and Guy P. Nason. ¡. Department of Statistics, University of Leeds. ¡. Department of Mathematics, University of Bristol. 1 Wavelets in one and two dimensions.
Denoising Real Images Using Complex-Valued Wavelets Stuart Barber and Guy P. Nason
Department of Statistics, University of Leeds Department of Mathematics, University of Bristol
1 Wavelets in one and two dimensions Wavelets are a special type of basis function which are localised in both time and frequency. To generate a wavelet basis, a mother wavelet is chosen which has an associated father wavelet . Further wavelets are derived via the equations and . A function can be written in terms of this basis as
(1)
Two dimensional data such as images can be decomposed with respect to a more compli cated wavelet basis which is still determined solely by the choice of mother wavelet . A bivariate function can be expressed as !" !"
(2)
Here # $ % & ' ( represents the wavelets in the diagonal, horizontal and vertical directions which are defined by
!* "
and
!) "
!+ "
Although the details are more complicated in two dimensions, the basic ideas behind (1) and (2) are the same: a function is described in terms of scaled and shifted copies of the “building blocks” and . More details can be found in (e.g.) Vidakovic (1999). The two families of compactly supported wavelets described by Daubechies (1992) are by far the most commonly used. These “extremal phase” and “least asymmetric” wavelets have been applied to statistical problems such as nonparametric regression, density estimation, time series analysis, signal processing and image analysis. Less well known are the complex-valued Daubechies wavelets (cDws) introduced by Lawton (1993) and Lina and Mayrand (1995). A selection of cDws are shown in figure 1. These cDws have seen relatively little use in the literature compared to the real-valued Daubechies’ wavelets. Belzer, Lina and Villasensor (1995) have used the real parts only of “nearly real” cDws in image compression, while Lina and MacGibbon (1997) and Lina (1997) 91
psi(t)
psi(t)
psi(t)
psi(t)
t
psi(t)
t
psi(t)
t
t
t
t
Figure 1: Complex Daubechies’ wavelets with real and imaginary parts shown as solid black and dashed gray lines respectively. have applied Bayesian wavelet shrinkage methods with complex-valued wavelets to image data. Sardy (2000) has used cDws in denoising one-dimensional complex-valued radar data. In section 2 we describe how these complex Daubechies’ wavelets can be applied to the estimation of one-dimensional real signals. This approach has been shown to be startlingly effective when compared to methods using real Daubechies’ wavelets (Barber and Nason, 2003). This approach can be directly extended to the denoising of real-valued images, which we discuss in section 3. Some concluding remarks are made in section 4.
2 Complex multiwavelet style thresholding Barber and Nason (2003) demonstrated the superior performance of these cDws compared to real Daubechies’ wavelets in denoising one-dimensional signals. Consider estimating a function ( corrupted by independent Gaussian noise: from a set of observations %
#
#
(3)
By taking the wavelet transform of the data, the problem of estimating becomes one of estimating its wavelet coefficients. Due to the unitary nature of the DWT, the simple structure of the data is retained in the wavelet domain:
The % ( and % ( are the wavelet coefficients of the data and the signal respectively. If wavelet is complex-valued, so are these coefficients. For most functions, the the decomposing wavelet representation is sparse; in other words, the majority of the are zero or nearly zero and most of the information about is contained in just a few . This is true for both real-valued and complex-valued wavelets. 92
One of the methods proposed by Barber and Nason (2003) is a multiwavelet style thresholding rule. In this “CMWS” method, the complex-valued wavelet coefficients of the data are treated as bivariate real random variables and used to estimate the corresponding coefficients of
the underlying signal. If we identify a complex number , the # with the vector empirical wavelet coefficient has a bivariate Gaussian distribution . The covari the choice of wavelet up to multiplication by the noise variance ance matrix is determined by , which we estimate from the data. Following the multiwavelet approach of Downie and Silverman (1998), Barber and Nason
construct “thresholding statistics” . Small values of indicate that the is dominated by noise, while large values of indicate the presence of signal. Hence, the true wavelet coefficients are estimated either by hard thresholding
or by soft thresholding
from the asymptotic maxDownie and Silverman (1998) derive the choice of threshold
imum of the distribution; this is the distribution of the in the absence of signal.
