomes this problem and still allows excellent reconstruction of the image. These properties make the CDWT better in denoising than the normal DWT. Comparison of CNR, ENL, and SNR values for DWT and CDWT verifies this; however, we can also visually recognize the advantage of shrinkage denoising using the CDWT.

6.

Conclusion

The wavelet shrinkage denoising technique was used for speckle noise reduction in the OCT images of cavernous nerves in the rat prostate. Two different wavelet transform procedures were used, DWT and CDWT, and their denoising results were compared for identification of the cavernous nerves. The CDWT was used to take advantage of this transform, which is shift-invariant and directionally selective. An SNR increase of $14\mathrm{dB}$ with significant image quality improvements was obtained. This algorithm for wavelet denoising may be a useful technique for clinical OCT applications.