Wavelets for SAR Image Smoothing

Wavelets are an increasingly widely used tool in many applications of signal and image processing. This paper reviews the basic ideas of wavelets for representing the information in signals such as time series and images, and shows how wavelet shrinkage may be used to smooth these signals. This is illustrated by application to a synthetic aperture radar image. Some guidelines on using wavelet shrinkage are given. Introduction Wavelet analysis has recently been recognized as a tool with important applications in time series, function estimation, and image analysis. Applications in remote sensing have included the combination of images of different resolutions (Garguetduport et al., 1996), image compression (Werness et al., 1994), the provision of edge detection methods (Li and Shao, 1994), and the study of scales of variation (Lindsay et al., 1996). As the development of wavelet methods is recent, the fundamentals are not yet widely understood, and guidance on their practical use is hard to find. Much of the literature is not easily accessible without much mathematical sophistication. This paper explains the fundamental ideas of wavelets in a non-mathematical way. Their use for smoothing data and images, particularly by wavelet shrinkage, will be reviewed. This will be illustrated by application to a synthetic aperture radar image. Wavelets Wavelets arose from signal processing theory, a signal being the variability of some quantity over time. They can be generalized to two-dimensional signals, of which images are a special case. Wavelet decomposition is an alternative way of presenting the details of a signal which differs from specifying the value of the signal at successive times, the so-called time domain representation. Other ways of doing this have been used for a long time. The best known is the Fourier series. This represents a signal in terms of sine waves with frequencies which are multiples (harmonics) of a basic frequency. In many applications this is a useful way of decomposing a signal, and its properties can be understood in terms of these oscillations at different frequencies. The sine waves used are orthogonal to each other (two functions At) and g(t) are orthogonal if 5 : f(t]g(t]dt = 0). For a signal sampled at n points, a full reconstruction can be made from n Fourier components. This is termed the Fourier or frequency domain representation of a series. A drawback to the Fourier representation is that the frequency components apply to the signal as a whole, and the way that the signal variability changes over time may be important. However, it impossible to say what the frequency components are at a time point, but only in a region about it. To put it another way, a signal cannot be highly localized in Rowett Research Institute, Biomathematics & Statistics Scotland, Bucksburn, Aberdeen AB21 9SB, Scotland. I PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING both the time and frequency domains (this limitation gives rise in physics to the Heisenberg Uncertainty Principle). One approach is to estimate Fourier components locally, in a set of short intervals, or convolved with a decaying weight function about each point. To do this, however, is to lose the orthogonality and economy of representation of the Fourier method. Wavelets preserve these advantages, while enjoying good spatial and frequency resolution. The wavelet approach to signal representation is based on representing the signal at different scales or resolutions. At a particular resolution, the signal is approximated by a sum of scaling functions. The difference between the resolutions (termed the detail at the finer resolution) can be expressed by a sum of wavelet functions. For certain scaling and wavelet functions, this hierarchical or multiresolution representation can be constructed using scaled versions of the same functions at each resolution. For example, consider the simplest multiresolution representation, based on the Haar scaling function and wavelet. These are illustrated in Figure 1. They are piecewise constant, and it is clear how they may be used to approximate signals. Figure 2 shows a signal crudely approximated by the Haar scaling function at two different resolutions. It is clear that the signal could be approximated in this way at ever higher resolutions until it is as close to the original signal as we wish. If this is only available at discrete time points, the finest resolution approximation is exact. The wavelet representation consists of specifying an approximation to the signal at some coarse resolution, and then specifying the refinements needed at each stage to achieve the next resolution. In Figure 2, we see that the approximation (a] can be refined to (b) by the addition of two Haar wavelets (Figure 2b). One covers the first half of the time domain, and has a negative coefficient. The other covers the second half, and has a slightly smaller positive coefficient. A further four wavelets could then be added to (b) to refine it further. In the wavelet representation, we specify the coefficients of the scaling functions at the coarsest resolution (which could be just the average of the signal) and the coefficients of all the wavelets needed to refine it to the finest resolution. If the signal is of length Zm, then 1 + 1 + 2 + 4 +...+2"-l = 2" coefficients are needed. (The wavelet representation is most convenient for 2" time points, and some modifications are needed when this is not so see Discussion below.) Among the most important benefits of a wavelet representation are Single wavelet coefficients provide information about how the signal is changing over time: for example, the coefficient of the wavelet which refines the coarsest (overall average) approximation to obtain the next resolution tells us about the overall increase or decrease of the signal; Some wavelet coefficients can be discarded (for economy of Photogrammetric Engineering & Remote Sensing, Vol. 64, No. 12 , December 1998, pp. 1171-1177. 0099-1112/98/6412-1177$3.00/0 O 1998 American Society for Photogrammetry and Remote Sensing