Hyperspectral satellites collect imagery in hundreds of spectral bands simultaneously, covering wavelengths from the near-UV to the short-wave-IR regimes. They can provide direct identification of surface materials and are used in a wide variety of remote-sensing applications. The large number of spectral bands returned by hyperspectral sensors presents a challenge to traditional remote-sensing processing techniques. Preprocessing is often required to reduce the number of bands prior to deriving information from the images (see Figure 1).

**Figure 1. **Preprocessed dimensionality reduction prior to information derivation from hyperspectral imagery.

Techniques have been developed to mitigate the effects of high dimensionality on information extraction from hyperspectral imagery, including principal-components analysis (PCA),^{1} minimum-noise fraction (MNF),^{2} and linear discriminate analysis.^{3} They use linear projection and may result in a loss of nonlinear properties after dimensionality reduction. We introduced locally linear embedding (LLE)^{4}—which is used in arranging human facial images and words in semantic space—to the dimensionality reduction of satellite imagery. LLE projects high-dimensional data into low-dimensional space while preserving local topological structures. It approximates images in high-dimensional space with small, flat patches and stitches them together in low-dimensional space to retain nonlinear structures.

We developed novel, nonlinear dimensionality-reduction techniques using LLE.^{5,6} The computational complexity and memory requirements of LLE are challenges because it calculates the Euclidian distance between a given pixel and every other pixel in each of the band images in search for the *k* nearest neighbors. To overcome this problem, we introduced a spatial-neighborhood window by taking advantage of spatial correlation in the imagery, and we search for the *k* nearest neighbors of the relevant pixel only within this small window. Although LLE preserves local topological structures while projecting data to low-dimensional space, it may not optimally maintain the pixel distance in the dimension-reduced space to the same accuracy as in the original. Laplacian eigenmaps can preserve pixel separations. We combined LLE with such eigenmaps to achieve better locality preservation for nonlinear dimensionality reduction.^{6}

A particular dimensionality-reduction technique normally works best only for certain remote-sensing applications. We evaluated the performance of three widely used methods—PCA, wavelet transformation, and MNF—and a band-selection technique based on three popular remote-sensing applications, including classification, endmember extraction, and mineral detection. Our evaluation results indicate that PCA generally performs better than the other methods^{7} (see Figure 2).

**Figure 2. **Classification maps derived from original imagery and after dimensionality reduction from 224 to 22 dimensions using principal-components analysis (PCA), wavelet transformation, minimum-noise fraction (MNF), and band selection, respectively. The values in parentheses are correct classification rates with respect to the classification map of the original imagery.

After dimensionality reduction using PCA, the output channels often contain a significant amount of noise. Therefore, we simultaneously conducted denoising and dimensionality reduction.^{8,9} We applied a 2D wavelet-packet transform in the spatial domain to each of the images. The coefficients of the wavelet packet are then shrunk by employing a neighborhood wavelet-thresholding scheme. An inverse 2D wavelet-packet transform performed on the shrunk coefficients created the denoised product. We applied PCA to the latter in the spectral domain to produce dimension-reduced imagery. We also applied 2D bivariate wavelet shrinkage to the individual images to create denoised imagery. (Bivariate wavelet shrinkage takes into account the parent-child coefficient relationship in the wavelet domain and produces more effective noise removal.^{9})

Our experimental results show that our LLE-based techniques produce high-quality dimension-reduced imagery for remote-sensing applications and require less computational resources and memory than alternative approaches. Our techniques of simultaneously denoising and dimensionality reduction concentrate more energy in the first few output channels of the dimension-reduced imagery than PCA alone, so that better application results are derived. Dimensionality reduction is critical in deriving application products from multidimensional satellite imagery. We have been evaluating our techniques for other remote-sensing applications using more hyperspectral imagery. Our next step will be to apply these approaches to the compact and low-distortion hyperspectral-imaging sensors designed for the future Mars sample-return mission and airborne aerial vehicles that are currently under joint-venture development with the Canadian defense department.

Shen-En Qian

Canadian Space Agency

Saint-Hubert, Canada

Shen-En Qian is a senior scientist and the scientific authority for Canadian government contracts related to the development of space technologies and satellite missions. He leads a research and development team, holds six patents, and is author or co-author of over 100 papers.

References:

1. T. Jolliffe, *Principal Component Analysis*, Springer, 2002.

2. A. Green, A transformation for ordering multispectral data in terms of image quality with implications for noise removal, *IEEE Trans. Geosci. Rem. Sens*. 26, no. 1, pp. 65-74, 1988.

3. K. Fukunaga, *Introduction to Statistical Pattern Recognition*, Acad. Press, San Diego, CA, 1990.

4. S. T. Roweis, L. K. Saul, Nonlinear dimensionality reduction by locally linear embedding, *Science* 290, pp. 2323-2326, 2000.

5. G. Chen, S.-E. Qian, Dimensionality reduction of hyperspectral imagery using improved locally linear embedding, *J. Appl. Rem. Sens*. 1, pp. 013509, 2007.

6. S.-E. Qian, G. Chen, A new nonlinear dimensionality reduction technique with application to hyperspectral image analysis, *Proc. IEEE Int'l Geosci. Rem. Sens. Symp*., pp. 270-273, 2007.

7. G. Chen, S.-E. Qian, Evaluation and comparison of dimensionality reduction techniques and band selection, *Can. J. Rem. Sens*. 34, no. 1, pp. 26-36, 2008.

8. G. Chen, S.-E. Qian, Denoising and dimensionality reduction of hyperspectral imagery using wavelet packets, neighbour shrinking and principal component analysis, *Int'l J. Rem. Sens*. 30, no. 18, pp. 4889-4895, 2009.

9. G. Chen, S.-E. Qian, Simultaneous dimensionality reduction and denoising of hyperspectral imagery using bivariate wavelet shrinking and PCA, *Can. J. Rem. Sens*. 34, no. 5, pp. 447-454, 2008.