In this paper, MMSE estimator is employed for noise-free 3D OCT data recovery in 3D complex wavelet domain. Since the proposed distribution for noise-free data plays a key role in the performance of MMSE estimator, a priori distribution for the pdf of noise-free 3D complex wavelet coefficients is proposed which is able to model the main statistical properties of wavelets. We model the coefficients with a mixture of two bivariate Gaussian pdfs with local parameters which are able to capture the heavy-tailed property and inter- and intrascale dependencies of coefficients. In addition, based on the special structure of OCT images, we use an anisotropic windowing procedure for local parameters estimation that results in visual quality improvement. On this base, several OCT despeckling algorithms are obtained based on using Gaussian/two-sided Rayleigh noise distribution and homomorphic/nonhomomorphic model. In order to evaluate the performance of the proposed algorithm, we use 156 selected ROIs from 650 × 512 × 128 OCT dataset in the presence of wet AMD pathology. Our simulations show that the best MMSE estimator using local bivariate mixture prior is for the nonhomomorphic model in the presence of Gaussian noise which results in an improvement of 7.8 ± 1.7 in CNR.
Optical coherence tomography (OCT) is an optical signal acquisition and processing method that captures 3D images from within optical scattering media such as biological tissues [
Transform domain techniques typically outperform the image domain techniques because incorporating speckle statistics in the despeckling process would be facilitated in sparse domains. Such techniques apply a sparse transform (such as wavelet and curvelet transforms) [
In fact denoising is the problem of obtaining the noise-free data from noisy data observation, which may be solved in a deterministic or probabilistic framework. In the first case, each voxel is considered as an unknown deterministic variable, and non-Bayesian techniques are employed to solve this problem. In the second case, the data is modeled as a random field, and Bayesian methods are used for the estimation of clean data from the noisy environment. Therefore, the proposed prior probability distributions for noise-free data and noise (i.e., proposed as speckle for OCT data) play a key role in the noise reduction problem.
Description of the statistical properties of natural signals can be facilitated in the wavelet domain [
The dependencies between wavelet coefficients are not restricted to the interscale dependency. There is another dependency between spatial adjacent coefficients in each subband, namely, intrascale dependency [
The wavelet based image denoising consists of the following steps. Signal transformation of the noisy observation. Modification of the noisy wavelet coefficients based on some criteria. Inverse signal transformation of modified coefficients.
As explained earlier, the second step depends on the type of estimator and for a minimum mean square error (MMSE) estimator, the proposed model for signal and noise (which we propose as a multiplicative model), the proposed pdf of noise-free wavelet coefficients (modeled, in this paper, as a mixture of bivariate Gaussian pdfs with local parameters), and the proposed pdf for noise (with which we test both Gaussian and two-sided Rayleigh distributions) define the performance of the algorithm. However, for the first and last steps of wavelet-based denoising algorithm, the type of transformation plays a key role. In this paper, we use DCWT [ In the neighborhood of an edge, the DWT produces both large and small wavelet coefficients. In contrast, the magnitudes of DCWT coefficients are more directly related to their adjacency to the edge. The main reason of this phenomenon is using bandpass filters that produce DWT coefficients which oscillate positively and negatively around the singularities, and this subject complicates wavelet-based processing. DWT is not shift invariant. It means that a small shift in the input signal of DWT makes the total energy of wavelet coefficients in subband completely differ. This shift greatly perturbs oscillation pattern around singularities of the DWT coefficient which complicates wavelet-domain processing. Since the DWT coefficients in each subband are produced via critical sampling after using nonideal low-pass and high-pass filters, substantial aliasing would be produced. If the wavelet coefficients are not changed, the inverse DWT cancels this aliasing. Applying any processing method on wavelet coefficients (such as thresholding) disarranges this balance between the forward and inverse transforms which leads to artifacts in the reconstructed signal. The directional selectivity of 2D DCWT has been explained in Appendix
A comparison between the idealized support of the Fourier spectrum of each standard and complex wavelet in the 3D frequency domain. (a) Isosurfaces of the 7 3D wavelets for a standard 3D wavelet transform. The blue and red colors have the same amplitude, but their phases are complement. (b) Isosurfaces of 7 of the 28 3D wavelets for a 3D DCWT. Each subband corresponds to motion in a specific direction [
In Section
One of the primary properties of the wavelet transform is compression. This property means that the marginal distributions of wavelet coefficients are highly kurtotic, and so long-tailed distributions are suitable models for marginal pdf. A zero-mean mixture model could have a large peak at zero and would be long tailed. For example, in [
Zero-mean Gaussian mixture model (left image) and empirical histogram of wavelet in a subband together with the Gaussian mixture model (right image) [
Empirical joint parent-child histogram of wavelet coefficients (computed from the Corel image database) [
In this paper, we assume a pdf as a mixture of two bivariate Gaussian pdfs with local parameters in order to model the distribution of wavelet coefficients of images as follows:
Our proposed model in this paper, that is a mixture of bivariate Gaussian pdfs with local parameters, is mixture, bivariate and local. Therefore, it is able to simultaneously capture the heavy-tailed property and inter- and intrascale dependencies.
