Denoising Images Using Wavelets

Motivation

In general, it is difficult to apply data mining techniques directly to raw scientific image datasets. A common approach is to first pre-process the images into a form suitable for the detection of patterns. Denoising is an example of such a pre-processing operation, geared towards removing instrumental (or other type of) noise from the data.

Denoising data by thresholding of the wavelet coefficients has been proposed by various researchers during the past two decades. The method consists of applying a discrete wavelet transform to the original data, thresholding the detail wavelet coefficients, then inverse transforming the thresholded coefficients to obtain the denoised data [1,2,3]. There are several ways of calculating and applying thresholds.

The simplest threshold is the Universal [4], , where n is the sample size, and is the noise variance. Threshold selection alternatives, based on minimizing certain optimization criteria, include the minimax [4], and the SURE [5] methods. Thresholds can also be based on hypothesis testing, cross-validation, and Bayesian estimation approaches [2]. The most flexible of the threshold calculation methods, the Top method, involves selecting the threshold as a quantile of the empirical distribution of the wavelet coefficients. By experimenting with different quantile values, the user can interactively explore the best threshold for a given application.

If unknown, the noise variance , can be estimated in various ways. Conventional estimates include the sample standard deviation, various norms, and the median absolute deviation (MAD). Level-independent estimates of , (i.e. one common estimate for all the multiresolution levels in the wavelet decomposition), can be obtained by either including all the detail coefficients, or by using detail coefficients only from a given multiresolution level in the chosen estimate formula. Level-dependent estimates can be obtained by plugging only the detail coefficients corresponding to the given level into the variance-estimating formula.

Threshold, or shrinkage, application functions include the hard, the soft, and the semisoft functions [6]. The hard function involves a keep or kill strategy: coefficients whose absolute values are below a positive threshold are all ``killed'' (set to zero), while the others are kept unchanged. The soft function is similar to the hard, except that it either shrinks or kills: the coefficients that are kept are modified by shrinking them towards zero. The semisoft function generalizes the hard and the soft functions by using two thresholds, and includes both the hard and the soft as special cases [7].

The combination of soft shrinkage with the universal threshold is referred to as VisuShrink [4], soft with the minimax threshold is called RiskShrink [4], and soft along with the SURE threshold is known as SureShrink [5]. More recent advances, BayesShrink [8,9], advocating soft shrinkage in a certain Bayesian framework, claim to outperform SureShrink estimates in the context of denoising images.

Experimental Results

We are applying wavelet denoising techniques to remove noise from image data. The images we consider are from the Faint Images of the Radio Sky at Twenty-cm (FIRST) survey. The FIRST survey is producing the radio equivalent of the Palomar Observatory Sky Survey. Using the Very Large Array (VLA) at the National Radio Astronomy Observatory (NRAO), FIRST is scheduled to cover more than 10,000 square degrees of the northern and southern galactic caps, to a flux density limit of 1.0 mJy (milli-Jansky). The latest data, released on July 17, 2000, includes observations from 1993 through February 2000, and covers 7377 square degrees in the north galactic cap and 611 square degrees in the south galactic cap. It contains about 722,000 radio emitting galaxies or sources.

Radio emitting galaxies exhibit a wide range of morphological types that provide clues to the source class, emission mechanism, and properties of the surrounding medium. Of particular interest to the astronomers are radio sources with a bent-double morphology, as they indicate the presence of large clusters of galaxies. Currently, FIRST scientists identify bent-doubles by manually looking through the images. The image dataset is ``only'' about 200 Gigabytes, moderate by today's emerging standards, but large enough to inhibit an exhaustive visual inspection by the astronomers. Our goal is to automate the detection of bent-doubles by using data mining techniques.

The data from FIRST, both raw and postprocessed, are readily available on the FIRST website. A user friendly interface enables easy access to radio sources at a given RA (Right Ascension, analogous to longitude) and Dec (declination, analogous to latitude) position in the sky. There are two forms of data available for use --- image maps and a catalog. For example, Fig.1 shows an 1550x1150 pixel image map that contains two bent-double galaxies, which are magnified in the two 64x64 sub-images. This typical image map is mostly ``empty'', that is, composed of background noise. Each map covers approximately 0.45 square degrees area of the sky, and has pixels which are 1.8 arc seconds wide. In addition to the image maps, FIRST also provides a source catalog. The catalog is obtained by fitting two-dimensional elliptic Gaussians to the radio sources in the image data. For example, one of the bent-doubles (shown on the 64x64 sub-image around the middle of the figure) in Fig.1 is approximated by more than seven Gaussians, while the other one (shown on the 64x64 sub-image in the upper right corner of the image) is approximated by three Gaussians. There is an upper limit to the number of Gaussians that are used to fit each radio source. As a result, highly complex sources are not approximated well using just the information in the catalog. Each entry in the catalog corresponds to the information on a single Gaussian. This includes, among other things, the RA and Dec for the center of the Gaussian, the major and minor axes, the peak flux, and the position angle of the major axis (degrees counterclockwise from North). The table in Fig.1 illustrates the three catalog entries (CE) for the three elliptical Gaussians fitted to the bent-double in the upper right corner of the figure. A CE thus refers to a single fitted Gaussian, while a radio source (RS) refers to the collections of CEs forming a galaxy.

