Wavelet based compression and denoising of optical tomography data
Introduction
In the context of tomographic imaging there could be at least two occasions when wavelet analysis would be worthwhile attempting. One is when data compression is required as in the case of telemedical diagnosis where radiological data is centrally archived and transmitted to various centres for examination. Considering the enormous volume of imaging data that needs to be stored and transmitted compression becomes quite essential. The second context is when a signal buried in noisy data has to be recovered for a reasonable quality reconstruction. This is of particular significance in optical tomography, where the presence of highly scattering tissue surrounding an object of interest makes the output data very noisy. Several experimental methods which make use of tools such as time-gating 1, 2, spatial filtering 3, 4, heterodyne interferometry 5, 6 and polarization [7] were developed in the past to recover the signal from noisy data which relies on separating the ballistic or near-ballistic photons from the multiply scattered photons. Recovery of a signal from noisy data can also be done using transform-based methods relying on signal processing, which have proved their usefulness in recovering ECG data [8], astronomical spectra [9] and voice and picture signals transmitted through noisy channels [10]. Many of these noise reduction methods are based on wavelet expansion which provide an extra flexibility of time (or space) localization compared to Fourier expansion [11]. Donoho and Johnstone 12, 13 demonstrated a wavelet domain thresholding and shrinkage technique for noise reduction. The reason for recourse to such processing is the ability of wavelet expansion to concentrate signal energy to a small number of large coefficients. The coefficients that are below a threshold are dropped, and those that are small but above the threshold are shrunk. Hard thresholding alone results in a smaller mean square error (MSE) compared to an ideal estimate [14], while soft thresholding achieves smoother estimates with near minimum MSEs.
Donoho and Johnstone's 12, 13 and related methods gave excellent noise reduction when applied to NMR spectra [15] synthetic aperture radar signals [15] and astronomical data [9]. Their ability to compress signals was only through the number of small coefficients dropped owing to the hard threshold limit. Coifman and Wickerhauser [16] have proposed an entropy based method to select the best basis from the wavelet packet tree into which noisy data are expanded which was used to compress ECG data as well as finger print data bank [8]. In this work, we make use of both Wickerhauser's algorithm as well as Donoho and Johnstone's method to compress and denoise optical tomography data. In 2 Optical tomography, 3 Denoising by wavelet shrinkage, 4 Wavelet based compression and denoising using Wickerhauser's method, we briefly describe both Donoho's method of denoising and Wickerhauser's method of entropy minimization used generally for signal compression. For completeness, we also introduce optical tomography. In Section 5, we describe the experimental set-up for data collection from light transmitted across a fiber in a liquid medium containing scattering particles. The data are compressed and denoised using the methods described in 3 Denoising by wavelet shrinkage, 4 Wavelet based compression and denoising using Wickerhauser's method. The denoised and compressed data are compared with the data uncorrupted by noise to assess the performance of the algorithms of 3 Denoising by wavelet shrinkage, 4 Wavelet based compression and denoising using Wickerhauser's method. The processed data are reconstructed using the standard filtered backprojection (FBP) algorithm. Section 6gives our concluding remarks.
Section snippets
Optical tomography
In optical tomography, objects, which are refractive index distributions, are scanned by light whose time delay or intensity loss is measured. Time delay reconstructs the non-absorbing (real) part of refractive index distribution and the intensity of the absorbing (imaginary) part, if the data are properly inverted. Even under geometrical optics approximations, refraction cannot be neglected if the refractive index variations are not small. Incorporation of refraction correction methods is the
Denoising by wavelet shrinkage
This section is based on the work of Donoho and Johnstone 12, 13. We have a finite length signal of observations xi of a signal si corrupted by an i.i.d, zero mean, white Gaussian noise ni with standard deviation σ, i.e. where i=1,2,3,⋯N. We aim to recover si from xi. Let W and W−1 11, 14 represent an N×N wavelet transform matrix and its inverse, respectively. In the wavelet domain, Eq. (1)becomeswhereand x, s and n are column vectors containing xi, si and ni,
Wavelet based compression and denoising using Wickerhauser's method
Wavelet packets are a generalization of wavelets. In the fast wavelet transform algorithm, the sampled data are passed through the scaling and wavelet filters [18] and down sampled, resulting in approximation and detail coefficients. The approximation coefficients are then filtered with the same scaling and wavelet filters, generating another set of detail and approximation coefficients. This process is continued until a desired level of decomposition is reached. This is Mallat's pyramidal
Experimental data collection and processing
The experimental set-up is shown in Fig. 3. The object, a 125 μm diameter (core diameter=50 μm) graded index single mode fiber with a transmission window at 830 nm was immersed in an index matching liquid in a cuvette C. This was trans-illuminated by a spatially noncoherent quasi-monochromatic light and the output wavefront at the exit plane, E, of the cuvette was imaged by a microscope objective onto a CCD array which was connected to a computer.
Light, which traveled through the object, was
Conclusions
We have brought out here the usefulness of two denoising algorithms to preprocess noisy optical tomography data. Both the algorithms fail when the scattering particle concentration is high, in which case an experimental method to capture the least scattered photons has to be used to separate the signal. Among the two denoising algorithms, it is notable that Donoho and Johnstone's method gave reasonably good results for comparatively higher noise levels, whereas Wickerhauser's method also
Acknowledgements
The support of DAAD (German Academic Exchange Service) to the first author for carrying out part of the work at University of Kaiserslautern is gratefully acknowledged. Comments from the referee have helped us correct a few mistakes in the original manuscript.
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