3 Description of the method

PCA is a technique generally used to enhance the information content of multiband data and to reduce the dimensionality of the data set (Davis 1986; Chevrel et al. 1994; Erard et al. 1991; Jaumann 1991). PCA uses a linear transformation of the data to translate and rotate them in a new coordinate system that maximizes the variance. The rows of the transformation matrix A are composed of the eigenvectors of the covariance matrix K_x between the bands. The matrix A performs a diagonalization of the covariance matrix K_x such that the covariance matrix of the transformed imagery

$$K_{y} = {\bf A}K_{x}\bf A^{T} = \Lambda$$ (1)

is a diagonal matrix whose elements are the eigenvalues of K_x arranged in descending value. The principal components decomposition therefore results in a set of decorrelated data planes and the information is contained in few PC images (the eigenvectors) while most of the noise is segregated in the others components. The first transformed image is approximately an albedo picture and depicts the average brightness of the surface while the others components contain the color information. These higher order components are used for our purposes. We applied this technique to an image cube of the Moon composed of 128 $\times$ 258 spectra of 96 bands each, obtained by means of a visible-near infrared imaging spectrometer ($0.4 - 1.0\ \mu$m spectral range, $\Delta \lambda$ = 7.5 nm). Details about the instrumentation can be found in Bellucci et al. 1998.

  

Table 1: Statistics of three principal components of the image cube discussed in the text. The units are in digital numbers

The PC transformation has not been directly applied to the raw image cube but to a relative image cube. Each spectrum of the original data set, has been normalized respect to an average spectrum of a small area (10 $\times$ 10 pixels) chosen within the image. This operation removes at the same time the instrumental and atmospheric transfer functions and allows to enhance spectral differences proper of the lunar surface only. Table 1 summarizes the relevant statistics of 3 PC images which contain most of the image cube variance. Eigenvalues for bands that contain some information are larger than those that contain only noise. The corresponding PC images are spatially coherent, while the noise images do not contain any spatial information.

  

Figure 2: Raw image of the study region at 0.7 $\mu$m. Main geologic features are indicated

  

Figure 3: Principal components of the image cube discussed in the text

Figure 2 shows a raw image at 0.7 $\mu$m of the Mare Serenitatis/Tranquillitatis, Montes Haemus and Plinius region on the Moon. The scale is 1.2 arcsec per pixel, while the seeing is about 2.5 arcsec. The small crater in the center of Serenitatis is Bessel and will be used in the following to evaluate the enhancement process. Figure 3 shows the PC images. The PC1 image (Fig. 3a) is the albedo picture. In the PC2 image (Fig. 3b) the Menelaus rim and some features inside the crater are visible; also the Al-Bakri rim and floor are well discernible. In general, the PC2 image shows more details than raw image. The PC3 image (Fig. 3c) still contains enough information to allow recognition of narrow features as the Menelaus and Plinius rims and the Al-Bakri crater. Very narrow features, like small craters, crater rims and ejecta, are also well visible. In order to enhance the spatial contrast of the raw image shown in Fig. 2 by utilizing the results of the PCA, we have applied on the raw image $F_{\rm R}(j, k)$ the following transformation:

$$\begin{eqnarray} F_{\rm En}(j, k) &=& {\rm stretch}[F_{\rm R}(j, k)]\nonumber \\ && - {\rm stretch}[PC_{n}(j, k) \times L(m, n)]\end{eqnarray}$$ (2)

where $F_{\rm En}(j, k)$ is the enhanced image, PC_n(j, k) is one of the three principal component images, L(m, n) is a Laplacian convolution array. The j and k indices are the spatial coordinates of a pixel within the image, while m=3 and n=3 are the dimensions of the convolution array L. The Laplacian filter is:

$$L(3, 3) = \left [ \begin{array} {rrr} 0 & -1 & 0 \\ -1 & 4 & -1 \\ 0 & -1 & 0 \end{array} \right ].$$ (3)

It is a second derivative edge enhancement filter that operates without regard to edge direction (Ross 1995). In order to display the result when both positive and negative pixel values rise, a stretch operation is applied to each of the right-hand terms of Eq. (2). In this way, brighter and darker values produced by the Laplacian can be seen. The stretch[I] is a linear scaling operation:

$${\rm stretch}[I]= \frac{I-{\rm min}(I)}{{\rm max}(I)-{\rm min}(I)} \times 255.$$ (4)

Figure 4 shows schematically the sharpening procedure.

  

Figure 4: Block scheme of the sharpening method described in the text