Fig.1. FIRST image map containing two bent-double galaxies, and catalog entries corresponding to the bent-double magnified in the upper right corner
Our approach to finding bent-doubles involves defining and extracting features for all radio galaxies, then using classification techniques to classify them as bent-doubles or as non-bent-doubles. As a first step, we have been defining and extracting features from the catalog. However, some information is lost when converting the image data to the catalog. Our next step, therefore, is to extract features directly from the original images. Due to the sensors used to collect the data, there is a pronounced noise pattern that appears as "streaks" in the images. Before feature extraction from the images can begin, it is desirable to remove the noise from the images.

The task of denoising these images is made challenging by the fact that some of the information necessary to identify a galaxy as a bent-double, could lie on a "streak" and be removed as "noise". To ensure that wavelet denoising can indeed be applied to our images without any significant loss of useful information, we are experimenting with smaller images extracted from the larger 1150x1550 pixel image maps.

Fig.2 presents an example FIRST image, and various denoised versions of it. All the examples were obtained using the Haar wavelet, and three multiresolution levels in the wavelet decomposition. We are currently experimenting with other wavelets and denoising options to find an optimal combination for the FIRST dataset. A good technique should remove the background noise effectively, but keep the important bent-double features intact. For example, in the bent-double shown in the figure, the two wavy lobes of the bent-double are connected by a fainter bridge. This bridge is an important feature as it contains information on the bentness of the galaxy. Losing the bridge in a denoising operation will complicate the subsequent feature extraction and classification steps. As the bridge lies on one of the noise streaks, we have to be careful in the use of denoising. This task is made more challenging by the fact that the number of radio galaxies precludes individual examination of the effects of denoising on each image.

(a) (b) (c) (d)
(e) (f) (g) (h)
Fig.2. Original and denoised images. (a) Original. (b) Top 0.1st and 0.9th quantile thresholds; semisoft. (c) Universal, as the norm of the detail coefficients; hard. (d) As (c); soft. (e) Universal, as the norm of all the detail coefficients; hard. (f) As (e); soft. (g) Universal, as the MAD of the detail coefficients; hard. (h) As (g); soft.

Future Directions

Our preliminary results indicate that wavelet denoising techniques can be used effectively on smaller images. We are currently comparing the performances of the various methods on the FIRST dataset, as well as on synthetic images.

References

1. I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.

2. R.T. Ogden, Essential Wavelets for Statistical Applications and Data Analysis, Birkhauser Boston, 1997.

3. B. Vidakovic, Statistical Modeling by Wavelets, Wiley Series in Probability and Statistics, 1999.

4. D. L. Donoho and I. M. Johnstone, "Ideal spatial adaptation via wavelet shrinkage", Biometrika, vol. 81, pp. 425-455, 1994.

5. D. L. Donoho and I. M. Johnstone, "Adapting to unknown smoothness via wavelet shrinkage", Journal of the American Statistical Association, vol. 90, pp. 1200-1224, December 1995.

6. A.G. Bruce and H.Y. Gao, "Understanding WaveShrink: Variance and Bias Estimation", Tech. Rep. 36, StatSci Division of MathSoft, Inc., 1995.

7. H.Y. Gao and A.G. Bruce, "WaveShrink with Semisoft Shrinkage", Tech. Rep. 39, StatSci Division of MathSoft, Inc., 1995.

8. G. Chang, B. Yu and M. Vetterli, "Spatially adaptive wavelet thresholding based on context modeling for image denoising", IEEE Trans. Image Processing, Accepted 1998.

9. G. Chang, B. Yu and M. Vetterli, "Adaptive wavelet thresholding for image denoising and compression", IEEE Trans. Image Processing, Accepted 1998.


For more technical information, contact:
kamath2@llnl.gov -- Chandrika Kamath, (925) 423-3768
Last modified: August 30, 2000.
UCRL-MI-131567 Rev. 1
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