After substitution of mixture model in the definition of
Interestingly, the marginal pdf of
It is easy to see that
See Appendix
To characterize the parameters in (
The
The variances
In this section, the denoising of a 3D OCT data is considered. We assume that dominant noise in OCT data is speckle. In this case as a common model, we propose multiplicative model as follows:
As explained in Introduction, reported transform-based OCT noise reduction methods in the literatures [
Recently, it has been reported [
Again we can write
Based on the persistence property, we need to have a bivariate model based on parent-child pairs. So, we can propose the following bivariate model:
Now our goal is the estimation of
If we employ the MMSE estimator for the estimation problem, we get the posterior mean as an optimal solution:
In order to solve (
In this case,
And so we can write
Similarly, if we choose two-sided Rayleigh pdf for noise distribution, the following estimator is obtained [
And so we can write
Suppose that the input noise variance is known. To implement (
A nonlinear shrinkage function for wavelet-based denoising, which is derived by assuming that the noise-free wavelet coefficients follow a bivariate Gaussian mixture model with local parameters given by (
In fact,
For two-sided Rayleigh noise (
Using (
We call the new obtained bivariate local shrinkage function as
A shrinkage function produced from BiGaussMixShrink for sample parameters.
Similarly, using (
We call this bivariate local shrinkage function as
A shrinkage function produced from BiGaussRayMixShrink for sample parameters.
For implementation of our denoising algorithm, we must estimate the parameters
Our denoising algorithm is summarized in Algorithm
(after obtaining noise (after obtaining
It has been shown that using anisotropic and shape adaptive window for local parameter estimation can extremely improve the modeling and processing results. For example, in [
To select the shape-adaptive window, we must take a look at the special structure of OCT data. In ophthalmology, the OCT data shows detailed images from within the retina. The automated analysis of OCT images can be used for the image-guided retinal therapy. Every year, many people become blind as a result of age-related macular degeneration (AMD) due to affecting the central retina where our central vision is perceived. The most sight-threatening form of AMD is called exudative or wet AMD. Choroidal neovascularization (CNV) is a common symptom of the degenerative maculopathy wet AMD. A wealth of powerful new treatments for CNV, especially anti-VEGF agents, have become available very recently to restore normal visual function. The risk of ocular adverse events, including the devastating intraocular infection, endophthalmitis, increases with repeated intravitreal treatment injections, and the effects of chronic treatment with anti-VEGF agents on the retina are unknown. Ideally a more cost-effective, patient-specific dosing strategy with the minimally necessary number of anti-VEGF injections is required. With all the promise, these novel treatments will only reach their full potential when objective and early indices of treatment response are developed. Prior to the introduction of retinal OCT imaging, clinical assessment of whether the preservation or restoration of visual function is successful, which indeed is the ultimate goal of treatment, could only be obtained by measuring visual function. Unfortunately, visual function lags structural response and is cumbersome and noisy, and its reproducibility is limited. Two-dimensional OCT imaging of the retina was introduced several years ago, and was rapidly adopted, among others, to qualitatively measure macular structure as an indicator of AMD treatment response and for guidance of retreatment in CNV recurrence. It is now becoming clear that these simplified structural measures though leading indicators of visual function are inadequate, as they are based on simplified interpretation of single transverse slices of the macula, some patients do not recover visual function even though their total macular thickness has become normally thin after treatment, and others paradoxically gain visual acuity while their macula is still thickened.
