Quantifying Planetary and Terrestrial Geologic Surfaces Using Wavelet Statistics

Over the past two decades the planetary and terrestrial scientific communities have seen a significant increase in airborne and space-based scientific monitoring and data acquisition. Laser altimetry, visible and microwave imaging sensors, and radar altimeters provide insights into fine-scale details and large-scale surface processes. Characterization of surface features and processes from such data, however, still poses significant challenges. To this end, we describe a quantitative approach to statistically characterize surface features and processes from gridded elevation data. The computer vision community has recently seen significant advances in modeling the statistics of natural images. These models consist of statistical measurements extracted from an image (e.g., a parametric description of Fourier energy). The model's descriptive power is verified by synthesizing a new image with matching statistics. If the synthesized image is visually similar to the original, then the model likely captured some inherent properties of the image. The model parameters can then be used as a quantitative similarity metric. The statistical model employed here is that of Portilla and Simoncelli, 2000. The model first decomposes an image using a complex wavelet transform. From this decomposition and the original image, a number of statistics are extracted: (1) marginal statistics that embody the basic pixel intensity distribution; (2) coefficient correlations that embody the salient spatial frequencies and local spatial regularities; (3) coefficient magnitude statistics that embody higher-order geometric structures; and (4) cross-scale phase statistics that embody long-range spatial correlations. Depending on the image size and wavelet parameters, approximately 1,000 to 10,000 statistics are extracted. We applied this model to a grayscale shaded relief image derived from a 2m lidar DEM. We extracted statistical measurements from each of five qualitatively different regions (fluvial, glacial and aeolian). Synthesized images based on these measurements qualitatively capture the underlying structure of each region. When coupled with pattern recognition techniques, the measurements are used to quantify the structural similarity between different regions. Further development is needed to apply this approach to surfaces imaged with different modalities and at different scales. These results, nevertheless, provide an encouraging first step in quantifying surface features and their underlying processes.