3 Complex multiwavelet-style denoising of real images We now describe some initial work done to generalise the “CMWS” method of Barber and
from Nason (2003) to two dimensions. We consider estimating a discretised image
noisy data corrupted by independent Gaussian noise:
(4)
At each resolution level and direction #, we regard the! "complex-valued empirical wavelet ! " !" coefficients as being bivariate Gaussian with mean and covariance matrix . We wavelet transform in terms of macomputed these covariance matrices by expressing the 2D trix operations ! " and then proceeded as in the one-dimensional case by computing “thresholding statistics” and comparing these to the threshold . (The threshold has doubled as
in (4) there are data points instead of the data values in (3).) We have applied this CMWS thresholding rule to several test images supplied with the Splus WAVELETS module; one example is shown in figure 2. Corrupted versions of the “phone” test image is in the top left panel. The other panels show the results of denoising these images using CMWS thresholding, universal thresholding (Donoho and Johnstone, 1994) and false discovery rate (FDR) thresholding (Abramovich and Benjamini, 1996). Table 1 shows brief simulation results obtained when denoising simulated data sets based on the “phone”, “Lenna”, “Daubechies” and “MRI” images from the Splus WAVELETS module.
In each case independent Gaussian noise with was added to produce 25 simulated data sets. Results were measured by the peak signal-to-noise ratio (PSNR), ( % PSNR
high PSNR values indicate more accurate signal reconstruction. For the simulation results reported in table 1, hard CMWS thresholding gives the highest PSNR value on each test signal. 93
Noisy data
CMWS
Universal
FDR
Figure 2: “Phone” test image corrupted by Gaussian noise with denoised image using hard CMWS, universal and FDR thresholding.
and , and
4 Conclusions Initial results indicate that the multiwavelet style thresholding with complex-valued wavelets holds promise as an image denoising technique. Barber and Nason (2003) found empirical Bayes techniques to be even more effective when analysing one-dimensional data and it remains to be seen whether this technique will be effective on images, although in the image setting the computationally intensive nature of the empirical Bayes methodology may be prohibitive. Further improvements will be obtained by using the nondecimated wavelet transform. There has been substantial recent work on alternative two-dimensional basis functions based on wavelets (e.g. Starck, Cand`es and Donoho, 2002; Do and Vetterli, 2003). It will be interesting to see how complex wavelets compare to these. Another exciting possibility is to produce versions of these basis functions based on complex-valued wavelets.
94
Policy Phone CMWS - hard 22.19 CMWS - soft 21.76 FDR - hard 21.03 FDR - soft 20.00 Universal - hard 20.35 Universal - soft 19.32
Lenna 22.58 22.05 21.65 20.06 20.72 18.91
Daubechies 27.37 26.89 26.32 25.16 25.85 24.66
MRI 24.00 23.40 22.72 20.93 21.63 19.63
Table 1: Average PSNR results from denoising 25 simulated data sets based on the indicated
test images. The data was corrupted with independent Gaussian noise with .
References Abramovich, F. and Benjamini, Y. (1996). Adaptive thresholding of wavelet coefficients. Computational Statistics and Data Analysis, 22, 351-361. Barber, S. and Nason, G.P. (2003). Real nonparametric regression using complex wavelets. Submitted for publication. Belzer, B., Lina, J.-M. and Villasensor, J. (1995). Complex, linear-phase filters for efficient image coding. IEEE Transactions on Signal Processing, 43, 2425-2427. Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia, SIAM. Do, M.N. and Vetterli, M. (2003). The finite ridgelet transform for image representation. IEEE Transaction on Image Processing, 12, 16-28. Donoho, D.L. and Johnstone, I.M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81, 425-455. Downie, T.R. and Silverman, B.W. (1998). The discrete multiple wavelet transform and thresholding methods. IEEE Transactions on Signal Processing, 46, 2558-2561. Lawton, W. (1993). Applications of complex valued wavelet transforms to subband decomposition. IEEE Transactions on Signal Processing, 41, 3566-3568. Lina, J.-M. (1997). Image processing with complex Daubechies wavelets. Journal of Mathematical Imaging and Vision, 7, 211-223. Lina, J.-M. and MacGibbon, B. (1997). Non-linear shrinkage estimation with complex Daubechies wavelets. Wavelet Applications in Signal and Image Processing (SPIE vol 3169), 67-79. Lina, J.-M. and Mayrand, M (1995). Complex Daubechies Wavelets. Applied and Computational Harmonic Analysis, 2, 219-229. Sardy, S. (2000). Minimax threshold for denoising complex signal with waveshrink. IEEE Transactions on Signal Processing, 48, 1023-1028 Starck, J.-L. and Cand`es, E.J. and Donoho, D.L. (2002). The curvelet transform for image denoising. IEEE Transactions on Image Processing, 11, 670-684. Vidakovic, B. (1999). Statistical Modelling by Wavelets. New York, Wiley. 95