True 3D spectral OCT imaging, that became available in 2007 is fast (1.5 s per volume scan), allows full 3D retinal coverage at a much higher resolution and offers improved imaging of subtle differences in retinal structure. In the recent years [
On this base, developing analysis methods and approaches for 3D spectral OCT image analysis in the presence of wet AMD pathology (Symptomatic Exudate Associated Derangements or SEADs, also known as AMD-related cysts, vessel leakages, etc.) and assessing their performance by comparison to expert analyses are of utmost interest. Another interesting subject is determining how well the quantitative SEAD- and layer-derived measures from 3D OCT predict the patient-specific outcome parameters in response to postinduction anti-VEGF treatment in patients with CNV in order to predict the timing of retreatment.
Figure
Macular OCTs and detected SEADs by an expert.
In [
For each
According to the ICI rule,
The largest
Figure
The red line shows the detected SEAD by an expert. The yellow circles show the isotropic windows with various radii. The green line illustrates the obtained anisotropic based on LPA-ICI rule.
Since applying LPA-ICI in each subband is a time consuming process, a fast version of the mentioned algorithm can be based on only applying LPA-ICI to low-pass subbands using
From left to right: imaginary LL subband of one slice of OCT data, the oriented (imaginary) subband around 45° (225°),
A similar manner can be proposed in 3D case [
Comparison between a circular sector for direction
Note that in order to incorporate the anisotropic window selection for each DCWT coefficient in our OCT denoising algorithm explained in Algorithm
In this section, we apply the proposed despeckling algorithm to OCT image noise reduction. For this reason, we use 20 three-dimensional OCT datasets in the presence of wet AMD pathology (SEAD) and use mean signal-to-noise ratio (MSNR) and contrast-to-noise ratio (CNR) as two quality measurements for OCT data. To calculate these measurements, we must define the region of interest (ROI). In this paper, we propose this region within the SEAD as illustrated in Figure
One slice from a sample OCT image and proposed ROIs for computation of MSNR and CNR reported in Table
Table
The results of MSNR and CNR using several ROIs, shown in Figure
Methods | MSNRROI1 | MSNRROI2 | CNR | ||
---|---|---|---|---|---|
Local (L) |
Homomorphic (H) Nonhomomorphic (NH) | Gaussian noise (G) | |||
L | H | G | 7.00 | 15.76 | 8.76 |
NL | H | G | 7.56 | 17.03 | 9.47 |
L | NH | G | 12.27 | 27.76 | 13.49 |
NL | NH | G | 10.77 | 22.73 | 11.95 |
L | H | R | 5.89 | 13.11 | 7.22 |
NL | H | R | 8.63 | 19.59 | 10.95 |
L | NH | R | 10.75 | 22.55 | 11.81 |
NL | NH | R | 10.88 | 23.05 | 12.17 |
| |||||
Original image | 2.56 | 5.30 | 2.74 |
The results of applying homomorphic methods on proposed image in Figure
The results of applying non-homomorphic methods on proposed image in Figure
A comparison between CNR curves for 156 selected ROIs from OCT dataset.
Another way for evaluating the effect of our despeckling algorithm is the investigation of the intralayer segmentation results. Figure
A comparison between the segmented layers of a
In this paper, we introduced a new noise reduction algorithm for 3D OCT data. We found new shrinkage functions employing a mixture of bivariate Gaussian for modeling wavelet coefficients in each subband of complex wavelets. The parameters of this mixture model are estimated locally using a shape-adaptive manner based on the special structure of OCT data. We also used this model for denoising of other kinds of noise. Experiments show that our model has better results than other methods visually and in terms of PSNR especially for the crowded images. In this paper, we suppose that the parameters of EM algorithm, in extracted windows are constant. It is possible to improve the EM algorithm, for example, by using recurrence equations. It is possible that we only propose the main section of data (between the first and last layers) containing retina layer information and apply our algorithm on the selected data to improve the speed and performance of denoising process.
Using 3D DCWT instead of other transforms such as 3D DWT is a main reason for improvement of the denoising results [
Since DWT in 2D domain is produced using separable (row-column) implementation, it has a poor directional selectivity. For example, the HH wavelet is the product of the high-pass functions along the first and second dimensions. Because DWT uses real filters, the HH wavelet mixes +45° and −45° orientations that results in the checkerboard artifact because it fails to isolate these orientations. In contrast, since the spectrum of the (approximately) analytic 1D wavelet is supported on only one side of the frequency axis, the spectrum of the DCWT in 2D domain is supported in only one quadrant of the 2D frequency plane. Figure
A comparison between subbands of DWT and DCWT. (a) The wavelets in the space domain (LH, HL, and HH). (b) The idealized support of the Fourier spectrum of each wavelet in the 2D frequency domain. We can see the checkerboard artifact of the third wavelet. (c) The complex wavelets in the space domain. (d) The idealized support of the Fourier spectrum of each wavelet in the 2D frequency plane. The absence of the checkerboard phenomenon is observed in both the space and frequency domains [
Figure
The pdf of a bivariate Gaussian mixture model for sample parameters and its marginal distribution.
In this appendix, we briefly explain the abilities of the proposed denoising algorithm in this paper for other kinds of noise.
We tested the shrinkage function
(a) shows a part of Barbara image denoised using BiShrink [
This section presents nonstationary noise reduction examples in complex wavelet domain. Although the stationary noise model is able to simplify the implementation of denoising algorithms, the statistical properties of the noise are not always accurately described with this assumption. For example, in some applications, the noise statistics are spatially varying and the noise power varies between pixels or samples. In these cases, the nonstationary noise assumption is more reasonable and can improve the denoising results. For example, we contaminate three
Since the variance of each noise component is spatially varying with the corresponding content of signal, the nonstationary processes are able to model the statistical properties of this noise. A comparison between the denoised image using soft thresholding, proposed method in [
PSNR (in dB) values of test images for different nonstationary noise levels.
Noise parameters |
Lena | Boat | Barbara | |||||||||
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Noisy Image | Soft thresh. [ |
Proposed method in [ |
Our method | Noisy image | Soft thresh. [ |
Proposed method in [ |
Our method | Noisy image | Soft thresh. [ |
Proposed method in [ |
Our method | |
|
27.72 | 34.13 | 34.61 |
|
27.51 | 32.44 | 32.59 |
|
27.94 | 31.99 | 32.20 |
|
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23.48 | 31.62 | 32.49 |
|
23.22 | 29.95 | 30.23 |
|
23.69 | 29.05 | 25.81 |
|
|
18.49 | 27.50 | 29.65 |
|
18.20 | 26.77 | 27.60 |
|
18.71 | 25.81 | 25.94 |
|
In [
Comparison between PSNRs (in dB) of denoised images with Fast TI Haar algorithm [
Lena | Boat | Barbara | Confocal Phantom | Shep Logan Phantom | Bowl | |
---|---|---|---|---|---|---|
Noisy image | 27.22 | 27.05 | 27.49 | 35.74 | 47.68 | 28.21 |
Fast TI Haar | 32.11 | 29.30 | 26.59 | 44.49 | 60.63 | 46.79 |
BiGaussMixShrinL |
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(a–d) show denoising results for Confocal Microscopy Phantom: from left to right: noise-free image, noisy image, and denoised image with our model and denoised image with Fast TI Haar algorithm. (e–h) show from left to right parts of denoised Barbara image with BiGaussMixShrinkL method and parts of denoised Barbara image with Fast TI Haar algorithm.